a978-3-7091-6711... · 2017-08-28 · a.6 reduced matrix, minor, cofactor, adjoint 611 the inner...

Appendix A

Matrix Algebra and Control

Boldface lower case letters, e.g., a or b, denote vectors, boldface capital letters, e.g., A, M, denote matrices. A vector is a column matrix. Containing m elements (entries) it is referred to as an m-vector. The number of rows and columns of a matrix A is nand m, respectively. Then, A is an (n, m)-matrix or n x m-matrix (dimension n x m).

The matrix A is called positive or non-negative if A>, 0 or A :2:, 0 , respectively, i.e., if the elements are real, positive and non-negative, respectively.

A.1 Matrix Multiplication

Two matrices A and B may only be multiplied, C = AB , if they are conformable. A has size n x m, B m x r, C n x r. Two matrices are conformable for multiplication if the number m of columns of the first matrix A equals the number m of rows of the second matrix B. Kronecker matrix products do not require conformable multiplicands.

The elements or entries of the matrices are related as follows Cij = 2::;;'=1 AivBvj 'Vi = 1 ... n, j = 1 ... r . The jth column vector C,j of the matrix C as denoted in Eq.(A.I3) can be calculated from the columns Av and the entries BVj by the following relation; the jth row C j ' from rows Bv. and Ajv:

column C j = L A,vBvj , /1=1

row C j ' = (CT),j = LAjvBv, 11=1

(A. 1)

A matrix product, e.g., AB = (c: c:) (~b ~b) = 0 , may be zero although neither multipli

cand A nor multiplicator B is zero. Without A or B being the nullmatrix the product AB only vanishes if both A and B are singular. The matrix B = A 1. is the (right) annihilator of A, i.e., AA 1. = O.

A.2 Properties of Matrix Operations

Distributivity: A(B + C) = AB + AC . Associativity of addition (A + B) + C = A + (B + C) and multiplication (AB)C = A(BC). Commutativity of addition, non-commutativity of multiplication and raise to higher powers:

A+B=B+A, AB f- BA, (A.2)

Exceptions: Consider, first, a multi variable control with transfer matrix G( s) in the forward path and unity feedback and, second, H in the forward path and F in the feedback where G = FH. The overall transfer matrix is given by

(I + FH)-IFH = FH(I + FH)-I . (A.3)

The inverse of the return-difference matrix and G commute unexpectedly. Another exceptional case is the product A exp (At), i.e. the coefficient matrix and the state transition matrix. Finally, suppose A and B nonsingular. Then, A and B commute if their product is the identity matrix: Both AB = I and BA = I yield A = B- 1 . Generally, the matrices A and B commute with respect to multiplication if B is a function of A , e.g. as given by a matrix polynomial or by the decomposition in Eq.(A.45).

610 A Matrix Algebra and Control

Note that F(I + HF)-I = (I + FH)-I F and F(I + HF)-I H = (I + FH)-IFH , particularly observe the change of order within the parentheses. 0

Properties when transposing or inverting a matrix product:

(AA)

Inverse and transpose operations (symbols) may be permuted: (AT)-I = (A-If. If A-I = AT is true then A is referred to as an orthogonal matrix.

An idempotent matrix A has the property A 2 = A . This result can be observed in least squares, estimation and sliding mode theory, e.g.,

or A = I - B(C,B)-IC, . (A.5)

A matrix A is nilpotent if A k = 0 for some k. Such a matrix appears in the case of the state-space representation of a k-tuples integrator.

A.3 Diagonal Matrices

A diagonal matrix A is a square matrix with non-zero entries A;; in the main diagonal, only, e.g.,

(A.6)

If these entries A;; are equal to each other A is a scalar matrix. The identity matrix Inis a scalar matrix with elements 1 and dimension n x n: In ~ dia~(1,I,I, ... ,1), In E nnxn .

Given a rectangular (n, r)-matrix B, premultiplying B by the identity matrix In or postmultiplying B by Ir yields InB = B or Blr = B .

Premultiplying [postmultiplying] a matrix A by a diagonal matrix yields

... )

... ... ) , (A.7)

i.e., a new matrix the rows [columns] of which are successively multiplied (scaled) by d; (i.e., i-th row [column] with di).

A.4 Triangular Matrices

A lower triangular matrix is a square matrix having all elements zero above the main diagonal, an upper triangular matrix only contains zero elements below the main diagonal. The product of two triangular matrices produces a triangular one again.

If A is given as a diagonal matrix or an upper or lower triangular matrix, the eigenvalues A[A] are already given by the entries in the main diagonal A;; Vi = L.n.

A.S Column Matrices (Vectors) and Row Matrices

The unit m-vector with k-th component 1 is termed ek . Defining this m-vector ek and the n-vector ei ,

the elementary matrix or Kronecker matrix E;k is given by the dyadic product

ek = (0,0, ... I, ... ,0, of = e~mx I) = (Im).k E;k E nnxm (A.8)

as an (n,m)-matrix with entry 1 only in the i,k-element and zero elsewhere. Thus, the (n,m)-matrix A can be established element by element: A = L7=1 L;'=I A;kEik where A E nnxm or dim A = n X m .

The identity matrix can be achieved by the sum In = L7=1 e;er = L7=1 E;; . The sum vector with elements throughout unity 1 = (I, 1, ... ,If serves as a summation operator for an m-vector: ITa= L~ai.

A.6 Reduced Matrix, Minor, Cofactor, Adjoint 611

The inner product of two vectors a and b is a scalar aTb = bT a . Orthogonal vectors have zero inner product.

Assume a vector output signal y given by the linear combination of a vector input signal x governed by the transfer equation

y=Cx where yEnr , xEnn , CEnrxn . (A.9)

The entry Gij of C is considered as an operational factor from the input component Xj to the output component Yi. Note that the output subscript i (effect) is written first and j (source) is written second.

A partitioned vector u is denoted by the "vec" symbol

u- (:~ ) =vec(uI,u2, ... uN)=vecui=(uf,uI, ... u~f uiEnm " uEnm , m=Lmi.

- u~ (A.lO)

The norm or length of a is the distance to the nullvector (origin) and is defined by the Frobenius norm lIaliF = .,;;T; . The inner product is always smaller than the product of the norms of the multiplicators (Schwartz inequality): laTbl ::; lIallFllbliF .

Triangle inequality (for any kind of norm): lIa + bll ::; Iiall + IIbll . The angle 0 between two vectors u and v is defined by cosO = uTv/(lluIlFllvIlF) . Mapping a matrix A

to a vector a is provided by the operator "col" (or by the operator "vec")

colA = vecA = a = (All ... AmI: AI2 ... Am2 ... Amnf

The operator col lists the entries of A column-wise.

(A.ll)

(A.I2)

Separating the ith column of A will be termed by (A).i. For abbreviation also A.i is used although, usually, only matrices are denoted by upper case boldface letters, irrespective of the subscript. A.i is defined as a column matrix (vector). The operator col (column string) can also be written as

col A = (A·f : A·I ... A.~f. (A.I3)

The inner product of two real matrices A and B of equal dimension m x n is a scalar and coincides with the trace of the matrix product: (A, B) = (coIA)T colB = trATB = L:::I L:j=1 AijBij ::; IIAIIFIIBIIF . The Frobenius or Euclidian norm of a real matrix A is given by

(A.I4)

A.6 Reduced Matrix, Minor, Cofactor, Adjoint

Given the (n, n)-matrix A , the reduced matrix A"d ik of the size (n - 1) x (n - 1) is obtained by cancelling row i and column k. Repeating for NI ... n, k'v'I ... n yields n2 different matrices. The minor on the i, k-component of A is defined as the determinant det Ared ik . The cofactor of the i, k-element of A is obtained by permuting the sign of the minor, precisely by multiplying the minor with (-1 )iH, that is, COfikA = (_1)iH det Ared ik . Given the (n, n)-matrix A on the elements Aik, then A = (Aik) = matrix[AikJ and the adjoint is given as the transposed matrix of cofactors: adjA = [matrix( COfikA)JT .

The determinant can be decomposed with respect to the row i , or with respect to the kth column, that is,

n

det A = L AikcofikA k=1

Vi = 1 ... n or det A = L AikcofikA i=l

Interpretation of the equations above as a matrix multiplication yields

'v'k=l. .. n.

I det A = A adjA = (adjA) A A -I = adjA . detA

(A.I5)

(A.I6)

612 A Ma.trix Algebra. and Control

A.7 Similar Matrices

Matrices A and A are similar, i.e., A ~ A, if A = TAT-I. The preceding equation is named similarity transformation. Similar matrices A and A are characterized by the property det A = det A, by the same eigenvalues, eigenvalue multiplicities and eigenvalue indices (and by the same number of generalized eigenvectors), see Eq.(B.I07). Examples of similar matrices are A and A = diag ~i[A) = T-I AT in Eq.(B.I3).

A.S Some Properties of Determinants

Multiplying an (n, n)-matrix A by a scalar Jl yields a (n, n)-matrix r = JlA = (JlAik) = matrix!JlAik) . The determinants of A and r comply with the relation det r = det JlA = Jln det A . Determinant of the transpose: det AT = det A . Determinant of the product AB if both A and B are square: det(AB) = (det A)(det B). Determinant of the inverse: det(A -I) = (det A)-I = 1/ det A .

In the case of an orthogonal matrix R the relation RT = R -1 holds and det R = ±I . The eigenvalues A[A) of an (n, n)-matrix A are related with the determinant and with the trace as

follows n n

det A = II ~i[A) trA = LAii = LAi[A). (A.I7)

Selecting two conformable matrices, the (m, n)-matrix A and the (n, m)-matrix B, the equivalence holds

det(Im + AB) = det(In + BA). (A. IS)

A.9 Singularity

The matrix U is singular and no inverse U- I exists if det U = 0 . The matrix U is nonsingular if its determinant does not vanish. A nonsquare matrix always is singular.

A.I0 System of Linear Equations

Solving a system of n linear algebraic equations Ax = b yields

x=A-Ib= adjA b detA

or 1

Xk = -d A L(cofikA)bi et . . where the transposed matrix of cofactors replaces the adjoint.

A.ll Stable Matrices

Consider the set A(A) of eigenvalues AdA) of the (square) matrix A

A[A) = P;(A) E C det(A - ~iI) = 0 } .

(A.I9)

(A.20)

If and only if A is a subset of C- (complex numbers with negative real part), i.e., A[A) ~ C- , the matrix A is said to be asymptotically stable.

A.12 Range Space. Rank. Null Space

The range (or image) of a linear operator (map) X -+ Y is the set of all linear combinations of the columns of the matrix A of this operator and is termed range space n[A), see Figs. A.I and A.2. The range space of A is the set of all vectors Ax where x ranges over the set nn .

n[A)~ {y:y EY, x E X, Y = Ax} C Y x E X Enn , A E nmxn . (A.2I)

A.12 Range Space. Rank. Null Space 613

Example: Consider m > n, m = 3, n = 2, A column-like

(A.22)

Within the three-dimensional Euclidian space y the range space of A is a plane through the origin spanned by both columns of A . In a case m < n the columns of A are linearly dependent which yields redundancy but is not in contradiction to the definition Eq.(A.2I). 0

The dimension of the range space is the rank of A : dim R[A] = rank A , i.e., the number of linearly independent columns of A . In any square or rectangular matrix the maximum number oflinearly independent columns is equal to the maximum number of linearly independent rows. The rank can be checked either from rows or columns of a matrix. Given the (m, n)-matrix A , if m ~ n and rank A = m the matrix is of full row rank. If m :::: n and rank A = n the matrix is of full column rank. In the case m = n = rank A the matrix A is nonsingular.

An (m, n)-matrix A is said to be of full rank if rank A is n or m whichever is less, thus

rank A = min {n,m}. (A.23)

A square matrix of full rank is nonsingular, a nonsingular square matrix is of full rank. If a matrix is nonsingular, the inverse of the matrix exists.

Since the rank of a matrix A is the maximum number of linear independent rows or columns of A the rank may be considered the order of its largest nonsingular square submatrix (non-vanishing minor).

Two matrices A and B, B = rAn, are equivalent if rank A = rank B and both rand n are nonsingular. The rank of a square matrix A equals the number of eigenvalues A[A] that are nonzero,

e. g., rank A = 1 A[A] = 0 ; -8 . 0 (A.24)

Usually, the input matrix B E R nxm in state-space representation is full rank: rankB = m . If not, the deficiency of rankB corresponds to redundant input signals U;

Additionally, rank (BBIL - In) = n - rank B .

Rank of the dyadic (outer) product:

Some more properties:

1 rank (abT) = { 0

if a i 0 and b i 0 if a = 0 or b = 0 .

(A.25)

rank a = 1 if a i 0, rank (A+B) ~ rank A+rank B, rank (AB) ~ min (rank A, rank B) . (A.26)

The null space (or kernel) of a matrix A is termed N[A] and is defined as a subspace

N[A] ~ {x: x E X where Ax = O} eX (A.27)

In other words: If Ax = 0, then x belongs to the null space of A , i.e., x E N[A], see Figs. A.I and A.2. The range space R[AT] and the null space N[A] are orthogonal since Y1 = ATx and AY2 = 0

yields YI Y2 = xT AY2 = 0 . The dimension of the null space plus dimension of column space of the (n, m)-matrix B is

dim N[B] + dim R[B] = n . (A.28)

Example: Consider a multi variable system

x(t) = Ax(t) + Bu(t) , (A.29)

its Kalman controllability matrix SK

(A.30)

and a subspace SK spanned by the columns of SK . Then SK is the controllable subspace of the system given in Eq.(A.29). Only intitial conditions x(O) E SK can be controlled (Kalman, R.E., 1963). Within a limited time interval any system state in SK can be reached. If any vector z satisfies z E R[SK] then also Az E R[SK]' Consider a special case (m = 2,n = 2)

A = (0 -1) -5 4 B=b= ( 1 ) ~ • = ( 0.707 )

1 a 1 0.707 (A.31)


y eYe 'R'"

Figure A.l: Range space (or image or column space) and null space (or kernel)

-1 ) -1

rank SK = 1 f. 2, hence (A, B) not completely controllable. (A.32)

SK =R[A] = {y y = a ( ~ ), a E (-oo,oo)} (A.33)

i.e., only the line a(l 1) is the controllable subspace of the x-plane. In this example with distinct eigenvalues

( 0.707 T = (aj : a;) = 0.707

-0.196 ) 0.981 '

T- I = ( 1.178 0.236) -0.850 0.850

and T-IB = T-Ib = C·~04) . (A.34)

As far as controllability is concerned, Eq.(B.20) shows that xio cannot be influenced by u since the second entry of T- I b is zero and the system equations in x mo are decoupled. There is a transient motion in xio , only, starting from the initial conditions, and decaying to zero. The first variable xjO is generated from u(t) via PTI-delay, conventionally. Invoking x = Txmo and omitting, for brevity, the homogeneous part of the solution, yields

x = (XI) = T(XjO) = T(XjO) = (0.707)xmo X2 xio 0 0.707 I

(A.35)

which corresponds to the line a(l : 1) . IfB = b coincides with an eigenvector b = a~ then the controllable subspace is reduced to this eigenvector. 0

A.13 Trace

The trace of a square matrix is defined as the sum of the elements Aii in the main diagonal. Moreover, the trace is identical to the sum of the eigenvalues. For comparison: The product of the eigenvalues yields the determinant. The trace of the dyadic product of two vectors gives the inner product: tr abT = aTb .

Taking the trace of a matrix product, the matrix factors are commutative tr AB = tr BA = L:~=I L:~=I AikBki although AB or BA and their dimensions are strongly different, A is an (n,p)matrix and B a (p, n)-matrix. This cyclic property for a product matrix in a trace argument is very convenient when matrix derivative operations with respect to matrices are studied. For applications see e.g. sensitivity theory. Further properties of the trace are

tr(A + B) = trA + trB, tr In = n . (A.36)

A.14 Matrix Functions 615

A.14 Matrix Functions

Consider a function I(A) and assume that its Taylor expansion

I(A) = f ;"(Qi I(A») (A - AiY j=O J! aA] A=A,

(A.37)

exists in the vicinity of Ai where Ai = Ai[A) is one of the p eigenvalues of A of multiplicity mi and degeneracy qi. Inverting the characteristic polynomial e(A) = det(AI - A) and expanding into partial fractions, using a numerator polynomial bi (A)

(A.38)

The definition of ai( A) is

a'(A) ~ bi(A)e(A) • - (A - Ai)m, (A.39)

Multiplying Eq.(A.38) with e(A) and invoking Eq.(A.39) yields

1 = t ai(A) = e(A) t bi(A) i=1 i=1 (A - Ai)m,

(A.40)

The matrix function I(A) associated with the scalar function I(A) is obtained by substituting A by A . From Eqs.(A.39) ,(A.40) and using Cayley-Hamiltion theorem etA) = 0

(A - M)m'ai(A) = bi(A)e(A) = 0

and 2:;=1 ai(A) = I . Multiplying I(A) with ai(A) yields

00 1(j)(A;) . I(A)ai(A) = L -'-1 -(A - AiI)]ai(A) .

j=O J.

(A.41)

(A.42)

The upper bound in the sum above can be replaced by mi. This results since one has (A-AiI)j b>m,-I = 0 in Eq.(A.41). If the multiplicity mmi of Ai in the minimal polynomial is mmi < mi then the upper bound can be replaced by mmi . Thus, taking the sum with respect to i yields

p p m,-l/(j)(A') L I(A)ai(A) = I(A) = L L -'-1 -' (A - AiI)j ai(A) . i:::l i=l j=o J.

(A.43)

With the abbreviation Di,HI ~ (A - AiI)j ai(A) = j! Zij (A.44)

where Zij denotes the interpolating matrix polynomial or component of A , one has

p m,-I/U)(A;) I(A) = L L -'-1 -Di ,j+1 .

i=l j=O J. (A.45)

Note that the matrices Di,HI or Zij do not depend on the function I but only on A . Hence, any f can be used to determine Di,HI or Zij for a given A. The derivatives are very simple if I is taken the rth power I(A) = A r

A r = t 1:' (r) A~-j Di,HI . i=1 j=O J

(A.46)

The last equation can be used to determine Di,HI . Now, defining column-like partitioned matrices

Dc = (DII : DIz : ... DJm p f and Ac = (I : AT : ... A Tn f, Eq.(A.46) is rewritten to V dmg Dc = Ac where V dmg is the generalized Vandermonde matrix. In the case mmi < mi one has Di,mm,+2 = 0 through Di,m,+1 = 0 (Frame, J.S., 1964; Ganlmacher, F.R., 1986).


A.15 Metzler Matrices

A Metzler matrix A E nnxn is characterized iii, j = [1, n] by

{ <0 i=j aij = 2: 0 i f- j

main diagonal elsewhere. (A.47)

A Metzler matrix is stable if and only if all leading principal minors Vk = [1, n] premultiplied by {_I)k

(

all

a2l {_I)k det :

akl

... alk 1

... a2k

. . > O.

... akk

(A.48)

Remark: To check the stability of an ordinary matrix one has to test all the principal minors. A Metzler matrix is stable if and only if it is quasi dominant negativ diagonal. A matrix A is said

quasi dominant negative dominant if all aii < 0 and Vi,j = [1, n] there exist positive numbers di such that

n

either dda .. I > L dj laij I j=l, iti

n

or dj lajj I > L dilaijl i=l, itj

is satisfied (Metzler, L.A., 1950; !iiljak, D.D., 1978; Mansour, M., 1987; Xin, L.X., 1987). Example:

A _ (-1 0.5) - 0.5 -3 Al,2[A] = -3.118; -0.882

leading principal minors x ( _1)k :

(A.49)

(A.50)

There exists another definition of M-matrices which is explained using the notation P . The matrix P is an M-matrix if Pij ::; 0 Vi f- j and ~e A[P] > 0 . Then, Eq.{A.48) has to be rewritten with aij := Pij and {_I)k must be omitted. For any non-negative matrix B the matrix P = vI - B is an M-matrix if and only if v> 1I'[B] where 11'[.] denotes the Perron root. Finally, if P and D are square and non-negative and 1I'[P] < 1 then all the matrices

I-P-D; I-D{I-P)-l; I-{I-P)-ID (A.52)

are M-matrices and the Perron root of all the following matrices is less than unity

P + D; D{I - p)-l; (I _ P)-ID . (A.53)

A.16 Projectors

Consider the n-dimensional space decomposed into two subspaces namely the range space and null space. The matrix P r is a projector ifit projects nn on the range space n[Prl ofPr along the null space N[P r ]

of Pro A matrix acts as a projector if it is idempotent, i.e. P; = Pro The projection operator function and the range space details can be seen by premultiplying a vector x of nn by P r

Prx = X . (A.54)

The above projection is essentially augmented by another projection: If the same vector x is considered to be projected by the projector In - P r the projection result

(I - Pr)x = 0 (A.55)

is the null space, in other words, In - P r projects x of nn on the null space N[P r 1 along the range space n[Pr ]. Summarizing,

P r projects nn on n[Pr ] along N[Prl (A.56)

A .17 Projectors and Rank 617

II .. - P,

- plan 4xI -f "l.:rl r·, = 0

Figure A.2: Range space and null space of a projector

1- P r projects nn on N[Prl along n[Prl . (A. 57)

Hence, n[Prl = N[I - P r land N[Prl = n[I - Prl .

Example: Range space and null space: Consider n = 3 and

(0 0 0) P r = 0 0 0 ; 4 2 1

(A.58)

Hence, XI = 0, X2 = 0, X3 undetermined, i.e. , the range space n[Prl is given by the x3-axis in this example (see Fig. A.2). Check

( 1 0 0) I-P r = 0 1 0 ;

-4 -2 0 (A.59)

The null space N[P r 1 is obtained from P rX' = 0 or from the range space of I - p .. i.e.,

n[I - Prl: (I - Pr)x' = x' (A .50)

(I-Pr)x'=x' ~ (i4 52 ~) (~~) (~~) (A.51)

Thus, x~ = x~; x~ = x~; -4x~ - 2x~ = x;, i.e. the null space N[Prl is the plane 4xI + 2X2 + X3 = 0 (see Fig. A.2). The matrix P r is a projector on the x3-axis along the plane. The matrix (I - Prj is a projector on the plane along the x3-axis and rank P r = 1, rank (I - Prj = 2, tr P r = 1. 0

A.17 Projectors and Rank

Note that rank P r = tr P r and rank (In - Prj = n - rank P r . If the matrices B, K, H, C have full rank, the following relations are true

n[BK] = n[Bl and N[HC] = N[Cl (A.52)

which is often used when treating the matter of sliding mode (EI-Ghezawi, O.M.E., et al. 1983 ). Summarizing: The rank of a projector matrix equals the dimension of its range space .

If Pr=B(CB)-IC ~ rankPr=rankB=m and rank(In-Pr)=n-rankPr=n-m. (A.53)


Premultiplying any square matrix by I,. - P r will cause the product to have the rank of n - m (or less):

rank (I,. - Pr)A = n - m . (A.64)

A.IS Projectors. Left-Inverse and Right-Inverse

Let C be any (m, n )-matrix then the following list of idempotent matrices, subdivided into left-inverse and right-inverse matrices, serve as projectors: CC Ll , C RI C, I". - CC Ll , I,. - C Rr C . For instance, the projector P r = B(CB)-IC := CRIC is considered as a product of C RI and C. Then B(CB)-I = C RI

is true since CC RI = CB(CB)-I = 1m.

A.19 Trigonal Operator

The permutation matrix P is characterized by the property that each row and column has exactly one element equal to unity while the other entries are zero. The matrix P can be derived from the identity matrix I by permuting columns or rows. Postmultiplying and premultiplying a matrix by P causes permutation of columns and rows of this matrix, respectively (Qianhua, W., and Zhongjun, Z., 1987). Finally detP = 1.

As a special case of the permutation matrix the rotation matrix is defined as the unity matrix with unity entries in the secondary diagonal, only,

(001) 1'" = 0 1 0 . For comparison, 1 0 0

I = I" = diag{l, 1, ... , I} . (A.65)

Postmultiplication of a matrix A with 1/ yields a reflection of A across the vertical symmetric axis

( :: :~ ::) 1/ = (:: :~ ::) (~ ~ ~) (:::~::) a7 as a9 a7 as a9 1 0 0 a9 as a7

(A.66)

Premultiplication determines a reflection ;'cross horizontal symmetric axis

(A.67)

The reflection operation holds for every kind of vector or matrix. The secondary diagonal matrix possesses the following properties: (1/)-1 = 1/ , (1/)2 = I ,1/ =

vi, (1/ C)T = CTI/T = CTI/ . With regard to the symmetry of 1/ and trigonal matrices

(1/ trig bf = (trig b)TI/T = (trig b)I/, trig a = trig(ai) = trig[I/ x (an+I_;)). (A.68)

For a given vector b the trigonal matrix is defined as trig b

o . . b4

) ~ ... trIgonal matrix (A.69)

The secondary diagonal of the original trigonal matrix is filled with bn , i.e., the entry hi of highest index n. Reflecting the vector b

(A.70)

The elements of the secondary diagonal of the result are troughout hi.

A.20 Transfer Function Zeros and Initial Step Transients 619

Further properties:

(" b3 b2

" ) Upper triangular matrix: (trig b)I/ = ~ b. b3 b2

0 b. 63

0 0 b4

(A.71)

t( transpose)

C' 0 0 n Lower triangular matrix: 1/ (trig b) = :: b. 0 b3 b.

b1 b2 b3

(A.72)

b2 b3 b4

) bl b2 63

0 bl b2

0 0 bl

(A.73)

!( transpose)

~ ) . b1

(A.74)

The trigonal matrices in Eqs.(A.71) through (A.74) are persymmetric and Toeplitz matrices because the elements along and parallel to the main diagonal are identical (Grenander, U., and SzegiJ, G., 1958, Makhoul, J., 1981; Brillinger, D.R., 1981). The remaining trigonal matrices are of Toeplitz type with respect to the secondary diagonal.

A.20 Transfer Function Zeros and Initial Step Transients

Consider the single-input single-output system with the transfer function O( s)

Y(s) = O(s) = L~ /ksk = Z(s) U(s) Lo aksk N(s)

(A.75)

and with input signal u(t) (ic 0 t 2': 0;= 0 t < 0). For the sake of abbreviation, the following vectors of constants, signal derivatives and powers of the Laplace operator s are defined

u(t) = (u(t) u(t) ... u(n-I)(t)f y(t) = (y(t) y(t) ... y(n-I)(t)f

fr = (to II··· In-If (A.76)

S = (1 S s2 ... sn-If. (A.77)

If O(s) has zeros, i.e., coefficients Ii exist (i > 0), the output signal y changes suddenly between 0- and 0+ (FijI/inger, 0., 1961; Wunsch, G., 1971 ). The initial conditions of y obey the relation

(A.78)

is a matrix, relating the initial condition step transients in a general view of a and f (Weinmann, A., 1988 ).

Note that the Laplace transform of the differential equation corresponding to Eq.(A.75) does not depend on y(O+) but only on y(O-) (Wunsch, G., 1971 )

Y(s)N(s) - yT(O-)(trig a) s = U(s)Z(s). (A.79)

The nth derivatives u(n)(o+) and y(n)(o+) comply with the following relation

y(n)(O+) = (t; - a;O)u(O+) - Inu(n)(o+) an

(A.80)


u(t) weighting

function get)

Figure A.3: System output analysis

yet)

Even in the case of non-existing zeros of O(s), i.e., t;li;tO = 0 and 0 = 0, the nth derivative y(n)(o+) has the non-zero amount

(A.8I)

Using the state-space representation for the single-input single-output system of Eq.(A.75)

itt) = Ax(t) + butt) y(t) = eT x(t) + du(t) (A.82)

and observing the definitions of boldface u(t) and y(t) in Eq.(A.77) which are different from the scalars u(t) and y(t), it results

y(O+)=(e ATe A 2,Te ... An-l,Tefx(O+)+L~u(O+)

where 0 = 1/ trig(I/ (d cTb cT Ab ... fJ is equivalent to Eq. (A.78).

A.21 Convolution Sum and Thigonal Operator

Definitions:

u(!l.t) y(!l.t) u(2!l.t) y(2!l.t)

u=(u;)~ u( i!l.t)

, Y = (y;) ~ y(i!l.t)

urN !l.t) y(N !l.t)

Convolution integral and convolution sum (see Fig. A.3):

k-l k-l

g(o) g(!l.t)

g(i!l.t)

g[(N - l)!l.tJ

(A.83)

(A.84)

y(t) = l' g(r)u(t - r)dr , y(Mt) = L g(i!l.t)u(Mt - i!l.t)!l.t = L u(i!l.t)g(Mt - i!l.t)!l.t (A.85) i=Q

System analysis: y = 1/ [trig (1/ u)) g!l.t or y = 1/ [trig (1/ g)) u !l.t (A.86)

System identification:

(A.87)

Input synthesis:

(A.88)

Compensator design (see Fig. A.4):

e = !l.~2 (trig 1/ r)-l 1/ (trig 1/ g)-l 1/ Y . (A.89)

Considering k -> 00 the equation above corresponds to infinite matrix equations in expanded form. For

stability considerations and norm definitions of multivariable infinite matrices etc. see Makhlouf, M.A.,

1972.

A.21 Convolution Sum and Trigonal Operator 621

compensator plant

reference r( t)

~I I v(t)

~I I output y(t) c(t) g(t) ~

Figure A.4: Compensator design

Appendix B

Eigenvalues and Eigenvectors

Assume a stable (n, n)-matrix A with the following definitions: Characteristic matrix of A: AI - A . Characteristic function (polynomial) in A: C(A) = det (AI - A). Characteristic equation of A : C(A) = det (AI - A) = (A - AI)(A - A2)'" (A - An) = 0 . Roots of

the characteristic equation yield the eigenvalues A[A). The coefficients of the characteristic polynomial

can be written as

C(A) = An + Cn_1 An- I + ... Co

K

where tr(.) A = L det (principal minors of order r), j=1

(B.1)

K= (~). (B.2)

Note that tr(n) A = det A and tr(1) A = tr A . The polynomial C(A) is a monic polynomial, i.e., a polynomial with coefficients one of the highest power in A . The coefficients Cn _. of the characteristic polynomial can be calculated from the trace of A up to the power n - i (starting with Cn = 1) according to the following formulas

-tr A and 'Vi = [1,2 ... n)

Cn-i 1 2' I .

-"7(Cn_.+1 tr A + Cn _.+2 tr A + ... + Cn_1 tr A'- + tr A') . I

(B.3)

(B.4)

The inverse matrix (AI - A)-I ~ iI'(A) is known as resolvent matrix for A and is identical to the fundamental or transition matrix iI'(A) .

Cayley-Hamilton Theorem: In the characteristic equation A may be substituted by A : c(A) = o. The matrix A satisfies its own characteristic equation.

Examples: A[In) = l(n times); A[dia&a;) = ai, a2, ... an; A[aA) = aA[A) . If any eigenvalue A[A) = 0 then det(A - AI) = detA = 0 and A -I does not exist. See Eq.(A.16). 0

B.l Right-Eigenvectors

The eigenvector a. is derived from Aa. = A.ai = AdA) a •. (B.5)

These eigenvectors are denoted as right-eigenvectors. The transpose AT possesses the same eigenvalues A •. The right-eigenvectors Pi of AT are defined by

(B.6)

B.2 Left-Eigenvectors

If eigenvectors a. are taken into consideration, they are interpreted as right-eigenvectors a. = a~ = aHA). Left-eigenvectors a~ are defined as a~T A = A.a~T . Transposing and comparing with Eq. B.6 shows the conformity with the right-eigenvector of AT. The left-eigenvector of A is identical to the right-eigenvector of AT

(B.7)

624 B Eigenvalues and Eigenvectors

B.3 Complex-Conjugate Eigenvalues

If the eigenvalues Ai[A] of the real matrix A are complex-conjugate note the following properties

Aai Aj8 i right-eigenvectors ai = ai of A, (B.8) aiTA ..\;a;T left-eigenvectors at of A, (B.g)

ATat ..\;a; right-eigenvectors a? . of AT, (B.lO)

arT ai == 1 and a;T a: = 0 or a;T a: == Dik . (B.ll)

The asterix superscript denotes the complex conjugation. It does not matter if these properties are applied in the case of real eigenvalues but not vice versa. The pair (Ai, ai) is denoted eigensolution of A. Then, the corresponding eigensolution of the transpose AT is (Ai, an . See also Eq.(6.70).

Example:

A= (0 -1) 2 -2

(B.12)

End of Example

Perturbation theory of eigenvalues see separate chapter. Numerical solutions see Wilkinson, J.H., 1965.

B.4 Modal Matrix of Eigenvectors

Assume all eigenvalues AdA] distinct. Then, the modal matrix T of a given square matrix A is built up by the right-eigenvectors T = (a"a2,'" ,an) . Multiplying T by A from the left and substituting Eqs.(B.8) and (A.7), right-hand side,

AT = A(al,a2, ... ,an) = T diagAi T- l AT = diagAi = A . (B.13)

In this expression A = A[A] = diag Ai[A] is the diagonal matrix of the eigenvalues Ai of A in the main diagonal.

The eigenvector matrix associated with AT is named P, i.e., P = (Pl, P2,'" ,Pn) . Combining the modal matrices T and P and referring to Eq.(B.13),

(B.14)

Taking the transpose and comparing with the left-hand side of the equation above yields

(B.15)

In other words: The modal matrices T and P associated with A and AT are orthogonal. Hence, the corresponding eigenvectors are also orthogonal:

(B.16)

Table B.1 presents an overview of the matrices cited above where double lines denote identities. If a matrix A is symmetric the eigenvectors associated with A and AT are identical. The right and

left-eigenvectors are identical, too. With regard to P = T and Eq.(B.15)

T= T[A) if (B.17)

Although the eigenvectors are normalized to unity the eigenvectors are only fixed with the exception of the sign. See also Eq.(6.70).

B.5 Complex Matrices 625

Table B.1: Modal matrices T = T[A] and P = T[AT]

T Ii T T I Tl

X X p-IT P p-I pT

In the case of real and complex conjugate eigenvalues and eigenvectors, the following relations are used. The definition of the modal matrix P!; (ai, a~ ... a~) associated with the transposed matrix AT corresponds to AT at = Ai at. Then, since at! at = 6ik

B.5 Complex Matrices

If A E en xn the following notation is used: Aai D. A.a. and A H a1 D. Ai a1 where aJ at af: ai = Oki .

B.6 Modal Decomposition

Transformation of the original system

x(t) '" Ax(t) + Bu(t), x(O) '" Xo y(t) '" Cx(t)

(B.lS)

(B.19)

into the diagonalized modal system by defining the modal state variable x mo is achieved by substituting x '" Txmo. Thus,

y(t) '" CTxmo . (B.20)

If the sign of eigenvector i is altered, the sign of the i-th component of the modal vector xmo is merely changed.

B.7 Linear Differential Equations and Modal Transformations

The interrelations between original and modal domain are listed in Table B.2. An eigenvector a., in any

case, is determined except a constant factor ai. To verify that the modal matrices T '" (al : a2 ... an)

and T '" (alaI: a2a2 ... anan ) '" T diag a. possess the same modal decomposition, T is substituted by T in the expression A '" TAT-I

Example:

( -3 -1) A", 2 0 '

(B.21)

T '" (1/..;5 1/V2) -2/V2 -1/V2 ' T- I '" (;:;: -:) , AI,2 '" -1; -2 (B.22)

- s~1 + s~2 2 2

s+1 - s+2

- S~I + S~2) . 0 2 I

s+1 - s+2

(B.23)


Table B.2: Transformation into the modal domain

Original domain Transformation Modal domain x=Ax x = Txmo x mo = Axmo

A T lAT=AorA=TAT 1 A x(t) = eAt x(o+) xmo(t) = eAt xmo(o+)

CI>(t) = eAt eAt = TeAt T 1 eAt = diag {exp( Ait)} CI>(s) = (sI - A) (sI - A) I=T(sI-A) IT 1 (sI - A) 1_ diag s~

B.B Eigenvalue Assignment

Defining a polynomial

PI(>') ;; Plo + PI 1>' + ...... + PI,n_l>.n-l + >.n = (pJ : 1)~

where PI ;; (Plo P/l" 'PI,n_dT and ~;; (1 >. ... >.n-l >.nf

(B.24)

(B.25)

consider an (n, n)-matrix A and an n-vector b and let (A, b) be an observable pair, i.e. det R;6 0 where R = (b, Ab, A 2b, ... An-1b). Then, the zeros of PI(>') are assigned to the eigenvalues of A + bkT if

(B.26)

The unique solution, given by Rissanen (Ackermann, J., 1980), is

(B.27)

the vector (R-1)n' being the last row of R- 1 . It can be rewritten

(B.28)

where en is the unit n-vector ek with k = n. Note that the matrix W has the property of transforming the system (A, b) to the control canonical form, i.e., W-l AW is the companion matrix and W-1b = en .

B.9 Eigensystem Assignment

The closed-loop system given by the plant and the state feedback, respectively,

x(t) = Ax(t) + Bu(t), x(O) = xo , u(t) = Kx(t) (B.29)

can be rewritten to x(t) = (A + BK)x(t) ;; Fx(t) . (B.30)

Using the diagonal matrix of distinct eigenvalues or the Jordan canonical form J , and the modal matrix of eigenvectors T, one has

F = A+BK = TJT- 1 AT-TJ= -BKT (B.31)

and a Lyapunov-type of closed-loop system equation is obtained. Let rna denote a non-negative integer, rna ~ 0 and a > IIAIIF a positive constant. Then, it will be

proved in what follows that choosing T according with the Lyapunov equation

- (A + rnaaIn)T + T[-(A + rnaaInW = -(rna + 1)BR-1BT where R> 0, T = TT > 0 (B.32)

B.10 Complete Modal Synthesis 627

makes all the closed-loop eigenvalues ..\[F] lie to the left of -mall' in the complex s-plane, i.e., the controller matrix K is

K == -(ma + I)R-lBTT- l

Proof: From Eq.(B.32)

..\[F] < -mall' .

- AT - TAT - 2maO'T == -(ma + I)BR-lBT

AT + T(AT + 2maO'In) - maBR-lBTT-lT == BR-lBT

(A-maBR-lBTT-l)T+T(AT +2maO'!n)==BR- l BT . , v '~

A, -J From Eq.(B.36) the following statements can be derived: First,

Al == A - maBR-lBTT- l ~ A + BKl Kl == -maR-lBTT- l

where Kl is a feedback gain to stabilize A. Second, comparing Eq.(B.36) with Eq.(B.31),

BKT == _BR-lBT K2 == _R-lBTT- l

where K2 is an additional feedback to shift ..\[F] to the left of -0'. Hence

F == Al + BK2 == A + B(Kl + K 2) == A + BK

K == Kl + K2 == -(ma + I)R-lBTT- l .

(B.33)

(B.34)

(B.35)

(B.36)

(B.37)

(B.38)

(B.39)

(B.40)

Third, in order to obtain all the eigenvalues of J == _(AT + 2maO'In) in the left of -mall' in the complex plane, the decay factor 0' must satisfy 0' > IIAIIF since

..\[J] == "\[_AT - 2ma O'I] == "\[_AT]_ 2ma O' < -mall'

- "\[A] < mall' 0' > -"\[A]/ma ¢= 0' > IIAIIF .

(B.4I)

(B.42)

End of the Proof

Example:

..\i[A] == -1;-2

ma == 1 IIAIIF == v'i4 == 3.7417

R==I

17m • x [A] == 3.7025 .

(B.43)

(B.44)

1 (1 -4) 0' > 17m • x [A] "" 0' == 4 . From Lyapunov equation T == 30 -4 22 ' T- l == (12100 250 )

(B.45)

K == -(ma + 1)R- lBTT- l == -2(0 1) (110 20) == (-40 -10) (B.46) 20 5

F==A+BK==(~2 ~3)+(~)(-40 -10)==(_~2 -~3) ..\i[F]==-6;-7. (B.47)

Check "\[-A]- 2maO' < -4.

End of Example

B.lO Complete Modal Synthesis

Consider the nth order open-loop system and its matrices A, B E nnxm, the closed-loop matrix with state feedback F == A + BK and the eigenvectors fi associated with F. Then,

(A + BK)fi == Ffi == ..\i[F] fi (A - ..\i[F] I)fi == -BKfi ~ - Bpi. (B.48)

With the aforementioned definition Pi ~ Kfi of n parameter vectors Pi E nm and a parameter matrix P E nmxn it results, using the modal matrix T[F] of the closed loop (Roppenecker, G., and Lohmann, B., 1989; Roppenecker, G., 1987 and 1988),

fi == -(A - "\i[F]Inl-lBpi (B.49)

P == (Pl : P2 ... Pn) == (Kfl : Kf2 ... Kfn) == K(fl : f2 ... fn) == K T[F] . (B.50)


B.ll Vandermonde Matrix

Consider a system with (n, n)-matrix A in companion form (regulator form, controllable canonical form) and distinct eigenvalues,

A= ( ;

-ao

1 o

o

o

(B.51)

o

Then, the eigenvectors (the eigenvector direction) and the modal matrix are given by the Vandermonde matrix Vdm

( " A2 '. 1 T = Vdm = ~I A~

,:~, (B.52)

An-I An-I 1 2

which simplifies computation to a high extent. Note the helpful relation that the determimant of the Vandermonde matrix is equal to rr~j=l;i<j (Aj - Ai) . If A is given as a companion matrix and the modal decomposition is to be calculated, note for computational purpose: T[A] can be calculated simply via Vandermonde matrix but Vandermonde matrix cannot be applied to T- 1 = pT = TT[AT] (Table B.l) since AT is not a companion matrix.

Consider another canonical form (Isermann, R., 1981) with the system matrix A'

( -".-, 1 0 ...

; 1 -an -2 0 ...

A' ;; 1/ ATI/ = A = (B.53)

-al 0 0 -ao 0 0 ..

where A' is reflected from A with respect to the secondary axis and A is taken from the aforementioned A in companion form. Then, the corresponding modal matrix is denoted T' = T[A/J, the eigenvalues are the same AdA] = AdA/]. Combining TAT-I = A and T/A(T/)-I = A' = 1/ ATI/ in order to eliminate A , and referring to 1/ = (1/)-1

Example:

Eq.(B.54) ""

Ai[A] = -1, 5;

A = T diagAi[A] T- 1

ai= CJG) (~1 ;) ( ~1

( 1 I)T (0 1) = (-1 1) -1 5 1 0 5 1

which is equivalent to T' or 1" , calculated directly from

A' = (: ~) yielding, e.g., 1"

0 5

(

T = Vdm = ( ~1 1 ) (B.55) 5

) ( 5 -1 ) /6 . (B.56)

T' = ( -1 ) /6 (B.57)

5

1 ) . 0 (B.58)

-5

B.12 Decompostion into Eigenvectors 629

Consider the nth order system as given in Eq.(B.20) but single-input (u := u) and distinct eigenvalues, only. The Vandermonde matrix V dm also plays an important role in decomposing the Kalman controllability matrix Sk as mentioned in Eq.(A.30).

(B.59)

Then, rank Sk < n occurs if one or more elements (T-I b)i should vanish and controllability is no longer given. Furthermore, rankSk is the dimension of the controllable subspace (Ackermann, J., 1988).

B.12 Decompostion into Eigenvectors

In this section the solution of a system of linear algebraic equations is decomposed into a linear combination of the eigenvectors of the system matrix. Consider the nonhomogenous equation Ax = b with the (n, n)-matrix A real and symmetric. The eigenvalues ~[Al are assumed to be distinct. The (right) eigenvectors a; of A are orthogonal. Moreover, ai are normalized to unity. Expanding the unknown solution x into vector components according with ai

n n n n

x= LaiR;, ;=1

Ax = A Laiai = LaiAa; = La; ~i[Al a; = b. ;=1 i=1

Premultiplying this equation with aJ yields

n

aj~j[AlaJ aj+ L ai~i[AlaJ ai = aj~j[Alx1+0 = arb i=l;i;;!:j

B.13 Properties of Eigenvalues

;=1

and

B.13.1 Smallest and Largest Eigenvalue of Symmetric Matrices

(B.60)

n arb x = L ,J[ laj .

j=1 I\J A (B.61)

Consider ~min and ~max , the smallest and largest eigenvalues of a symmetric (n, n)-matrix A , respectively. Then, the following relations hold for any vector x

(B.62)

If the matrix A is positive definite, the notation for the modulus can be omitted. All eigenvalues ~[Al are real and positive if A is positive definite. Assume all eigenvalues distinct.

Thus, all eigenvectors a are distinct, too. With regard to the symmetry A = AT the eigenvectors are orthogonal, see Eq.(B.16), and form an orthogonal basis in n-dimensional space. Any vector variable x can be assumed of the form

n

X= Laia;. ;=1

The quadratic forms xT Ax and xT x using Eq.(B.63) can be deduced as follows

n n

Ax = L aiAai = L a;~i[Al ai ;=1 ;=1

n n

xTAx = (LajaJ) Lai~i[Al a; = LLaiaj~iaJa; = La~~iarai . j=1 i=1 j i i

(B.63)

(B.64)

(B.65)

Using ~max leads to x T Ax = Li a~ ~iar a; $; Li a~ ~maxaT a; = ~maxxT x . Finally, an upper and a lower bound for the quadratic (Lyapunov) form is

~min[Al xT x$; xT Ax $; ~max[Al xT x . (B.66)

From Eq.(B.64) one can derive

(B.67)


If A and B are symmetric matrices and A > 0 then a nonsingular matrix S exists such that

ST(A + B)S = 1+ diagPi[A -IBn (B.68)

(Thrall, R.M., and Tornheim, L., 1957, p.188; Petkovski, D.B., 1985). For A, B, C E 1lnxn symmetric and A, B ;::: 0, C > 0 one has

Amax[AB] S Amax[AC] Amax[C-IB] . (B.69)

Proof: Setting C = TTT and using A;lM] = Ai[TMT- I ] and Eq.(22.25)

Amax[ATTT] Amax[T-IT-I,TB] = Amax[TATT] Amax[T-I,TBT-I] (B.70)

IITATTII,IIT-1,TBT-1II, ;::: IITABT-1II, = Amax[TABT-1] = Am.x[AB]

(Corless, M., and Da, D., 1989). 0

B.13.2 Eigenvalues and Trace

A similar property is given as follows. If G, H E 1lnxn where H ;::: 0 and G such that either the relations hold G ,H ;::: 0 (Kleinman, D.L., and Athans, M., 1968) or H = HT (Wang, S.D., et al. 1986), then,

Amin[G] tr H S tr [GH] S Am.x[G] tr H . (B.71)

Calculating bounds of the Riccati and Lyapunov matrix equation, these conditions are useful (Kwon, B.H., et al. 1985; Mori, T., et al. 1987).

B.13.3 Maximum Real Part of an Eigenvalue

For A E cnxn consider Aai = A[A]ai, the expression af1Aa, = Ai[A]af1ai and its conjugate transpose af1 A H ai = Ai[A]af1 ai. Adding both expressions above,

Since the sum A + AH is Hermitian, from Rayleigh's theorem it is known that

af1 A\AH ai A + AH max H = Amax[--2--] . a i ai 8i

Hence,

If A E 1lnxn then the symmetric part A, is used to evaluate 3?e Ai[A]

Another property is

Amin[A,] + Amin[B,] S 3?e A[A + B] S Amax[A,] + Amax[B,]

where A, ~ (A + AT)/2, A, BE 1lnxn 0 (Jiang, C.L., 1987).

B.13.4 Definiteness of As and Stability of A

(B.72)

(B.73)

(B.74)

(B.75)

(B.76)

The symmetric part of a matrix A is A, ~ (A + AT)/2. If the matrix A, is negative definite (xT A,x < 0 for any x) since Eq.(B.73) and since ai E {x} "" Amax[A,] < O. Since Eq.(B.75) Amax[A,] > 3?e Ai[A]. Hence,

A, < 0 3?e Ai[A] < 0 and A is stable. (B.77)

Further properties concerning norms of a matrix see separate chapter and H eitzinger, W., et al. 1985.

B.14 Rayleigh's Theorem 631

B.13.S Adding the Identity Matrix

Adding /-II (/-I positive) to a matrix A causes increased eigenvalues

(B.78)

The eigenvalues are uniformly shifted to the right. The amount of shifting is /-I. Frequently A serves as a system coefficient matrix in state-space approach. Then, the displacement follows towards the imaginary axis, towards instability. Adding the /-I-scaled identity matrix to A -1 results in .\fJ.II+ A -1] = /-1+ 1/ '\[A] .

Example: Inverting (sIn - A) , the determinant det(sIn - A) is needed. Note that the s-dependent eigenvalues of the combined matrix (sIn - A) are related to '\;[A] as follows .\;[sIn - A] = -'\;[A] + s . For the sake of completeness,

n

'\;[-A] = -'\;[A] det(-A) = (-1)" det A = (_I)n II '\;[A]. (B.79) ;=1

Characteristic equation and the sum and product of eigenvalues (Vieta's rule):

n

det(sIn - A) = 0 sn + sn-1 I: {-.\i[A]} + + II{-.\;[A]} = O. (B.80) i=1 i=l

End of Example

B.13.6 Eigenvalues of Matrix Products

Note the property '\[AB] = '\[BA] if A and B are square.

B.13.7 Eigenvalue of a Matrix Polynomial

(B.81)

For a matrix polynomial J(A) (or a function which can be represented by a matrix polynomial) there exists the relation between eigenvectors

.\[J(A)] = J('\[A]) . (B.82)

B.13.S Weyl Inequality

Let A and E be Hermitian matrices and let a perturbed matrix be defined as Ap = A + E where the largest eigenvalue is denoted by the subscript n etc.

(B.83)

Then, .\i[A] + .\max[E] 2: '\;[Ap] 2: A;[A] + Amin[E] Vi = [1, n] . (B.84)

If E > 0 then Ai[Ap] > Ai[A] Vi = [1, n] and

I A;[Ap]- A;[A] I::; IIEII, Vi = [1, n] (B.85)

(Franklin, J.N., 1968, p.157). For comparison see Eq.(15.43).

B.14 Rayleigh's Theorem

Consider G = G H E c nxn and let G n_r E c(n-r)x(n-r) be a principal submatrix of G Vr = 1 ... n-l .

If the eigenvalues of G n_r are A1[Gn_r]::; A2[Gn_r] ::;

and the eigenvalues of G are AdG] ::; A2[G] ::;

then (Lancaster, P., and Tismenetsky, M., 1985, p. 294)

::; An-r [Gn_r]

... ::; An[G] (B.86)

(B.87)


B.15 Eigenvalues and Eigenvectors of the Inverse

Let an eigenvalue of A and A -I be A and J-l, respectively,

det(AI - A) = 0 (B.88)

Multiplying by det A yields

det A det(J-lI - A -I) = det(J-lA - I) = 0 . (B.89)

Comparing with Eq.(B.88) yields J-l = 1/ A . If A['] is used as an operator for determining the eigenvalue, then generally, A[A -I] = A-I[A] . Denoting the eigenvectors of A -I as a· then A -Ia· = A-Ia· . Multiplying by A from the left shows that a· = a is the solution. If a square matrix A has eigenvalues Ai and eigenvectors ai , then, the inverted matrix A -I has inverted eigenvalues 1/ Ai but identical eigenvectors a: = ai. Eigenvalues are reciprocals, the eigenvectors are shared.

Example:

A = (~5 ~1)

A-I = (-~i8 -~.2)

J-lI,2 = -1; 0.2

detA =-5 (B.90)

det A -I = -0.2 (B.91)

• (0.7071) (-0.1961) a l ,2 = 0.7071 ' . 0.9806 . (B.92)

End of Example

B.16 Dyadic Decomposition (Spectral Representation)

Given the m-vector a and the n-vector b , the (m, n)-matrix abT is the dyadic product (or outer product). Using the modal matrices T[A] and P = T[AT] of A and AT, respectively, decomposed to columns (eigenvectors)

T = T[A] = (al,a2' .. an)

one can decompose the (n, n)-matrix A

A = T(diag Ai)T- I = T(diag Ai)pT

n n n n

A = L Ai(aipT) = L(Aiai)pT = L(Aa;)pT = A L aiPT = AI = A . ;=1 ;=1 ;=1

In this equation use is made of the following result

L aiPT = matrix[L aikPiI] = TpT = TT- I = I ;=1

(B.93)

(B.94)

(B.95)

(B.96)

caused by the orthonormality of T and pT. Both expressions of A , mentioned in the right-hand side of Eqs. (B.94) and (B.95), are equal to a square matrix the (k, I)-element of which is 2::7=1 Ai aik Pil .

Example:

A = -1; 5 T _ (0.707 -0.196) - 0.707 0.981

p = (T-I)T = (1.178 -0.850) 0.236 0.850

(B.97)

A = (-1) ( ~:~~~ ) (1.178 0.236) + (5) ( ~~g~;6 ) (-0.850 0.850)

A = (-1) (0.833 0.167) (5) (0.167 -0.167) 0.833 0.167 + -0.834 0.834 .

(B.98)

(B.99)

End of Example

B.17 Spectral Representation of the Exponential Matrix 633

Starting once more from Eq.(B.94), the decomposition of A is A = T(diag A;)T- 1 . Inverting yields

(B.100)

Comparison with Eq.(B.94) gives an expression corresponding with Eq.(B.95)

A-I = L Ail(aipTl . (B.I0l) ;=1

The dyadic product matrices (aiPTl are the same no matter if A or A -I is decomposed. The factors of these decomposition matrices are Ai and Ai I, respectively.

With respect to the quadratic structure of the dyadic decomposition in the variables ai and Pi , the notation spectml representation is also of common use.

B.17 Spectral Representation of the Exponential Matrix

Suppose A E nnxn, Ai[A] distinct, the eigenvectors ai and ai linear independent and, finally, ai normalized: aiT ak = Cik. Decomposing A = 2:7=1 aiaiT AdA] and applying the well-known Taylor expansion

for eA ,

B.18

n

eA = L aia;T e·qAI

i=l

n

c}(t) = eAt = Laia;Te·>.;[Alt

i=l

Perron-Frobenius Theorem

(B.102)

(B.103)

A non-negative (n, n)-matrix A always has a non-negative eigenvalue .. [A] , named Perron root or Perron eigenvalue or Perron-Frobenius radius of A, such that IAdA] I :s: .. [A] Vi = [1, n]. If A is irreducible then .. [A] is positive and simple and the corresponding eigenvector can be chosen as a positive vector.

A matrix is reducible if by identical row and column transpositions the matrix can be brought to an

upper trigonal block matrix (A~I ~~~) where Au and A22 are square.

B.19 Multiple Eigenvalues. Generalized Eigenvectors

Consider A E cnxn . The n eigenvalues AdA] Vi = 1 ... n are the zeros of the characteristic polynomial det(Ai[A] In - A) with multiplicities counted. Multiple eigenvalues Ai[A] of (algebraic) multiplicity ffii correspond to an ffii-order zero. If there are p distinct eigenvalues 2:;=1 ffii = n . In the case ffii > 1 a full set of n linearly independent eigenvectors need not exist. Assume qi linearly independent eigenvectors associated with Ai. The number qi is the degeneracy of AdA] In - A

qi ~ n - rank (Ai [A] In - A) where 1:S: qi :s: ffii . (B.104)

In addition to the qi distinct eigenvectors, ffii - qi generalized eigenvectors (principal vectors) associated with AdA] are needed.

There is the ambiguity which of the qi linearly independent eigenvectors the generalized eigenvector chain is associated with. Associated with an eigenvector means that the generalized eigenvector at equals a;, see Eq. (B.I07) and Fig. B.l.

These ffii - q; generalized eigenvectors appear in qi chains (see e.g. Fig. B.l). The generalized eigenvectors associated with Ai[A] are given as follows: Calculating (Ai [A] In - A)~ for I-' = 1,2 ... until for some I-' = k the relation

rank (Ai [A] In - A)k = rank (Ai[A] In - A)k+1 (B.105)

is obtained. Then, it can be verified that the highest generalized eigenvector afj of rank k is given by

(B.106)


see ~l in Fig. B.l. The superscript k is used to distinguish between various generalized eigenvectors within a chain. This superscript, of course, cannot be mistaken with an exponent. If k f. mi there are two or more chains of generalized eigenvectors. The entire chain of generalized eigenvectors of rank IJ < k is given by

VIJ= l. .. (k-l). (B.107)

Totally, there are k generalized eigenvectors a;j VIJ = 1 ... k within this chain. This manifold k is denoted k = 8ij. The generalized eigenvectors are linearly independent, as can be easily verified.

Each eigenvector or each chain of eigenvectors is labelled by an additional subscript j. The number of eigenvectors belonging to a chain is denoted 8ij. Following this procedure, j is affixed to every aij in the following Eqs. (B. lOS) up to (B.llI). Writing aij , the index j varies from 1 to qi. The overall sum of the manifold eigenvectors is E1~:; 8ij = m; . Note that Eq.(B.107) can be rewritten to

a~j-l = (Ai [A] In - A)afj or Aa~j = a~j-l + Ai[A]afj (B. lOS)

af;2 = (Ai [A] In - A)2a~j = (Ai [A] I,. - A)a~j-l or Aa~j-l = a~j-2 + Ai[A]a~-l (8.109)

alj = (Ai[A] I,. - A)k-lafj = (Ai [A] In - A)a~j or A~j = alj + A;[A]a~j . (8.110)

PremuItiplication of the last row by A;[A] In - A and referring to Eq.(B.I06) yields

(B.lll)

which reveals that alj (within the chain of generalized eigenvectors) corresponds to the ordinary eigenvector I1;j.

Synopsis of symbols:

mi ... algebraic multiplicity of the eigenvalue A;[A]

qi . . .. geometric multiplicity, degeneracy, number oflinear independent (distinct) eigenvectors associated with Ai [A] , dimension of the space spanned by the eigenvectors, number of chains

m; - qi··· number of generalized eigenvectors associated with A;[A] (total number for all j associated with Ai)

mmi.. index of the eigenvalue Ai in A given by the largest order of the Jordan blocks associated with Ai , Le., mmi = maxj 8lj

8ij . .. length of the chain containing the eigenvector aij and the generalized eigenvectors ~j through a:ji associated with Ai, Blj E [Sib sn ... Sill.,).

B.20 Jordan Canonical Form and Jordan Blocks

In order to obtain a canonical representation the following definitions are used: i) The Jordan canonical form J, i.e., a block diagonal matrix containing qi Jordan blocks for each

eigenvalue Ai[A] (one Jordan block for each distinct eigenvector aij) . ii) The Jordan block Jij is given by

Ai 0 0 0 0 0 Ai 0 0 0 0 0 Ai 1 0 0

Jij ~ 0 0 0 Ai 0 E C'ijX6 i j (B.1I2)

0 0 0 0 0 Ai

i.e., a matrix with the same Ai[A] for all main diagonal elements and ones on the diagonal just above the main diagonal. The size of the Jordan blocks within the Jordan canonical form is 8ij X 8ij and is not given merely by mi or qi. (In the example of Fig. B.I there are three Jordan blocks of dimension

B.20 Jordan Canonical Form and Jordan Blocks 635

generalized eigenvectors 3

I \ "";!1. __ ", ~ 11""'"""-----_ ..,........ j = 1 a;l first chain multiplicity mj

of the eigenvalue AdA]

••••• ___ "'i'i'""~a/;'·2 .a.._- a;2 .. -------~--~ i = 1 second cham qj distinct eigenvectors

~ j = 2

Figure B.l: Brief sketch to obtain orientation about eigenvectors and generalized eigenvectors (associated with Ai = 2, i = 1) and their multiplicity. Two chains (qi = 2), length

of eigenvector chains is 3 and 2 vectors (Si1 = 3, Si2 = 2), mi = 5

3 x 3,2 x 2,1 x 1 associated with Ai') The order of the Jordan blocks within the Jordan canonical form is given by choosing Sil > Si2 > ... > Siq, > 1. Of course, ordering the eigenvectors within the modal matrix T, see Eq.(B.117), must correspond to the Jordan canonical form J.

The Jordan canonical form J is given by

J ~ block diag (Jll, J 12 ,··., J lq" J 2l ... J ij ... J 2q" ... J pqp ) E cnxn .

Rewriting the right-hand equation block of Eqs.(B.108) to (B.110), one has

Aialj

aIj + Aia;j

(B.113)

(B.114)

(B.115)

(B.116)

which can be easily abbreviated to and identified as a special part of AT = TJ given in Eq.(B.117). The nonsingular modal matrix T E cnxn is composed by the eigenvectors and generalized eigenvectors. With the help of Jordan canonical form J and the modal matrix T the matrix A can be decomposed as follows

A = TJT- l . (B.117)

Example (corresponding to Fig. B.1 if the single-input case is selected), n = 6,p = 2

T-1AT = J (B.118)

generalized eigenvectors associated with Al

'" '0 3 '" 2 a2) T = (all all all a12 a12

/~ \ (B.119)

eigenvectors associated with Al eigenvector associated with A2

T{;5 0 0 0.876 0 0

) ( -1.142 0 0

0 0 1 0 0 0 -2 1.142 4 0 0 0 -0.438 1

T- l = 4 0 0 0 0.25 0 0.5 0.25 0 1.142 0 0 0

0 0 0 -0.438 -1 0 0 -1.142 0 0 0.25 -0.876 0 0 0 0 0.5 0

1.~42 ~) -1~142 ~ -0.5 0

(B.120)


ql = n - rank (A - All) = 6 - 4 = 2 (B.121)

i AdA] qi mi mmi

1 2 2 5 3 2 0 1 1 1

p

C(A) = (A - 2)5 A = II (A - Ai)m; characteristic polynomial (B.122) ;=1

p

m(A) = (A - 2)3 A = II(A - Ai)m~; minimal polynomial (B.123) ;=1

( ! 1 0 0 0

! ) 2 1 0 0

= block diag [( ~ 1

~ ) , ( ~ 0 2 0 0 ~ ) ,0]. J= 0 0 2 1

2 (B.124)

0 0 0 2 0

0 0 0 0

B.21 Special Cases

In the special case qi = mi (full degeneracy), there are mi separate eigenvectors and Sij = 1 'Vj = 1 ... mi'

The Jordan blocks become 1 x 1 matrices, identical to the scalar Ai , the diagonalization of A by similarity transformation is still available if qi = mi 'Vi, see Eq. (B.117). Whenever qi '" mj for any i , diagonalization is substituted by the Jordan canonical form.

In the special case of qi = 1 (simple degeneracy), one has Sijlj=l = Sil = mi and J ij E cm,xm;, i.e.

only one Jordan block for AdA]. If this should be true for each Ai then J = diag (J I , J 2 ... J p). This case

arises, e.g., if the (n, n)-matrix A is given in companion form, irrespective of the multiplicity of the roots Ai since rank (AiIn - A) = n - 1 is always true. Various examples are given by Chen, C. T., 1970.

B.22 Fundamental Matrix

Calculating the fundamental matrix (state transition matrix) yields

(B.125)

The matrix exponential of the Jordan canonical form can be calculated block by block, thus

eJ , = block diag (811 ... 8ij ... 8pqp ) eblock diag (J", .. J,j' J pqp ) (B.126)

where the trigonal matrices as given below are used

( j t t 2 /2! t3 /3' ,' •• -' Ii'" - 1)' 1 1 t t 2 /2!

Sij = 0 1 t ... E CJI,X$IJ .

0 0 1

(B.127)

For comparison, in the case of distinct eigenvalues there is Sij ;: 1 and the simple version

exp( diag Ait) = diag e~;' . (B.128)

B.23 Eigenvector .;\ssignment 637

Table B.3: Multiple eigenvalues and degeneracy

Only distinct Multiple eigenvalues Ai with multiplicity mi

eigenvalues simple degeneracy degeneracy, general case full degeneracy

mi = 1 qi = 1 qi = 1 qi = n - rank(A - Ail,,) qi=mi Sij = Sil = mi(> 1) Sij E [Sib ... Siq.] Sij = 1

mmi = 1 mmi=mi mmi = maxj Sij mmi = 1 m(A) = C(A) m(A) = C(A) m(A) = niP - A;}mm, m(A) = ni(A - A;) diagonalization diagonalization diagonalization diagonalization feasible where impossible impossible feasible J = diag Ai only if qi = mi Vi one eigenvector one eigenvector qi eigenvectors qi eigenvectors per eigenvalue irrespective of mi(> 1)

mi - 1 generalized mi - qi generalized no generalized eigenvectors eigenvectors eigenvectors only one Jordan block qi Jordan blocks mi Jordan blocks mixmi Sij x Sij 1 x 1 identical per eigenvalue per eigenvalue to the scalars Ai

e.g., A in companion e.g., identity form regardless of mi matrix I"

B .23 Eigenvector Assignment

B.23.1 Assignable Subspaces. Parametrization of Controllers

Applying state feedback by (m, n)-matrix K in muItivariable systems, the degree of freedom is mn in order to assign n closed-loop eigenvalues. The remaining degree of freedom (m - I)n Can be utilized to assign eigenvectors. Assigning eigenvectors in addition to eigenvalues is an important matter because the eigenvectors are responsible for the relation between the state variables and the modal state variables (see Table B.2) and thus the eigenvectors determine the strength of eigenvalue dominance in the state variables. For a given control system in state space representation, i.e., (A, B) and state feedback K with the closed-loop behaviour (A + BK) = F, there exists a subspace within which the (right) eigenvectors fi of the closed-loop system are located and can be assigned arbitrarily.

B.23.2 Single Real or Complex-Conjugate Eigenvalues

From the eigenvector definition (A + BK)f = Ff= ~[F) f

BKf= (~[F) In -A) f.

Referring to Eq.(C.55), by substitution of M := K, P := B, Q := f, L:= (A[F) In - A)f

BBI(~[F) In - A) f flf = (A[F) In - A)f .

(B.I29)

(B.I30)

(B.I3I)

Noting that the pseudoinverse BI of the column-like matrix B is BIL and that f flf = f a solution K exists if (Sinswat, V., and Fa/lside, F., 1977)

(BBIL - In)(~[F) In - A)f = 0 , (B.132)

i.e., the vector [(A[F) In - A)f) is in the null space of (BBIL - In). Since

BBILB - B = 0 "" (BBIL -In)B = 0 (B.I33)


and the column space of B equals the null space of (BnIL - In). Hence, the eigenvector f to be assigned premultiplied by a modified characteristic matrix lies within the column space n[B)

(A[F) In - A)f E n[B) . (B.134)

Note that the modified characteristic matrix uses the eigenvalues of the closed loop and, thus, differs from the ordinary characteristic matrix (A[A) In - A).

B.23.3 Multiple Eigenvalues and Linearly Independent Eigenvectors

Find a controller gain matrix K to a given system A, B where a number of k eigenvalues and k linearly independent eigenvectors should be assigned. The numbers At[F) ... AdF) are multiple real or conjugate complex closed-loop eigenvalues. The vectors fi must be linearly independent (qi = mi, Sij = 1). If A1 ... At and f1 ... ft belong to an assignable set

(A + BK)(f1 : f2 ... ft ) = (f1 : f2 ... ft) diag Ai[F) or (A +BK) TJ = TJ diag Ai[F) (B.135)

BKTJ = TJ diag Ai[F)- ATJ . (B.136)

The matrix diag Ai[F) is a (k, k)-diagonal matrix with A;[F) in the main diagonal. Referring to Eq.(C.55) and applying the following substitutions yields

Since T~TJ = It

M:= K, P:= B, Q:= TJ, L:= TJ diag Ai -ATJ

BBIL(TJ diag Ai[F)- ATJ)T}TJ = TJ diag Ai[F)- ATJ .

(BBIL - In)(TJ diag Ai[F)- ATJ) = 0 .

Resubstituting the definition T J ' one has

Postmultiplication by a diagonal matrix yields column-wise multiplication with Ai

(B.137)

(B.138)

(B.139)

(B.140)

(B.141)

(B.142)

From the above equation, a necessary and sufficient condition is obtained for fi being in the nullspace of (BBIL - In)(A;In - A)

fi E N[(BBIL - In)(Ai[F) In - A») . (B.143)

Referring to Eq.(C.56) and applying the same substitutions as mentioned above, the solution K of Eq.(B.136) is given by

(B.144)

where Z is any (m, n)-matrix. Eq.(B.144) points out the parametrization of the result. In the case of complex conjugate eigenvalues Ai[F) , the matrices TJ and diag Ai[F) can be modified

in order to obtain a real-valued controller gain matrix1. Consider A1 and A2 complex-conjugate and the remaining k - 2 eigenvalues real,

(B.145)

f1 ~ f1re + jflim , f2 ~ f1re - jflim . (B.146)

It can be easily verified that the result as given by Eq.(B.144) is obtained by the modified real matrices TJ and diag Ai (Korn, U., and Wilfert, H.H., 1982; Schwarz, H., 1971)

T J := (f1re : f1im : f3 ... f k )

diag Ai[F):= block diag [( A,lre ~lim) ,A3 ... Ak). -"lim "lre

(B.147)

(B.148)

1The modification can be applied but need not be applied if complex controller gain K is overcome by any other programming facility.

B.23 Eigenvector Assignment 639

B.23.4 Multiple Eigenvalues and Generalized Eigenvectors

First, the conditions for the existence of K are derived. The desired set of eigenvalues Ai[F] of the closedloop system should comprise p distinct eigenvalues, with multiplicity mi each. The design of K should provide qi linear independent eigenvectors fij with manifold Sij each (see Fig. B.I but a replaced by f). The overall number of linear independent eigenvectors is d. Thus,

qi

LSij = mi, j=1

(B.149)

If the set of eigenvalues A;(F] and eigenvectors fi is preassumed assignable, then, referring to Eq.(B.1l7), the closed-loop system matrix F can be decomposed using the nonsingular modal matrix T, and the Jordan canonical form J,

F = (A + BK) = TJ,T-1 (B.150)

where the Jordan canonical form is defined by Eq.(B.1l3)

J,=blockdiag(Jll , J I2 ... J 1q" J21 ... J2q" ... Jij ... ,Jpq.)ECnxn. (B.151)

The Jordan blocks are given by Eq.(B.1l2) with Ai = Ai[F]. The modal matrix Tis

T = (Tll : T I2 ... T 1q, T 21 ... T 2q, ... T rq.) E cnxn where Tij = (fi) : fi~ : ... f:t) E CnX$i; . (B.152)

Note that the vectors f~ are linearly independent eigenvectors and generalized eigenvectors, to be assigned to F = A + BK satisfying

(A + BK)fi)

(A +BK)f~

Ai[F] Cij

Ai [F] ftj + fC I "IJl = 2 ... Sij .

Applying the condition Eq.(C.55) to Eqs.(B.153) and (B.154), there result d conditions

"Ij=I,2 ... qi; i=I,2 ... p

(B.153)

(B.154)

(B.155)

associated with the linearly independent eigenvectors Cij == Ci~ , and n - d conditions associated with the generalized eigenvectors

"IA: = 2,3 . .. Sij (B.156)

(Sinswat, V., and Fallside, F., 1977). In the case of a given system (A, B) and A: predetermined eigenvalues and eigenvectors (A: < n)

is used. Referring to Eq.(C.56),

K=BUL(T,JJ -ATJ)T~L+Z(I-T,T~L).

In the case A: = n, K = BU(TJ,T- 1 - A) + Z - Z = BUL(TJ,T- 1 - A)

all degrees of freedom are required and the solution K is unique.

B.23.5 Assignable Subspace. Concluding Remarks

(B.157)

(B.158)

(B.159)

In order to determine n eigenvalues and n eigenvectors, the degree offreedom must be n for the eigenvalues and n - I for each given eigenvector since the direction of the eigenvectors is sufficient. This leads to the (m, n)-matrix K of the controller. One has to satisfy the condition

n+(n-I)A: ~ nm. (B.160)

The designer is free to predetermine a complete set of eigenvalues and/or an incomplete set of additionally given eigenvectors. Determining the null space N[H] = N[(BB'L - In)(AiIn - A)] for the given A, B


and the desired Ai , a matrix can be established whose columns span a subspace (or whose column space is) identical to the assignable eigenvector space. It has to be checked if the desired set of eigenvalues and eigenvectors belongs to the assignable eigenvalues and the assignable eigenvector subspace.

If the open-loop system has an uncontrollable eigenvalue, then the closed-loop system must comprise this eigenvalue. Regardless of the fact that this eigenvalue cannot be moved the eigenvector associated with this uncontrollable eigenvalue can be arbitrarily assigned. The only condition is that the eigenvalue to be assigned lies within the assignable subspace N[H] associated with the uncontrollable eigenvalue.

The eigenvector assignment (or eigenstructure method) is carried out by finding the right and lefteigenvectors for the closed loop such that the eigenvalues meet the expected closed-loop values. The eigenvector assignment has the disadvantage that the eigenvalue sensitivity with respect to eigenvector errors may be very high, resulting in a non-robust approach. In the case of state feedback, the sensitivity can be reduced by choosing the eigenvectors as orthogonal to each other as possible.

Prespecifying the closed-loop eigenvectors associated with unchanged open-loop eigenvalues is denoted partial eigenstructure assignment (Jin Lu et al., 1991).

The output feedback case is investigated by Ho, w.e., and Fletcher, L.R., 1988. Norm bounds are

given on the closed-loop eigenvalues, caused by the perturbation of the assigned closed-loop eigenvectors.

Moreover, the influence of the plant and controller matrices A, B, C and K is estimated by using their

spectral norm.

Appendix C

Matrix Inversion

In addition to the simple inversion as given by Eq.(A.16), the control engineer is involved in many problems associated with matrix inversion such as the right and left-inverse, the pseudo-inverse and the inverse of partitioned matrices. Furthermore, in many applications the inversion of matrices is related to the operations of conditioning, scaling and orthogonalizing.

C.I Matrix Inversion Using Cayley-Hamilton Theorem

The Cayley-Hamilton theorem gives a tool to calculate the matrix inverse via powers of the same matrix. In detail, a matrix polynomial up to the power An-I yields the result. Defining the characteristic polynomial C(A) = An + Cn_IAn- 1 + ... + Co where Co = det A the Cayley-Hamilton theorem is given by

c(A) = An + cn_IAn- 1 + ... + CoIn = O.

Postmultiplying by A -I (assuming A -I exists) yields

(C.I)

(C.2)

(C.3)

E.g. A = (~ !), n = 2, c(A) = A2 - 4A - 5 ; CI = -4, Co = -5 = det A (CA)

A-I = -[G!) -4G ~)1I(-5) = (~5 -;)/(-5) = (-0;8 ~.2) .0 (C.5)

C.2 Matrix Inversion Lemma

The matrix inversion lemma proves very helpful in inverting complicated matrices and economizing computational effort. Take into consideration: The quadratic (n, n)-matrix A, its inverse A -I and two rectangular matrices, the (n, r)-matrix B and the (r, n)-matrix C. These matrices are given. The dimensions are related as r < n or r < n . The inverse of A + BC is to be calculated, i.e.,

r ~ (A + BC)-I (BC) E nnxn, (CB) E nrxr

r x I r- I

In A-I

A-IB

A-IB(Ir +CA-IB)-I

A+BC

rA+rBC I x A-I

r+rBCA- I IxB

rB + rBCA -IB = rB(Ir + CA -IB)

rB.

(C.6)

(C.7)

(C.8)

(C.g)

(C.1O) (C.ll)

PostmuItiplying by (-CA- I) yields -A-IB(Ir + CA-IB)-ICA-I = -rBCA- I . Adding A-Ion both sides,

(C.12)

642 C Matrix Inversion

The last equating operation follows directly from Eq.(C.9). So one has the result

(C.13)

If the inverse A -I is already given, then the matrix inversion lemma Eq.C.13 requires the inversion of an (r x r)-matrix (Ir + CA -I B), only. Eq.(C.13) can easily be proved: Examining riA + BC) results in the identity matrix In.

EX8Dlple: If IIFII < IIAII, from Eq.(C.13) it results

(A + F)-I A-I _ A -IF[I + A -IFt l A-I = A-I _ A -IF[I _ IA -1(1+ FA -1)-IFjA- I

A-I _ A -IFA -I + A -IFA -1(1 + FA -I)-IFA -I = etc. (C.14)

Using (I + B)-I = I - B + B2 - B3 + ... where IIBII < 1

[A(I+ A -IF)j-1 = [I - A -IF + (A-IF? - (A -IF)3 + ... jA-I

A-I _ A -IFA -I + A -IFA -IFA -I _ A -IFA -IFA -IFA -I + ... (C.15)

(C.16)

End of Example

C.3 Simplified Version of the Matrix Inversion Lemma

Given the (n, r)-matrix B and the (r, n)-matrix C where r < n the inverse (In + BC)-I is derived as follows

B + BCB = (In + BC)B = B(Ir + CB) I x (Ir + CB)-I (C.17)

(In + BC)B(Ir + CB)-I = B I X C (C. IS)

(I" + BC)B(Ir + CB)-IC = BC I + In (C.19)

(In + BC)-I X I (C.20)

(C.21)

C.4 Matrices in Partitioned Form

CA.1 Algebraic Properties

The transpose of a partitioned matrix is given by

(C.22)

Partial matrices or partial vectors within matrices and vectors (in partitioned form) may be multiplied or added as though the submatrices were scalar elements (numerical or scalar functions) provided the submatrices or sub vectors are conformable. The matrices can be treated as follows

( A B) (u) = (AU + BV) C D v Cu+Dv

( AC DB) (E F) _ (AE + BG G H - CE+DG

AF+BH) CF+DH .

(C.23)

(C.24)

C.4 Matrices in Partitioned Form 643

C.4.2 Inversion of a Partitioned Matrix

Preassaming the matrix M in partitioned form is a square (n + q, n + q)-matrix containing the (n, n)submatrix A, the (q, q)-submatrix D and corresponding rectangular (n, q)-matrix Band (q, n)-matrix C, then, find M-1. On condition det A "# 0 multiply the first submatrix row of M with -CA -1 and add the result to the second row in order to get a block triangular matrix. These operations can be combined by premultiplying M with a certain matrix as follows

(C.25)

Inverting this equation,

(C.26)

With regard to the upper trigonal property in the right-hand side of Eq.(C.26), it is easy to postulate the structure of the inverse containing a specific submatrix S and the inverse of the submatrices

(~ B D-CA-1B (C.27)

In order to obtain I(n+q) by transferring the left-hand side of Eq.(C.27) to the right side, i.e., to get zero matrices outside the main diagonal, the condition is

AS + B(D - CA -IB)-1 = 0

Combining this result with Eqs.(C.26) and (C.27),

-1_ (A-1 -A-1B(D-CA-I B)-I) ( I M - 0 (D-CA-1B)-1 _CA-1 ~ )

-A -IB(D - CA -IB)-1 (D - CA- 1B)-1

(C.28)

(C.29)

) . (C.30)

This result is the Frobenius formula for inverting a partitioned matrix. Eq.(C.30) can be rewritten to

M-1 _ (In _A-1B) ( A-I Onxq - O,xn In Oqxn (D - CA -IB)-1 ) .

If det D "# 0 can be preassumed instead of det A "# 0

) . C.4.3 Inversion of a Partitioned Matrix. Nonsingular Submatrices

If both det A "# 0 and det D "# 0 can be presupposed, then

-1 ( (A - BD-IC)-I -A-1B(D - CA-1B)-1 ) M = -D-1C(A _ BD-IC)-I (D _ CA -IB)-1 .

(C.31)

(C.32)

(C.33)

The result can be confirmed by applying the matrix inversion lemma to the expression (D - CA -IB)-I in the left column of the matrix in Eq.(C.30). Deriving the above result by constituting

M ( Xy) __ (AC BD ) (Xy) -_ (vu) or Ax + By = u and Cx + Dy = v ,

it follows

x

y

(A - BD-1C)-1 (u - BD-Iv)

(D - CA -IB)-1 (v - CA -Iu) .

(C.34)

(C.35)

(C.36)


Rearranging yields

(C.37)

The elements in the secondary diagonal of Eq.(C.33) can simply be checked as equal to those in Eq.(C.37), e.g., -A-IB(D - CA-IB)-I == -(A - BD-IC)-IBD-I . Applying the Frobenius formula in any version, the inversion of a (n + q, n + q)-matrix can be reduced to the inversion of (n, n)-matrices and (q,q)-matrices. To the former class A,A- 1 and BD-1C belong, to the latter D,D-1 and CA-IB.

C.4.4 Inversion of a Block-Diagonal Matrix

M = (~ ~) = block diag (A,D) M- 1 = ( o

D- 1 ) = block diag (A-1,D- 1) .

(C.38)

C.4.S Determinants of Matrices in Partitioned Form

A determinant remains unchanged if any row (column), multiplied by a certain constant factor, is added to another row (column). Thus, presuming A square and det A # 0, premultiply the first sub matrix with -CA -I and add the result to the second submatrix row

D _ ~A -I B ) = [det A)[det(D - CA -IB)] (C.39)

since submatrices in triangle determinants may be treated as scalar numbers. Similarly, if D is square and det D # 0

( A B) ( A - BD-1C 0) det C D = det C D = [det D)[det(A - BD-1C)] . (C.40)

When a partitioned matrix with submatrices A through D of equal dimensions is considered, the result detM = det(AD - ACA-1B) is obtained. If the matrices A and C commute (e.g., if A or C is the identity matrix) the result is det(AD - CB), regardless if det A = 0 . Considering Eq.(C.39) in the special case C = 0 and A, D square of any order, the result is [det A)[det D] .

C.4.6 Reducible Matrix

If a partitioned decomposition (~ ~) = PHP- I with C = 0, A and D square can be deduced from

a matrix H where P is the permutation matrix then the matrix H is denoted reducible.

C.S Right-Inverse

Define a rectangular matrix WRI as the right-inverse of the nonsquare matrix W

WWRI =1. (C.4l)

The right-pseudo-inverse (C.42)

satisfies Eq.(C.4l) WWRI = 1 and is, therefore, a possible solution WRI = WIR but not the only one. Given an np-vector p, an nm-vector y and an (nm, np )-matrix M in the relation np > nm the equation

Mp = y has more unknowns than knowns. There exist solutions in a large variety p = MRly . The matrix MRI is an (np, nm)-matrix and the right-inverse (inverse to the right). The matrix MRI exists if rank M ~ nm and obeys MMRI = Inm . The right-pseudo-inverse MIR is the minimum right-inverse (Momri, M., 1983), i.e.,

(C.43)

G.6 Left-Inverse 645

C.6 Left-Inverse

Given an op-vector p, an nm-vector y and an (n m , np)-matrix M the formula Mp = y is inadmissible if the relation Om > op holds, if more knowns exist than unknowns, if the number of scalar equations exceeds the number of unknown variables in p.

The left-inverse (or inverse form the left) MLI does not supply the solution. Although a variety of MLI exists, none of the associated products p? = MLly may be declared as a solution that is compatible to the overdetermined system, completely.

Example:

M=(

0 °t) ( 0.6 ) -0.5 y= 1.3 (C.44) -1.1506 0.5

MLI = ( 4 + 4.60240 -2 - 2.3012 0 ~ ) MLlM =12 '10,(3. (C.45) 2 + 4.6024 (3 -2.3012 (3

Selecting, e.g., 0 = (3 = 0 then MLly = (-0.2 1.2JT = p? . Computing Mp? = (0.6 1.3 0.230)T, this result is not consistent with y in Eq.(C.44). 0

The formulation by addition (subtraction) of e, i.e., Mp = y - e , is permissible even in the case Om > op. Within the infinite variety of MLI the minimum left-inverse is defined

(C.46)

C.7 Pseudo-Inverse

C.7.1 General Pseudo-Inverse

Let the (n, m)-matrix M with rankM = r contain only real elements. With regard to rank r, the product form can be attained

r

( m ... -) rank V = r

M=VW= V W rank W = r (C.47)

0 r r < n, r< m.

The general pseudo-inverse MI is defined by the following four equations

Explicitly, MI is given by

The pseudo-inverse M' is an (m, D)-matrix.

(C.48)

(C.49)

(C.50)

Inserting Eq.(C.50) into Eq. (C.48), i.e., VWWT(WWT)-I(VTV)-IVTVW = VW makes evident that it is not permissible to omit M neither in the left nor in the right half of Eq.(C.48).

Some more properties for the pseudoinverse:

(MI)I = M (M'l = (MT)I ((3M)1 = (lj(3)M' (if (3 =1= 0) (C.51)

(MTM)I = MI(MI)T (M'M)' = MIM (MMI)' = MM' (MB)' =1= BIMI (C.52)

rank M = rank MI = rank MTM = rank M'M tr M'M = rank M . (C.53)


C.7.2 General Pseudo-Inverse and a General Matrix Equation

Given the matrix equation with known matrices P, Q, L and unknown M

PMQ=L. (C.54)

All matrices are assumed of adequate dimensions. If the condition

(C.55)

is satisfied, the solution of Eq.(C.54) is (Rao, C.R., and Mitra, S./(., 1971)

(C.56)

for any matrix Z. This result can easily be proved by substitution.

The condition Eq.(C.55) is necessary and sufficient for Eq.(C.54) to have a solution.

(i) Necessity: It is unimaginable that the equation to be solved, Eq.(C.54), and the condition Eq.(C.55) for the existence of the solution are incompatible. Hence, it is a necessary condition. Consistency of Eq.(C.54) and (C.55) requires that L of Eq.(C.54) substituted into the left-hand side of Eq.(C.55) gives a true statement

ppl(PMQ)QIQ = pplpMQQIQ = PMQ = L . (C.57)

(ii) Sufficiency: Substituting the solution Eq.(C.56) into Eq.(C.54) yields the expression Eq.(C.55) irrespective of the value of Z. Thus, Eq.(C.55) is sufficient.

C.7.3 Right-Pseudo-Inverse

If n = r and V = I,. then M = Wand the right-pseudo-inverse is

(C.58)

Example: Specializing Eqs.(C.54) and (C.56), namely Mp = y, M E nn~xnp, np > nm . If MM'y = y exists (M' = MIR) then the solution p is

(C.59)

where z is any np-vector. This can easily be proved by substituting the solution into Mp = y and using Eq.(C.48)

Mp = MMly + M(lnp - MIM)z = y + Mz - MMIMz = y . (C.60)

Differentiating pTp with respect to z yields z = 0, i.e. the special case p = Mly has minimal norm and M' = MIR. The solution Eq.(C.59) can. be generalized to solve Eq.(C.54) by rewriting Eq.(C.54) into vector form using Eq.( 4.40)

(QT 0 P)col M = col L

col M = (QT 0 p)IR col L + [I - (QT 0 p)IR (QT 0 P)Jcol Z

col M = (QT 0 p)IR col L + col Z - (QTIR 0 pIR)(QT 0 P)col Z

M = plLQI + Z - plpZQQI .

(C.BI)

(C.62)

(C.63)

(C.64)

End of Example

C.8 General System Inverse 647

C.7.4 Left-Pseudo-Inverse

Considering m = rand W = Ir then M = V, the left-pseudo-inverse is

(C.65)

and M'LM = Ir . The left-pseudo-inverse is frequently applied in linear regression. Applying the leftpseudo-inverse, only, the superscript L is often omitted.

Example: Specializing the general pseudo-inverse to the left-pseudo-inverse,

Q = B (input matrix), P = 1m, BE cnxm , then L = 1m .

Eq.(C.55) yields BIB = 1m or BU = BIL. From Eq.(C.56) one has

M = BI + Z(ln - BBI) = BUL + Z(ln - BBIL) . 0

Note that for M E nnxm , n > m ,

det(sln - MMU) 1 1

sn det(ln - -MM') = sn det(lm - -MUM) s s

sn-m det(slm _ 1m) = sn-m(s _ l)m .

C.7.5 Projector Properties of MMtt and MttRM

(C.66)

(C.67)

(C.68)

(C.69)

The matrix product, in this order, possesses the property of a projector matrix, in both cases left-pseudoinverse and right-pseudo-inverse.

C.S General System Inverse

Consider the system x(t) = Ax(t) + Bu(t)

y(t) = Cx(t) + Du(t)

C.S.l Case r < m

x(t) E nn, u(t) E nm

y(t) E nr, rank D = min(r,m) .

(C.70)

(C.7l)

Eq.(C.71) is solved with respect to the input vector using right-pseudo-inverse and an arbitrary m-vector z as shown in Eq.(C.59)

where

(C.72)

(C.73)

Deriving z from x by defining an arbitrary (m, n)-matrix N, i.e., z = Nx, from Eqs.(C.70) and (C.72), it results

Ax + B[DIR(y - Cx) + (1m - DIRD)Nx]

[A - BDURC + B(Im - DIRD)N]x + BD"Ry .

(C.74)

(C.75)

Eqs.(C.75) and (C.72) as a system with left-hand vectors (x, u) is called general system inverse of the system with left-hand vectors (x, y).

C.S.2 Case r = m

Equating DUR = D-1 yields a unique inverse

u D-1y _ D-1Cx

X (A - BD-1C)x + DD-1y . (C.76)

(C.77)


C.S.3 Case r > m

With regard to r > m, Eq.(C.71) is overdetermined and incompatible in this version. Considering a dual system (Sinswat, V., 1976)

X' = AT x' + CT u ' y' = BTx' +DTu' (C.78)

with an r-vector y' as an input and an m-vector u' as an output, the aforementioned calculus can be applied again. Using the right-pseudo-inverse of D T , i.e.,

(C.79)

the transpose of the left-pseudo-inverse comes into action, corresponding to the fact that the system matrices are defined as transposed matrices in the case of Eq.(C.7S).

C.9 Pseudo-Inverse and Singular-Value Decomposition

If the singular-value decomposition of a complex matrix G is given by G = U1:VH , the pseudo-inverse amounts to

where 1: = (1:r a) a a ' r = rankG

since the basic properties of the pseudo-inverse are satisfied by the decomposition cited above.

C.lO Pseudo-Inverse of a Matrix Partitioned into Submatrices

If a matrix M is partitioned into matrices A through D

( A B) (A) -1 .' (A) . -1 ( A M = C D = C A (A: B) = C (I: A B) = C

(C.SO)

(C.SI)

and the submatrix rows in M are linearly dependent, i.e., the second row results from the first one by multiplying with CA -1 (A square and nonsingular) then D = CA -IB , and rankM = rank A . In this case, the pseudo-inverse can be calculated in partitioned form

(C.82)

C.l1 Pseudo-Inverse of a Matrix Partitioned into Columns

Using Greville's method, the pseudo-inverse can be calculated in a recursive way. The matrix M is disassembled into columns M = (mcl.mc2, ...• mcn) . On the other hand, the matrix M is recursively composed by defining

With the definitions

M ~a -I A' {aT 1 j(aTl al) 1 = 1 - aT

(ak - Mk_1dk)'

if al 1: a if al = a

hi = { (1 + df dk)-ldfML if ak - Mk-ldk 1: a if ak - Mk_ldk = a .

Taking these abbreviations, the pseudo-inverse associated with Mk V k = 2, ... ,n is obtained

(C.S3)

(C.S4)

(C.S5)

(C.S6)

C.12 Successive Application of Right and Left-Pseudo-Inverse Operator

C.12 Successive Application of Right and Left-Pseudo-Inverse Operator

According to the relation np and nrn , the following identities hold

Checking, e.g., Eq.(C.87)

(MIL)IR = [(MTM)-IMTf ([(MTM)-IMTJ[(MTM)-IMTf)-1

(MIL)IR = M(MTM)-I(MTM)-IMTM(MTM)-Ifl = M

649

(C.87)

(C.88)

(C.89)

(C.90)

(Barnett, S., 1971; Rao, C.R., and Mitra, S.K., 1971; Lancaster, P., and Tismenetsky, M., 1985) .

C.13 Conditioning and Scaling

C.l3.l Condition Number of a Matrix

The condition number ",,[A] of a matrix A is defined for any unit vector u and v

",,[A] '" max IIAu[[F maxllullp=1 IIAuliF urn,x[A]

lIullF=I, II v IlF=1 IIAvllF minllvlIF=1 IIAv[[F urnin[A]

",,[A] IIAII, IIAv[[F = 111~1[:i;1 = IIAII,IIA-III, ?: 1 .

minIlA-'AvIIF=1

Using the relation between norm and maximum/minimum modulus of eigenvalues,

",,[A] = ~ > m~", [.AdA] [ _1_ - mmi [.AdA] [ IIA-III,

(C.91)

(C.92)

(C.93)

Solving, e.g., Ay = b where A is nonsingular, b and A are only known to be in the limit of a tolerance 6b and 6A, an estimate of the error 6y of the solution y is (Franklin, J.N., 1968)

(C.94)

A matrix A is ill-conditioned if "" [A] is large. Then, the solution of the equation Ay = b or the computation of the inverse of A or of the eigenvalues .A[A] becomes inaccurate, especially as far as the smaller eigenvalues are concerned. Conditioning and scaling methods help to overcome this problem.

A general approach transforming a given (n, n )-matrix A into a decomposed version D is considered as D = T[A] . Some transformations T are listed in Table C.l. With regard to their quadratic structure, they are referred to as spectral decompositions.

C.l3.2 General Spectral Decomposition

In the general case of spectral decomposition A = EDET, the matrices D and A are congruent matrices. Substituting x' = ET X to the quadratic form a(x)

(C.95)

If D is desired diagonal, D = diag di, the quadratic form turns out a sum a(x) = L~ di:ct 2 •


Table C.l: Various decompositions (matrix transformations)T

T I name, details, main properties

A = EDEl general spectral decomposition (congruent matrix transformation)

A = TDT-1 eigenvalue decomposition, D ... diagonal (A=AT --... T-l = TT --... E=T) (orthogonal matrix transformation)

A=EDE scaled decomposition A=D2 square root decomposition

A=DD1 Cholesky decomposition

C.13.3 Eigenvalue Decomposition

Assume the matrix A symmetric, only. The modal matrix T = T[A] associated with A then becomes TT = T-1. The matrix T has orthogonal property. Using T for transformation E := T, ET = E-1 . Since AT = A Pi = ai and from Eq.(B.95)

n

A = T( diag Ai[A])TT = L: Ai(aiai) (C.96) i=l

n

A-I = (TT)-l(diag A;1[A])T-1 = T(diag A;l)TT = L: A; 1 (aiai) . (C.97) i=l

Since E = T, D = T-l AT and det D = det A, the matrices D and A are similar, in this special case.

C.13.4 Orthogonal Transformation

Applying an orthogonal transformation T to second-order polynomials or quadratic forms, it can be arranged that the transformed variables Vi are non-interacting. Consider a quadratic polynomial in the vector-valued variable x

(C.9S)

By the linear transformation x = TT v, the scalar J(x) is transformed to

(C.99)

The transformation matrix T is defined as the modal matrix associated with Q . Invoking Eq.(B.16),

TT = T-1 and TQTT = TQT- 1 = diag Ai[Q] = A (C.lOO)

(C.lOI)

The gradient of J(v) with respect to v is 8J(v)/8v = 2Av + 2Tb . Since A is diagonal, the gradient component [8J(v)/8v]i only depends on the component Vi. In order to find the optimum of J(v), the variables Vi may be adjusted independently of each other. Moreover, the optimum is given by (Roberts, P.D., 1967)

v* = -A-1Tb, (C.I02)

Important applications are given in the field of parameter identification and optimal system design.

0.13 Conditioning and Scaling 651

C.13.S Scaled Decomposition

Consider A symmetric, only, and E diagonal, E := diag ei, E = ET, then

A =EDE. (C.103)

The matrix D is the scaled version of A. The elements ei are assigned to

e· - { .jjA;";T ,- 1 if Aii # 0 if Aii = 0 (initial data).

(C.104)

In the following paragraphs, Aii has to be replaced by 1 if the initial data Aii are zero. SUbstituting Eq.(C.104) into Eq.(C.103) and rearranging yields

(C.105)

The matrix D is symmetric since A is symmetric. The modal matrix associated with D is T = T[D] . With regard to the symmetry of D it results TT = T-1 and D = T( diag Ai[D])TT . Note that T[D] is used, not T[A]. Finally, the scaled decomposition of A is

A = EDE = ET(diag A;[D])TTE = ET(diag Ai[D])(ETf .

Inverting Eq.(C.106), the scaled decomposition of A- 1 is achieved

A-1 = E- 1T(diag Ai1)(E-1Tf .

(C.106)

(C.107)

Partitioning the matrices ET and E- 1T into columns (ET).i and (E-1T).i, respectively, and denoting the kth element with the additional subscript k, the relations can be proved

(C.10S)

where di is the eigenvector of D . The scaled decomposition of A and A- 1 can be written as the sum of Ai-weighted dyadic products

n n

A = L Ai[D] (ET).i(ET)·T A -1 = L Ai1[D] (E-1T).i (E- 1T).[ . (C.109) i=1 .=1

Example:

A = (1010000 \0) E = (1000 01 ) n,(A) = I A[A]lmax/l A[A]lmin = 10000.01 I 0.99 = 10101

Scaled version of A: D = (O.gl ~) (1010000 \0) (O.gl ~) = (0\ Oil) Eigenvalues A[D]: A1 = 1.10, A2 = 0,90; n,(D) =1.10 I 0.90 = 1.22

M d I . T (0.7071 0.7071) (dO dO) o a matnx: = 0.7071 -0.7071 = 1 2

( 70,71 70,71) ( . ) ET = 0.71 -0.71 = (ET).l: (ETh

. ( 70 71 70,71) (1.1 0) ( 70,71 0.71) Matnx decomposed: A = 0.'71 -0.71 0 0.9 70,71 -0.71

-1 (0.0071 0.0071) ( -1) : ( -1 ) ) E T = 0.7071 -0.7071 = E T.1 . E T.2

(C.llO)

(c.m)

(C.1l2)

(C.113)

(C.114)

(C.115)

(C.116)

d -1 (0.0071 0.0071) ( 0.9091 0 ) (0.0071 0.7071) Inverse decompose: A = 0.7071 -0.7071 0 1.1111 0.0071 -0.7071 .

(C.1l7)

End of Example


C.13.6 Square Root Decomposition

Considering nonsingular A = DDT and symmetric D, the square root decomposition is obtained. The matrix D is named the square root of A . That is, A = D2 and D = .,fA . In view of the eigenvalue decomposition Eq.(C.96),

A = T(diag ~;)TT = T Jdiag ~; TTT Jdiag ~; TT . (C.lIS)

The term TTT in Eq.(C.lIS) is 1 . Thus, it results D = T ~ TT .

C.13.7 Cholesky Decomposition

Consider nonsingular A, only, and a lower triangular matrix L with elements L;j = 0 Vi < j . The decomposition

or n

A;j = LL;.L;. Vi,j = 1 ... n v=l

is chosen. Then, L;j can be found out by a simple recursion formula. If a new variable y is defined when solving Ax = b then

LLT x = b and LT x = Y Ly=b.

(C.1I9)

(C.120)

Both equations for y and x can easily be solved on account of the triangular nature of L and its simple inverse.

C.14 Orthogonalizing

A given set of data Z;, i.e., a set of n-vectors, possibly contains linear dependent information. Composing these data Z; to a matrix Z = (z;, Z2, ... ), the information of Z should be transferred into the column information of a new matrix X. The aim is to receive orthogonal columns x;, only. The matrix X;-l is the submatrix of X, containing columns Xj (j:$ i-I) already transformed by previous operations

X; = (Xl,X2 ... x;) X = (Xl,X2 ... Xq) X; E nnx;, X E nnx q , x; E nn, i < q:$ n. (C.121)

Remember the orthogonality properties treated when deriving Eq.(D.37). Thus, the operator matrix (premultiplication matrix) to be applied to Z; , in order to obtain x; orthogonal to each column of Xi_l , is (I - X;-lX!:l)' i.e.,

x; = (I - Xi-1X!:1)Zi .

Multiplying xT with X i_ l yields the null row OT

zT(1 - Xi-1X!:lfxi-l = zT (Xi_ l - Xi_l(xLXi_l)-lxLxi_l) = zTo = OT .

An alternative way to achieve Xi is to apply the pseudo-left-inverse X!: 1 to the vector Zi

Pi = (XLl Xi_d-lxL1Zi .

(C.122)

(C.123)

(C.124)

The parameter vector Pi fits Zi optimally into the Xi_l-space. The vector Xi-1Pi is the optimal least square regression of Zi. The residual Zi - Xi-1Pi is orthogonal to all previous vectors from Xl until Xi-l' This residual is declared as x;

(C.12S)

and is chosen to complete Xi-l to X; . The algorithm is started with Xl = Zl , without any regression manipulation. After having regressed

all input vectors Zi and having computed the residuals, all the vectors Xi are orthogonal.

Example:

(C.126)

C.14 Orthogonalizing 653

( 1) (-0.02) 5) 1 -0.36 2 0.30 1. Xl . (C.127)

End of Example

If a zero column Xi appears, the linear dependence is obvious. This datum Xi = 0 has to be cancelled to avoid singularity of xLlxi- 1 . Linear dependence occurs if at least one column of Xi-l is a linear combination of the other ones. Then, rank(XLIXi- l ) < n, detXLIXi- 1 = 0 and xLlxi- 1 is singular.

Example:

XrX2 = C~ ~). (C.128)

End of Example

Appendix D

Linear Regression and Estimation

Linear regression is considered as the deterministic problem of minimizing the error between measurement sequences and a deterministic model. The aim is to find the optimum model parameters based on least squares. Linear models are taken into account, only. The model is defined as

Mp=y (0.1)

The matrix M is an (nrn,np)-matrix with fixed elements, p (usually) a vector with np elements of unknown parameters and y an observation vector with dimension nrn (the number of measurements or observations).

D.l Parameter Demarcation

When measurements are available to state nrn linear equations with np parameters and if nrn < np ,

the system is underdetermined and a solution does not exist. The shape of the matrix M is row-like. Although there is no solution the matrix M defines a demarcation which relation between the elements of p exists.

Within the scope given by M and y, find p such that the least square pT p or the Frobenius norm !!p!!} is minimum. This investigation can be formulated with the help of a vector-Lagrange-multiplier .\

(0.2)

8 8p [pTp+.\T(Mp_y)]=O p = -0.5 MT.\ . (0.3)

In view of Eq.(O.l) and (0.3),

M( -0.5 MT.\) = y and .\ = -2(MMT)-ly . (0.4)

The matrix MMT is preassumed nonsingular. Combining Eq.(0.3) and (0.4), the result is

(0.5)

The starred parameter p* is the optimal (minimal least square) parameter vector, complying with Eq. (0.1).

Eq.(0.5) shows the linear relationship between y and p*. The matrix operator to be applied to y is the right-pseudo-inverse MIH It is a right-inverse of M because M has to be multiplied by MIR from the right to yield the identity matrix In m.

Example 1: Suppose that nrn = 2 measurements are available and np = 3 parameters are to be determined, one has two linear equations with three variables PI, P2 and P3. Each equation corresponds to a plane in Fig. 0.1.

M=( o -0.5 -1.01506)

0.5 (MMT)-I = (0.7279 0.2911)

0.2911 0.9165 (0.6)

656 D Linear Regression and Estimation

} equation

Figure D.l: Illustration of two linear equations with three variables

MIR = MT(MM1')-1 = -0.0729 0.7710 , y = ( 0.5 ) ( 0.1456 0.4583)

-0.8375 -0.3349 1.1506 p* = MIRy = 0.8507 (

0.6001 )

Vectors perpendicular to the two planes given by M:

( 0) -05 -1.1506

and

Vector g in the direction of the intersection of both planes:

( -0.5 x 0 + 1.1506 Xl)

g= -1.1506xO.5-0xO o x 1 + 0.5 x 0.5

Checking if g is perpendicular to p*:

( 1.1506 )

-0.5753 . 0.25

-0.8041 (0.7)

(0.8)

(0.9)

gT p* = 0.6001 x 1.1506 + 0.8507( -0.5753) - 0.8041 x 0.25 = 4.23 x 10- 5 = O. 0 (0.10)

Example 2: Optimal initial conditions for observers. The initial condition is usually chosen X(O) = 0 . This default choice does not reduce x(t) as much as

possible and is not compatible with the initial output y(O) . The optimal choice is given by IIx(O)IIF -+ min, subject to Cx(O) = y(O) (Johnson, C.D., 1988) which is exactly the problem posed in this section. Substituting M := C, y := y(O), p:= X(O), the optimal initial condition is

(0.11)

End of Examples

D.2 Interpolation

Consider the case nm = np' The number of measurements is supposed equal to the number of parameters to fit the model of Eq. (0.1). The measurements are preassumed linearly independent. Eq.(D.1) with square matrix M can be solved

(0.12)

The inverse exists with regard to the independence of the measurements. The unique solution in the parameter vector p* provides a unique interpolation of the observation y.

D.3 Weighted Least Squares Approximation 657

D.3 Weighted Least Squares Approximation

In the case nm > np , more measurement information is given than one needs at minimum to solve Eq.(D.l), see Fig. D.3. The system of equations of this model is overdetermined. Hence, Eq.(D.l) must be rewritten to

Mp=y-t: (D.13)

in order to have a consistent system of equations. The solution of Eq.(D.13) yields an optimal approximation of the measurements without rejecting

any information. Optimal approximation is gained by postulating p* in such a way that an index of performance is minimized. In most cases this index of performance e is defined as the sum of squares t:~ produced by a special p

"-e= L t:r = t:Tt: min. (D.14)

1'=1

This case is shown in a subsequent section in detail. The function e can also be established in a more general way, using different weighting factors for t:; and o.Oj thus forming a weighting matrix W

(D.15)

where W denotes an arbitrary positive definite matrix, normally symmetric. Following the necessary condition for the minimum of e with respect to p yields

oe 0 0 - = -[(-Mp+yfW(-Mp+y)] = _[pTMTWMp - yTWMp - pTMTWy+yTWy] = 0 op op op

(D.16) [MTWM + (MTWMf]p - (yTWMf - MTWy = 0 (D.17)

p = p* = (MTWM)-IMTWy . (D.18)

Calculating the second derivative of e with respect to p, this derivative matrix has to be positive definite. IfW=WT

02e opTop = MTWM > O. (D.19)

Both Eqs.(D.18) and (D.19) together are a necessary and sufficient condition for achieving a minimum of e with respect to p. The condition Eq.(D.19) is satisfied if W is a positive definite matrix.

If the matrix H = MTWM is singular, linear independent information must be cancelled. If cancelling leads to nm < np then Eq.(D.5) has to be employed. After cancelling, the matrix H red plays the same

role as M in Eq.(D.l). Using the right-pseudo-inverse H~~d yields the optimal solution as in Eq.(D.5). A necessary condition to uniquely estimate p in the model Mp = y when applying the least square

method is the nonsingula.rity of MTM or det MTM '" O. It is a necessary condition even in the weighted least square case. This can easily be realized by inverting the assertion: If one has detMTM = 0 then also det MTWM = O. It is not a sufficient condition: If det MTM '" 0 then det MTWM '" 0 or = 0 is possible, depending on W. Thus, for the model Mp = y the statement det MTM '" 0 is denoted identifiability condition.

Example: Four simple cases are chosen to illustrate various assumptions on nm , np and rank M. To obtain definite solutions rank M 2: np must be satisfied.

Case a) nm = 3, np = 2 :

M=(~ !) (D.20)

Third equation (row) linear dependent on the first and second one: rank M = 2

MTM _ (14 16) - 16 21 ' p = (1 2f. (D.21)


Case b) Rm = 2, Rp = 2 :

M=(: :) (0.22)

Second equation linear dependent on the first: rank M = 1

T (4560) MM= 6080' (0.23)

Case c) Rm = 3, Rp = 2 :

M= (: :) -3 -4

(0.24)

Second and third row linear dependent on the first one: rank M = 1

(0.25)

Case d): Case c) continued: The equations

H" = MT = (198) p y 264 (0.26)

are linear dependent just as Eq.(0.24). The system contains the same information as the first equation in Eq. (0.24) or any other one: (3 4)p = 11 . After cancelling useless information in Eq.(0.26),

H •• d = (54 72), p = H;.d(H.ed H;ed)-'(MT y) •• d = H~~d (MT y) •• d (0.27)

" = (54) [(54 72) (54)]_, 198 = (0.0067) 198 = (1.32) p 72 72 x 0.0089 1.76 . (0.28)

End of Example

The error vector e at the minimum p*, e(p*) = e* = e is named residual e . In statistic estimation theory the expectation E[P*] turns out to be the optimal estimation p . Anticipating this fact, the superscript * is frequently omitted and the optimal parameter p* is equated with p, i.e. p* = p . The quantity Mp = jis the regression value of y . Fig. 0.2 gives an impression in three dimensional sample space.

The weighted squared error index of performance C = eTWe = C(p) depends on p. After having minimized C, there remains the residual sum of squares C* = e*TWe* = C(p). In view of this, substituting e = e* = -Mp + y and pinto C yields

GtvLS = (-Mp + y?W( -Mp + y) = pTMTWMp - yTWMp - pTMTWy + yTWy

GtvLS = yTWTM(MTWM)-"TMTWM(MTWM)-'MTWy

_ yTWM(MTWM)-'MTWy _ yTWTM(MTWM)-"TMTWy + yTWy .

The first and second term above can be cancelled. Supposing W symmetric, simplifying yields

(0.29)

(0.30)

By substituting y = Mp + e, the expression Eq.(0.30) is rewritten by elementary algebraic operations

Gtv LS (Mp + e)TW[I - M(MTWM)-'MTW](Mp + e)

= (pTMT + eT)[W - WM(MTWM)-'MTW](Mp + e) = [pTMTW + eTW _ pTMTWM(MTWM)-'MTW

-eTWM(MTWM)-'MTW](Mp + e) eTWMp - eTWM(MTWM)-'MTWMp

+eTWe _ eTWM(MTWM)-'MTWe

GtvLS = eTW[I - M(MTWM)-'MTW]e . (0.31)

The matrixW[I-M(MTWM)-'MTW] weights the squares ofy and e in such a way that equal results GtvLS are obtained (Rosen, J.B., 1960).

DA Ordinary Least Squares Approximation 659

(residual vector in error space)

Figure D.2: Sample space, estimation space and error space in the case of nm = 3 measurements and np = 2 parameters

D.4 Ordinary Least Squares Approximation

Setting W = I, the weighted least squares solution is specialized to the ordinary least squares one. The ordinary least squares solution is

(0.32)

The matrix MIL is the left-pseudo-inverse of the matrix M and is of dimension np x nm (vice versa to the dimension nm x np of the model matrix M).

For the sake of comparison, previous results are repeated: If solutions of Mp = y exist then either M-1y is the unique solution or p = MIRy is the solution of minimum norm (length) IIpIiF. If solutions of Mp = y do not exist then the sum of squares £T £ of deviations £ = -Mp + y is minimized by p = MILy.

For automatic control purposes the left-pseudo-inverse MIL plays a dominant role. If only the leftpseudo-inverse MIL is used within a section the superscript L is omitted in MI , for simplicity. In the ordinary least squares case the resu lting residual least squares becomes more simple

C1s = yT(I - MMI)y = eT(I - MMI)e .

Corresponding with

Mp

and with y = Mp

y - £ (arbitrary p)

y - e (optimal parameter P)

(0.33)

(0.34)

(0.35)


I vector

M~p=Mp

Mp=y e(= y)

Table D.l: Vectors and sum of squares

I squared amount of the vector (distance)

C(p) = cl·c (sum of squares in sample space representing the squared distance from any point Mp in estimation space to point y)

C(p) - C(p) regression sum of squares residual sum of squares Crnin = ~ S = C(p) = e T e

the squared amount of the distances Ilyll}, lIyll} and lIell} is investigated. With regard to the fact that e is minimum in quadratic sense, y = Mp and e = c(f» are vertical. Applying Eq. (0.32),

(0.36)

Hence, the above-mentioned distances y,y and e satisfy Ilyll} = lIyll} + lIell} (see Fig. 0.2). The regression sum of squares is defined by the Frobenius norm lIyll} = yT y . In terms of the observation vector y the regression sum can be expressed as

(0.37)

Referring to the orthogonality between y and e, the same result is obtained from (MPVy or (Mf»T y. The projection of the vector y on to the plane spanned by ml and mz in Fig. 0.2 is identical to y. The squared length of the vector y is given by yT y, the optimal C by G!s in Eq.(0.33). Combining Eqs. (0.37) and (0.33),

(0.38)

This expression G!s is named the residual sum of squares. Thus, the residual sum G!s plus the regression sum lIyll} yields the squared sum of the observation vector lIyll}

G!s + lIyll} = Ilyll} . (0.39)

The vector quantity y = y + e = y + Y is used to define y = e. The residual e frequently is declared as the minimum estimation error y (deterministic equation error vector), see Table 0.1.

Combining Eq.(0.18) and y = Mp* ,

(0.40)

where TM is an idempotent transformation matrix. Applying TM twice, three times etc., the same result is to be expected. This can easily be verified in Fig. 0.2, by inspection.

In Fig. 0.2 one has to distinguish between three spaces (Draper, N.R., and Smith, H., 1966). An additional space is sketched in Fig. 0.4.

(i) The urn-dimensional sample space containing the observation vector y and all columns of M. Each of the up columns of M and the vector y is represented by a single point in the sample space.

(ii) The up-dimensional subspace named estimation space defined by the Dp column vectors of M.

(iii) The error space assembled by the error vectors" and the residual vector e. The dimension of the error space is Urn - up. In Fig. 0.2 it is a cone with the apex pointed out by y.

(iv) The parameter space assembled by the components of p.

Example: Linear regression of two parameters on the basis of three observations Drn = 3, up = 2. a) Non-faulty observations:

( 0

M = -0.5 -1.1506

( 0.5 )

y = 1.1506 0.3466

(0.41)

D.5 Left Inverse and Right Inverse. Mnemonic Aid 661

MTM = (0.7279 0.2911) 0.2911 0.9165

MIL = (MTM)-IMT = (0.1456 -0.0728 -0.8375) (D.42) 0.4582 0.7709 -0.3350

* = M'L = (-0.3012) p y 1.0000· (D.43)

b) Faulty observations:

(0.6 )

y = 1.3 , 0.5

• _ MIL _ (-0.426) p - y - 1.1096 ' (

0.5548 ) Y = Mj) = 1.3226 ,

0.4902 e = y - Mj) = ( ~O~~;;6 )

0.0098 (D.44)

Check if e and sample space are orthogonal: e™ == (0 0). Residual sum of squares: eT e = Cis = C{j») = 0.0026, lIeliF = 0.0515 . Regression sum of squares: yTy = 2.2974, IIYIIF = v'2.2974 = 1.5157 . Norm of observation: yT y = 2.3000 , lIyllF = 1.5166 . Check: eT e + yT Y =? = yT y 0.0026 + 2.2974 = 2.3000 . Parameter space (see subsequent section):

!:!..G = 0.7279 P~ + 0.5822 P1P2 + 0.9165 P~ . (D.45)

End of Example

D.S Left Inverse and Right Inverse. Mnemonic Aid

Considering Mp = y, M E nnm xnp and comparing (i) the least squares approximation nm > np with (ii) the parameter demarcation nm < np , note the following facts when solving Mp = y, i.e. when "separating" p as a result.

(i) Least squares approximation: j) = MILy can be achieved by "multiplying" Mp = y by MIL from the left although none of the scalar equations Mp = y is true (t: omitted) but the premultiplication by MIL is a good mnemonic aid.

(ii) Parameter demarcation: p = MIRy cannot be obtained by a multiplication operation from the statement Mp = y although the result can easily be proved by multiplying the result p = MIRy by M from the left. Note that each scalar equation ofMp = y is true but multiplication ofMp = y by M,L from the left is inadmissible although one is induced to do this: MTM in the case nm < np

has dimension np x n p , rankM™ = nm . Hence, the inverse (MTM)-l and MIL do not exist.

D.G Complex Matrix M

If the matrices involved are complex but only a real parameter vector p is admissible the result

is given by

p* = argminGwLs subject to !;)<m p = 0 p

w>o.

(D.46)

(D.47)

D.7 Sum of Errors and Residual Sum in Parameter Space

The parameter space is defined by the np-dimensional parameter vector p. The sum of squared errors eT e is

(D.4B)

The smallest value Cis = G(I» = yT y - yTMMly is subtracted from G(p), thus obtaining

(D.49)

662

Least squares approximation. Left-pseudo-inverse. System overdetermined if E is neglected:

Left- ~ pseudo-inverse, u-'

Result:B

D Linear Regression and Estimation

Parameter demarcation. Right-pseudo-inverse. System underdeterminecl:

n, ~ Given d.t.:~ x P = ~ nm<n,'

nm ~ lJ IIpIiF-> min

Figure D.3: Ordinary least squares approximation and parameter demarcation. Graphic interpretation: left-pseudo-inverse and right-pseudo-inverse

It can easily be checked that the formulations

~c = (p - p)T(MTM)(p - p) = [M(p - pW[M(p - p)l = (M~pf(M~p) = (Mpf(Mp) (D.50)

are identical to the previous version. The vector ~p = p is the parameter difference. The error sum difference to the residual sum expressed by ~C is equal to the squared sum of ~p weighted by the matrix MTM, as known from identifiability condition. The gradient of ~C in the parameter space reads as follows

(D.51)

If the columns mei of the matrix M = (mel, m e2 .. . m en.) are orthogonal 'tip, /I = 1 ... np then

if and if 1'=/1. (D.52)

The matrix MTM becomes diagonal, MTM = diag a • . The increment ~C and its gradient turn out as

n. ~C = ~pT(MTM)~p = ~pT(diag a.)~p = E a.(~p.)2

11=1

~~~ = 2(diag a.) ~p. (D.53)

The shape ~C does not contain cross products. Hence, the shape is symmetrical to the axes ~P •. The component /I of the gradient only depends on ~P •.

If MTM is diagonal recognize the following fact: Fixing P2 at an arbitrary value and minimizing ~C with regard to PI , the minimum PI min = PI is always the same, no matter which arbitrary P2 was initially chosen (see Fig. DA).

D.S Successive Estimation in Large-Scale Systems

In the field of large-scale systems, the parameter estimation is based on a decomposition technique. For small-scale systems, the model y = Mp+e is applied as given in Eq.(D.13). For a large-scale system, with regard to computational difficulties, it is troublesome to apply Eq.(D.13). Decomposing into L submodels yields

L

Yi = EMi;p;+ei ;=1

L

Yi E'R,nmi , pi,ei E 1lnpi , Mij E 1l"mi Xn,i, E npj = n p ,

;=1

L

E nmj =nm -

;=1 (D.54)

D.S Successive Estimation in Large-Scale Systems 663

P2 --------

np = 2 parameter space

ih = plmin

Figure D.4: Shape C(p) and contour lines of constant amount b.C in parameter space

The parameter vector P in decomposed form is pT = (pT pI ... pI) . The least squares estimation is obtained by minimizing

(D.55) i=l

with respect to Pi. Furthermore, try to find Pi in such a way that the interaction between the L sub models is minimized. Thus, an interaction variable

L

Wi= L i=l,i¢i

(D.56)

is defined, providing the influence from all subsystems j -I i to the subsystem i. Following Eq.(D.55),

L

C= L(Yi - MiiPi - Wi? (Yi - MiiPi - Wi) (D.57) i=l

has to be minimized with respect to Pi and Wi within the constraint of Eq.(D.56). By application of the method of the Lagrange multiplier, the new criterion function I is

L L

1= L(Yi - MiiPi - Wi?(Yi - MiiPi - w;) + >o.T(Wi - L Mijpj). (D.58) i=1 j=l.j;!i

Differentiating with respect to Pi, >o.i, Wi yields

L

Pi = (MEMii)-I[M[;(Yi - Wi) + L MJ,>o.j] (D.59) j=I,j;!i

L

Wi = L MijPj (D.60) i=l,j¢i

respectively, Vi = 1 ... L. The previous equations can only be solved by iteration using appropriate starting values for w;O) and >0.;0) in Eq.(D.59). Eq.(D.59) gives p;O). Taking this value, Eqs. (D.60) yield

w;l) and >0.;1), respectively. This interaction-prediction algorithm is carried out until some preset stopping

condition in >o.;k) and w;k) is satisfied, e.g., in the norm of the difference between consecutive quantities.


The above algorithm can be modified by substituting Eq.(D.60) into Eq.(D.59). In this way, a successive algorithm in pIt) is formulated. In order to process on-line measurements, a recursive version of this algorithm also was developed (see Sultan, M.A., et al. 1988), corresponding with the recursive algorithm given in the next section.

D.9 Recursive Least-Squares Estimation

In practice, measurements are recorded successively with time. An estimator version updating the latest parameter state is of interest. By processing recent data this recursive estimator provides an incremental complement to the hitherto existing parameter result. Up to the kth sample assume to process the measurement matrix M(k) and the measurement vector y(k). Advancing from sample instant k to k + 1, the measurement matrix M( k) is supplemented by a new row m T (k + 1) and M( k) turns out as the new measurement matrix M(k + 1).

Assume the model specified by the difference equation of nth order

y(k) + aly(k - 1) + ... + any(k - n) = blu(k - 1) + ... + bnu(k - n) (D.61)

and define the parameter vector p ~ (al ... an bl ... bn)T . Let L + 1 measurements be available from sampling instant k - L through k. Without measurement errors this equation is rewritten, applying the definitions m(k), M(k) and y(k)

y(k - L) - mT(k - L)p = 0, y(k-L+ 1)-mT(k-L+ l)p = 0, ..... y(k)-mT(k)p = o. (D.62)

Taking into account some measurement error e,

y(k) - M(k)p = 0 y(k) - M(k)p = e

-y(k -1)

m(k) = ------y(k - n)

u(k - 1) (

Y(k-L») y(k) = : E "R(L+l)

y(k)

u(k - n)

(

-y(k-L-l) -y(k - L)

-y(k - L - 2).. -y(k - L - n) u(k - L - 1).. u(k - L - n) ) -y(k - L -1) ..

M(k)= -Y(k~L+l)

-y(k - 1) u(k - 1) u(k - n)

M(k) = [m(k - L) m(k-L+l) M(k) E "R(L+I)x(2n) .

(D.63)

(D.64)

(D.65)

(D.66)

Starting the recursive algorithm, care must be taken of negative arguments and k 2:: L + n must be provided. Advancing from sample instant k to k + 1 and assuming enlarged measurement data from L + 1 to L + 2, the dimension of y(k) to y(k + 1) is increased by one

(y(k-L»)

y(k) = : E "R(L+l),

y(k)

( y(k - L) 1

: y(k) y(H 1) = y(·k) = ( y(k + 1) )

y(k + 1)

E "R(L+2) . (D.67)

Within the vectors m(k) and m(k+ 1) only a shift operation is carried out. Thus, the dimension ofm(k)

D.9 Recursive Least-Squares Estimation 665

does not increase when proceeding to m(k + 1).

-y(k-I) -y(k)

-y(k - n) -y(k - n + 1)

u(k - 1) u(k)

m(k) E n2n , m(k + 1) E n2n. (0.68) From m(k) = - - - - - get m(k+I) = - - - - - --

u(k - n) u(k-n+I)

From M(k) to M(k + 1) proceed by enclosing a row mT(k + 1). Thus,

M(k + 1) = ( M(k) )

mT(k + 1)

M(k + 1) E n(L+2)x2n . (0.69)

The least-squares estimator P(k) and p(k+ 1) resulting from (L+ 1) or (L+2) measurements between samples k and k + L or k + L + 1 is obtained by applying Eq.(0.32)

p(k + 1) = [MT(k + I)M(k + I)tlMT(k + I)y(k + 1), (0.70)

respectively. Making use of Eq.(C.23) and substituting

A := MT(k), B:= m(k + 1), C = D := 0, u:= y(k), v:= y(k + 1),

one has

MT(k + l)y(k + 1) = ,( m(k - L) m(k.- L + 1) ... m(k), m(k+ 1)) ( Y(~~)I) )

MT(k)

MT(k + l)y(k + 1) = MT(k)y(k) + m(k + l)y(k + 1).

Invoking Eq.(C.24) and substituting C = D = F = H := 0

A := MT(k), B := m(k + 1), E := M(k), G := mT(k + 1),

T _ ( M(k) )T ( M(k) ) _ (T: ) ( M(k) ) M (k + I)M(k + 1) - mT(k + 1) mT(k + 1) - M (k). m(k + 1) mT(k + 1)

MT(k + I)M(k + 1) = MT(k)M(k) + m(k + I)mT(k + 1)

is achieved. Taking Eqs.(0.73), (0.70) and (0.76) into consideration,

p(k + 1)

p(k + 1)

p(k + 1)

p(k + 1)

[MT(k + I)M(k + I)]-I[MT(k)y(k) + m(k + I)y(k + 1)]

[MT(k + I)M(k + I)]-I[MT(k)M(k)p(k) + m(k + I)y(k + 1)] [MT(k + I)M(k + 1)]-I[MT(k + I)M(k + I)p(k)

-m(k + l)mT (k + I)p(k) + m(k + l)y(k + 1)] P(k) + [MT(k + I)M(k + l)tlm(k + l)[y(k + 1) - mT(k + 1)P(k)] .

(0.71)

(0.72)

(0.73)

(0.74)

(0.75)

(0.76)

(0.77)

(0.78)

(0.79)

(0.80)

/ / -1---old estimate I "-- one sample step

. new meas,urement ahead prediction new estimate - (k Ik) correcting gain vector measurement y + 1 including I

(Kalman gain factor) ,(k) ~ sample k + 1 prediction error

Applying the matrix inversion lemma Eq.(C.13) to the inverse of Eq.(0.76), substituting and defining

B = CT := m(k + 1), (0.81)


the result may be considered as a covariance matrix

P(k + 1) = P(k) - !,(k)m(k + 1)[1 + mT(k + I)P(k)m(k + 1)tI,mT(k + I)P(k). (D.82)

"I,

Note that the bracket expression in the above equation is a scalar. Hence, it can be shifted to any other position within the matrix product

P( k + 1) __ P-,-( k-,-,Hc-I...c+_m_T....:.( k_+-,--1 );-P....:.( k--".)"m;7( k;-+-:,-;-;1 )~] "'-;"7P"":'( k-;);-m....:.( k-;-;+-,1 ),---m_T....:.( k_+_l ),---P....:.( k---,-) - 1 + mT(k + I)P(k)m(k + 1)

(D.83)

Postmultiplying the above numerator by m(k + 1) yields

P(k)m(k+ 1) + P(k)m(k + l)mT(k+ I)P(k)m(k + 1) - P(k)m(k + l)mT(k + I)P(k)m(k+ 1) . (D.84)

The second and third term are cancelled. Hence, by Eq.(D.83),

P(k)m(k + I) P(k + l)m(k + 1) = 1 + mT(k + I)P(k)m(k + 1) . (D.85)

The left side above corresponds to the correcting gain vector "I(k) in Eq. (D.80), the right side above equals "II (k), as defined in Eq.(D.82): "II (k) = "I(k). Finally, the resulting recursive algorithm is expressed by the following three equations

P(k)m(k + 1) "I(k) = 1 + mT(k + I)P(k)m(k + 1)

P(k + 1) = P(k) - "I(k)mT(k + I)P(k)

i>(k + 1) = i>(k) + "I(kHy(k + 1) - mT(k + 1)i>(k)] .

(D.86)

(0.87)

(0.88)

Note that the matrix M increases its dimension from sample to sample but the product MTM does not. The matrix P as its inverse also keeps the dimension (2n) x (2n) unchanged. For the sake of comparison with block-pulse function approximation, see Eq.(37.58).

D.lO Recursive Instrumental Variable Method

The recursive (and one-shot) least-squares method only yields an unbiased parameter vector if the error signal e is uncorrelated both with the input and output signal of the process u and y, respectively. If uncorrelated error signals cannot be preassumed the method of instrumental variables is used since it also works with correlated e. The instrumental variable vector Yaux is synthesized as the output of the instrumental model, see Fig. D.5

-Yaux(k - 1)

IDl1tu:(k) = - - - ----Yaux(k - n)

u(k - I)

_ ( Yaux(~-I) ) Yaux - : .

Yaux(k - n)

(0.89)

u(k - n)

The instrumental variable Yaux is generated by exciting the instrumental model with u and by declaration Yaux = mIux i>aux using an auxiliary parameter vector i>aux. To avoid stochastic dependence between Yaux and e, the auxiliary parameter i>aux is taken from the real i> after having passed PTI-delay. A memory is employed to generate the vector Yaux from the past scalars Yaux( k - i) . The recursive leastsquares algorithm runs as given by Eqs.(D.87), (0.88) and, additionally, by

(D.90)

D.ll Linear Estimation 667

~

" • instrumental model >: =' I

~ maux Your

~'== maux(k) m;uxPauz r-------. II memory

I ~ instrum ental variable "I;' t '/ • . a II Paux

u= (*-1) ) maux Y.- ~ (

u(k- n) f>aux(O)~ PTl

~ P(O) ~ ,(0) first-order low pass filter

recursive least squares algorithm u m=(-:) In

,(k)

F== P(k + 1) f>(k + 1)

Y p(k + 1)

Figure D.5: Recursive instrumental variable method

The matrix P(k) is a modified covariance P(k) = [M~ux(k)M(k»)-l and Maux(k) is the instrumental variable matrix

(0.91)

(Jakoby, W., 1985). The aim is to establish a signal Yaux strongly correlated with the undisturbed process output but weakly correlated with measurement noise (Isermann, R., 1988).

D.ll Linear Estimation

For x( k) scalar, the set {x( k)} with k = l...N is referred to as a scalar random process. The vector-valued variable {x( k)} denotes a vector process.

Linear expectation (mean): x ~ E{x(k)} ~ limN_oo k L~=l x(k) .

Variance: IT; ~ E ([xi k) - xF} ~ limN _00 k L~=l [xi k) - xF· Autocorrelation function: Rxx(-r) ~ E{x(k)x(k + r)} ~ limN_oo k L~=l x(k)x(k + r) . Autocovariance function:

cov [x(k)x(k+r») ~ cov (x,x,r) ~ E{[x(k)-x)[x(k+r)-x)} ~ Qxx(r) ~ X(r). (0.92)

If the signal is Gauss distributed (= normally distributed), then the process is completely determined by x and cov [x(k )x(k+r»). The process is denoted stationary if x and cov [x(k)x(k+r») are time-invariant.

Cross covariance function: cov [x, y, r) ~ E{[x(k) - x)[y(k) - jj]} = Rxy( r) - xjj ~ Qxy( r) . A random discrete-time process is named white if the signal x(k) has no stochastic interrelation to


earlier signals x(k - p). The covariance function of a white process (white noise) is

{ I if r = 0 OT ( r) = 0 if r t 0 (0.93)

where oT(r) is the Kronecker delta. The power density function is constant versus frequency V Iwl ::; 7r/T where T is the sampling interval.

For comparison: Continuous-time white random processes are characterized by a vanishing stochastic interrelation, even between instants of infinitesimally small time-shift. The power density function is constant for all frequencies up to infinity and signals may grow to infinity. Hence, a white continuoustime process does not exist in reality and is only imaginable as a limes. Kronecker delta in Eq.(0.93) is replaced by the dirac function o( r) .

If the stochastic signal contains several components x(k) = [XI (k), x2(k) ... xn(k)Y a vector process {x(k)} is established and a covariance function matrix is given by

cov [x, r] l>. cov x ~ E{[x(k) - x][x(k + r) - xV}

( cov [x1o XI, r] cov [XI, X2, r] coy [X2J Xl, r] coy [X2' X2, r]

· . · . · .

::: ) g Q •• (d g X(,).

(0.94)

(0.95)

Note that Qxx is positive semidefinite symmetric for r = 0 and that the main diagonal positions of Qrr( r) are given by the autocovariance functions.

D.I1.1 Parametric Models. Markov Processes

Markov signal processes (of first order) are defined by the conditioned probability distribution function p[x(k)lx(k - 1), x(k - 2) ... x(O)] = p[x(k)lx(k - 1)] depending only on the preceding x(k - 1). A signal process obeys the probability distribution above if it is generated by the first order difference equation x(k + 1) = aMx(k) + bMV(k + 1) . Higher-order time-invariant Markov signal processes are given by

(0.96)

If AM(k) and/or BM(k) are time-dependent the Markov process is time-varying. The process {x(k)} is a white Markov process if v( k) is white. Oefining

cov [x(k + 1), x(k + 1), r = 0] = cov [x(k + 1)] ~ E{[x(k + 1) - x][x(k + 1) - xf} ~ X(k + 1) (0.91)

and substituting Eq.(0.91) into (0.96) yields (calculations omitted) in the case of white noise v(k)

X(k + 1) = AMX(k)A1 + BMV(k)B1 . (0.98)

If AM is constant and stable and BM is constant for k --+ 00 then X(k + 1) = X(k). Using Eq.(4.40)

col X = (I - AM ® AM )-I(BM ® BM)col V . (0.99)

Assuming a positive semidefinite weighting function matrix R, the Markov signal process x( k) can be assessed globally by 1= E{xT(k)Rx(k)} "" 1= xTRx + tr [RX] .

D.I1.2 Observation as a Random Process

Techniques of linear regression yield deterministic linear models with weighted least squares 11£112 where £ is a vector of distinct errors. If there is a high number of data the error space is of high dimension. With regard to the high amount of data it is suitable to consider e: as a stochastic process.

The observation vector y now is regarded as the sum of the model function Mp plus an additive zero mean noise £ where the matrix M is given by y = Mp + £ . The parameter p should be estimated optimally with respect to a risk function cov p . Supposing a linear relationship p* = ry , in order to derive the optimal estimation p* from the observation y, and using the model e = -Mp + y

p = E{p*} = E{r(Mp + e)} = E{rMp} + E{re}. (0.100)

The second term in Eq.(O.lOO) is named bias b = E {re} . The bias can be separated into E {r} and E{e:} if both processes are uncorrelated. Regardless of E{r}, the bias b vanishes if e has zero mean as preassumed. The condition of being uncorrelated is met if e is a white noise (see Bard, Y., 1974; isermann, R., 1988). If b = 0 then only the first term remains in Eq. (O.lOO) P = E{rMp} = E{rM}p . In order to obtain p = E{p*} = p, the condition rM = 1 must be satisfied. Hence, r is a left inverse of M.

D.ll Linear Estimation 669

D.11.3 Minimum Variance Estimator. Gauss-Markov Theorem

Which linear estimator p = ry or which r matches the minimum of some scalar valuation of the covariance cov p ? The variance of the estimation error vector

cov p = E{[P - E{P}][P - E{pW} (0.101)

could be reduced since E{p} = 0 (unbiased estimator). The estimation error p = p - p, the optimum estimate p = ry and the model y = Mp +" are substituted in Eq.(0.101). Thus,

cov p = E{(p-p)(p_p)T} = E{[P-r(Mp+,,)][P-r(Mp+.Y} = E{(r,,)(r,,)T} = r(cov ,,)rT = rvrT (0.102)

where V is assumed a given function. As scalar valuation of cov p, the determinant is chosen. Hence, the objective is to minimize det cov p subject to the matrix condition p = ry or rM = I . Introducing a matrix of Lagrange multipliers A and selecting the trace as a scalar measure of the constraint, the resultant form det (rVrT) + tr A(rM - I) is achieved. The minimum with respect to r is obtained by differentiation

:r {det (rVrT) + tr A(rM - I)} = 0 (0.103)

Comparing Eq.(0.103) with the optimum weighted least squares estimation in Eq.(0.18) yields the following result. If the weighting matrix W is defined as W = V-I (the inverse of the covariance matrix V = cov ,,), the optimal linear estimator p = ry in accordance with Eq.(0.103) yields an unbiased estimate, additionally a minimal weighted risk function C = "TW" = "TV-I" and, finally, a minimum determinant of the variance of the parameter cov p = r(cov ,,)rT . Using biased and nonlinear estimators better results are available. The linear estimator discussed above is the best linear one and is unbiased but does not supply the least squares estimate at all (Hoerl, A.E., 1962; Hoerl, A.E., and Kennard, R. W., 1970 ).

The minimum variance estimator can only be realized if the covariance matrix of the equation error is known a priori. Otherwise it is known after a recursive process. Substituting Eq.(0.103) into (0.102), the variance in the minimum variance estimation case is (cov p)MV = (MTV-IM)-I . Then, substituting the identity matrix V = I in Eqs.(0.103) and (0.102) yields the variance in the least squares estimation case (cov P)LS = (MTM)-IMT(cov ,,)M(MTM)-I which cannot be simplified any further for a general matrix M. The covariance (cov P)MV is smaller than (cov p)LS.

The expectation of the ordinary least squares risk function C = "T" is now investigated, provided the standard assumptions are valid (additive zero mean non correlated measurement errors with constant variance cov" = ".21 ; nonrandom parameters; independent variables; no other prior information) (Beck, J. V., and Arnold, K.J., 1977). In the ordinary least squares case r is the left pseudo-inverse MIL. Thus, the expectation of C turns out

D.H.4 Estimation Sensitivity

Consider the case that the exact value of the covariance matrix V = cov " is not known but an erroneous (perturbed) symmetric matrix V p = V + .:l V is available as a basis for the linear regression formula Eq.(0.103). Then, rp is obtained and the erroneous estimation is Pp = rpy . Using Eq.(0.102) with the true value V but with the erroneous rp yields cov Pp = rp vr;1 .

A degree of inefficiency f/e can be defined. This efficiency factor is always greater than or equal to unity because p is the minimum variance estimation. An upper bound for f/e can be formulated

f:>. det cov Pp (1 + a)2 1 <f/e = • < ---

- det cov p - 4a where and H = ,;v; V ,;v;. (0.105)

The condition number a is given by the ratio between largest and smallest eigenvalue of H with Hermite

property, see Hoerl, A.E., and Kennard, R. W., 1970; Wilkinson, J.H., 1965.

Appendix E

Notations

E.1 General Conventions

• For detailed definition see boldface page in subject index.

• Boldface lower and capital letters denote vectors and matrices, respectively. Calligraphic letters denote sets.

• Closed set ~ or [) , e.g., 1 ~ a ~ 5 or a E [1,5) for a real number.

• Open set < or ( ), e.g., 2 < b < 4 or bE (2,4) for b real number.

• Set of integers { }, e.g., i = 1,2, ... 8 or i = {I, 2, ... 8} for i integer.

• 0 end of the proof, end of the example, end of discussion.

•

E.2

=, -I, t:>.

LO > >.0

II· lip or 0[·) II II 1.

Dots stand for undesignated variables.

Abbreviations and General Symbols

== equal, is different from, approximately equals, respectively equality by definition equality by simple substitution identity angle corresponding to the argument of the complex number positive definite (matrix) (P > Q means P - Q positive definite)

~. 0 positive and non-negative (matrix), respectively, element-by-element relation, ratio of size comparison p-norm boundary of the region r degree modulus (absolute value) of a complex number modulus (matrix), e.g., IMI =matrix[ IMijl) perpendicular to leads to

~e , ~m real part and imaginary part of a complex number, respectively e field of complex numbers en n-dimensional linear vector space over e, en == enXI e- set of complex numbers with negative real part (open complex left-half plane) e+ closed right-half complex plane o the empty space {ail set (with elements ai) V for all E is an element of, belongs to ¢ does not belong to ~ is a subset of (is contained in) or identical to u union (of sets)

672

(- .. ) F , F- I

£ £-1 Z', Z-I

H2

L~ L~. Loo N R.['] R Rn

R+ R V'p bX

logbx o o $

*

adj arg circ

E Notations

intersection (of sets) A=> B means: If the statement A (e.g. equation, inequality) is true then B is also true or, equivalently, "A is sufficient for B" or "A implies B", e.g., lRe A;[A) < -4 => A is stable. N ~ T means "N is implied by T' or, equivalently expressed as a necessary condition, "only if N then T', e.g., the system with state space coefficient matrix A is stable only if trA < 0 . implies and is implied, if and only if. E.g., B <;;; A ~ "Ix : (x E B) => (x E A) . inner product of two functions or vectors Fourier transformation and its inverse, respectively. Fy(t) = y(jw) or, if necessary, y(jw) Laplace transformation and its inverse, respectively. £y(t) = y(s) or, if necessary, y(s) z-transformation and its inverse, respectively Hardy space satisfying H 2-norm,

(i) for continuous-time systems: Hardy space of functions analytic in the open right-haif plane and square-integrable in the closed right-half plane

f(s)EH2 ~ suPjOO f*(u+jw)f(u+jw)dw < 00, 0'>0 -00

(ii) for discrete-time systems: Hardy space of functions analytic in the open unit disc (Izl < 1) and square-integrable in the closed unit disc

g(z) E H2 ~ sup f2~ g*(r ej9 ) g(r ej9 )dlJ < 00 . r<l Jo

Hardy space of complex-valued functions of a complex variable, (i) for continuous-time systems: Hardy space of functions analytic in the open right-half plane and bounded in the closed right-half plane f(s) E Hoo ~ suplf(u + jw)1 < 00,

0>0 or functions on Loo with bounded analytic continuation in the right-half plane, (ii) for discrete-time systems: Hardy space of functions analytic in the open unit disc and bounded in the closed unit disc g(z) E Hoo ~ suplg(r ej9 )1 < 00 ,

r<1 set of asymptotically stable transfer functions with IIGlloo < 00 . subspace of functions in Hardy space H 00 that are real rational space of functions Hilbert space of functions square-integrable on jR f(s) E L2 ~ fr Coo f*(jw)f(jw)dw ~ IIf(s)ll~ < 00 space of vector-valued functions extended space L~ Banach space essentially bounded on jR null space (kernel) range space (image) field of real numbers n-dimensional linear vector space over R, R n == R nx1

field of non-negative real numbers used as a prefix denotes real rational Nabla operator (gradient) with respect to p, 8()/8p exponential of base b logarithm of a real number x to base b generalized product corresponding to a polynomial product Kronecker product Kronecker sum convolution e.g., M(s) 1.=1 means function M(s) selected for s = 1 {a; EX: f(a;) = O} the set of elements of X having the property f(ai) = 0 adjoint of a matrix argminx(.), that is, value of x that minimizes () clockwise encirclements

E.3 Superscripts

cof cofactor (of a matrix) col column string cony convex hull, convex combination coy covariance matrix det determinant (of a matrix) diagn {ai} diagonal matrix of dimension n x n with entries ai dim dimension of a vector or a matrix (number of rows x number of columns) E{x} (expected) mean value of a stochstic process {x}, expectation operator expAt matrix exponential function of the matrix At gradp gradient operator, 80/8p h.o.t. higher-order-terms infu infimum over u, largest lower bound over u log logarithm to base 10 matrix[aij j arranging a matrix with entries aij min{a, b} selecting the minimum of a and b rank rank of a matrix row sign supu tr tr(i) trig vec

E.3

-1

* *

row string signum function (sign x = ±1 for x > 0 and x < 0, respectively, and = 0 for x = 0) supremum over u, least upper bound over u trace of a matrix generalized trace trigonal matrix stacking operator

Superscripts

inverse (of a matrix) (star) optimal (asterix) complex conjugate

673

+ +

e.g., 0+, i.e., 0 + € where € > 0, € --.. 0, e.g., t = 0+ means instant immediately following zero a+(s) is a polynomial free of zeros in the closed right-half s-plane

.L

.L o (n x m)

(k) Uj 8 ~

"

n

# #L, #R

a-(s) is a polyomial free of zeros in the open left-half s-plane all-pass extension (right) annihilator (of a matrix) corner matrix matrix M of dimension n x m (overline dot) first derivative with respect to time k-th derivative with respect to time jth Kronecker power specified bounds according to Eq.(12.80) specified bounds according to Eq.(12.66) (overline bar) mean value, linear expectation (overline bar) signals in the case of unperturbed plant (written as a superscript) piecewise linear (function expansion coefficient) (overline hat) (if necessary) referring to the frequency domain or the Laplace domain (overline hat) estimate (overline tilde) estimation error (casually) characterizing a modified signal block-pulse (function expansion coefficient) general orthogonal (coefficient vector) general pseudo-inverse left-pseudo-inverse and right-pseudo-inverse, respectively. BUL left-pseudo-inverse where B E cnxm and rankB = m < n. KIR right-pseudo-inverse where K E cmxn and rankK = m < n . right (-eigenvalue) left (-eigenvalue)

674

D Hadamard product h homogeneous solution H conjugate transpose (AH = A·T )

LI (L), RI left-inverse, right-inverse, respectively mo modal variable, modal coordinate op operator p particular solution R reciprocated (polynomial) R para-Hermitian transpose (of a matrix) T transpose w Walsh function (expansion coefficient)

E.4

er

00

(n x m)

OJ ();. c c cp CL d D e eq

f F H

ij I k L L L8 I, r

L, R m M MV nom o o

Subscripts

e.g., Fer means transfer matrix from the input r to the output e sum vector norm, largest absolute column sum vector infinity norm, largest absolute row sum matrix M of dimension n x m jth column of a matrix ith row of a matrix observer based compensator servo-compensator combined servo-compensator and plant closed-loop dead-time delay diagonally weighted or D-weighted Holder norm element by element equivalent sliding mode condition final Frobenius norm Hankel (singular value or norm) inner, see Eq.(30.166) (i, j)-partition of a matrix interval index denoting sampling instant L-step ahead prediction least favourable least squares left-coprime and right-coprime, respectively left and right polar decomposition, respectively measurement Markov process minimum variance nominal (value) (casually 0) initial outer plant perturbed system

p-norm 1I·lIp , p-measure I'p 1I·lIp function norm e.g., Ap abbreviates 8A/8p

E Notations

p p p p p pr r

index denoting a system with combined plant state and dynamic controller state dynamic feedback controller

red ij ref 8

reduced (matrix), obtained by cancelling row i and column j reference, setpoint, particularly Y reJ symmetric part of a matrix, e.g., A.

E.5 Glossary of Symbols in Alphabetic Order

E.5

a(5) a,(5) aik(s) ap (5)

Ilnom

A

spectral norm, e.g., IIAII, slow model, neglecting high-frequency dynamics (Sobolev) S2-bounded truncation of a function variable multiplied by e"'

Glossary of Symbols in Alphabetic Order

unperturbed polynomial in s extreme polynomial edge polynomial perturbed polynomial a( s) (right) eigenvector of A associated with Ak[A], abbreviation for ilk left-eigenvector of A associated with Ak[A] nominal coefficient vector associated with the nominal polynomial a( s) matrix of coefficients of a system in state-space representation, having elements aij in row i and column j

A average matrix Ab bias matrix Ap perturbed matrix A AI real interval matrix dA perturbation matrix B input 'matrix in state-space representation C performance, index of performance C, C (matrix) transfer function of the controller C output matrix C, matrix expressing sliding mode condition cmxn set of complex matrices with m rows and n columns db decibels, e.g., a db means a gain of 10(;'0)

d path in the s-plane, see Eq.(21.113) d[SI,52] chordal metric between the points 51 and 52

dG(G 1 , G 2 ) graph metric distance d,i stochastic parameter D Nyquistcont~r enclosing the r.ight half of the s-plane DR Nyquist contour enclosing the right half of the s-plane with indentations

Do(w) D D

of radius 1/ R along the imaginary axis performance deterioration, see Eq.(24.43) matrix in the dynamic-free part of the observer diagonal scaling matrix

Del, Dcr common left-divisor and common right-divisor, respectively Dd operational matrix for dead-time delay D/, Dr left-divisor and right-divisor, respectively D M operational matrix for differentiation

optimal p-norm weight

e max

e e e; E E E

set of structured and unstructured perturbations, respectively base of natural logarithm (e = 2.71828) perturbation factor (dA = eE) maximum perturbation parameter tracking error, see Eq.(1.5) residual error error vector (e = Yr,! - y) ith standard basis vector (unit i-vector) with 1 only in the ith position matrix in the dynamic-free part of the observer error matrix unidirectional perturbation matrix upper bound for structured perturbation of continuous-time systems

675

676

Ei Ep Eij J(z) fi F F F F Fo Fuw Fz g(t) g g(x, t) G,G G Gil Gn

GN GOp

p h h hi hi, h H H H H H i, 'l

I I I,. 1° 1/ 0 On j, J j J J ij k k

ki km

k, ku k K, K KK Kv Kg I I Ii lo(w)

and discrete-time systems, respectively constant perturbation matrix perturbation bound Kronecker matrix polynomial in z eigenvector of F associated with Ai(F] matrix of closed-loop coefficients in state-space description closed-loop system transfer matrix aggregated matrix power series expansion matrix of orthogonal functions, see Eq.(36.41) return-ratio matrix, open-loop transfer matrix transfer matrix of a system with output u and input w system matrix of the observer continuous impulse response function, weighting function perturbation vector nonlinear time-varying perturbation vector (matrix) transfer function, plant (matrix) transfer function aggregated input matrix invertible and non-invertible part of the matrix G nearest normal approximation operator associated with the perturbed plant high-frequency signal, dither degree of stability (" high") upper limit of a coefficient ai

E Notations

general shifted orthogonal polynomial and polynomial function vector, respectively Hamiltonian function Hurwitz testing matrix observer input matrix associated with the control variable upper bound of an interval matrix measurement transfer matrix integer numbers index of performance identity matrix of appropriate dimension identity matrix of dimension n x n symplectic matrix rotation matrix null matrix of appropriate dimension null matrix of dimension n x n integer numbers

=H matrix in its Jordan canonical form, Jordan form, Jordan canonical form Jordan block integer number number of terms of generalized shifted orthogonal polynomials in t which are taken into consideration dimension of ~i, see Eq.(26.15) stability margin stability margin with respect to block-structured uncertainty attenuation factor output feedback controller (matrix) transfer function of the controller, matrix gain factor steady-state Kalman gain matrix equivalent sliding mode state feedback controller output feedback controller matrix integer number dimension of the aggregated state vector z lower limit of (a coefficient ai) bound for ~Lo

E.5 Glossary of Symbols in Alphabetic Order 677

lq dimension of the parameter vector q L number of submodels L observer input matrix associated with the output variable of the plant L lower bound of an interval matrix L scaling matrix Le, La controllability and observability gramian, respectively AL scalar multiplicative uncertainty ALo , Lo , ALi, Li output and input associated uncertainty, respectively LJ proportional-integral observer input matrix associated with the output of the plant LA, Lv matrix-Lagrange-multipliers m dimension of the input variable u m scalar function replacing the norm of the perturbation matrix Mi mi number of partitions ai, see Eq.(26.15) mi mUltiplicity of the eigenvalue Ai mm order of the minimal polynomial mmi index of the eigenvalue Ai, equivalent to the

multiplicity of the eigenvalue in the minimal polynomial mM measure of stabilty robustness, see Eq.(23.2) maux

M M M M

Mi MR

auxiliary measurement vector maximal bias matrix, deviation matrix measurement matrix matrix transfer function of the model matrix partition of the unperturbed control system interacting with the uncertainty which is pulled out and is considered as an external part of the control system instrumental variable matrix feedback controller or sensor transfer matrix perturbation matrix associated with the sensor partitioned Riccati coefficient matrix

M u , My component connection model input matrices n dimension of state variable, order of a dynamical system, state dimension,

order of the characteristic polynomial nE number of fixed perturbation matrices nm dimension of the observation vector y np dimension of the parameter vector P n measurement noise N number of zeros in the closed right half s-plane N number of corners N number of local subsystems N. smoothed nonlinearity No resolvent matrix of L N°P operator associated with a nonlinear system N u , Ny component connection model output matrices p scalar parameter varying the interconnection p, P parameter, parameter vector p[x(k)lx(k - 1)] conditioned probability distribution function p( s) polynomial in s Po(w) low-frequency performance PM(jW) modified polynomial for Mikhailow hodograph Pi right-eigenvector of AT P number of unstable poles of the open-loop system P permutation matrix P operational premultiplication matrix P solution matrix of the Lyapunov equation P solution matrix of the Riccati equation P matrix weighting the state vector x(t) P Pick matrix P covariance matrix PK matrix solution of the Riccati equation determining a Kalman-Bucy filter

678

projector matrix solution of the Lyapunov equation when Q = I operational matrix of integration using block-pulse functions operational matrix of integration using piecewise linear polynomial functions

Pr

PI P M &

PMI

PM(,

PMp

PM I, PM general operational matrices of integration

q

operational matrix of integration associated with the power series expansion vector, see Eq.(36.102) stretched operational matrix of integration operational matrix of integration using Walsh functions operator substituting the multiplication by eal

truncation operator family of perturbed polynomials family of perturbation polynomials degeneracy modified spectral radius, see Eq.(28.17) real uncertain parameter vector which allows for linear dependent coefficient perturbations

Q positive definite matrix Q weighing matrix, weighting the state vector x( t) Qa weighting matrix, weighting the differential sensitivity vector IT;

r dimension of the parameter vector p r dimension of the output vector y re['], rR['] complex and real stability radius, respectively rL, r 0 uncertainty radius bounding I~L(jw) I R R[x] R R ROP

n,(mxn) 8

Sij

8ij

8 w

So

Su

S Sj 8(8) S; sf; t, to tm t, T T Td TI 11 T T T(8) TF T u, u il

region Rayleigh quotient scaling matrix weighting matrix, weighting the control vector u( t) retardation operator set of real matrices with m rows and n columns complex Laplace variable entry of the corner matrix length of the chain containing the eigenvector and the generalized eigenvectors sliding mode condition output noise of a plant (measurement noise) input noise of a plant intersection of hyperplanes Sj hyperplane sensitivity matrix of the nominal system normalized sensitivity function of the closed-loop transfer function F( 8) with respect to the characteristic polynomial a( s) of the plant differential sensitivity real time variable, initial time, respectively discrete sampling instants sliding mode initial time sampling interval time constant time delay time constant of integration mapping given by the matrix sum weighted with powers of time matrix relating the plant state variable and the observer state variable transformation matrix (modal matrix) complementary sensitivity matrix function modal matrix associated with F = A + BK transformation control variable, controlling variable (vector), input vector input signal in the case of unperturbed plant singular vector of GGH

E Notations


U unitary matrix U matrix of singular vectors U;, see Eq.(22.96) U e normalized perturbation bound Ukl permutation matrix U kl self-derivative matrix U~ orthogonal spectrum of U

v vector of measurement noise v plant state vector applying the orthogonal transformation T, see Eq.(32.54) v state vector of the dynamic feedback controller v; singular vector of G H G V gain constant V, Vk Lyapunov function, Lyapunov function at the sampling instant k, respectively VN norm-like Lyapunov function V(cPp , d) normalized value set of perturbation polynomials, see Eq.(21.113) V matrix of singular vectors v; , see Eq.(22.97) V prefilter matrix V dm, V dmg Vandermonde matrix, generalized Vandermonde matrix, respectively w weighting function w distortion, disturbance vector w vector of process noise W weighting matrix W A Kalman controllability matrix X,Xo x~

x(t) Xin, Xout

Xr X X(t) X" X~

Xn X Xoo Y y Y YouX'

Yre! y z z z z z z, Z Z Z;j Z

1 (t) 1

a a a, Po PG

state vector, initial state vector, respectively general orthogonal coefficient vector signal vector after having multiplied x(t) by the operator polynomial <1>(8)

output and input ( Note the order !) of the block-diagonal uncertainty, see Fig. 26.3 state of the dynamic feedback controller covariance matrix state variable matrix piecewise linear spectrum orthogonal spectrum of x block-pulse spectrum set of matrices X set of block-diagonal matrices with no restriction on the norm measurement vector, output vector output signal in the case of unperturbed plant residual sum of squares instrumental variable vector reference, set point set of matrices Y complex z-transform variable variable characterizing the E-contour observer state vector (fast) state vector of parasitic dynamics aggregated model state vector observer state and observer estimation error, respectively (preferably) any matrix interpolating matrix polynomial, component of a matrix set of matrices Z

unit step function at t = 0 (= 1 for t 2: 0, and = 0 for t < 0) sum vector

degree of stability positive constant determining the radius of a disc spectral abscissa robustness measure constant, bounding the plant operator, see Eq.(2.26)

679

680

I

constant, bounding the nonlinearity, see Eq.(2.26) sector bound

I maximum gain IG gain term, bounding the plant operator, see Eq.(2.26) IN gain term, bounding the nonlinearity, see Eq.(2.26) -y(k) correcting gain vector (Kalman gain vector) r stability region in the s-plane r c complement of the region r in the s-plane

E Notations

r system matrix of a singularly perturbed system including dynamic feedback controller r aggregation matrix r generalized system transfer matrix r A , rv matrix constraints {j e.g., ox, first-order infinitesimal difference (first variation) of x o(t) unit Dirac delta function, unit impulse occuring at t = 0 Oij Kronecker delta (= 1 for i = j, and = 0 for i -# j) oT(r) Kronecker delta (= 1 for r = 0, and = 0 for r -# 0) 6( G 1 , G 2) directed gap between two systems G 1 and G 2

~ e.g., ~t, (small) increment of t Abd block-diagonal uncertainty matrix Ai partition of the block-diagonal uncertainty matrix Abd

€ small quantity € small scalar parameter representing high-frequency parasitic dynamics € quantity preferably moving from 0 to 1 € parameter determining the size of the polytope £

8 8(s) 1<, []

A, .x A Ai [A] Ap .x A A A

P PeN PD[] pp[] pp p,[] v ~lik ,

1r

error vector inverse matrix of power series expansion of orthogonal functions, see Eq.(36.46) norm-bound uncertainty associated with an additive perturbance, see Eq.(26.1) spectral condition number scalar and vector-Lagrange-multiplier, respectively coefficient of stretched time scale ith eigenvalue of the matrix A eigenvalue of the perturbed plant transfer matrix costate variable, adjoint variable set of eigenvalues diagonal matrix of the eigenvalues Ai matrix-Lagrange-multiplier, costate matrix integer number upper bound for nonlinear time-varying perturbation, see Eq.(13.75) structured singular value, see Eq.(26.18) matrix measure factor Amin[QJ/Amax[P] , see Eq.(13.41) spectral matrix measure integer number

6ik multiplicative noise 1r = 3.14159265

1r[] Perron-Frobenius radius, Perron eigenvalue (root) P scalar parameter characterizing the p-system, see Eq.(25.173) PH distance from Hurwitz stability, see Eq.(21.100), radius of largest hypersphere Popt maximum radius of robust stabilizability, see Eq.(30.1l6) PH numerical radius P, spectral radius Pi modified differential sensitivity variable ot] , (Ti['] singular value (THi['] Hankel singular value (T" standard deviation, square root of the constant variance of observation errors (Tyv, (Tuv maximum plant sensitivity and maximum controller sensitivity, respectively (Ti differential sensitivity vector Epj region in k-space where r-stability is satisfied for a fixed plant parameter vector


T

T

q,( s)

'P;{() 'Pm 'P; 41(t, T) 41cl(t) 41cL(T) ~

w

(n, m)-matrix of singular values, see Eq.(22.88) time shift fast time scale, treating singular perturbation behaviour least common multiple of the minimal polynomials of Aw and Ar where Aw and Ar are system matrices associated with the disturbance and reference, respectively orthogonal polynomial phase margin right-eigenvector of 41(T) transition matrix for a time-varying system transition matrix of the closed-loop system closed-loop coefficient matrix in state-space description of a discrete-time system input matrix of a discrete-time system (real) frequency in radians per second crossover frequency ellipsoidal set transfer matrix from the input Zj to the internal signal V;

681

Appendix F

Author Index

Abdul- Wahab, A.A., 1989 Robustness measure bounds for generalized dynamic output feedback controllers, Int.J.Systems Sci. 20, pp. 2095-2105

Abdul- Wahab, A.A., 1990, Lyapunov-type equations for matrix root-clustering in subregions of the complex plane, Int.J.Systems Sci. 21, pp. 1819-1830

Abdul- Wahab, A.A., 1990a, Lyapunov bounds for root clustering in the presence of system uncertainty, Int. J. Systems Sci. 21, pp. 2603-2611

Abdul- Wahab, A.A., 1990b, Lyapunov stability robustness measures for multivariable, continuous, timeinvariant, linear systems, Int. J. Systems Sci. 21, pp.2577-2587

Abdul- Wahab, A.A., 1991, Perturbation bounds for root-clustering of linear continuous-time systems, Int. J. Systems Sci. 22, pp. 921-930

Abdul- Wahab, A.A., and Zohdy, M.A., 1988, dynamic output feedback controllers,

Abdul- Wahab, A.A., and Zohdy, M.A., 1989, Int.J.Control 50, pp. 1619-1634

Generalized linear transformations on the design of robust Int.J.Control 48, pp. 1241-1266

Eigensystem assignment by feedback control,

Ackermann, J.,1972, Der Entwurf linearer Regelungssysteme im Zustandsraum, Rege/ungstechnik 20, pp. 297-300

Ackermann, J., 1980, Parameter space design of robust control systems, IEEE-Trans. AC-25, pp. 1058-1072

Ackermann, J., 1984, Robustness against sensor failures, Automatica 20, pp. 211-215 Ackermann, J.,(Ed.) 1985, Uncertainty and Control (Springer, Berlin New York) Ackermann, J.,1985a, Multi-model approaches to robust control system design, In: Ackermann, J.,{Ed.)

1985, Uncertainty and Control (Springer, Berlin New York) Ackermann, J., 1988, Abtastregelung, 3. Auflage (Springer, Berlin) Ackermann, J., and Barmish, B.R., 1988, Robust Schur stability of a polytope of polynomials, IEEE

Trans. AC-33, pp. 984-986 Ackermann, J., Hu, H.Z., and Kaesbauer, D., 1990, Robustness analysis: a case study, IEEE-Trans.

AC-35, pp. 352-356 Ackermann, J., and Hu, H.Z., 1990a, Robustness of sampled-data control systems with uncertain phy

sical parameters, 11th IFAC-Congress Tallinn, Vol. 5, pp. 194-199 Ackermann, J., Kaesbauer, D., and Muench, R., 1991, Robust Gamma-stability analysis in a plant

parameter space, Automatica 27, pp. 75-85 Adamjan, V.M., Arov, D.Z., and Krein, M.C., 1978, Infinite Hankel block matrices and related exten

sion problems, Am. Math. Soc., Transl. Series 2, 111, pp. 133-156 Aida, K., and Kitamori, T., 1990, Design of a PI-type state feedback optimal servo system,

Int.J.Control 52, pp. 613-625 Aly, C.M., and Ali, W.C., 1990, Digital design of variable structure control systems, Int.J. Systems

Sci. 21, pp. 1709-1720 Anagnost, J.I., Desoer, C.A., and Minichelli, R.I., 1989, Generalized Nyquist tests for robust stabi

lity: Frequency domain generalizations of Kharitonov's theorem In: Milanese, M., Tempo, R., and Vicino, A., (Eds.) 1989 Robustness in Identification and Control (Plenum Press, New York London) pp. 79-96

Anderson, B.D.O., 1969, Stability results for optimal systems, Electronics Letters 5, pp. 545. Anderson, B.D.O., 1973, Exponential data weighting in the Kalman-Buey filter, Information Sci-

ences 5, pp. 217-230

684 F Author Index

Anderson, B.D.D., Dasgoupta, S., Khargonekar, P., Kraus, F.J., and Mansour, M., 1989, Robust strict positive realness: characterization and construction Proc. 28th IEEE-Conference on Decision and Control, Tampa, Florida pp. 426-430

Anderson, B.D.D., Jury, E.!., and Mansour, M., 1987, On robust Hurwitz polynomials, IEEE-Trans. AC-32, pp. 909-913

Anderson, B.D. Dc, and Moore, J.B., 1971, Linear Optimal Control (Prentice-Hall, Englewood Cliffs/New Jersey)

Anderson, B.D.D., and Moore, J.B., 1989, Optimal Control. Linear Quadratic Methods. (PrenticeHall, Englewood Cliffs)

Aoki, M., 1968, Control of large-scale dynamic systems by aggregation, IEEE-Trans. AC-13, pp. 246-253

Argoun, M.B., 1986a, Allowable coefficient perturbations with preserved stability of a Hurwitz polynomial, Int.J.Control 44, pp. 927-934

Argoun, M.B., 1986b, On sufficient conditions for the stability of interval matrices, Int.J.Control 44, pp. 1245-1250

Argoun, M.B., 1986c, Allowable coefficient perturbations with preserved stability of a Hurwitz polynomial, Int.J.Control 44, pp. 927-934

Argoun, M.B., 1987, Stability of a Hurwitz polynomial under coefficient perturbations: necessary and sufficient conditions, Int. J. Control 45, pp. 739-744

Arkun, Y., 1987, Dynamic block relative gain and its connection with the performance and stability of decentralized control structures, Int. J. Control 46, pp. 1187-1193

Asada, H., and Siotine, J.J., 1986, Robot Analysis and Control (John Wiley, New York) Astrom, K.J., 1985, Adaptive control- a way to deal with uncertainty, In: Ackermann, J.,(Ed.) 1985,

Uncertainty and Control (Springer, Berlin New York) Athans, M., 1968, The matrix minimum principle, Information and Control 11, pp. 592-606 Ball, J.A., and Helton, J. W., 1983, A Beurling-Lax theorem for the lie group U(m, n) which contains

most classical interpolation theory, J. Operator Theory 9, pp. 107-142 Banks, S.P., 1988, Mathematical Theories of Nonlinear Systems (Prentice-Hall, New York London) Bard, Y., 1974, Nonlinear Parameter Estimation (Academic Press, New York) Barmish, B.R., 1983, Stabilization of uncertain systems via linear control, IEEE- Trans. AC-28, pp.

848-850 Barmish, B.R., 1984, Invariance of the strict Hurwitz property for polynomials with perturbed coeffi

cients, IEEE-Trans. AC-29, pp. 935-936 Barmish, B.R., Corless, M., and Leitmann, G., 1983, A new class of stabilizing controllers for uncer

tain dynamical systems, SIAM J. on Control and Optimization 21, pp. 246-255 Barmish, B.R., and Hollot, C. V., 1984, Counter-example to a recent result on the stability of interval

matrices by S. Bialas, Int.J.Control 39, pp. 1103-1104 Barmish, B.R., and Leitmann, G., 1982, On ultimate boundedness control of uncertain systems in the

absence of matching assumptions, IEEE-Trans. AC-27, pp. 153-158 Barmish, B.R., and Shi, Z., 1990, Robust stability of a class of polynomials with coefficients depending

multilinearly on perturbations, IEEE-Trans. AC-35, pp. 1040-1043 Barmish, B.R., and Tempo, R., 1990, The robust root locus, Automatica 26, pp. 283-292 Barnett, S., 1971, Matrices in Control Theory (Van Nortrand Reinhold, London) Barnett, S., 1973, Matrix differential equations and Kronecker products, SIAM J.Appl.Math. 24, pp.

1-5 Bartlett, A.C., Hollot, C. V., and Lin, H., 1987, Root locations of an entire polytope of polynomials: it

suffices to check the edges, Proc. American Control Conf. pp. 1611-1616 Bauer, F.L., 1962, On the field of values subordinate to a norm, Numer. Math.4, pp. 103-113 Bauer, F.L., 1963, Optimally scaled matrices, Numer.Math.5, pp. 73-87 Bauer, F.L., and Fike, C. T., 1960, Norms and exclusion theorems, Numer. Math.2, pp. 137-141 Beale, S., and Shafai, B., 1989, Robust control system design with a proportional integral observer,

Int.J.Control 50, pp. 97-111 Beck, J. V., and Arnold, K.J., 1977, Parameter Estimation in Engineering and Science (John Wiley,

New York London) Bell, D.J., (Ed.) 1973, Recent Mathematical Developments in Control. Proceedings of a conference,

Bath 1972, (Academic Press, London New York) Bellman, R., 1964, Perturbation Techniques in Mathematics, Physics and Engineering (Holt, Rinehart

and Winston, New York)

685

Bellman, R., 1967, Introduction to the Mathematical Theory of Control Processes, Vol. 1, Linear Equations and Quadratic Criteria (Academic Press, New York)

Bellman, R., 1970, Introduction to Matrix Analysis, Second Edition, (Mc Graw-Hill, New York London)

Benidir, M., and Picinbono, B., 1988, Comparison between some stability criteria of discrete-time filters, IEEE-Trans. ASSP-36, pp. 993-1001

Benninger, N.F., 1986, Eine gut konditionierte Transformation auf Sensorkoordinaten, Automatisierungstechnik 34 , pp.41~411

Bernussou, J., Peres, P.L.D., and Geromel, J.C., 1990, Stabilizability of uncertain dynamical sy-stems: the continuous and the discrete case, 11th IFAC-Congress Tallinn, Vol. 5, pp. 135-140

Bernussou, J., and Tit/i, A., 1982, Interconnected Dynamical Systems: Stability, Decomposition and Decentralization (North-Holland, Amsterdam New York Oxford)

Bhattacharyya, S.P., 1987, Robust Stabilization Against Structured Perturbations, (Springer, Berlin) Bhattacharyya, S.P., del Nero Gomes, A.C., and Howze,J. W., 1983, The structure of robust distur

bance rejection control, IEEE-Trans. AC-28, pp. 874-881 Bialas, S., 1983, A necessary and sufficient condition for the stability of interval matrices,

Int.J.Control 37, pp.717-722 Bialas, S., and Garloff, J., 1985, Convex combinations of stable polynomials, J. of the Franklin Insti

tute 319, pp. 375-377 Billings, S.A., and Chen, S., 1989, Extended model set, global data and threshold model identification

of severely nonlinear systems, Int.J. Control 50, pp. 1897-1923 Birdwell, J.D., and Laub, A.J., 1987, Balanced singular values for LQG/LTR design, Int-J.Control

45, pp. 939-950 Bistritz, Y., 1984, Zero location with respect to the unit circle of discrete-time linear system polyno

mials, Proc. IEEE 72, pp. 1131-1142 Bocker, J., Hartmann, I., und Zwanzig, Ch., 1986, Nichtlineare und adaptive Regelungssysteme

(Springer, Berlin New York) Bode, H. W., 1945, Network Analysis and Feedback Amplifier Design, (Van Nostrand, New York) Bongiorno, J. J. Jr., 1969, Minimum sensitivity design oflinear multi variable feedback control systems

by matrix spectral factorization, IEEE-Trans. AC-14 pp.665-673 Bose, N.K., 1977, Implementation of a new stability test for two-dimensional filters, IEEE-Trans.

ASSP-25, pp. 117-120 Bose, N.K., 1985, A system-theoretic approach to stability of sets of polynomials, Contemporary Ma

thematics 47, pp. 25-34 Bose, N.K., and Delansky, J.F., 1989, Boundary implications for interval positive rational functions:

preliminaries, In: Milanese, M., Tempo, R., and Vicino, A., (Eds.) 1989 Robustness in Identification and Control (Plenum Press, New York London) pp. 287-291

Bose, N.K., Jury, E.I., and Zeheb, E., 1988, On robust Hurwitz and Schur polynomials, IEEE-Trans. AC-33, pp. 1166-1168

Bose, N.K., and Shi, YQ., 1987, A simple general proof of Kharitonov's generalized stability criterion, IEEE-Trans. CAS-34, pp. 1233-1237

Boyd, S., and Yang, Q., 1989, Structured and simultaneous Lyapunov functions for system stability problems, In: Milanese, M., Tempo, R., and Vicino, A., (Eds.) 1989 Robustness in Identification and Control (Plenum Press, New York London) pp. 243-262

Brammer, K., and Sijjling, G., 1975, Kalman-Bucy-Filter (Oldenbourg, Miinchen Wien) Breinl, W., 1980, Entwurf eines parameterunempfindlichen Reglers am Beispiel der Magnetschwebe

bahn, Regelungstechnik 28, pp. 87-92 Brewer, J. W., 1977, The derivative of the exponential matrix with respect to a matrix, IEEE-

Trans. AC-22, pp. 656-657 Brewer, J. W., 1977a, The derivative of the Riccati matrix with respect to a matrix, IEEE- Trans. AC-

22, pp. 980-983 Brewer, J. W., 1978, Kronecker products and matrix calculus in system theory, IEEE-Trans. CAS-25,

pp.772-781 Brewer, J. W., 1978a, Matrix calculus and the sensitivity analysis of linear dynamic systems, IEEE

Trans. AC-23, pp. 748-751 Bri/linger, D.R., 1981, Time Series, Data Analysis and Theory (Holden-Day, San Francisco) Brogan, W.L., 1985, Modern Control Theory, 2nd edition (Prentice-Hall, New York London)

686 F Author Index

Biihler, H., 1986, Minirnierung der Norm des Riickfiihrvektors bei der Polvergabe, Automatisierungstechnik 34, pp.152-155

Byers, R., and Nash, S.G., 1989, Byrne, P.C., and Burke, M., 1976,

7rans. AC-21, pp. 282-283

Approaches to robust pole assignment, Int.J.ControI49, pp. 97-117 Optimization with trajectory sensitivity considerations, IEEE-

Cao, C. T., 1989, Entwurf eines robusten pradiktiven Reglers fiir totzeitbehaftete Regelsysteme, Automatisierungstechnik 37, pp. 104-110

Carlucci, D., and Donati, F., 1975, Control of norm uncertain systems, IEEE- Trans. AC-20, pp. 792-795

Carlucci, D., and Val/auri, M., 1981, Feedback control of linear multivariable systems with uncertain description in the frequency domain, Int.J.Control 33, pp.903-912

Cavallo, A., Celentano, G., and De Maria, G., 1991, Robust stability analysis of polynomials with linearly dependent coefficient perturbations, IEEE-7rans. AC-36, pp. 380-384

Chang, B.C., and Pearson, J.B., Jr., 1984, Optimal disturbance reduction in linear multi variable systems, IEEE-Trans. AC-29, pp. 880-887

Chang, R. Y., Yang, S. Y., and Wang, M.L., 1987, Solution of a scaled system via generalized orthogonal polynomials, Int.J.Systems Sci. 18, pp. 2369-2382

Chang, R. Y., Yang,S, Y., and Wang, M.L., 1988, Parameter identification of time-varying systems via generalized orthogonal polynomials, Int.J.Systems Sci. 19, pp. 471-485

Chang, Y.F., and Lee, T. T., 1986, Application of general orthogonal polynomials to the optimal control of time-varying linear systems, Int.J.Control 43, pp. 1283-1304

Chang, S.S.L., and Peng, T.K.C., 1972, Adaptive guaranteed cost control of systems with uncertain parameters, IEEE-Trans. AC-17, pp. 474-483

Chapellat, H., and Bhattacharyya, S.P., 1989, A generalization of Kharitonov's theorem: robust stability of interval plants, IEEE- Trans. AC-34, pp. 306-311

Chapel/at, H., and Bhattacharyya, S.P., 1989a, Robust stability and stabilization of interval plants, In: Milanese, M., Tempo, R., and Vicino, A., (Eds.) 1989 Robustness in Identification and Control (Plenum Press, New York London) pp. 207-229

Chapel/at, H., Bhattacharyya, S.P., and Dahleh, M., 1990, Robust stability of a family of disc polynomials, Int.J.Control 51, pp. 1353-1362

Chapel/at, H., Dahleh, M., and Bhattacharyya, S.P., 1990a, Robust stability under structured and unstructured perturbations, IEEE- Trans. AC-35, pp. 1100-1108

Chapellat, H., Dahleh, M., and Bhattacharyya, S.P., 1991, On robust nonlinear stability of interval control systems, IEEE-Trans. AC-36, pp. 59-67

Chen, B.S., 1984, Controller synthesis of optimal sensitivity: multivariable case, Proc.IEE Pan D 131, pp.47-51

Chen, B.S., and Chou, H.H., 1988, Optimal robustness via constrained controller in SISO discrete-time feedback systems, Int.J.Control 48, pp. 1391-1408

Chen, B.S., and Dong, T. Y., 1988, Robust stability analysis of Kalman-Bucy filter under parametric and noise uncertainties, Int.J.Control 48, pp. 2189-2199

Chen, B.S., and Dong, T. Y., 1989, LQG optimal control system design under plant perturbation and noise uncertainty: a state-space approach, Automatica 25, pp. 431-436

Chen, B.S., and Kung, J. Y., 1988, Robust stability of structured perturbation system in state space model, Proc. 27th IEEE Conference on Decision and Control, Austin, pp. 121-122

Chen, B.S., and Lin, C.L., 1990, On the stability bounds of singularly perturbed systems, IEEE- Trans. AC-35, pp. 1265-1270

Chen, B.S., and Lo, C.H., 1989, Necessary and sufficient conditions for robust stabilization of perturbed observer-based compensating systems, Int.J.Control 49, pp. 937-960

Chen, B.S., Wang, S.S., and Lu, H.C., 1989, Minimal sensitivity perfect model matching control, IEEE-Trans. AC-34, pp. 1279-1283

Chen, B.S., and Wang, w.J., 1990, Robust stabilization of nonlinearly perturbed large-scale systems by decentralized observer-controller compensators, Automatica 26, pp. 1035-1041

Chen, B.S., and Wong, C.C., 1987, Robust linear controller design: Time domain approach, IEEE-7rans. AC-32, pp. 161-164

Chen, C.F., and Hsiao, C.H., 1975, Design of piecewise constant gains for optimal control via Walsh functions, IEEE- Trans.AC-20, pp. 596-603

Chen, C. T., 1968, Stability of linear multivariable feedback systems, Proc. IEEE 56, pp. 821-828

687

Chen, C. T., 1970, Introduction to Linear System Theory (Holt, Rinehart and Winston, New York London)

Chen, Ii., 1961, Analysis and design of feedback systems with gain and time constant variations, IRE'Irans. AC-6, pp. 73-79

Chen, K.H., and Chen, W.L., 1989, Robust analysis of multivariable systems with plant perturbation and sensor uncertainty, Int.J.Systems Sci. 20, pp. 2329-2333

Chen, M.J., and Desoer, C.A., 1982, Necessary and sufficient condition for robust stability of linear distributed feedback systems, Int. J. Control 35, pp. 255-267

Chen, S., Billings, S.A., and Luo, W., 1989, Orthogonal least squares methods and their application to nonlinear system identification, Int. J. Control 50, pp. 1873-1896

Chen, Y.H., 1986, On the deterministic performance of uncertain dynamical systems, In/.J.Control 43, pp. 1557-1579

Chen, Y.H., 1987, Robust output feedback controller: direct/indirect design Int. J. Control 46, pp. 1083-1103

Chen, Y.H., 1989, Modified adaptive robust control system design, Int.J.Control 49, pp. 1869-1882 Cheng, B., and Hsu, N.S., 1982, Single-input-single-output system identification via block- pulse func

tions, Int.J.Systems Sci. 13, pp. 697-702 Cheok, K.C., Hu, H.X., and Loh, N.K., 1988a, Optimal output feedback regulation with frequency-

shaped cost functional, Int.J.Control 47, pp. 1665-1681 Cheok, K.C., Hu, H.X., and Loh, N.K., 1989, Representation of dynamic output feedback control sy

stem variables via orthogonal functions, Int. J.8ystems Sci. 20, pp.1163-1171 Cheok, K.C., Loh, N.K., and Ho, J.B., 1988, Continuous-time optimal robust servo-controller design

with internal model principle, Int. J. Control 18 , pp. 1993-2010 Cheres, E., Gutman, S., and Palmor, Z.J., 1989, Stabilization of uncertain dynamic systems including

state delay, IEEE-mns. AC-34, pp. 1199-1203 Cheres, E., Palmor, Z.J., and Gutman, S., 1989a, Quantitative measures of robustness for systems

including delayed perturbations, IEEE- Trans. AC-34, pp. 1203-1204 Chiang, C.C., and Chen, B.S., 1988, Robust stabilization against non-linear time-varying uncer-

tainties, Int.J.Systems Sci. 19, pp. 747-760 Cho, Y.S., and Narendra, K.S., 1968, An off-axis circle criterion for the stability of feedback systems

with a monotonic nonlinearity, IEEE-'Irans. AC-13, pp. 413-416 Chou, H.H., Chen, B.S., and Lin, Y.P., 1990, Necessary and sufficient condition for stability robust

ness under nonlinear time-varying uncertainties, Int.J.Systems Sci. 21, pp. 305-316 Chou, H.H., Chen, B.S., and Lin, Y.P., 1990a, Robust pole placement: a frequency-domain ap-

proach, Int.J.Systems Sci. 21, pp. 317-333 Chu, C.C., Doyle, J.C., and Lee, E.B., 1986, The general distance problem in Hoo optimal control

theory, Int. J. Control 44, pp. 565-596 Chung, H. Y, 1987, System identification via Fourier series, blt.J.Systems Sci. 18, pp. 1191-1194 Cieslik, J., 1987, On possibilities of the extension of Kharitonov's stability test for interval polynomials

to the discrete-time case, IEEE- Trans. AC-32, pp. 237-238 Cobb, J.D., 1988, The minimal dimension of stable faces required to guarantee stability of a matrix

polytope: D-stability, Proc. IEEE Conf. on Decision and Control pp. 119-120 Cobb, J.D., 1990, The minimal dimension of stable faces required to guarantee stability of a matrix

polytope: D-stability, IEEE-Trans. AC-35, pp. 469-473 Cobb, J.D., and DeMarco, C.L., 1988, The minimal dimensionality of stable faces required to guarantee

stability of a matrix polytope, American Control Conference pp. 818-819 Cook, P.A., 1972, Modified multivariable circle theorems. In Bell, D.J.JEd.) 1973, pp. 367-372 Corless, M., 1985, Stabilization of uncertain discrete-time systems, In: Skelton, R.E. and Owens, D.H.

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New criteria for robust stability In: Milanese, M., Tempo, R., and Robustness in Identification and Control (Plenum Press, New York

Corless, M., Garofalo, F., and Leitmann, G., 1989, Guaranteeing ultimate bounded ness of exponential rate of convergence for a class of uncertain systems In: Milanese, M., Tempo, R., and Vicino, A., (Eds.) 1989 Robustness in Identification and Control (Plenum Press, New York London) pp. 293-301

Corless, M.J., and Leitmann, G., 1981, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE-Trans. AC-26, pp. 1139-1144

688 F Author Index

Cruz, I., (Ed.) 1973, System Sensitivity Analysis (Dowden, Hutchinson and Ross, Strouds-burg/Pennsylvania) ,

Curtain, R.F., and Pritchard, A.J., 1977, Functional Analysis in Modern Applied Mathematics (Academic Press, New York)

Dahleh, M.A., 1990, BIBO stability robustness for coprime factor perturbations, 11th IFAC-Congress Tallinn, Vol. 5, pp. 201-204

Daniel, R. W., and Kouvaritakis, B., 1985, A new robust stability criterion for linear and nonlinear multivariable feedback systems, Int.J.Control 41, pp. 1349-1379

Darwish, M.G., and Soliman, H.M., 1988, Design of decentralized reliable controllers for large-scale systems, Int.J.Systems Sci. 19, pp. 1529-1538

Dasgupta, S., 1987, A Kharitonov-like theorem for systems under nonlinear passive feedback, Proc. IEEE-Con! on Decision and Control, Los Angeles pp. 2062-2063

Dasgupta, S., 1988, Kbaritonov's theorem revisited, Systems and Control Letters 11, pp. 381-384 Dasgupta, S., Parker, P.J., Anderson, B.D.O., Kraus, F.J., and Mansour, M., 1988, Frequency do-

main conditions for the robust stability of linear and nonlinear dynamical systems, Proc. American Control Conf., Atlanta pp.1863-1868

Davision, E.J., 1976, The robust control of a servomechanism problem for linear time-invariant multivariable systems, IEEE-Trans. AC-23, pp. 25-34

Davison, E.J., and Ferguson, I.J., 1981, The design of controllers for the multivariable robust servomechanism problem using parameter optimization methods, IEEE-Trans. AC-26, pp. 93-110

Dawson, D.M., Qu, Z., Lewis, F.L., and Dorsey, J.F., 1990, Robust control for the tracking of a robot motion, Int.J.Control 52, pp. 581-595

De Carlo, R.A., and Saeks, R., 1981, Interconnected Dynamical Systems (Marcel Dekker, New York) de la Sen, M., 1990, Use of Gronwall's Lemma for robust compensation of time-varying linear systems

via synthesis of augmented exciting signals, Int. J. Systems Sci. 21, pp. 2317-2335 Delsarte, P., Genin, Y., and Kamp, Y., 1981, On the role of the Nevanlinna-Pick problem in circuit

and system theory, Int. J. on Circuit Theory and Application 9, pp. 177-187 Desoer, C.A., and Gustafson, C.L., 1984, Algebraic theory of linear multivariable feedback sy-

stems, IEEE-Trans. AC-29, pp. 909-917 Desoer, C.A., Liu, R. W., Murray, J., and Saeks, R., 1980, Feedback system design: The fractional re

presentation approach to analysis and synthesis, IEEE- Trans. AC-25, pp. 399-412 Desoer, C.A., and Vidyasagar, M., 1975, Feedback Systems: Input-Output Properties (Academic

Press, New York) Desoer, C.A., and Wang, Y. T., 1978, On the minimum order of a robust servocompensator, IEEE-

Irans. AC-23, pp. 70-73 Dickman, A., and Sivan, R., 1985, On the robustness of multivariable linear feedback systems, IEEE

Irans. AC-30, pp. 401-404 Diduch, C.P., and Doraiswami, R., 1988, Role of sample period in transients, robustness and sensiti

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raum, Automatisierungstechnik 38, pp.134-143 Djaferis, T.E., 1991, Simple robust stability tests for polynomials with real parameter uncertainty, Int.

J. Control 53, pp. 907-927 Doetsch, G., 1967, Guide to the Applications of the Laplace and z-Transforms, (Van Nostrand Reinhold,

London) Doraiswami, R., 1990, Time-domain approach to quantification of overshoot, speed and robust-

ness, Int.J.Systems Sci. 21, pp. 1057-1075 Doraiswami, R., and Bordry, F., 1987, Robust two-time-level control strategy for sampled-data servo

mechanism problem, Int.J.Systems Sci. 18, pp. 2261-2277 Dorato, P., (Ed.) 1987, Robust Control, (IEEE-Press, New York) Dorato, P., Park, H.B., and Li, Y., 1989, An algorithm for interpolation with units in HOC>, with ap-

plications to feedback stabilization, Automatica 25, pp. 427-430 . Dorling, C.M., and Zinober, A.S.I., 1986, Two approaches to hyperplane design in multivariable va

riable structure control systems, Int. J. Control 44, pp. 65-82 Dorling, C.M., and Zinober, A.S.I., 1988, Robust hyperplane design in multivariable variable structure

control systems, Int. J. Control 48, pp. 2043-2054 Douce, J.L., and Roberts, P.D., 1964, Orthogonal functions, Control 8, pp. 457-461

689

Doyle, J.C., 19S2, Analysis offeedback systems with structured uncertainties, lEE Proc. Part D 129, pp.242-250

Doyle, J.C., 19S3, Synthesis of robust controllers and filters, Proc. 22nd IEEE Conf. on Decision and Control pp. 109-114

Doyle, J. C., Glover, K., Khargonekar, P., and Francis, 8., 19S5, State-space solutions to standard H2 and Hoo control problems, Proc. American Control Conference, Atlanta, pp. 1691-1696

Doyle, J.D., Glover, K., Khargonekar, P.P., and Francis, B.A., 19S9, State-space solutions to standard H2 and H 00 control, IEEE- Trans. AC-34, pp.831-847

Doyle, J.C., and Stein, G., 1979, Robustness with observers, IEEE-Trans. AC-24, pp.607-611 Doyle, J.C., and Stein, G., 19S1, Multivariable feedback design: Concepts for a classical/modern syn

thesis, IEEE-Trans. AC-26, pp. 4-16 Doyle, J.C., Wall, J.E., and Stein, G., 19S2, Performance and robustness analysis for struc~ured un

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694 F Author Index

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Leitmann, G., Ryan, E.P., and Steinberg, A., 1986, Feedback control of uncertain systems: robustness with respect to actuator and sensor dynamics, Int.J. Control 43, pp. 1243-1256

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696 F Author Index

Limebeer, D.J.N., 1982, The application of generalized diagonal dominance to linear systems stability theory, Int.J.Control 36, pp. 185-212

Limebeer, D.J.N., and Hung, Y.S., 1987, An analysis of the pole-zero cancellations in Hoo-optimal control problems of the first kind, SIAM J. on Control and Optimization 25, pp. 1457-1493

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Lin, C.M., and Chou, W.D., 1989, Robust optimization of feedback systems with time-varying nonlinear uncertainties, Int.J.Systems Sci. 20, pp. 1011-1018

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Lin, S.H., Juang, Y. T., Fong, I.K, Hsu, C.F., and Kuo, T.S., 1988, Dynamic interval systems analysis and design, Int. J. Control 48, pp. 1807-1818

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697

Martin, J.M., and Hewer, G.A., 1987, Smallest destabilizing perturbations for linear systems, Int.J.Control 45, pp. 1495-1504

Martin, R.H., Jr., 1976, Nonlinear Operators and Differential Equations in Banach Spaces (Wiley, New York)

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698 F Author Index

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Index

Page references in boldface type indicate definitions

A

abscissa see spectral abscissa absolute bound, 301 absolute value, 364, 394

~ of an interval matrix, 178 absolute-value-integral theorem, 121 actuator, 196, 520 adaptation, 41, 42, 490 adaptive and robust, 222 additional feedback, 138 additive uncertainty, 38, 181, 301, 385, 492, 497,

515 adjoint, 79, 611 affine coefficient function, 335, 340 affine linear dependence, 340 aggregable, 554 aggregation

~ algorithm, 552 ~ conditions, 554 ~ model, 547

algebra, 609 algebraic Lyapunov equation, 202 algebraic multiplicity, 633 all-pass, 365, 501

~ extension, 490, 496 ~ matrix, 516 ~ transfer function, 496

amplitude distribution, 520 amplitude margin, 371 analysis using matrices, 79 angle between two vectors, 611 annihilating polynomial, 466 annihilator, 530, 609 antimirror-image, 330 antimirror polynomial, 330 antisymmetric part, 378 antisymmetric property, Hurwitz ~, 354 aperiodicity condition, 354 approximate

~ aggregation, 549 ~ model, 36, 523 ~ system, 557

approximation, optimal ~, 657 arbitrary connected region, 334 assignable subspace, 637, 639 assignment, 98, 117 associativity, 73, 609 asymmetric

~bounds, 308 ~ characteristic, 313 ~ part of a polynomial, 329 ~ polynomial, 330

asymptotically stable, 244 asynchronous model, 577 attenuation factor, 318 attraction, 212 augmentation, 354 autocorrelation, 667 autocovariance, 667 autonomous system, 200 average

~ degree, 338 ~ gain, 308 ~ matrix, 169, 193, 209

axis-parallel box, 330

B

backward time direction, 268 Banach space, 63, 502 Bauer-Fike theorem, 164, 207, 256 Bellman-Gronwall lemma

~ for continuous-time systems, 148, 155, 162,206,293,296,298

~ for discrete-time systems, 232 Bezout identity, 249, 476, 485, 491, 511 bialternate product, 170 bias, 668

~ matrix, 193 BIBO stability, 150 BIBO system, 504 bilinear form, 199 bilinear system, 212, 558, 580 bilinear transformation, 328 block-diagonal, 218, 264, 272, 274, 401, 446

~ identity matrix, 441 ~ matrix, 74, 438, 551, 553, 634 norm of a ~ matrix, 54 ~ structure, 98, 267 ~ unitary matrix, 441

block problem, 517 block-pulse function, 582 block-structured uncertainty, 433, 437 block-triangle matrix, 140 block-triangular matrix, 643 Bacher formula, 110 Bode plot, 38, 395

708

bound, 153, 157, 160, 183, 195, 200, 203, 204, 212, 216, 223, 231, 245, 255, 277, 292, 301, 325, 327, 351, 353, 378, 382, 387, 391, 394, 398, 412, 422, 438, 441, 452, 493, 510, 519, 550, 559, 569, 629

asymmetric ~, 308 ellipsoid ~, 186 exponentially decaying ~, 150 ~ for Lyapunov derivative, 221 hyperellipsoid ~, 187

boundary, 189, 323 ~ condition, 592, 605 ~ layer, 541 ~ representation theorem, 342 ~ theorem, 343 ~ value problem, 592

bounded,37 bounded-input bounded-output system, 504 bounded-real, 504 bounding differential equation solution, 57 box of coefficients, 326 Box theorem, 337 break point, 393, 482, 588 bypass, 482

C

cactus approach, 196 calculus of variations, 592 canonical equation, 260 canonical form, 626, 628 canonical form and sliding mode, 531 Cauchy's index, 331 Cauchy's residue theorem, 513 Cauchy sequence, 63 causal operator, 304, 494 causal system, 67 Cayley-Hamilton theorem, 615, 623, 641 center of perturbed motion, 277 centralized control, 97 chain rule, 88 characteristic

~ equation, 92, 168, 249, 623 ~ function, 323, 623 ~ locus, 394,407,419,434 ~ ~ inclusion region, 417

~ matrix, 110, 373, 623 ~ polynomial, 92, 109, 120, 186, 189, 241,

252,323,347,408,623,633,636 chattering, 535 Chebychev polynomial, 332, 565, 567, 570 Cholesky decomposition, 650, 652 chordal metric, 430 circle criterion, 307, 313, 500 circle region, pole assignment in a ~~, 254 closed-loop

~ eigenvalue sensitivity, 95 ~ singular values, 393

~ system, 314 coefficient parametrization, 323 cofactor, 80, 611 col operator, 573, 611, 646 collocation point approximation, 587 column compression, 368 column-like partitioned matrix, 518 column

~ operator, 76 ~ string, 76 ~ sum, 49,176 ...... sum norm, 51

combination of two polynomials, 326 combining plant and controller, 154 common left-divisor, 475 common right-divisor, 460, 476,479 commutativity, 609

Index

companion form, 111, 117, 128, 169, 185, 189, 190,381, 628

companion matrix, 626, 628, 637 comparison theorem, 152 compatible, 50 compensator, 463 complementary sensitivity, 397, 499, 508 complete modal synthesis, 627 complex-conjugate, 357 complex-conjugate eigenvalues, 624 complex matrix, 357 complex matrix and real parameter regression,

660 complex polynomial coefficients, 324 complex root boundary, 339, 340, 345 complex stability radius, 376 component

~ connection framework, 97, 274, 553 ~ detection (failures), 607 ~ of a matrix, 615

composite system, 553 composite matrices, 378 compression, 368 computation delay, 236, 240 conditioning, 405, 649 condition number, 216, 256, 405, 649, 669 cone-bounded transfer function, 433 conformable, 609 congruent matrices, 649 conjugate transpose, 358 conjugation, 624 conservatism, 43, 164, 180, 189, 218, 308, 329,

391,399,410,416,427,438,440,443 consistent norm, 48, 511 constraint, 131, 278, 300, 470 control canonical form, 626, 628 see a/so com

panion form control energy, 61 control signal, 482

~ ~ magnitude, 398 control uncertainty, 318

Index

control variable constraint, 404 controllability, 544

~ gramian, 60 ~ matrix, 548, 613, 629

controllable subspace, 614, 629 controlled signal, 482 controller, 96, 123, 181

~ gain constraint, 470 ~ matrix, 116 ~ norm, 135 ~ parameter, 250 maximum ~ sensitivity, 316

convergence condition, 64 convex, 416,441

~ combination, 353 ~~ of polynomials, 327

~ hull, 182, 196, 324, 333 ~ performance, 252 ~ polygon, 343

convolution, 61, 67-70, 149, 157, 297, 312, 313, 563, 620 .

~ infinity norm, 68 coprime, 491, 515

~ factorization, 457 ~ polynomials, 476

core vector, 354 corner, 328

~ matrix, 182, 184, 202 ~~ norm, 182

~ points, 253, 328, 332 reduced number of ~~, 183

~ polynomial, 324 corona, pole assignment in a~, 255 cost function, 300 cost matrix, 269

~~ difference, 270 costate matrix, 260 costate variable, 268 covariance, 266, 292, 667, 668

perturbed ~, 294 cross-condition number, 423, 424 crossover frequency, 35, 36, 38, 392 crossover slope, 38 cyclic property of the trace, 614

D

damping factor constraint, 470 D-contour, 391, 419 dead time, 35, 432, 574 decay rate, 158 decentralized control, 98, 172, 264, 272 decomposition, 39, 210, 650

~ into orthogonal functions, 567 ~ into piecewise linear functions, 589

decoupling property, 568 definiteness, 202, 630 degeneracy, 634, 636, 637

degree, 476 ~ of inefficiency, 669 ~ of interpolation, 496 ~ of polynomials, 565 ~ ofrobust stability, 175 ~ of robustness, 41, 400 ~ of stability, 181, 244, 273

degree-dropping boundary, 339, 345 delay, 574

computation ~, 236, 241 delayed perturbation, 210 demarcation, 562 denominator, least common ~, 407 denominator matrix, 476 derivative, 68, 79, 204, 363

~ of a singular value, 366 ~s overview, 79

709

~ with respect to a vector or a matrix, 79 ~ with respect to time, 88

desensitized controller, 132, 137 design of a time-varying system, 576 detection observer, 607 determinant, 391, 408, 612

~ derivative, 83 ~ of a partitioned matrix, 644

deviation matrix, 193 diagonal

~ dominance theorem, 168 ~ elements, 183 ~ feedback, 420 ~ matrix, 54, 610 ~ perturbation, 59 ~ scaling, 411 ~ weighting, 164

diagonalization, 192, 637 diagonally dominant, 371, 532 diagonally perturbed system, 387 diagonally weighted norm, 59, 387 diagonizable, 206, 207, 212

~ transition matrix, 231 diamond, 325 differential equation, 91, 107

time-varying ~~, 145, 148 differential operator, 475, 463 differential recurrence, 569 differential sensitivity, 33, 71, 79, 95, 137, 395,

535 ~~ feedback, 138 minimize ~~, 256 ~~ of characteristic polynomial, 115 ~~ of state variables, 111 performance ~~, 123 transfer function ~~, 119

differentiation, 574 dimension, 196, 613 Diophantine see Bezout identity Dirac function, 668 Dirac input, 297

710

direct matrix norm, 48 directed gap metric, 520 direction, most sensitive ~, 369 disc, 174,188,254,310,325,371,417

~ bounding eigenvalues, 175 open unit ~, 502 ~ polynomial, 335

discontinuous see discrete-time discrete lossless positive real function, 331 discrete-time interval system, 184, 352 discrete-time system, 69, 69,112, 165, 177, 179,

188, 190, 195, 225, 249, 351, 374 distance from Hurwitz stability, 339, 342 distance, graph metric ~, 519 distance problem, 517 distinct eigenvalues, 354, 624, 628, 633, 636 distributed control, 323 distributed time-varying system, 604 distribution, 520 distributivity, 73, 609 disturbance, 34,42,397,482

~ differential equation, 459 ~ rejection, 31,390,403,459,464,483,507,

527 dither, 520 divergence, 81 divisive feedback structure, 435 domain of attraction, 212 dominant, 466

~ eigenvalue, 151,547,552 ~ pole location problem, 335 ~ state, 537, 539, 544, 550

double Laguerre expansion, 604 doubly-coprime, 486 D-weighted norm, 59, 164,414 dyadic

~ decomposition, 632 ~product, 199,259,572,610,613,614,632 ~ representation, 78

dynamic ~ control factor, 121 ~ controller, 154, 156, 218, 400, 540 ~ interval system, 167 ~ modelling uncertainty, 491

dynamically bounded, 149

E

E-contour, 417 edge, 196,328,344

~ perturbation, 443 ~ polynomial, 333 ~ theorem, 334, 338,443

eigenlocus, 380 eigensolution, 624 eigenstructure assignment, 640 eigensystem assignment, 626

Index

eigenvalue, 55, 82, 167, 170, 173, 180, 184, 190, 200,214,227, 256, 273, 298, 394, 417, 615, 623

~ assignment, 117, 532,626 ~ bound, 371 ~~ of an interval matrix, 174

~ decomposition, 650 ~ differential sensitivity, 95 ~ distribution, 190 dominant ~, 151 ~ exclusion circle, 417 ~ exclusion lemma, 383 generalized ~, 192 ~, Hankel norm, 61 ~ inclusion region, 418 ~ increment, 104, 164 largest. modulus of an ~, 49 multiple ~s, 102 non-distinct ~s, 359 ~ of a complex matrix, 357 ~ of the Kronecker product, 75 ~ of the Kronecker sum, 76 ~ of the Lyapunov matrix, 204, 230 ~ of the symmetric part, 210 ~ of the transition matrix, 225 ~ properties, 629 ~ sensitivity, 640

eigenvector, 78, 90, 127, 216, 257, 275, 410, 533, 623

~ assignment, 98, 637, 640 ~ chain, 623 ~ derivative, 105 ~ differential sensitivity, 95, 103, 107 ~ increment, 104, 106 normalized ~, 104,360 ~ of Hermite matrices, 359 ~ of the Kronecker product, 75 ~ of the Kronecker sum, 76 ~ of the transition matrix, 227

element by element bound, 162, 165, 167, 181, 188

elementary matrix, 610 ellipsoid sets, 277 ellipsoidal constraints, 278 elliptic norm, 47 encirclement, 310, 391, 408, 419, 435 enclave, 335 energy

~ bounded, 510 ~ gain factor, 504 ~ of the error, 509 ~ of the signal, 64

entropy function, 509 envelope, optimum ~, 300 equivalent controller, 526 equivalent perturbed system, 428 error, 511, 658

~ energy, 509

Index

~ estimate, 649 generalized ~, 483 ~ matrix see perturbation matrix ~ polynomial, 323, 351, tracking ~, 36

estimate, 281, 291, 294 ~ of a function, 66

estimation, 607, 655, 658, 669 ~ error, 285, 292, 669 ~ in large-scale systems, 662 linear~, 667 ~ sensitivity, 669

Euclidian norm see Frobenius norm even part, 325 even-power term, 345 excess stability margin, 515 exclusion circle, 417 exogenous system, 463 expansion

~ integral of the product, 572 power series ~ of orthogonal functions, 571 ~ product of two functions, 572 Tay lor series ~, 557

expectation, 102, 667 expected function of Frobenius norm, 292 exponential matrix, 107,633, exponential L~-stability, 307, 310 exponentially decaying bound, 150 exponentially stable, 244, 247, 477 exposed edges, faces, 334 extended controllability matrix, 548 extended state controller, 139 external signal, 482 externally skew-prime matrix, 461, 476 extrapolation, 129 extreme coefficients, 187 extreme polynomial, 324

F

face, 196 factorization, 457, 475, 491

fractional ~, 515 inner-outer ~, 497 normalized ~, 488 spectral ~, 504, 514

Faddeev method, 110 failure, 34, 607 family of polynomials, 328, 333 fast mode, 525, 534, 537, 539, 541, 544 fast time scale, 539 feedforward, 465 fictitious uncertainty, 447 field of values, 358 final boundary condition, 270 finite energy, 504, 506 finite horizon case, 263 first-order expansion, 265

first variation, 132 fixed mode, 98 forcing phasor, 452 forgetting factor, 586 Fourier transformation, 69, 307 fractional representation, 477, 484, 515 fractional uncertainty, 39, 493

711

free of zeros in the closed right-half s-plane, 501 frequency-dependent uncertainty, 439 frequency domain, 52, 69, 407 frequency richness, 42, 293 Frobenius formula for inversion, 643 Frobenius norm, 47, 81, 95, 101, 149, 162, 175,

186, 202, 215, 256, 311, 365, 412, 423, 559,611,655,660

~~ derivative, 90 expected function of the ~~, 292

full-order observer, 284 full rank, 613 full-state loop transfer recovery, 287, 289 function norm, 53, 63 function of a matrix, 108 function space, 63, 301 functional analysis, 63 functional differential equations, 576 fundamental matrix (see transition matrix) 623,

636 future outputs, 61

G

g~n, 36, 227, 231,316, 399 average ~, 308 ~ and singular value, 52 ~ factor, 226, 504, 665 ~ margin, 289, 435, 491 ~ matrix, 284, 291, 292 ~ range, 308 ~ scheduling, 42

gap metric, 520 Gastinel-Kahan theorem, 195, 373 Gauss distributed, 294, 667 Gauss-Markov theorem, 669 Gaussian noise, 291 general

~ distance problem, 517 ~ matrix equation, 646 ~ orthogonal polynomial, 565 ~ pseudo-inverse, 646 ~ stability bounds, 146 ~ system inverse, 647

generalized ~ eigenvalue, 192 ~ eigenvector, 633 ~ Nyquist criterion, 419 ~ observer, 288 ~ Parseval theorem, 70 ~ plant, 508

712

- polynomial, 331, 334 - product, 557 - resolvent matrix, 385 - signal, system, 482 - stability, 170

geometric multiplicity, 634 Gershgorin's theorem, 176, 188, 313, 371 globally stable, 201 globally weighted modulus matrix, 208 gradient,81,84,126,133,221,251,260,416,568,

592,650 gramian, 59, 60 graph metric perturbation, 519 greatest common divisor, 476 Greville's method, 648 grey matrix, 193 guardian map, 170

H

Hadamard product, 412 Hamiltonian, 279

- function, 260 - matrix, 379 - property, 125

Hammerstein model, 563 Hankel

- approximation, 517, 519 - norm, 59, 490 - operator, 60 - singular value, 491

Hardy space, 64, 499, 502, 672 harmonic function, 570 Hermite

- form, 363 - matrix, 46, 53, 358, 630 - polynomial, 565, 567

Hessian matrix, 128 hierarchical controller, 274 high-frequency

neglected - dynamics, 436 - oscillations, 473 - range, 392 - signal, 520

Hilbert norm, 50 (see also spectral norm) Hilbert space, 503 H2-norm, 64, 65

minimum weighted -, 514 Hoo-norm, 64, 299, 339, 380, 435, 443, 480, 490,

496,500,505,509,516,518,521,672 -, invariance re inner matrix, 497

hodograph, 349 Holder

- inequality, 65, 307, 311 - norm, 47, 49, 256

homegeneous solution, 145, 478 Hurwitz

- antisymmetric, 354

Index

- determinant, 252 - polynomial, 325, 516 - stability, 176, 183, 194, 324, 339, 385 - testing matrix, 327, 349

hyperellipsoid, 346 hyperplane design, 525 hyperplane motion, 530 hyperplane, supporting -, 196 hyper-rectangle, 343 hypersphere, 219, 339, 345, 346 hysteresis, 31

idempotent matrix, 526, 610, 616, 618, 660 identifiability condition, 657 identification, 42, 491, 537, 581, 602, 605, 607,

620 identity matrix, 610, 618, 631, 637

eigenvalues of the --, 359 ill-conditioned matrix, 649 imaginary axis, 167 imaginary eigenvalues, 191 imaginary part of the eigenvalue, sensitivity of

the --,100 impulse input, 297 impulse response, 312 inaccurate calculation, 43 inclusion principle, 359 inclusion region, 417 increment, 133, 135, 263

eigenvalue -, 164 incremental

- bound, 301 - notation, 90 - sensitivity, 270 - updating, 125

incrementally conic nonlinearity, 305 indentation, 419, 427 independent information, 652 index of an eigenvalue, 634 index of performance see performance index, robust stability -, 66 individually weighted modulus matrix, 208 induced

- gain, 399 - norm, 48, 49, 53, 374

inefficiency, 669 inertia, 190 infimum, 164, 412 infinite horizon case, 263 infinitely sensitive, 256 infinity function norm, 351 infinity norm, 46, 51, 62, 63, 68, 174, 193, 313,

416, 511, 516 initial condition, 117, 279, 461, 479, 595, 599,

605, 619 optimal -- for observers, 656

Index

initial function, 151 initial state, 60, 129, 140, 153,221,266,400,577,

581,586,619 random ~~, 134

initial value, 91 inner, 365

~ matrix, 496, 497 inner loop stability, 461 inner-outer factorization, 497 inner product, 46, 81, 104, 233, 357, 363, 369,

502,611 input noise, 283, 291 input-output mapping, 66 input-output relations, 505 input space, 505 insensitive, 256, 535 instrumental variable, 666 integral of squared error, 582 integration by parts, 262, 598 integrity, 470 interconnected system, 259, 553 interconnection, 247, 264, 274 interdependent perturbation, 336 interelement dependency, 80 interlacing property, 325, 332 internal

~ model principle, 459, 464 ~ parallel model, 402 ~ stability, 461, 481, 499

interpolating matrix polynomial, 615 interpolation, 110, 495, 656 intersection, 525

~ of parameter regions, 252 ~ theorem, 344

interval ~ boundaries of orthogonal polynomials,

566 ~ matrix, 38, 167, 171, 174, 176, 187, 188,

193,196,229 ~ plant, 336 ~ polynomial, 167, 185, 252, 323 ~ scalar, 178 ~ vector, 185, 250

inverse, 77,365,610,611,632,641 ~ complementary sensitivity function, 508 ~ matrix derivative, 88 ~ of a partitioned matrix, 643 ~ return difference, 432, 482

invertible matrix, 486 involutory matrix, 362 irreducible matrix, 416 iteration, 129, 251, 263, 271, 276,401,550,663

J

Jacobi formula, 100 Jacobi polynomial, 565, 566 Jacobian matrix, 85, 111

Jordan ~ block, 109, 634, 637 ~ canonical form, 370, 626, 634, 639

K

Kalman-Bucy filter, 291 Kalman controllability matrix, 548 Kalman inequality, 425 kernel see null space Kharitonov

~ polynomial, 252, 324, 337, 349 ~ segment, 338, 339

713

~ theorem for continuous-time systems, 184,324,339

~ theorem for discrete-time systems, 328 Kleinman lemma, 84, 263 Kronecker

L

~ algebra, 73, 79 ~ delta, 668 ~ matrix, 610 ~ power, 89, 601 ~~ model, 559

~product, 73, 78, 96,601,609 ~~ derivative, 87 ~~ of unitary matrices, 360 spectral norm of ~~, 51

~ sum, 75, 78, 170, 195,378

Lachmann model, 563 Lagrange multiplier, 131, 133, 141,252,267,339,

592,663,669 Laguerre polynomial, 565, 567, 604 Laplace transformation, 70, 108, 127, 545, 573,

619 large-scale system, 244, 264, 554, 662 largest

~ absolute column sum, 154 ~ hypersphere, 339, 340, 345 ~ modulus of an eigenvalue, 49 ~ stability box, 355

leading coefficient, 325 leading principal minor, 349 least-favourable noise, 295 least squares, 658 least upper bound, 49 Lebesque space, 503 left

~ convergent, 178 ~-coprime matrix, 477, 479 ~-eigenvector, 623 ~-fractional, 480 ~-inverse, 479, 529, 618, 645 ~-prime factorization, 461 ~-prime matrix, 475

714

-pseudo-inverse, 274, 548, 586, 599, 647, 652, 659, 673, 669

~ sector, 325 ~ singular vector, 367

Legendre polynomial, 565, 567, 570 Leverrier's algorithm, 110, 186 L2-gain factor, 60 limited-time exciting case, 470 L~-induced norm, 67 line segment, 337 linear

~ algebraic equations, error estimate, 649 ~ causal stable operator, 494 ~ dependent information, 652 ~ optimal control via orthogonal functions,

591 ~ quadratic Gaussian theory, 504 ~ quadratic regulator, 213, 288 ~ regression, 655 ~ system solution estimate, 151

L2-norm, 69, 307, 311, 316 localization of component failures, 607 loop

~ gain, 36, 315, 508 ~ shaping, 393, 442 ~ transfer function, 287, 393 ~ transfer recovery, 287

lossless positive real function, 329, 331 low-frequency

~ parameter errors, 436 ~ performance, 393

lower bound, sensitivity norm, 102 LQ regulator see Riccati controller LQG control, 43, 294 LR-Perron scaling, 424 L2-S2-gain, 68 L;-stability, 63, 301,310, 313, 399 Lyapunov

M

~ approach for unidirectional perturbation, 171

~ equation, 78,135,139, 158, 160, 161, 189, 194, 202, 214, 225, 235, 244, 262, 266, 299,381,626

~ function, 173, 225 ~~ dynamics, 200

~ stability, 199, 201

main diagonal element, 176 manifold, 525 mapping, 66

~ theorem, 443 Markov

~ parameter, 355 ~ process, 668 ~ theorem on determinants, 354

Martin's theorem, 374

matricial gradient, 82, 86, 221, 260 matrix

aggregation~, 547 ~ algebra, 609 ~ analysis, 79 ~ component, 108 ~ decomposition, 650 ~ differential equation, 91, 202, 278 ~ exponential, 75, 76, 107,633 ~ function, 615 ~ infinity norm, 49, 51 ~ inversion lemma, 641 ~ measure, 45, 56, 201, 212, 217 ~ norm, 45 ~ I-norm, 49 ~ polynomial, 420, 631 ~ product approximation, 594 ~ product rule, 86 ~ set, 557 symmetric ~, 57 ~ transpose, 642 ~-valued function, 86

max norm, 46 maximally robust pole placement, 251 maximizing robustness region, 221 maximum

~ absolute column/row sum, 193 ~ bias matrix, 193 ~ eigenvalue, 227 ~ gain, 227 ~ modulus, 210 ~-modulus theorem, 64, 502 ~ perturbation, 41, 185, 348, 349 ~ positive real eigenvalue, 171 ~ real part of an eigenvalue, 630

Index

~ singular value, 210, 312, 376, 392, 398, 409,439

~ stability bound, 342 mean value, 667 measure, matrix~, 45, 56 measurement noise, 283, 291, 390 measurement signal, 482 Metzler matrix, 229, 248, 616 Mikhailow robustness criterion, 349 minimal

~ dimension of stable faces, 196 ~ distance to Hurwitz stability, 339 ~ Frobenius norm, 412 ~ least square, 655, 659 ~ polynomial, 108,465, 469, 615, 636

minimax ~ frequency optimization, 457 Lyapunov ~ controller, 242 ~ optimization of the spectral radius, 370 ~ problem, 278, 295

minimum ~ control energy, 61 ~ degree of stability, 273

Index

- differential sensitivity, 100 - distance from the critical point, 421 - distance problem, 490 - intersection theorem, 344 - negative real eigenvalue, 171 - norm controller, 252, 402 - number of edges, 333 --phase, 478, 496 - sensitivity, 516 - singular value, 365, 389,404,409,417 - variance estimator, 669 - variance scaling, 413

Minkowski inequality, 66 minor, 349,408,611 mirror-image/polynomial, 330 mixed product rule, 74, 601 M-matrix, 616 modal

- decomposition, 625 - matrix, 165, 188, 206, 207, 257, 360, 363,

370, 624, 650 - state, 614 - transformation, 162

model --following approach, 142 --matching, 509, 516, 517 - plant mismatch, 404 - uncertainty gain, 317

modified spectral radius, 451 modulus, 394

- matrix, 153, 167,207,229,381 - sensitivity, 101

monic polynomial, 185, 333, 381, 623 most sensitive direction, 369,409 m-tuple of polynomials, 337 multi-model approach, 252 multilinear coefficient function, 335 multiple eigenvalues, 256, 633, 637, 638 multiplicative uncertainty, 38, 301, 366, 386, 390,

392,493,508 multiplicity, 76, 92, 109, 364, 513, 615, 633, 639 multivariable circle criterion, 313 multivariable system, gain of the --, 52

N

nabla operator, 81 nearest normal approximation, 418 necessary condition, 199, 329 negative definite, 173, 182, 199, 210, 630 negativity, 183 Nehari extension, 490 Nevanlinna-Pick theory, 496, 497 Newton-Raphson method, 129 nice stability, 252 nilpotent matrix, 610 node, 484

715

noise, 31, 43, 234, 274, 283, 284, 291, 389, 436, 668

--free output, 283 least-favourable - uncertainty, 292, 295 measurement -, 479, 482 - reduction, 318 - rejection/suppression, 397, 508 - uncertainty, 294, 295

nominal, 36, 137, 159, 243, 249, 302, 324, 345, 392,399,408,418,449

- parameter, 124 - polynomial, 324

non-anticipative, 67 non-autonomous system, 200 non-defective, 78 non-diagonal scaling, 412 nonlinear

- control, 222, 301 - estimator, 669 - perturbation, 159, 204, 211, 214, 225 - plant, 155, 520 - regression, 129 - singularly perturbed system, 541 - system, 599, 601, 557

--, block pulse expansion approach, 585 - uncertainty, 301

non penalizing control input, 471 non-similarity Perron scaling, 415 nonsingular, 56, 169,202,374, 613 nonsingular Kronecker sum of perturbed matri-

ces, 195 nonsingularity approach, 169 nontruncated function, 67 norm, 45, 178, 180, 184

- bounds, 153 column sum -, 51 consistent ""', 48 diagonally weighted -, 59 direct matrix -,47 Euclidian -, 48 Frobenius -, 48 function -, 63 function p--, 502 Hankel-,59 Holder -, 47 H2-, Hoo--, 503, 672 induced matrix -, 49 infinity matrix -m 51 infinity -, 63 L~-induced -, 67 matrix 1-~, 49 - of a matrix-valued function, 65 - of a vector-valued function, 63 - of mth power of a function, 65 - of the controller, 135 operator -, 67 row sum~, 51 Schur ~, 48

716

sensitivity matrix -, 101 Sobolev -, 68 spectral -, 50, 234 supremum -, 63 - uncertain plant, 315 vector -, 46 weighted -, 59

norm-bounded matrix, 438 norm-bounded perturbation, 164 norm-like Lyapunov function, 244, 246 normal matrix, 205, 206, 228, 273, 300, 359, 360,

364,384,418 normalized, 75, 104

- factorization, 488 normally distributed, 667 null space, 257, 525, 613, 616, 637 number of

- corners, 332 - edges, 334 - negative eigenvalues, 201 - segments, 337

numerator matrix, 116 numerical radius, 45 numerical range, 45, 418 Nyquist criterion, 64, 122, 242, 308, 335, 350,

351,391,407,419,452

o

observability, 470, 544 - gramian, 60

observation vector, 668 observer, 281,454, 607

- pole allocation, 288 observer-based

- compensator, 472, 491 - control, singularly pert. syst., 546 - controller, 454

odd part, 325, 326 odd-power term, 345 on-off-control, 534 one-parameter family of matrices, 196 one-sided Laplace transformation, 70 open-loop

- property, 285 - singular values, 393 - system, 313 - transfer function, 121 - uncertainty gain, 317

open right-half s-plane, 503 operational matrix

bidiagonal -, 605 - for delay, 574 - for differentiation, 574, 597, 605 - for integration, 569, 576, 580, 582, 588,

592,594,606 stretched -, 577

operational product vector, 572

Index

operator, 43, 66, 149, 301, 315, 399, 443, 463, 469,494

column -, 76 Hankel-,60 - norm, 67, 304, 313, 505 trigonal -, 618

optimal control, 123, 138, 160,238,259,262,265, 294,581

optimization, 62, 267, 279, 480 optimum gain, 228 order reduction, 526 orthogonal, 73, 257, 339, 624

- basis, 568 - coefficients, 567 - component coefficients, 591 - expansion, 40, 599 - matrix, 551 - polynomial, 565 - projection, 518 - transformation, 531, 650

orthogonality, 359, 566 orthonormal basis, 368 orthonormality, 568, 632 outer matrix, 496 outer product (see also dyadic product)

increment of the --, 105 outermost boundary, 178 output, 390

- energy, 61 - feedback, 97 - feedback controller, 130, 160, 164, 168,

252,259,538 - noise, 283, 291 - norm estimate, 52 - power, 505 - regulation see disturbance rejection - space, 505 - spectral density, 511 - uncertainty, 452

overall closed-loop transfer matrix, 398 overdetermined, 657 overshoot, 297

P

para-Hermitian transpose, 511 parallel model, 402 parallelotope, 343, 344 parameter, 37,250,657

- demarcation, 655 - estimation, 576, 585, 595, 599 - region, 252 - sensitivity, 111 - space, 203, 660

parametrization, 257, 274, 323, 475, 515, 637, parasitic dynamics, 537 Parseval theorem, 69, 70, 298, 307, 312 particular solution, 145, 478,

Index

partitioned matrix, 433, 487, 518, 642, 648 partitioned vector, 611 partitioning into columns, 648 pathwise connected, 335 penalizing closed-loop poles, 254 penalty near a root, 471 performance, 34,39, 123, 142, 172, 181,236,252,

n4,200,2~,~8,~4,~,W7,~,

389,392,395,400, 4lO, 436, 447, 505, 510,532,549,558,560,592,657,

~ deterioration, 394, 397 ~ gradient, 126 ~ kernel difference, 270 ~ index, 213 minimizing~, 141 ~ of qI-transformed signals, 471 ~ robustness, 445, 517 ~ robustness theorem, 416 ~ sensitivity, 123, 130

permutation matrix, 73, 77, 126,416,618,644 Perron eigenvector, 387, 414, 416 Perron (-Frobenius) eigenvalue, 164, 168, 169,

172, 385, 412, 416, 422, 424, 442, 616, 633

Perron-Frobenius theorem, 633 persymmetric matrix, 619 perturbation, 34, 71, 95,101,161,164,171,179,

185, 188, 191, 195, 199, 209, 215, 232, 234, 238, 277, 324, 336, 365, 369, 373, 377,399,408,417,488

~ bound, 204, 206 ~ by weighted sum of matrices, 218 ~ coefficients, 219 delay ~, 211 diagonal ~, 59 ~ factor, 205,211 graph metric ~, 519 initial condition ~, 297 linear ~, 159 ~ matrix, 219, 256 maximum~, 348 nonlinear ~, 159,211,225,234,247 ~ parameter, 170 plant~, 293 ~ polynomial, 324, 343

random ~~, 234 singular ~, 537 structured ~, 238, 422 time-varying~, 238 unidirectional ~, 168, 239 unstructured ~, 238, 519 ~ via constant matrices, 216

phase margin, 35, 289, 425, 491 phase variable form see companion form phasor, 452 Pick matrix, 496, 497 PI-controller, 253, 454, 455 piecewise-constant suboptimal controller, 271

piecewise linear functions, 579, 587 plant sensitivity, maximum ...... , 316 plant uncertainty principle, 490 p-norm weight, 414 point matrix, 178 polar decomposition, 2lO, 369

717

pole assignment/allocation/placement, 98, 118, 154, 182, 249, 284, 289

pole excess, 121 pole zero cancellation, 476 polygon, 328 polynomial, 37, 74, 249, 323, 407, 420, 623

~ coefficient, 339 ~~ function, 335

corner ~, 324 extreme ~, 324 generalized ~, 331 ~ matrix, 408, 459, 475, 484 minimal~, 108 orthogonal ~, 565 ~ product, 557 vertex ~, 324

polytope, 196, 203, 324, 333, 342, 353 Popov criterion, 3lO positive definite, 55,159,189,190,197,199,202,

206, 216, 225, 359 positive matrix, 180 positive real function, 329 positive realness and Schur stability, 331 positivity, 183, 252

~ test, 195 postmultiplication, 610 power

~ density, 668 ~ expansion, 121 ~ of a matrix, 641 ~ series domain, 599 ~ series expansion, 557, 571, 576, 592 ~ spectral density, 505

prediction, 235, 665 prefil ter, 283 premultiplication, 6lO prescribed eigenvalue, 258 prespecified controller, 134 primeness, 461 principal

~ direction, 367 ~ gain see singular value ~ minor, 349, 616, 623 ~ phase, 2lO ~ vector, 633

principle of argument, 419 probability distribution, 668 process noise, 294 product

~ of interval matrices, 178 ~ of singular values, 365 ~ of two functions, 589

718

~ rule, 74 projection, 430, 518

~ of a matrix, 369 ~ operator, 251

projector, 526, 529, 616, 647 proper, 64, 244, 477, 493, 503, 545 proportional-integral

~ controller, 455 ~ observer, 288

proportional observer, 284 pseudo-commutativity property, 461 pseudo-inverse, 74,99, 135,637,645,648 pulse functions, 579 purely fast variable, 540

Q

quadratic form, 199, 629 quality see performance quasi-diagonal form, 239 quasi-dominant, 616

R

radius numerical ~, 45 ~ of robust stabilizability, 489 spectral -, 49

random, 102, 667 - initial state, 134, 266 - sampling, 225

range, numerical -, 45 range space, 257, 526, 529, 612, 616

-- of stability, 328 rank, 47, 531, 613, 617

- reduction, 369 rational transfer matrix, 479 Rayleigh's principle, 45, 204, 205, 228, 255 Rayleigh quotient, 45, 47, 216 Rayleigh's theorem, 364, 630, 631 reachable, 469 real part of an eigenvalue, 630

sensitivity of the -~, 100 real polynomial, 334 real-rational functions, 64, 503 real root boundary, 339, 340, 345 real stabilty radius, 377 real-time process, 577 realizability, 403 reciprocated polynomial, 330 recovery

loop transfer -,287, 289 - matrix, 288 robustness -, 287 sensitivity ~, 287, 291

rectangular ~ box, 329 ~ coefficient space, 326

- polytope, 324 - representation, 358

recurrence - coefficients, 565, 567 - equation, 584

recurSIve

- algorithm, 558, 578 - estimation, 664

reduced matrix, 611 reduced-order

- model, 31, 60, 540 - observer, 281

reduced sensitivity, 429 reducible matrix, 644

Index

reference, 117,284,398,465,469,479,482,511 - differential equation, 459 - energy, 510 ~ input, 389 - norm, 318

reflected argument, 496, 511, 519 regression, 274, 549, 561,595, 647, 655

nonlinear ~, 129 - sum, 660

regulator form, 628 relatively left-prime/right-prime, 476 relatively prime, 461 rescaling operation, 442 residual, 658 residue, 117,513 resilience, 35, 403 resolvent matrix, 167, 180,373, 623 retardation operator, 66 return difference, 237, 240,287,508, 609 return gain, 315 return ratio matrix, 449 RH2 , RHoo, 64,486 Riccati

- coefficient matrix, 125 - controller, 123, 134, 173, 214, 238, 294,

426, 532, 592 - differential equation, 264, 269, 472, - equation, 160, 191,214,272,549,561,595 - matrix sensitivity, 125

Riemann sphere, 335,429 right

- convergent, 178 --coprime factorization, 519 ~-coprime matrix, 477, 479 --divisor, 460 --eigenvector, 623 --fractional, 479 --half-plane zero, 506 --inverse, 618, 644 --prime matrix, 475 --pseudo-inverse, 562, 646, 655, 673 - singular vector, 367

robust ~ dynamic controller, 154

Index

- observation, 607 - pole placement, 250 - rout locus, 348 - stability, 153, 175, 195 - stability index, 66

robustification, 210, 218, 348 robustness, 31, 33, 40,42, 145, 167, 172, 181,

195, 203, 206, 208, 210, 211, 213, 219, 222, 248, 253, 272, 277, 297, 301, 307, 352, 373, 377, 383, 389, 400, 409, 416, 429, 436, 445, 459, 469, 483, 490, 495, 499,506,517,523,527,538,558

- bounds, 244 - in the frequency domain, 321 - in the steady state, 289 - in the time domain, 145 - measure, 344 Mikhailov - criterion, 349 Nyquist - criterion, 350 - of Kalman-Bucy filters, 291 - of proportional integral observers, 288 - of proportional observers, 281 - recovery, 287 small-scale -, 33 stability -, 167 - using gap metric, 520 - via orthogonal decomposition, 607

root clustering, 254 root distribution, 189 root locus, 167, 334, 348 Rosenbrock matrix, 118 rotation matrix, 618 Routh's criterion, 328 row

- compression, 368 --like, 517 - rank, 496 - string, 77 - sum, 49, 59,176

S

saddle-point problem, 295 sampling interval, 225 saturating actuator, 520 saturation-like structure, 224 saw-tooth signal, 520 scalar product, 46, 81, 104, 233, 357, 369, 502,

611 scalar product rule, 88 scaled

- argument, 577 - decomposition, 651 - perturbation matrix, 172

scaling, 59, 187,217,225,410,422,442,443,515, 649

- matrix, 172, 180, 379, 386,433, 438

Schiiflian form, 170 Schur

- matrix norm, 48 - polynomial, 351

719

- stability (see also discrete-time systems) 178,182,196,328,331,336,353,385

Schwartz inequality, 611 second-derivative sensitivity, 128 secondary diagonal matrix, 618 sector, 521

- condition, 235 - gain, 307, 313, 499 --type inequality condition, 197

segment, 337 self-derivative matrix, 73 semidefinite, 58, 199 sensitive to a single component, 607 sensitivity, 34, 41, 121, 390, 395, 397,427, 429,

480, 482, 499, 504, 506, 515, 516, 535, 640

differential -, 33, 71, 79 estimation differential -, 669 - function, 242 - matrix, 101,302 maximum plant -, 316 - minimization, 483 - minimizing in H2 ,511 - norm, 102 - of eigenvalues, 256 performance -, 123 - recovery, 287 - vector, 139

sensor, 196 - noise, 31, 293, 436 - output, 482 - perturbation, 172

separation principle, 288 servo-compensator, 459, 464 servo-controller, 464 set-theoretic approach, elliptic -, 277 setpoint of the performance, 134 shape, 187, 278 shaping, 393, 442

- signal, 470 shifted orthogonal polynomial, 566 shifting eigenvalues, 631 sign, 324, 325

- function, 531 signal energy, 64, 69, 506 similar matrices, 612 similarity scaling, 411, 422 similarity transformation, 542 simple

- connectedness, 335 - L2-stability, 306 - perturbation, 340

single-input single-output system, 392 single parameter perturbation, 192

720

singleton, 196 singular

- matrix, 195 - perturbation, 38, 537

singular value, 163, 187, 205, 214, 217, 233, 238, 239, 245, 287, 357, 364, 389, 396, 445, 505

-- decomposition, 163, 361, 367, 494 --- and pseudo-inverse, 648

largest --, 50 maximum/minimum --, 90 -- properties, 365 structured --, 433, 438, 439

singular vector, 163,220, 366,423 singularity, 612 skew-Hermite, 358

- matrix, 190 skew-symmetric, 358 sliding mode, 40, 525 slow mode, 537, 539 slow sliding mode, 534 slow state, 541 Small gain theorem, 304, 315, 339, 399, 490 Smallil-theorem, 439 small-scale perturbation, 71 small-scale robustness, 33, 95 smallest but one singular value, 410 smallest singular value and rank reduction, 369 Smith predictor, 241 smoothed nonlinearity, 521 Sobolev-boundedness, 521 Sobolev space, 68 solution error estimate, 649 spectral abscissa, 49, 162, 164, 169,205,206,298,

374,402 spectral condition number, 164, 206, 214, 231,

239, 240, 535, 649 derivative of the --, 90

spectral - cross-condition number, 423 - decomposition, 112, 649 - density, 511 - factorization, 504, 514

spectral norm, 38, 48, 50, 55, 65, 173, 178, 208, 210, 230, 234, 255, 256, 297, 302, 311, 364,375,435,559,640

-- bound, 207 spectral radius, 49, 168, 172, 174, 178, 180, 184,

227, 229, 244, 245, 370, 374, 387, 422, 439, 449

spectral - representation, 78, 632 - richness, 398, 468

spectrum, 358, 360,379, 583 speed requirements, 297 square root decomposition, 652 stability, 153,226,391,405,408,430,435

- bounds, 146, 149

Index

- condition, 195,301,399 - degree, 308 generalized -, 170 - hyperellipsoid/sphere, 345, 346 - improvement, 293 - index, 66 - margin, 188, 247, 284, 338, 339, 346, 387,

~9,~2,400,4n,4~,«3,4~,OO8,

515 - preserving perturbation, 191

stability radius, 172, 226, 366, 376 - of parametrized system, 380 - of polynomials, 381

stability - region, 213 - robustness, 158, 167, 181,445 -- measure, 298

transformation of - to nonsingularity, 169 - with time delay, 151

stabilizability quadratical -, 222 radius of robust -, 489 robust -, 492

stabilization, 487 strong -, 495

stable face, 196 stable matrix, 55, 64, 168, 170, 171, 202, 612,

630 stacking operator, 76 standard form of the actual plant, 538 state-dependent noise, 293 state feedback, 117, 134, 138, 160, 163, 213, 252,

256, 283, 325, 525, 573, 626 state-space uncertainty, 161 state variable

-- differential sensitivity, 111 -- matrix, 259

stationary stochastic process, 667 statistical perturbation, 102 step input, 297 stereographic projection, 430 stiff dynamic system, 55 stochastic

- parameter, 234 - perturbation, 34, 234 - process, 668 - signal, 505

stretched - operational matrix, 578 - time scale, 577

strict aperiodicity, 354 strictly proper, 64,503, 545 strong Kharitonov theorem, 324, 332 strong stabilization, 481, 495 structural

- constraint, 264 - information, 416 - model error, 157

Index

structured ~ E-contour, 425 ~ Lyapunov function, 201 ~ perturbation, 179, 180 ~ singular value, 433, 438, 439 ~ stability radius, 379 ~ uncertainty, 153, 161, 229, 255, 410, 422,

447 sub determinant, 199 submatrix, 643 submodel, 663 suboptimal controller, 271 subpolytope, 196 subspace, 526 successive estimation, 662 successive matrices, 189 sufficient condition, 169, 183, 193, 199,202,207,

208,215,329,395 sufficient stability condition, 153, 158, 175, 217,

230,233,235,399,409,435,452 sum

~ norm, 46, 154 ~ of matrices, 202 ~ of matrix polynomials, 420 ~ vector, 610

summation junction, 461 supporting hyperplane, 334 supremum, 64, 67, 149, 303,312,397, 503 switching

~ hyperplane, 525 ~ mode, 525, 527

Sylvester inequality, 183 symmetric

~ bounds, 308 ~ interval matrix, 168 ~ matrix, 57, 183, 185

symmetric part, 55, 183, 210, 359, 375, 630 ~~ of a polynomial, 329 ~~ of corner matrices, 182

symmetric polynomial, 330 ~~ coefficients, 332

symmetrizing nonlinear charact.eristic, 309 symplectic matrix, 480 system inverse, 647

T

Taylor expansion, 89, 98, 123, 128, 135,222,615, 633

Taylor series ~~ approximation, 596 ~~ expansion, 557 ~~ model, 557

terminal condition, 219, 260, 279 testing matrix

Hurwitz ~~, 327 Kalman controllability ~~, 548

721

tight bounds, 442 time delay, 151 time-optimal,534 time scale, 539 time-variant system, 263 time-varying

~ differential equation, 145 ~ exponent, 162 ~ matrix differential equation, 93 ~ perturbation, 39, 155,203,207,211,301 ~ state-feedback gain, 268 ~ system, 145,221,315,558,575,585,589

Toeplitz matrix, 619 tolerance (see also perturbation, uncertainty)

391 ~ of desired poles, 250 ~ intervals, 268

trace, 82, 90, 105, 126, 260, 266, 295, 363, 513, 549,612,614,623,630

tracking, 31, 36, 291, 389, 452, 459, 464, 509 trajectory differential sensitivity, 111, 137 transfer function

~~ and chordal metric, 432 ~~ and L2-norm, 307 ~~ sensitivity, 119

transfer matrices, overview of ~~, 485 transfer zero, 115 transformation, "'-~, 465 transition matrix, 93, 149, 151, 190, 206, 225,

268,293,297,558,623,636 ~~ differential sensitivity, 107

transmission zero, 470 transpose, 623

conjugate ~, 358 transversality condition, 260 triangle

~ determinant, 644 ~ inequality, 48, 397 ~ signal, 520

triangular matrix, 610, 619, 652 triangular structure, 551 trigonal matrix, 636 trigonal operator, 618 truncation, 67, 304 two-frequency-scale decomposition, 544 two-level computational structure, 266 two-matrix combination, 176 two-parameter family of matrices, 196 two-polynomial combination, 326, 327, 331 two-sided Laplace transformation, 70 two smallest singular values, 369 two-time-scale decomposition, 539, 542

u

ultraspherical polynomial, 565, 567 unbiased estimator, 669

722

uncertainty, 31, 37, 250, 292, 404, 446, 499 additive~, 301, 399, 409, 433, 450, 492, 497,

515 block-structured ~, 433, 437 dynamic ~, 490 fractional ~, 493 identifying ~ via fuzzy logic, 294 ~ in norm, 315 induced norm of ~, 384 input associated ~, 392, 396,400, 409 ~ margin, 336 multiplicative~, 301,366,390,409,493,508 noise ~, 293, 294 nonlinear ~, 301 ~ of polynomials, 323 output associated ~, 392, 396, 400, 404, 452 ~ parameter, 339 parametric ~, 490 plant ~, 293, 336 plant ~ isolating, 434 ~ plot, 410 ~ radius, 395, 405 ~ reducibility, 456 spectral-noise-bounded ~, 255 structured ~, 153, 161, 255, 410, 422, 447 unstructured ~, 153, 410 various structures, 436 varying~, 345 ~ with common factor n, 211

underdetermined, 655 undesired region, 323 unexcited case, 469 unidirectional perturbation, 37, 168, 169, 179 unidirectional uncertainty margin, 336 unimodular, 475 unit circle, 331 unit vector control, 534 unitary

~ eigenvectors, 409 ~ matrix, 163,206,228,360,367,369,422,

439, 494 ~ transformation, 364 ~ vector, 360

unity feedback, 424 un modelled high-frequency dynamics, 537 unobservable disturbance model, 462 unstructured

~ stability margin, 339 ~ uncertainty, 153, 164, 179, 410

updating parameters, 664 upper left corner, 580 U -stable, 323

v

Vandermonde matrix, 615, 628 variable structure

~~ control, 525

~~ mode, 562 variance, 102,235,667, 669 variation box, 329

Index

variation of parameters, method of ~~, 145 variation of the performance index, 131 variational calculus, 512 varying uncertainty bounds, 345 vee operator, 611 vector

~ infinity norm, 416 ~ norm, 45, 152 ~ process, stochastic, 668 ~ product, 81 ~-valued function, 84

vertex, 196, 344, 353 ~ polynomial, 324

Vieta's rule, 631 Volterra series, 563

W

Walsh functions, 579 weak

~ control of parasitics, 537 ~ Kharitonov theorem, 324, 328, 332 ~ L; -stability, 303 ~ observation of parasitics, 537 ~ robustness, 469

weighted ~ linear regression, 586 ~ matrix norm, 416 "'" norm, 59 ~ sensitivity, 515 ~ square deviation, 568

weighting ~ function, 565 ~ matrix, 142, 158, 164, 199, 209, 287, 297,

400,483,505,511,657,669 Weyl inequality, 631 white noise/stochastic process, 294, 505, 667 whiting matrix, 169, 193

Y

Youla parametrization, 479, 490

z

zero, 341,408,495,496,619 ~ of a polytope, 334 ~ placement, 117 ~~ robustness, 118

transfer ~, 115 z-transfer function and L2-norm, 307 z-transform, 69

by the same author:

Alexander Weinmann Regelungen - Analyse und technischer Entwurf

Band 1: Systemtechnik linearer und linerarisierter Regelungen auf anwendungsnaher Grundlage

Zweite, iiberarbeitete und erweiterte Aufiage 1988. 184 Abb. und 51 Beispiele, XIII, 298 Seiten. Gebunden DM 85,-, oS 590,-. ISBN 3-211-82075-2

Band 2: Nichtlineare, abtastende, multivariable und komplexe Systeme; modale, optimale und stochastische Verfahren.

Zweite, iiberarbeitete und erweiterte Aufiage 1987.116 Abb. und 27 Beispiele, XIII, 278 Seiten. Gebunden DM 85,-, oS 590,-. ISBN 3-211-81977-0

Band 3: Rechnerische LOsungen zu industriellen Aufgabenstellungen

1986.103 Abb. XIII, 247 Seiten. Gebunden DM 78,-, oS 540,-. ISBN 3-211-81925-8

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