chen_ch5

I

182 CONTROLLABILITY AND OBSERVABILITY OF LINEAR DYNAMICAL EQUATIONS

Definition 5-4

The dynamical equation E is said to be uniformly controllable if and only if there exist a positive u, and positive ai that depend on u, such that

O B, we mean that the matrix (A - B) is a positive definite matrix (see Section 8-5). Uniform controllability ensures that the transfer of the states can be achieved in the time interval U c' The concept of uniform controllability is needed in the stability study of optimal control systems. See References 56 and 102.

Instantaneous controllability and uniform controllability both imply controllability. However, instantaneous controllability neither implies nor is implied by unifor.m controllability.

Example 4 Consider the one-dimensional linear dynamical equation

x e-Itlu

Since p(Mo(t)) = p(e -Itl) = 1 for all t, the dynamical equation is instantaneously controllable at every t. However, the dynamical equation is not uniformly controllable because there exists no althat depends on u, bilt not on t such that

rt + l1 , 211W(t, t +u,) = Jr e- 2t dt =0.5e- 2t(l- e- ,) > a I (uc;) for alIt> O.

Example 5

Consider the one-dimensional dynamical equation

x b(t)u with b(t) defined as in Figure 5-5. The dynamical equation is not instantaneously controllable in the interval (-1, 1). However, it is uniformly controllable in ( 00, oo). This can be easily verified by choosing Uc = 5.

A remark is in order concerning controllability, differential controllability, and uniform controllability. If a dynamical equatipn is differentially controllable, a state can be transferred to any other state in an arbitrarily small interval of time. However, the magnitude of the input may become very large; in the extreme case, a delta-function input is required. If a dynamical equation is merely controllable, the transfer of the states may take a very long interval of time. However, if it is uniformly controllable, the transfer of the states can be

~

CON'

b(t)

-3 -2

-I o

Figure 5-5 The function b(t).

achieved in the time interval U : c

will not be arbitrarily large [see ( W - 1]. In optimal control thea sometimes required to ensure th

Time-invariant case. In thi~ n-dimensional linear time-invaril

FE: where x is the n x 1 state vector, and n x p real constant matrices, equations, the time interval of i that is, [0, (0),

The condition for a time-vary that there exists a finite t I such that in t, on [to, t 1]. In the time-im As discussed in Chapter 4, all eler terms of form IkeA.t: hence they are sequently, if its rows are linearly independent on [to, t IJ for any to time-invariant state equation is c( and the transfer of any state to an~ time interval. Hence the referenc trollability study of linear time-im

Theorem 5-7

The n-dimensional linear time-inva if and only if any of the following ec

1. All rows of e-AtB (and conseqt: [0, (0) over C, the field of comp

1', All rows of (sI _A)-IB are line.

6 Although all the entries of A and Bare real of the field of complex numbers.

AR DYNAMICAL EQUATIONS

ycontrollable if and only if there 1Ue such that

S Ct2(Ue)I ~*(t +Ue, t) ~Ctiuc)I .tion matrix and W is as defined

is a positive definite matrix (see I that the transfer of the states Dcept of uniform controllability :rol systems. See References 56

controllability both imply conlIability neither implies nor is

1equation

lical equation is instantaneously lical equation is not uniformly :nds on Ue but not on t such that

tion

al equation is not instantaneousver, it is uniformly controllable Dosing Uc 5.

lility, differential controllability, equation is differentially con

ler state in an arbitrarily small le input may become very large; luired. Ifa dynamical equation may take a very long interval of the transfer of the states can be

CONTROLLABILITY OF LINEAR DYNAMICAL EQUATIONS 183

b(t)

-3 -2 -1 o 2 3

Figure 5-5 The function b(t).

achieved in the time interval Uc ; moreover, the magnitude of the control input will not be arbitrarily large [see (5-14) and note that the input is proportional to W 1]. In optimal control theory, the condition of uniform controllability is sometimes required to ensure the stability of an optimal control system.

Time-invariant case. In this subsection, we study the controllability of the n-dimensional linear time-invariant state equation

FE: x= Ax +Bu (5-23) where x is the n x 1 state vector, u is the p x 1 input vector: A and Bare n x n and n x p real constant matrices, respectively. For time-invariant dynamical equations, the time interval of interest is from the present time to infinity; that is. [0. (0).

The condition for a time-var.ying state equation to be controllable at to is that there exists a finite t 1 such that all rows ofcb(t0, t )B(t) are linearly independent, in t, on [to, t l ]. In the time-invariant case, we have (,1)(10' t)B(t) eA(to-t)B. As discussed in Chapter 4. all elements of eAlto -1)8 are linear combinations of terms of form Ike;'/; hence they are analytic on [0. 00) (see Appendix B). Consequently, if its rows are linearly independent on [0. (0). they are linearly independent on [1o, tlJ for any to and any 11 > to. In other words, if a linear time-invariant state equation is controllable, it is controllable at every to? 0 and the transfer of any state to any other state can be achieved in any nonzero time interval. Hence the reference of 10 and t I is often dropped in the controllability study of linear time-invariant state equations.

Theorem 5-7

The n-dimensional linear time-invariant state equation in (5-23) is controllable if and only if any of the following equivalent conditions is satisfied:

1. All rows of e-AtB (and consequently of eAtB) are linearly independent on [0, (0) over C, the field of complex numbers.6

1'. All rows of (sI - A)-l B are linearly independent over C.

6 Although all the entries of A and B are real numbers, we have agreed to consider them as elements of the field of complex numbers.

l

~ DYNAMICAL EQUATIONS

vcontrollable if and onl y if there (J'csuch that

:;; (l2((J'c)1 ,*(t +(J'c' t)~(li(J'c)1 lion matrix and W is as defined

ls a positive definite matrix {see that the transfer of the states !lcept of uniform controllability rol systems. See References 56

:ontrollability both imply conlability neither implies nor is

equation

cal equation is instantaneously ical equation is not uniformly lds on (J'c bilt not on t such that

lon

Iequation is not instantaneouser, it is uniformly controllable osing (J'c = 5.

Iity, differential controllability, ~quation is differentially con~r state in an arbitrarily small : input may become very large; lired. Ifa dynamical equation (lay take a very long interval of he transfer of the states can be


b(t)

-3 -2 -1 o 2 3

Figure 5-5 The function b(t}.

achieved in the time interval (J'c; moreover, the magnitude of the control input will not be arbitrarily large [see (5-14) and note that the input is proportional to W- 1]. In optimal control theory, the condition of uniform controllability is sometimes required to ensure the stability of an optimal control system.

Time-invariant case. In this subsection, we study the controllability of the n-dimensional1inear time-invariant state equation

FE: x= Ax +Bu (5-23) where x is the n x 1 state vector, u is the p x 1 input vector; A and Bare n x n and n x p real constant matrices, respectively. For time-invariant dynamical equations. the time interval of interest is from the present time to infinity: that is. [0. (0).

The condition for a time-varying state equation to be controllable at to is that there exists a finite t 1 such that all rows of (1)(10' t)B(t) are linearly independent, in t. on [to. t l ]. In the time-invariant case. we have $(10' t)B(t)=eA(to-t)B. As discussed in Chapter 4. all elements of eA(to-tlB are linear combinations of terms of form tke;'t; hence they are analytic on [0. XJ) (see Appendix B). Consequently. if its rows are linearly independent on [0. cJ). they are linearly independent on [to. t l ] for any to and any tl > to. In other words. if a linear time-invariant state equation is controllable. it is controllable at every to ;:::0 and the transfer of any state to any other state can be achieved in any nonzero time interval. Hence the reference of to and t I is often dropped in the controllability study of linear time-invariant state equations.

Theorem 5-7

The n-dimensional linear time-invariant state equation in (5-23) is controllable if and only if any of the following equivalent conditions is satisfied:

1. All rows of e-AtB (and consequently of eAtB) are linearly independent on [0, (0) over C, the field of complex numbers. 6

I'. All rows of (sl A)-l B are linearly independent over C.

6 Although all the entries of A and B are real numbers, we have agreed to consider them as elements of the field of complex numbers.


2. The controllability grammian

W,,~ J~ e"BB'eA', dT is nonsingular for any t > 0.7

3. The n x (np) controllability matrix U~[B: AB: A2B : ... : A"-IB] (5-24 )

has rank n. 4. For every eigenvalue Aof A (and consequently for every). in C), the n x (n +p)

complex matrix [)J - A : B] has rank n. 8

Proof The equivalence of statements 1 and 2 follows directly from Theorems 5-1 and 5-4. Since the entries of e- AtB are analytic functions, Theorem 5-3 implies that the rows of e - AtB are linearly independent on [0, 00) if and only if

p[e- AtB : - e- AtAB : . . . : ( 1)"- 1e - AtA11- I B : ...] n for any tin [0,00). Let t =0; then the equation reduces to

p[B: - AB: .. 0 : 1)" - 1 A" - I B : ( 1)nAnB : ...] = n( From the Cayley~-Hamilton theorem, we know that Am with m ~n can be written as a linear combination of I, A, ... , A"-I; hence the columns of AmB with m ~n are linearly dependent on the columns of B, AB, ... , Ail-lB. Consequently, p[B: -AB: ... : (_l)"-IAn- t B : .. 0] p[B: -AB: .00: ( l)"-IA"- IB] Since changing the sign will not change the linear independence, we conclude that the rows of e- AtB are linearly independent if and only if pCB : AB : 0 0 0 : An - I B] n. This proves the equivalence of statements 1 and 3. In the foregoing argument we also proved that the rows of e - AtB are linearly independent if and only if the rows of eAtB are linearly independent on [0, 00) over the field of complex numbers. Next we show the equivalence of statements 1 and 1'. Taking the Laplace transform of eAtB, we have

.se[eAtB] (sI - A)-l B Since the Laplace transform is a one-to-one linear operator, if the rows of eAtB are linearly independent on [0, 00) over the field of complex numbers, so are the rows of (sI - A)-l B, and vice versa.

The proof of statement 4 will be postponed to Section 5-5 (page 206). Q.E.D.

The matrix is in fact positive definite. See Problem E-11. 8 This condition implies that lsI A) and B are left coprime. See Appendix G.

COl'

Example 6

Consider the inverted pendulL equation is developed in (3-42) 2g(A1 +m)j(21W +m)/ 5,2/(21 (3-42) becomes

-1

*=[1 5 We compute

U [B AB A2B

This matrix can be readily shO\ Th us, if x 3 = eis different from, to bring it back to zero. In fact, derivatives back to zero. Thh balancing a broom on our han(

. Example 7

Consider the platform system st of the platform is zero and the systems. The spring constants coefficients are assumed to be 2 a X2 +x2=u,or

[~J=[ This is the state-variable descrip

Now if the initial displacem( platform will oscillate, and it ~ platform to come to rest. No~ and x 2(0) = - 1, is it possible to seconds? The answer does not applied to the two spring system

For the state equation in (5-~

p[B: AB

hence the state equation is contr

)YNAMICAL EQUATIONS

dt

(5-24)

)r every}, in C), the n x (n +p)

~ctly from Theorems 5-1 and fns, Theorem 5-3 implies th~t 00 ) if and only if

j4.n- 1B: ...J=n ~duces to

,Am with m ;? ncan be written :e the columns of AmB with B, AB, ... , An-lB. Conse

-AB:"': (-l)n-1An-1BJ . independence, we conc1ude f and only if p[B : AB : ... tatements 1 and 3. In the ; of e- AtB are linearly indeindependent on [0, (0) over equivalence of statements 1 have

!ar operator, if the rows of ield of complex numbers. so

~tion 5-5 (page 206). Q.E.D.

.ee Appendix G.


Example 6 Consider the inverted pendulum system studied in Figure 3-15. Its dynamical equation is developed in (3-42). For convenience, we assume 2mg/(2M +m) = 1, 2g(M +m)j(2M +m)l 5,2f(2M +m)= 1, and 1/(2M +m)I=2. Then Equation (3-42) becomes

0 -1 y [1 o 0 OJx (5-25)0 0x=[1 1 0

llx+ [J}0 5

We compute

0 2U = [B AB A2B A3BJ -2 0[j 1 0

-Ill0 -10 This matrix can be readily shown to have rank 4. Hence (5-25) is controllable. Thus, if X3 = 8 is different from zero by a small amount, a control u can be found to bring it back to zero. In fact, a control exists to bring x I = y, X3 = 8, and their derivatives back to zero. This is certainly consistent with our experience of balancing a broom on our hand.

, Example 7 Consider the platform system shown in Figure 5-2. It is assumed that the mass of the platform is zero and the force is equally divided among the two spring systems. The spring constants are assumed to be 1, and the viscous friction coefficients are assumed to be 2 and 1as shown. Then we have Xl +2Xl = u and X2 =U, or

(5-26)

This is the state-variable description of the system. Now if the initial displacements XI(O) and X2(O) are different from zero, the

platform will oscillate, and it will take, theoretically, an infinite time for the platform to come to rest. Now we pose the following problem: If Xl (0) = 10 and X2(0) = - 1, is it possible to apply a force to bring the platform to rest in 2 seconds? The answer does not seem to be obvious because the same force is applied to the two spring systems.

For the state equation in (5-26), we compute

p[D: AD] p [~.5 =~25J =2 hence the state equation is controllable. Consequently, the displacements can


be brought to zero in 2 seconds by applying a proper input u. Using Equations (5-13) and (5-14), we have

O Sf O Sf _ J.2 [e . J[0.5J [e . J r _ [1.6 6.33JW(O, 2) - ,f 1 [0.5 1] d - 6:B,f 27

o ( (.-u(t)

40

X2 (t)

10

2 3 4 o I ~

-s

Figure 5-6 Behavior of Xl(t) and X2(t) and the waveform of u.

CON

eo.Sand Ul(t)= -[0.5 1][

for t in [0, 2]. If a force of the at t = 2. The behavior of Xf, x lines in Figure 5-6.

In" Figure 5-6 we also plot ~ x1(0) = 10 and x 2(0) = -1 to Zl in transferring x(O) to zero, the s of the input. If no restriction x(O) to zero in an arbitrarily sn the input may become very lafJ imposed, we might not be ablf interval of time. For example might not be able to transfer x((

Example 8

Consider again the platform syst the viscous friction coefficient all all equal to 1. Then the state-v

[~J= Clearly we have

pU=

and the state equation is not con an input to transfer x(O) to the zc X2(0), no input can transfer both

*Controllability indices

Let A and B be n x nand n x

U k =[B:AB: It consists of k +1 block column The matrix U ~ Un - 1 is the cont then Un - 1 has a rank of n and con: Note that there are a total of np co ways to choose these n linearly ind ing the most natural and also the m Let b, ; = 1, 2, ... ,p, be the ith coli

DYNAMICAL EQUATIONS

lper input u. Using Equations

T 0J [1.6 6.33Jdr = Te 6.33 27

~form of u.


and u,(t)= -[0.5 It:" ~,J W-'(O, 2) [ ~~J = -44.le5< +20.7e' for t in [0,2]. If a force of the form Ul is applied, the platform will come to rest at t = 2. The behavior of Xb X2 and of the input Ul are plotted by using solid lines in Figure 5-6.

In Figure 5-6 we also plot by using dotted lines the input U2(t) that transfers x1(0) = 10 and X2(0) = 1 to zero in 4 seconds. We see from Figure 5-6 that, in transferring x(O) to zero, the smaller the time interval the larger the magnitude of the input. If no restriction is imposed on the input u, then we can transfer x(O) to zero in an arbitrarily small interval of time; however, the magnitude of the input may become very large. If some restriction on the magnitude of u is imposed, we might not be able to transfer x(O) to zero in an arbitrarily small interval of time. For example, if we require !u(t)!.::s; 5 in Example 5, then we might not be able to transfer x(O) to zero in less than 4 seconds.

Example 8

Consider again the platform system shown in Figure 5-2. Now it is assumed that the viscous friction coefficient and the spring constant of both spring systems are all equal to 1. Then the state-variable description of the platform system is

Clearly we have

and the state equation is not controllable. If Xl (0) = x 2(0), it is possible to find an input to transfer x(O) to the zero state in a finite time. However, if Xl(O) i= X2(0), no input can transfer both Xl(O) and X2(0) to zero in a finite time.

*Controllability indices

Let A and B be n x nand n x p constant matrices. Define

k =0,1,2, ... (5-27)

It consists of k +1 block columns of the form AiB and is of order n x (k + l)p. The matrix U~ Un _1 is the controllability matrix. If {A, B} is controllable, then Un - 1 has a rank ofn and consequently has n linearly independent columns. Note that there are a total ofnp columns in Un - 1 ; hence there are many possible ways to choose these n linearly independent columns. We discuss in the following the most natural and also the most important way of choosing these columns. Let b, i 1,2, ... ,p, be the ith column of B. Then the matrix Uk can be written

188 CONTROLLABILITY AND OBSERV ABILITY OF LINEAR DYNAMICAL EQUATIONS

explicitly as

Uk=[b l b2 bp Ab l Abp A 2 b l . A 2 bp I. .

v v ro rl

: Akb l Akb p] (5-28) "---...'V,.-----'

rk (no. of dependent columns) Now we search linearly independent columns of Uk in order from left to right; that is, if a column can be written as a linear combination of its left-hand-side columns, the column is linearly dependent; otherwise, it is linearly independent. This process of searching can be carried out by using, for example, the columnsearching algorithm discussed in Appendix A. Let rj be the number of linearly dependent columns in AiB, for i =0,1, ... ,k. IfB has a full column rank, then ro O. We note that if a column, say Ab2 , is linearly dependent on its left-handside columns, then all Ai b2, with j = 2, 3, ... , will be linearly dependent on its left-hand side columns. Indeed, if

Ab2 CX l b l +cx2b2 + ... +cxpbp +cxp+ IAb1 then A2b2= cx 1Ab 1 +cx2Ab2 + . " +cx~bp +cxp+ lA2b 1 We see that A2b2 is a linear combination of its left-hand-side columns. Proceeding similarly, we can show that Ai b2, for j 3,4, ... , are linearly dependent on its left-hand-side columns. Because of this property, we have

0'::;rO'::;r1'::;r2'::;'" .::;p

Since there are at most n linearly independent columns in U 00' there exists an integer J1- such that

0'::;rO'::;r1'::;'" '::;rJl-I

l DYNAMICAL EQUATIONS

,] (5-28) mns) Uk in order from left to right; nbination of its left-hand-side wise, it is linearly independent. sing, for example, the column,et ri be the number of linearly B has a full column rank, then rly dependent on its left-handII be linearly dependent on its

+ap + 1Ab l .Abp +ap + lA2 b1 left-hand-side columns. Pro,4, ... ,are linearly dependent operty, we have

-s;p

olumns in U em there exists an


For this problem, we can readily show

pUO =2 Ab l, ... , AJl1 lb l , b 2 , Ab 2 , ... , AJl2- l bz, ... , bp, Abp,"" AJlp-lbp

The integer Ili is the number of linearly independent columns associated with bi in the set, or is the length of the chain associated with bi' Clearly we have

Il =max{llb 1l2, ... ,Ilp } and III +112 + ... +II p S n. The equality holds if {A, B} is controllable. The set {Ill' 1l2, .. ,Ilp } will be called the controllability indices of {A, B}.

Now we shan establish the relationship between the controllability indices and the r/s defined in (5-28). In order to visualize the relationship, we use an example. We assume p = 4, III = 3, 112 1, 113 5, and 114 3. These independent columns are arranged in a crate diagram as shown in Figure 5-7. The (i,j)th cell represents the column Ai-lbj . A column which is linearly independent of its left-hand-side columns in (5-28) is denoted by "x"; otherwise denoted by "0". The search of linearly independent columns in (5-28) from left to right is equivalent to the search from left to right in each row and then to the next row in Figure 5-7. Hence, the number of zeros in the ith row of Figure 5-7 is equal to ri-l as shown. From the crate diagram, we can deduce that ri is equal to the number of {b k , k 1, 2, ... , p} with controllability indices equal to or smaller than i. Hence, we conclude that

rj - ri-l no. of {bk, k = 1, 2, ... ,p} with controllability index i (5-33) with r-l~O. For example, rl-rO 1, and b2 has controllability index 1; r2 - 1'1 0 and no bi has controllability index 2; 1'3 - r2 2, and b l and b4 have controllability index 3; 1'5 I'4 = 1, and b3 has controllability index 5. Hence,

b i b2 b3 b4 X X X x ro 0

A x 0 x x rl 0 x x 1'2 = 1 0 x 0 r3 =3 0 x 0 r4 3 0 0 0 r5 4

Ili 3 5 3

Figure 57 Crate diagram of Ai-1bj .

the controllability indices of ~ 1, .. ,1l}.

Theorem 5-8

The set of the controllability in transformation and any orderi

Proof Define

Uk = where A=PAP-l, B PB an easily verified that

Uk=F which implies

pUk=P Hence, the rj defined in (5-2~ indices are invariant under an)

The rearrangement of the c

where M is a p x p elementary forward to verify that

irk~[D : AD : ... where diag {M, M, ... , M} cor singular. Hence, we have

pUk pl

Consequently, we conclude tha the ordering of the columns of:

Now we discuss a different umns in U =[B AB '" All

b l , Ab1, A 2 b b ... , An-lbl ; b;

and then search its linearly in< In terms of the crate diagram i are searched in order from top t column and so forth. Let

b l , Ab l, ... , Aill- 1 b1 ; b2, AI

DYNAMICAL EQUATIONS

=pU3 =" .

We assume that the linearly right have been found. We

, .. , b , Ab , .. , AlJ.p-l bp p p nt columns associated with b i lth bi' Clearly we have

!p} f {A, B} is controllable. The ity indices of {A, B}. ~en the controllability indices ze the relationship, we use an , and 114 3. These indepenLS shown in Figure 5-7. The mn which is linearly indepented by "x" ; otherwise denoted lUS in (5-28) from left to right ach row and then to the next in the ith row of Figure 5-7 is we can deduce that ri is equal ~ol1abi1ity indices equal to or

controllability index i (5-33) : has controllability index 1; 3 - r2 2, and bI and b4 have mtrollability index 5. Hence,

LINEAR INDEPENDENCE OF TIME FUNCTIONS 191

the controllability indices of {A, B} are uniquely determinable from {rj' i =0, 1, ... , 11}.

Theorem 5-8 I

The set of the controllability indices of {A, B} is invariant under any equivalence transformation and any ordering of the columns of B.

Proof

Define

Uk [D: AD : ... : AkDJ where A PAP - 1, D= PB and P is any nonsingular matrix. Then it can be easily verified that

for k 0, 1, 2, ...

which implies

for k = 0, 1, 2, ...

Hence, the rj defined in (5-28) and, consequently, the set of controllability indices are invariant under any equivalence transformation.

The rearrangement of the columns of B can be represented by

D=BM

where M is a p x p elementary matrix and is nonsingular. Again it is straightJorward to verify that

ih~ [D : AD : ... : AkBJ = Uk diag{M, M, ... , M} where diag {M, M, ... ,M} consists of k +1 number of M, and is clearly nonsingular. Hence, we have

for k = 0, 1, 2, ...

Consequently, we conclude that the controllability indices are independent of the ordering of the columns of B. Q.E.D.

Now we discuss a different method of searching linearly independent columns in U [B AB ... An-l BJ. We first rearrange the columns of U as b I , Ab b A 2 b1, ... , An- 1 b1 ; b 2 , Ab 2, ... , An- 1 b2 ; . ; bp , Abp,"" An- 1 b p

(5-34)

and then search its linearly independent columns in order from left to right. In terms of the crate diagram in Figure 5-7, the linearly independent columns are searched in order from top to bottom in the first column, then in the second column and so forth. Let


be the resulting linearly independent columns. If {A, B} is controllable, we have iiI +Jiz + ... +Jip=n. The lengths of these chains are {Jib Jiz,, Ji p}. Unlike the controllability indices, these lengths of chains depend highly on the ordering of {bj, i = 1, 2, ... ,p}. Example 10

Consider the state equation in (5-32). lfwe search linearly independent columns from left to right in

b l , Ab l , A 2 b!> A 3 b1 ; b2, Ab2 , AZbz, A 3 b2 the resulting linearly independent columns are

b l , Ab b A 2 b l , b2 Its lengths are {3, I}. If we search from left to right in

b2 , Ab2 , A 2 bz, A 3 bz; b l , Ab b A 2 b!> A 3 b I

the resulting linearly independent columns are

bz, Ab2, A 2b2, A 3bz

Its lengths are {4, O}. The lengths are indeed different for different ordering of b l and b2 The controllability indices of (5-32) can be computed as {2, 2} and are independent of the ordering of b i and b2

Let ii =max{Jib Ji2' ... , Ji p}. It is clear that Ji can never be smaller than {L, the controllability index. Since {L =max{{Lt. {Lz, ... , {Lp}, we may conclude that {L is the smallest possible maximum length of chains obtainable in any search of linearly independent columns of U. This controllability index will play an important role in the design of state feedback in Chapter 7 and the design of compensators in Chapter 9.

5-4 Observability of Linear Dynamical Equations

Time-varying case. The concept of observability is dual to that of controllability. Roughly speaking, controllability studies the possibility of steering the state from the input; observability studies the possibility of estimating the state from the output. If a dynamical equation is controllable, all the modes of the equation can be excited from the input; if a dynamical equation is observable, all the modes of the equation can be observed at the output. These two concepts are defined under the assumption that we hC}ve the complete knowledge of a dynamical equation; that is, the matrices A, B, C, and E are known beforehand. Hence, the problem of observability is different from the problem of realization or identification. The problem of identification is a problem of estimating the matrices A, B, C, and E from the information collected at the input and output terminals.

o

Consider the n-dimensiona

E:

where A, B, C, and E are n x n, continuous functions of t defin

Definition 5-5

The dynamical equation E is if there exists a finite t I > to suc of the input u[to. td and the out} to determine the state xo. Ott unobservable at to.

Example 1

Consider the network shown in the initial voltage across the capi the output is identically zero. , cally zero), but we are not able to hence the system, or more preci: system, is not observable at any

Example 2

Consider the network shown in F reduces to the one shown in Fig

In~WHI+A IF

u(rv)

Inc>

.L

105-jy ~-..... ----.J-

Figure 5-8 An unobservable netwo

IH

l-L.. yIn? I~J

..

(a)

Figure 5-9 An unobservable networ

l DYNAMICAL EQUATIONS

{A, B} is controllable, we have chains are {.u1, .u2, . . ., .up}. )f chains depend highly on the

llinearly independent columns

ight in

A2bb A3b1

fferent for different ordering of can be computed as {2, 2} and

.u can never be smaller than 11, ~2' ... , IIp}, we may conclude h of chains obtainable in any This controllability index will ack in Chapter 7 and the design

cal Equations

bility is dual to that of controllies the possibility of steering le possibility of estimating the s controllable, all the modes of dynamical equation is Qbserv'Ved at the output. These two e have the complete knowledge B, C, and E are known beforedifferent from the problem of identification is a problem of e information collected at the

OBSERVABlLITY OF LINEAR DYNAMICAL EQUATIONS 193

Consider the n-dimensional linear dynamical equation

E: x A(t)x(t) + B(t)u(t) (5-35a) Y C(t)x(t) + E(t)u(t) (5-35b)

i where A, B, C, and E are n x n, n x p, q x n, and q x p matrices whose entries are continuous functions of t defined over (- 00, 00).

Definition 5-5

The dynamical equation E is said to be (completely state) observable at to if there exists a finite t 1 > to such that for any state Xo at time to, the knowledge of the input u[to,td and the output Y[to,td over the time interval [to, t 1] suffices to determine the state Xo. Otherwise, the dynamical equation E is said to be unobservable at to

Example 1

Consider the network shown in Figure 5-8. If the input is zero, no matter what the initial voltage across the capacitor is, in view of the symmetry of the network, the output is identically zero. We know the input and output (both are identically zero), but we are not able to determine the initial condition of the capaci~or; hence the system, or more precisely, the dynamical equation that describes the system, is not observable at any to.

Example 2

Consider the network shown in Figure 5-9(a). Ifno input is applied, the network reduces to the one shown in Figure 5-9(b). Clearly the response to the initial

In +

1 1

Yu'"'-'

In

Figure 5-8 An unobservable network.

IH I H

In 1Y 1 '---__----4-----JF (a) (b)

Figure 5-9 An unobservable network.

194 CONTROLLABILITY AND OBSERV ABILITY OF LINEAR DYNAMICAL EQUA nONS

current in the inductor can never appear at the output terminal. Therefore, there is no way of determining the initial current in the inductor from the input and the output terminals. Hence the system or its dynamical equation is not observable at any to

The response of the dynamical equation (5-35) is given by

y(t)=C(t)l>(t. to)xUo) +C(t) If (t, r)B(r)u(r)dr +E(t)u(t) (5-36) to

where (t. r) is the state transition matrix of x= A(t)x. In the study of observability. the output y and the input u are assumed to be known. the initial state x(to) is the only unknown; hence (5-36) can be written as

y(t) C( t)(t. t o)x(t0) (5-37)

where y(t)~y(t) - CU) It (t. r)8(t)u(r) lit - E(t)uU) (5-38) to

is a known function. Consequently, the observability problem is a problem of determining x(to) in (5-37) with the knowledge of y. C, and (t. to). Note that the estimated state x(to) is the state not at time t, but at time to. However, if x(t o) is known, the state after to can be computed from

x(t) = (t, to)x(to) + It (t, t )B( t)u( t) dr (5-39) to

Theorem 5-9

The dynamical equation E is observable at to if and only if there exists a finite t 1> to such that the n columns of the q x n matrix function C(')(', to) are linearly independent on [to, t 1]. Proof

Sufficiency: Multiplying *(t, to)C*(t) on both sides of (5-37) and integrating from to to t 1, we obtain

Itl *(t, to)C*(t)y(t) dt [L *(1, 10lC*(t )C(t )(1, (0) dlJ Xo ~V(lo, (1)xo to (5-40)

Itl where V(to, t d~ *(t, to)C*(t)C(t)(t, to) dt (5-41 ) to From Theorem 5-1 and the assumption that all the columns of C(')(', to) are linearly independent on [to, t 1], we conclude that N(to, td is nonsingular. Hence, from (5-40) we have

Itl Xo = V- 1(to, td *(t, to)C*(t)y(t) dt (5-42) to Thus, if the function y[to, ttl is known, Xo can be computed from (5-42). Necessity:

o

Prove by contradiction. Sup t1> to such that the columns ( Then there exists an n x 1 non:

C(t}fj Let us choose x(to) IX; then

y(t)=C(t)fj Hence the initial statex(to) IX C tion that E is observable. Th t 1> to such that the columns 0

We see from this theorem tt tion depends only on C(t) and j can also be ded uced from Defi] servability study, it is sometim x= A(t)x. y = C(t)x.

The controllability of a dy independence of the rows of (t by the linear independence of between these two concepts is e

Theorem 5-10 (Theorem of dl Consider the dynamical equati4 defined by

E*:

where A*, B*, C*, and E* are thl E in E. The equation E is COl equation E* is observable (contI

Proof

From Theorem 5-4, the dynamic rows of (to, t)B(t) are linearly i 5-9, the dynamical equation E' B*(t)a(t, to) are linearly indepeOl [B*(t)a(t, to)]* :(t, to}8(t} ar I>a is the state transition matrix of (to, t) (see Problem 4-8); hence J

We list in the follOWing, for 01 tions 5-6 to 5-8, which are dual' 5-4 for controllability. Theorem

IYNAMICAL EQUATIONS

mtput terminal. Therefore, the inductor from the input s dynamical equation is not

is given by

r)u(r) elr +E(t)u(t) (5-36)

[t)x. In the study of observo be known, the initial state .ten as

(5-37)

:lr E(t)u(t ) (5-38)

lity problem is a problem of i, C and (f)(t, to). Note that but at time to. However, if from

r)u(r) elr (5-39)

d only if there exists a finite lction C(')(f)(', to) are linearly

es of (5-37) and integrating

~t, to) dtJxo~ V(to, ttl"o (5-40)

(5-41 )

e columns of C(')(f)(', to) are at V(to, t 1) is nonsingular.

t)y(t) elt (5-42)

Ited from (5-42). Necessity:

OBSERVABILITY OF LINEAR DYNAMICAL EQUATIONS 195

Prove by contradiction. Suppose E is observable at to, but there exists no tl > to such that the columns of C(')(f)(', to) are linearly independent on [to, tIl Then there exists an n x 1 nonzero constant vector (I such that

Qt)(f)(t, to)(I =0 for all t> to Let us choose x(to) (I; then

y(t) = C(t}(f)(t, to)(I 0 for all t> to Hence the initial state x(to) (I cannot be detected. This contradicts the assumption that E is observable. Therefore, if E is observable, there exists a finite tl > to such that the columns ofC('}(f)(" to) are linearly independent on [to, tIl

Q.E.D.

We see from this theorem that the observability of a linear dynamical equation depends only on C(t) and (f)(t, to) or, equivalently, only on C and A. This can also be deduced from Definition 5-5 by choosing u O. Hence in the observability study, it is sometimes convenient to assume u 0 and study only x = A(t)x, y C(t)x.

The controllability of a dynamical equation is determined by the linear independence of the rows of (f)(to, ')B(,), whereas the observability is determined by the linear independence of the columns of C(')(f)(', to). The relationship between these two concepts is established in the following theorem.

Theorem 5-10 (Theorem of duality) Consider the dynamical equation E in (5-35) and the dynamical equation E* defined by

E*: i A*(t)z +C*(t)v (5-43a) "( B*(t)z +E*(t)v (5-43b)

where A*, B*, C*, and E* are the complex conjugate transposes of A, B, C, and E in E. The equation E is controllable: (observable) at to if and only if the equation E* is observable (controllable) at to.

Proof

From Theorem 5-4, the dynamical equation E is controllable if and only if the rows of (f)(to, t)B(t) are linearly independent, in t, on [to, tIl From Theorem 5-9, the dynamical equation E* is observable if and only if the columns of B*(t)4it, to) are linearly independent, in t, on [to, tlJ, or equivalently, the rows of [B*(t)4it, to)J* = 4:(t, to)B(t) are linearly independent, in t, on [to, t IJ, where 4a is the state transition matrix ofi - A*(t)z. It is easy to show that 4:(t, to) (f)(to, t) (see Problem 4-8); hence E is controllable if and only if E* is observable.

Q.E.D.

We list in the following, for observability, Theorems 5-11 to 5-14 and Definitions 5-6 to 5-8, which are dual to Theorems 5-5 to 5-8 and Definitions 5-2 to 5-4 for controllability. Theorems 5-11 to 5-14 can be proved either directly or


by applying Theorem 5-10 to Theorems 5-5 to 5-8. The interpretations in the controllability part also apply to the observability part.

Theorem 5-11

Assume that the matrices A(') and C() in the n-dimensional dynamical equation E are n - 1 times continuously differentiable. Then the dynamical equation E is observable at to if there exists a finite t1> to such that

NO(t 1 ) 1N1(tl) (S-44 ) [

P . =n

Nn-1(t 1) d

where Nk+ 1(t) = Nk(t)A(t) +dt Nk(t) k=0,1,2, ... ,n-1 (S-4Sa)

WIth No(t) = C(t) (S-4Sb)

* Differential observabilitv, instantaneous observability, and uniform observability

Differential, instantaneous, and uniform observabilities can be defined by using the theorem of duality; for example, we may define {A, C} to be differentially observable if and only if { - A*, C*} is differentially controllable. However, for ease of reference we shall define them explicitly in the following.

Definition 5-6

The dynamical equation E is said to be differentially observable at time to if, for any statex(to) in the state space L, the knowledge of the input and the output over an arbitrarily small interval of time suffices to determine x(to).

Theorem 5-12

If the matrices A and C are analytic on ( - 00, 00), then the n-dimensional dynamical equation E is differentially observable at every t in ( - 00,00) if and only if, for any fixed to in ( - 00, 00),

No(t o) N 1(to)

PI I =n N n - 1(to)

*Sections noted with an asterisk may be skipped without loss of continuity.

Definition 5- 7

The linear dynamical equati4 ( - 00, 00) if and only if

No(t) N1(t)

[ p N"-l(t)

where the N/s are as defined i

Definition 5-8

The linear dynamical equatior if and only if there exist a posit

0< Pl(aJI ~V(t, t 0< P3((Jo)I ~4l>*(t,

for all t, where is the state tl

Time-invariance case. equation

FJ

where A, B, C, and E are n x n time interval of interest is [0, C1. time-invariant dynamical equ to ;;::: 0, and the determination 0 time interval. Hence, the refen bility study of linear time-inval

Theorem 5-13

The n-dimensional linear time-i able if and only if any of the fo

1. All columns of CeA! are lim complex numbers.

I'. All columns ofC(sI-Atl 2. The observability grammian

is nonsingular for any t > O.

v

'NAMICAL EQUATIONS

The interpretations in the )art.

lsional dynamical equation l the dynamical equation E that

(5-44 )

1,2, ... ,n 1 (5-45a)

(5-45b)

Jbility,

lties can be defined by using ;: {A, C} to be differentially Ily controllable. However, in the following.

lily observable at time to if, ~ ofthe input and the output determine x(to).

then the n-dimensional dyery t in 00,(0) if and only

of continuity.

OBSERVABILITY OF LINEAR DYNAMICAL EQUATIONS 197

Definition 5-7

The linear dynamical equation E is said to be instantaneously observable in ( - 00, (0) if and only if

I,

p [ for all t in ( 00, (0)~:~:~ ]=n Nn~l(t)

where the N/s are as defined in (5-45).

Definition 5-8

The linear dynamical equation E is said to be uniformly observable in ( 00, (0) if and only if there exist a positive (10 and positive {3i that depends on (10 such that

0< {31((10)1 sV(t, t +(10)s{32((10)1 0< {33((10)1 sf1l*(t, t +(1o)V(t, t +(1o)cl>(t, t +(10) s{34((10)1

for all t, where (J) is the state transition matrix and V is as defined in (5-41).

Time-invariance case. Consider the linear time-invariant dynamical equation

FE: X Ax +Bu (5-46a) y=Cx +Eu (5-46b)

where A, B, C, and E are n x n, n x p, q x n, and q x p constant matrices. The time interval of interest is [0, (0). Similar to the controllability part, if a linear time-invariant dynamical equation is observable, it is observable at every to 2:::0, and the determination of the initial state can be achieved in any nonzero time intervaL Hence, the reference of to and t 1 is often dropped in the observability study of linear time-invariant dynamical equations.

Theorem 5 -1 3

The n-dimensional linear time-invariant dynamical equation in (5-46) is observable if and only if any of the following equivalent conditions is satisfied:

1. All columns of CeA! are linearly independent on [0, (0) over C, the field of complex numbers.

I'. All columns of C(sl - A)-l are linearly independent over C.

2. The observability grammian

w"'~ I /'C'CeA, dr is nonsingular for any t > 0.


3. The nq x n observability matrix

C CA

V CA2 (5-47) [ CAn-l

has rank n. 4. For every eigenvalue AofA (and consequently for every Ain C), the (n +q) x n

. complex matrix

[AICA]

has rank n, or equivalently, (sI A) and C are right coprime.

*Observability Indices Let A and C be n x nand q x n constant matrices. Define

~::11 "0 (no. of dependent rows) C Jq

CJ CIA}V. _ CA czA (5-48) ,-[C~. = _C_q~J ,.,

~;Akl }

c2Ak

I'k

clJAk I It consists of k + 1 block rows of the form CAi and is of order (k + 1)q x n. The matrix V = Vn _ 1 is the observability matrix. The q rows of C are denoted by Cj, i = 1, 2, ... ,q. Let us search linearly independent rows of Vk in order from top to bottom. Let rj be the number of linearly dependent rows in CA i , i =0, 1, ... ,k. Similar to the controllability part, we have

'I

1'0 '::;;;1'1 :::;1'2:::;" :::;q

Since there are at most n linearly independent rows in V 00' there exists an integer v such that

0'::;;;1'0:::;1'1:::;'" :::;rv -l

AMICAL EQUATIONS

(5-47)

very A. in C), the (n +q) x n

1t coprime.

efine

:nt rows)

(5-48)

f order (k + 1)q x n. The ows of C are denoted by rows of V k in order from :lependent rows in CA i , have

, 00' there exists an integer

(5-49a)

CANONICAL DECOMPOSITION OF A LINEAR TIME-INVARIANT DYNAMICAL EQUATION 199

and q (5-49b) Equivalently, v is the integer such that

pV 0 < PV 1 < .. i < PV v-I PV v PV v + 1 =' . . (5-50) The integer v is called the observability index of {A, C}. Similar to (5-31), we have

n . (_-svsmm n,n q+ 1) (5-51 ) q

where ii is the degree of the minimal polynomial of A and ij is the rank of C.

Corollary 5-13 The dynamical equation FEin (5-46) is observable if and only if the matrix Vn - q, where ij is the rank of C, is of rank n, or equivalently, the n.x n matrix V:-qVn _q is nonsingular.

Consider the matrix V n _ l' It is assumed that its linearly independent rows in order from top to bottom have been found. Let Vi be the number of linearly independent rows associated with C i . The set of {Vi, i = 1,2, ... ,q) is called the observability indices of {A, C}. Clearly we have

v = max [ Vi' i 1, 2, ... , q1

and VI +V2 + ... +Vq Sll. The equality holds if [A, C: is observable. Theorem 5-14

The set of observability indices of tA, Cj is invariant under any equivalence transformation and any ordering of the rows of C.

5-5 Canonical Decomposition of a Linear Time-Invariant Dynamical Equation

In the remainder of this chapter, we study exclusively linear time-invariant dynamical equations. Consider the dynamical equation

FE: x=Ax +Bu (5-52a) y=Cx +Eu (5-52b)

where A, B, C, and E are n x n, n x p, q x n, and q x p real constant matrices. We introduced in the previous sections the concepts of controllability and observability. The conditions for the equation to be controllable and observable are also derived. A question that may be raised at this point is: What can be said if the equation is uncontrollable and/or unobservable? In this section we shall study this problem. Before proceeding, we review briefly the equivalence transformation. Let x= Px, where P is a constant nonsingular matrix. The

200 CONTROLLABILITY AND OBSERVABlLITY OF LINEAR DYNAMICAL EQUATIONS

substitution of x Px into (5-52) yields FE: x Ax +Du (5-53a)

y=Cx +Eu (5-53b) where A= PAp-I, D= PB, C= Cp-I, and E E. The dynamical equations FE and FE are said to be equivalent, and the matrix P is called an equivalence transformation. Clearly we have

iT~ [D : AD : ... : An-ID] = [PB : PAB :' ... : PAn-IB] =P[B: AB: ... : An-IB]~PU

Since the rank of a matrix does not change after multiplication of a nonsingular matrix (Theorem 2-7), we have rank U rank U. Consequently FE is controllable if and only if FE is controllable. A similar statement holds for the observability part.

Theorem 5-15

The controllability and observability of a linear time-invariant dynamical equation are invariant under any equivalence transformation.

This theorem is in fact a special case of Theorems 5-8 and 5-14. For easy reference, we have restated it as a theorem.

In the following, c will be used to stand for controllable, cfor uncontrollable, o for observable, and 0 for unobservable.

Theorem 5-16

Consider the n-dimensional linear time-invariant dynamical equation FE. If the controllability matrix of FE has rank n1 (where nl < n), then there exists an equivalence transformation :X = Px, where P is a constant nonsingular matrix, which transforms FE into

(5-54a) FE: [~~J [Ae ~~. 2J [~:J + [DcJ u Xc 0 At xe 0 Y= [Cc (5-54b)C,] [:;J +Eu

and the ni-dimensional subequation of FE FEe: Xe Acxe +Dcu (5-55a)

y =Cc:Xc +Eu (5-55b) is controllable9 and has the same transfer function matrix as FE.

9 I t is easy to show that if the equation Fit is observable. then its sUbequation Fit, is also observable. (Try)

CANONICAL DECOMPOSITION OF

Proof

If the dynamical equation F1 have

pU~p[B Let q1, q2, ... , qn be any nl Ii

1

each i= 1, 2, ... , nb Aqj c. {qb q2,'" qnJ. (Why?) De

P-l~Q~ where the last n ni columns Q is nonsingular. We claim tl into the form of (5-54). Rec: x Px we are actually using th state space. The ith column of Aqi with respect to {qb qz, ... , linearly dependent on the set {~ given in (5-54a). The columns B with respect to {qb q2, ... , {qj, q2" . " q!!.1}; hence B is of

Let U and U be the controlla we have pU = piT =nl (see The

V = [Be A) o 0

=[~c X: where Vc represents the contre A!B with k '2:nl are linearly de pV = n1 implies pVc n l . Heno

We show now that the dynarr. FE and FEe' have the same traI

[ SI - Ac A~..z ] [(Sl -.Aet o sl Ac 0 Thus the transfer-function matri

-[Cc - [SI-X,Cia 0 -X12 J sl-Ai.'

-

=[Cc - [(SICc] 0 =Ce(sl A)-IB -t e e

which is the transfer-function rna

(5-53a) (5-53b)

The dynamical equations x P is called an equivalence

:' ... : PAn-1B] ,.: An-lB]~PU

ltiplication of a nonsingular Consequently FE is con

lar statement holds for the

time-invariant dynamical formation.

ms 5-8 and 5-14. For easy

)llable, cfor uncontrollable,

l dynamical equation FE. :re n1 < n), then there exists is a constant nonsingular

u (5-54a)

(5-54b)

(5-55a) (5-55b)

matrix as FE.

lubequation FEe is also observable.


Proof

If the dynamical equation FE is not controllable, then from Theorem 5-7 we have

pU~p[B : AB : ... : An-lB] nl < n Let qb q2' ... , qnl be any nl linearly independent columns of U. Note that for each i = 1, 2, ... , nl , Aqj can be written as a linear combination of {ql' q2' .. ,qnJ. (Why?) Define a nonsingular matrix

(5-56) where the last n - nl columns of Q are entirely arbitrary so long as the matrix Q is nonsingular. We claim that the transformation x Px will transform FE into the form of (5-54). Recall from Figure 2-5 that in the transformation x Px we are actually using the columns ofQ~ p-l as new basis vectors of the state space. The ith column of the new representation A is the representation of Aqj with respect to {ql' q2,"" qn}. Now the vectors Aqh for i 1,2, ... , nl, are linearly dependent on the set {q t, q2, ... , qnl} ; hence the matrix A has the form given in (5-54a). The columns of Bare the representations of the columns of B with respect to {ql' q21 ... ,qn}' Now the columns of B depend only on {qt, q2,"" q!!J}; hence B is of the form shown in (5-54a)._

Let U and U be the controllability rna trices of FE and FE, respectively. Then we have pU pU = nl (see Theorem 5-15). It is easy to verify that

- [Bc: AJic! ! A~-lBc]U= : I"',0: 0: I 0

I I

Uc: A~lBc! L~n-lB ]}n rows= ! I' "1 c c I[ 0: 0 : : 0 }(n - nd rows

where Uc represents the controllability matrices of FEe- Since columns of A~B with k :?:nl are linearly dependent on the columns of Uc, the condition pU n1 implies pUc = n1. Hence the dynamical equation FEc is controllable.

We show now that the dynamical equations FE and FEe' or correspondingly, and have the same transfer-function matrix. It is easy to verify that

Sl OAc -A~] [(SI -.Ac)-l (sl -Ac)-1A~?(SI1-Ac)-1] I (5-57) [ sl - Ac 0 (sl - Ad-Thus the transfer-function matrix of FE is

-A12]-l[Bc] +E sl -Ac 0

(.'II= [C, C,] [

202 CONTROLLABILn', AND r,BSERVABILITY OF LINEAR DYNAMICAL EQUATIONS

In the equivalence transformation x = Px, the state space L of FE is divided into two subspaces. One is the n1-dimensional subspace of L, denoted by L 1,

which consists of all the vectors [~'1the other is the (n - nIl-dimensional subspace, which consists of all the vectors [:J Since FE, is controllable, all the vectors Xc in L1 are controllable. Equation (5-54a) shows that the state variables in Xc are not affected directly by the input u or indirectly through the state vector Xc; therefore, the state vector is not controllable and is dropped in the reduced equation (5-55). Thus, ifa linear time-invariant dynamical equation is not controllable, by a proper choice of a basis, the state vector can be decomposed into two groups: one controllable, the other uncontrollable. By dropping the uncontrollable state vectors, we may obtain a controllable dynamical equation of lesser dimension that is zero-state equivalent to the original equation. See Problems 5-22 and 5-23.

Example 1

Consider the three-dimensional dynamical equation

1 1 0] [0 1Jx= 0 1 0 X+ IOu y=[1 l]x (5-58)[o 1 1 0 1 The rank of B is 2; therefore, we need to check U 1 = [B : AB] in determining the controllability of the equation. Since

0 11 1]p[B : AB] = p 1 0 1 0 = 2 < 3[

o 1 1

the state equation is not controllable. Let us choose, as in (5-56),

0 1: 1]p- 1 Q 1 0: 0I[o 110

The first two columns of Q are the first two linearly independent columns of U1; the last column of Q is chosen arbitrarily to make Q nonsingular. Let X Px. We compute

1 0][1 1 0][0A PAP-I~[~ o 0 0 1 ~, ~J = l~--~ L~J1 1 1 0 0 0: 1o -1 0 1 1 0 B 0PB~[~ 1

-:lG ~] [~- -~]0

CANONICAL DECOMPOSITION OF

and C Cp-l [

Hence, the reduced controllab

x, G~J Dual to Theorem 5-16, w

dynamical equations.

Theorem 5-17

Consider the n-dimensional If the 0 bservability matrix of 1 valence transformation X Px

[~~ Xii and the nz-dimensional subequ

FEo

is observable and has the same

This theorem can be readil~ The first n2 rows of P in Theon the observability matrix of {A, arbitrary so long as P is nonsi xii does not appear directly in t vector xij is not observable and

Combining Theorems 5-16 I theorem.

Theorem 5-18 (Canonical de Consider the linear time-invaria

FE

By equivalence transformation

10 This is a simplified version of the can References 57.60. and 116. See also

IAMICAL EQUAnONS

e space L of FE is divided pace of L, denoted by I: I,

I the (n nd-dimensional

lce FEe is controllable, all

54a) shows that the state lor indirectly through the trollable and is dropped in ariant dynamical equation ,tate vector can be decom:her uncontrollable. By obtain a controllable dy)-state equivalent to the

1Jx (5-58)

: [B : ADJ in determining

independent columns of ake Q nonsingular. Let

~] [1 0: OJO ~- -~ ;-~-


1 and C=CP-l=[l 0 2 : 1J1] [~ ~] =[1

1

Hence, the reduced controllabfe equation is

Xc y=[1 2Jxc (5-59)[: ~}, +[~ ~}

Dual to Theorem 5-16, we have the following theorem for unobservable dynamical equations.

Theorem 5-17

Consider the n-dimensional linear time-invariant dynamical equation FE. If the observability matrix of FE has rank n2(n2 < n), then there exists an equivalence transformation i = Px that transforms FE into

FE: (5-60a)[x~oo_J [Ao 0 J[XoJ [DoJA21 Ao Xii + Do u y [C. 0] [:;J +Eu (5-60b)

and the n2-dimensional subequation of FE

Xo = Aoxo +Dou (5-61 a) y. =Coxo+Eu (5-61 b)

is observable and has the same transfer-function matrix as FE. This theorem can be readily established by using Theorems 5-16 and 5-10.

The first n2 rows of P in Theorem 5-17 are any n2 linearly independent rows of the observability matrix of {A, C}; the remaining n - n2 rows of P are entirely arbitrary so long as P is nonsingular. Equation (5-60) shows that the vector x(j does not appear directly in the output y or indirectly through xO. Hence the vector Xi) is not observable and is dropped in the reduced equation.

Combining Theorems 5-16 and 5-17, we have the following very important theorem.

Theorem 5-18 (Canonical decomposition theorem) 10 Consider the linear time-invariant dynamical equation

FE: x=Ax +Bu y=Cx+Eu

By equivalence transformations, FE can be transformed into the following

10 This is a simplified version of the canonical decomposition theorem. For the general form, see References 57,60, and 116. See also Reference S127.


canonical form

-. [~eoJ_rAeo ~12 ~ ~13][~eoJ -I-[~eoJFE. Xeo - 0 Aeo An xeo ' Beo U (5-62a)Xc -0 ---0- , A~ - Xc 0

y = [0 Ceo: Cc]X +Eu (5-62b) where the vector xeo is controllable but not observable, xeo is controllable and observable, and Xc is not controllable. Furthermore, the transfer function of FE is

Ceo(sl - Aeo)- 1Beo +E which depends solely on the controllable and observable part of the equation FE.

Proof

If the dynamical equation FE is not controllable, it can be transformed into the form of (5-54). Consider now the dynamical equation FEe which is the controllable part of FE. If FEe is not observable, then FEe can be transformed into the form of (5-60), which can also be written as

[~eoJ = [AeD ~12J[~eoJ +[~eoJ UXeo 0 Aco Xeo Beo y = [0 Cco]X +Eu

Combining these two transformations, we immediately obtain Equation (5-62). Following directly from Theorems 5-16 and 5-17, we conclude that the transfer function of FE is given by Ceo(sl Aco)-1Beo +E. Q.E.D.

u y

,-------,

Il~lco I I I

I I

I - I

leG:I -- I I co I

I I

I IL ______ --.J

Figure 5-10 Canonical decomposition of a dynamical equation. (c stands for controllable, cfor uncontrollable, 0 for observable, and 0 for unobservable.)

CANONICAL DECOMPOSITION C

This theorem can be illl which the uncontrollable p, observable parts. We see . response matrix of a dynan and observable part of the matrix (the input-output des( controllable and observable. ' input-output description and us why the input-output de system, for the uncontrollabl appear in the transfer-functic

Example 2

Consider the network shown source, the behaVior due to 1 detected from the output. f L1 are not observable (they rn the state variable associated, metry, the state variable assocl By dropping the state variable the network in Figure 5-11(a) the transfer function of the ne

Before moving to the next of Theorem 5-7.

C1

.l Ll In? ?If.

II r t r In..? J:

(a)

IJ In In

I

NAMICAL EQUATIONS

,] [BCD]+ ~D U (5-62a)

(5-62b) rvable, XeD is controllable Dore, the transfer function

vable part of the equation

III be transformed into the ion FEe which is the conEe can be transformed into

. U'0] :0

:ly obtain Equation (5-62). conclude that the transfer

Q.E.D.

mtion. (c stands for controlervable.)


This theorem can be illustrated symbolically as shown in Figure 5-10, in which the uncontrollable part is further decomposed into observable and unobservable parts. We see that the transfer-function matrix or the impulseresponse matrix of a dynamical equation depends solely on the controllable and observable part of the ~quation. In other words, the impulse-response matrix (the input-output description) describes only the part of a system that is controllable and observable. This is the most important relation between the input-output description and the state-variable description. This theorem tells us why the input-output description is sometimes insufficient to describe a system, for the uncontrollable and/or unobservable parts of the system do not appear in the transfer-function matrix description.

Example 2 Consider the network shown in Figure 5-11(a). Because the input is a current source, the behavior due to the initial conditions in C 1 and Ll can never be detected from the output. Hence the state variables associated with C 1 and Ll are not observable (they may be controllable, but we don't care). Similarly the state variable associated with L2 is not controllable. Because of the symmetry, the state variable associated with C2 is uncontrollable and unobservable. By dropping the state variables that are either uncontrollable or unobservable, the network in Figure 5-11(a) is reduced to the form in Figure 5-11(b). Hence the transfer function of the network in Figure 5-11(a) is g(s) 1.

Before moving to the next topic, we use Theorem 5-16 to prove statement 4 of Theorem 5-7.

C1 R

Ll L2In In

u yt In In

(a)

In

In

(b)

In

In

+

y

Figure 5-11 An uncontrollable and unobservable system with transfer function 1.


Proof of statement 4 of Theorem 5-7 The matrix (sI A) is nonsingular at every sin C except at the eigenvalues ofA; hence the matrix [sI - A : B] has rank n at every s in C except possibly at the eigenvalues of A. Now we show that if {A, B} is controllable, then p[sI - A : B] = n at every eigenvalue ofA. Ifnot, then there exist an eigenvalue A and a 1 x n vector ex =1= 0 such that

ex[.H A : B] 0 or exA = exA and exB =0

which imply exA2 = )"exA =)"2ex

and, in general, exAi = Aiex 1,2, ...

Hence we have

ex[B AB ... An-lB] [exB AexB ... An -lexB] = 0 This contradicts the controllability assumption of {A, B}. Hence, if {A, B} is controllable, then p[sI A : B] n at every eigenvalue of A and, consequently, at every s in C.

Now we show that if {A, B} is not controllable, then p[AI - A : B] < n for some eigenvalue A of A. If {A, B} is not controllable, there exists an equivalence transformation P that transforms {A, B} into {A, B} with

A=PAP- 1 [Ac A!2] B PB [~,]o Ac Let A be an eigenvalue of Ac. We choose p =1=0 such that pAc = AP. Now we form a 1 x n vector ex [0 p]. Then we have

.. [ AI -A i B] [0 II] [AI ~A, -A l.: Bc] =0 ),,1 Ac 0I which implies

o=ex[P{AI _A)P- l : PB] =exP[(AI _A)P- 1 : BJ Since ex =1=0, we have ~~exP =1=0. Because p-l is nonsingular, ~(AI - A)P- l =0 implies a(AI A) O. Hence we have

ti[AI - A : BJ = 0 In other words, if {A, B} is not controllable, then [sI A: BJ does not have a rank of n for some eigenvalue of A. This completes the proof. Q.E.D.

Irreducible dynamical equation. We have seen trom Theorems 5-16 and 5-17 that if a linear time-invariant dynamical equation is either uncontrollable or unobservable, then there exists a dynamical equation of lesser dimension that has the same transfer-function matrix as the original dynamical equation. In other words, if a linear time-invariant dynamical equation is either uncontrollable or unobservable, its dimension can be reduced such that the reduced

CANONICAL DECOMPOSITION 0

equation still has the same ZI ing definition.

Definition 5-9

A linear time-invariant dym only if there exists a linear ti sion that has the same transi state equivalent to FE. Otht

Theorem 5-19

A linear time-invariant dyna is controllable and observabl

Proof

If the dynamical equation FE is reducible (Theorems 5-16 a FE is controllable and obser contradiction. Suppose tha1 and observable and that there FE,

of lesser dimenSion, say nl < 11 Theorem 4-6, we have E =E;:

CAkB

Consider now the product

VU~ [ ~~ 1 CAn-l

CB CAB

[ CAn-IE

By (5-64), we may replace CAk

where Vn - 1 and Un- l are defi lable and observable, we have 2-6 that p(VU) n. Now V-n matrices; hence the matrix VJ

t\.MICAL EQUAnONS

)t at the eigenvalues of A; I C except possibly at the I} is controllable, then l there exist an eigenvalue

},n-l IXBJ 0 "B}. Hence, if {A, B} is nvalue of A and, conse

then p[AI-A : BJ


(5-66) implies that p(Yn-lUn-d n>nl' This is a contradiction. Hence, if FE is controllable and observable, then FE is irreducible. Q.E.D.

Recall from Section 4-4 that if a dynamical equation {A, D, C, E} has a prescribed transfer-function matrix G(s), then the dynamical equation {A, D, C, E} is called a realization of G(s). Now if {A, D, C, E} is controllable and observable, then {A, D, C, E} is called an irreducible realization of G(s), In the following we shall show that all the irreducible realizations ofG(s) are equivalent.

Theorem 5 -20

Let the dynamical equation {A, D, C, E} be an irreducible realization of a q x p proper rational matrix G(s). Then {X, B, C, E} is also an irreducible realization ofG(s) if and only if {A, D, C, E} and {X, B, C, E} are equivalent; that is, there exists a nonsingular constant matrix P such that X PAP - 1, B PD, C =CP-l, and E E.

Proof

The sufficiency follows directly from Theorems 4-6 and 5-15. We show now the necessity of the theorem. Let U, V be the controllability and the observability matrices of {A, D, c, E}, and let U, Y be similarly defined for {X, B, C, E}. If {A, D, C, E} and {X, B, C, E} are realizations of the same G(s), then from (5-64) and (5-65) we have E = E,

VU YU (5-67) and VAU=YXU (5-68) The irreducibility assumption implies that pY n; hence the matrix (y*y) is nonsingular (Theorem 2-8). Consequently, from (5-67), we have

U = (y*y)-l y*VU ~ PU (5-69) where P~(y*y)-ly*V. From (5-69) we have pU .::;;;min (pP, pU), which, together with pU = n, implies that pP = n. Hence P qualifies as an equivalence transformation. The first p columns of (5-69) give B = PD. Since pU = n, Equation (5-69) implies that

P = (UU*)(UU*) - 1 With P = (y*y)-ly*V (UU*)(UU*)-l, it is easy to derive from (5-67) and (5-68) that V=YP and PA=XP, which imply that C=CP and X=PAP- l .

Q.E.D.

This theorem implies that all the irreducible realizations of G(s) have the same dimension. Physically, the dimension of an irred....cible dynamical equation is the minimal number of integrators (if we simulate the equation in an analog computer) or the minimal number of energy-storage elements (if the system is an RLC network) required\ to generate the given transfer-function matrix.

We studied in this section only the canonical decomposition of linear time-

CONTROLLABILITY AND oBsa

invariant dynamical equatior is referred to References 106;

*5-6 Controllabilityal Dynamical Equations

The controllability and ob equation are invariant unde] ceivable that we may obtain by transforming the equati4 equation is in a Jordan form, almost by inspection. In th

Consider the n-dimensio equation

where the matrices A, D, and The n x n matrix A is in the J4 Am' Ai denotes all the Jorda the number ofJordan blocks i

Ai = diag (Ail' Ai2' ... ,

Table 5-1 Jordan-Fonn Dynal

AI

A (n x n) [

C [C l Ail

(nj~in;) = [ Cj =[Cil

(q x ni) A.i 1

A. i Ai} =

(nij xnij) [

Clj [elij

NAMICAL EQUATIONS

. contradiction. Hence, if cible. Q.E.D.

ion {A, B, C, E} has a prelical equation {A, B, C, E} mtrollable and observable, .G(s), In the following we are equivalent.

reducible realization of a E} is also an irreducible ~ B, C, E} are equivalent; P such that A PAP-I,

.d 5-15. We show now the dlity and the observability

~fined for {A, B, C, E}. If arne G(s), then from (5-64)

(5-67) (5-68)

lence the matrix CV*V) is i7), we have

(5-69) J:s;min (pP, pU), which, lualifies as an equivalence

~ B=PB. Since pU =n,

o derive from (5-67) and C CP and A PAP- 1

Q.E.D.

iizations of G(s) have the educible dynamical equamlate the equation in an ,storage elements (if the

COI'>l'TROLLABlLlTY AND OBSERVABILlTY OF JORDAN-FORM DYNAMICAL EQUATIONS 209

invariant dynamical equations. For the time-varying case, the interested reader is referred to References 106 and 108,

*5-6 Controllability and Observability of Jordan-Form Dynamical Equations

The controllability and observability of a linear time-invariant dynamical equation are invariant under any equivalence transformation; hence it is conceivable that we may obtain simpler controllability and observability criteria by transforming the equation into a special form. Indeed, if a dynamical equation is in a Jordan form, the conditions are very simple and can be checked almost by inspection, In this section, we shall derive these conditions,

Consider the n-dimensional linear time-invariant Jordan-form dynamical equation

JFE: x=Ax +Bu (5-70a) y=Cx +Eu (5-70b)

where the matrices A, B, and C are assumed of the forms shown in Table 5-l. The n x n matrix A is in the Jordan form, with m distinct eigenvalues AI, }.z' . .. , Am' Ai denotes all the Jordan blocks associated with the eigenvalue }.i; r(i) is the number of Jordan blocks in Ai; and Ai} is thejth Jordan block in Ai' Clearly,

and

Table 5-1 Jordan-Form Dynamical Equation

A (n x n)

[A! A2 AJ (n ~p) {j]

C=[C1 C2 Cm]

~["Ai (ni x n;)

Ai2

A.J [Bil 1B. _ Bi2 (nixlp) : Bir(i)

C i =[Cil Ci2 Cir(i)] (q x n;)

Aij [,1,1Ai 1

= '. '.

1 Bij [b ll ] b2ij = .

(nu x nil) A, ~. (n;jxp) . blij le given transfer-function I

...Cij [CUi C2ij CliJ Imposition of linear time


Let ni and nij be the order of Ai and Aij, respectively; then m m rU)

n= I ni I I nij i= 1 i= 1 j= 1

Corresponding to Ai and Aij' the matrices Band C are partitioned as shown. The first row and the last row of Bij are denoted by b1 ij and b lij, respectively. The first column and the last column of Cij are denoted by C1 ij and c1ij.

Theorem 5-21

The n-dimensional linear time-invariant Jordan-form dynamical equation JFE is controllable if and only if for each i 1, 2, ... , m, the rows of the r(i) x p matrix

Bl~ b~:2bnj (5-71 a)[

b,ir(i)

are linearly independent (over the field of complex numbers). JFE is observable if and only if for each i = 1, 2, ... , m, the columns of the q x r(i) matrix

I .. ]C i = LCli1 C1i2 CUrti) (5-71 b) are linearly independent (over the field of complex numbers). Example 1

Consider the Jordan-form dynamical equation

000Al 1: 0 0 0 0 0 o AI: 0 0 0 0 0 1 0 0 I ~b111 -0 --6 -I :-l~: 0 0 0 0 o 1 0 I ~b112 o 0 --0-;-1-: 0 0 0JFE: X= ___ .1 ________ _ x+IO 0 11u~b113 (5-72a)

1 1 2o 0 0 0: A2 1 0 010o 0 0 0 0 A2 1

o 0 0 0: 0 0 A2 o 0 1 I ~b'21 1 2: 0: 0[ 2 01y ~ 0:1:2:0 1 1 x (5-72b)I I I

1 0:2:3:0 2 2

iii i

C l11 C l12 C l 13C 121

The matrix A has two distinct eIgenvalues Al and A2 There are three Jordan blocks associated with AI; hence r(1) 3. There is only one Jordan block associated with A2 ; hence r(2) = 1. The conditions for JF E to be controllable are that the set {bill' b,12, b/13 } and the set {b12 d be, individually, linearly independent. This is the case; hence JF E is controllable. The conditions for JF E to be

CONTROLLABILITY AND OBSEJ

observable are that the set { linearly independent. Alth dent, the set {C12I}, which Hence JFE is not observabl

The conditions for COt req uired that each of the m pendence. The linear depe Furthermore, the row vecto determining the controllabil

The physical meaning of the block diagram of the Jc studying the general case, , Jordan-form dynamical eqm an integrator and a feedback block, or more precisely th( variable. Each chain of bID< Consider the last chain of F variables in that chain can b that chain can be observed. same eigenvalue, then we r vectors of these chains. The be studied separately.

~~b -~ -~ ~l~~~~

Figure 5-12 Block diagram of

\MICAL EQUATIONS

then

re partitioned as shown. )Uj and blij, respectively. d by CI ij and CUj'

dynamical equation JF E , the rows of the r(i) x p

(5-71 a)

bers). JFE is observable eq x r(i) matrix

(5-71 b) nbers).

3 0 3 0 ~b111 1 0 ~b112 3 1 U~b113 (5-72a)

1 0 ) ~b121

(5-72b)

There are three Jordan one Jordan block associ) be controllable are that lly, linearly independent. Inditions for JFE to be

CONTROLLABILITY AND OBSERVABILITY OF JORDAN-FORM DYNAMICAL EQUATIONS 211

observable are that the set {Cll I ' Cll2, C1l3} and the set {c12 d be, individually, linearly independent. Although the set {CIII , C112 ' C113} is linearly independent, the set {c12d, which consists of a zero vector, is linearly dependent. Hence JFE is not observable.

t

The conditions for controllability and observability in Theorem 5-21 required that each of the m set of vectors be individually tested for linear independence. The linear dependence of one set on the other set is immaterial. Furthermore, the row vectors of B excluding the bli/s do not play any role in determining the controllability of the equation.

The physical meaning of the conditions of Theorem 5-21 can be seen from the block diagram of the Jordan-form dynamical equation JFE. Instead of studying the general case, we draw in Figure 5-12 a block diagram for the Jordan-form dynamical equation in (5-72). Observe that each block consists of an integrator and a feedback path, as shown in Figure 5-13. The output of each block, or more precisely the output of each integrator, is assigned as a state variable. Each chain of blocks corresponds to a Jordan block in the equation. Consider the last chain of Figure 5-12. We see that if bl21 1= 0, then all state variables in that chain can be controlled; if C l2l 1=0, then all state variables in that chain can be observed. If there are two or more chains associated with the same eigenvalue, then we require the linear .independence of the first gain vectors of these chains. The chains associated with different eigenvalues can be studied separately.

y

y

y

y

Figure 5-12 Block diagram of the Jordan-form equation (5-72).


x ~ Figure 5-13 Analog computer simulation of 1/(s - AJ

Proof of Theorem 5-21 We use statement 4 of Theorem 5-7 to prove the theorem. In order not to be overwhelmed by notation, we assume [sI - A : B] to be of the form

s - A1 -1 o b ll1 o S -A1 -1 b211 o 0 S A1 , bl1 1

----------------------: - --i: -----...: j - - -:~ b11 2 : s A1 : , b

'12 - - - - - - - - - - - - - - - -: -s- -)~ - - - - --=..-( - -,- -bl-2~

S A2 bl21

(5-73) The matrix A has two distinct eigenvalues A1 and A2. There are two Jordan blocks associated with Ab one associated with A2 If SAl, (5-73) becomes

0 -1 0 b ll1 0 0 -1 b 211

,0 0 0 , bill - - - - - - - - - - - - - - - T - - - - - - - - - - -,

,0 -1, (5-74 )b l12 '0 0, , bl12-----------T----------------T-----

,, A1 -A2 -1,, b121

:A1 A2 b/21,

By a sequence of elementary column operations, the matrix in (5-74) can be transformed into

o 1 0 o

o o -1 o

o o 0 billI

- - - - - - - - - - - - - - i - 0- ---=--{ -: :--0-- (5-75) : 0 0 I , bll2------------:-i: =--i - - - - - 0- ---:- -0 --

, 2 ,

A1 - A2 : 0 Note that A1 -A2 is different from zero. The matrix in (5-75), or equivalently, the matrix in (5-73) at S = Ab has a full row rank if and only if bIll and bl12 are

CONTROLLABItITY AND OBSEl

linearly independent. By p theorem can be established.

Observe that in order f independent, it is necessary that is, p = I-it is necessar linearly independent. In wo form dynamical equation t4 block associated with each d to a vector. Thus we have 1

Corollary 5-21

A single-input linear time-i: troll able if and only if ther distinct eigenvalue and all tl spond to the last row of eac

A single-output linear tl observable if and only if th( distinct eigenvalue and all th to the first column of each J,

Example 2

Consider the single-variable

0 1 0: I I

o 0 1 I ~ X= I [ ~--~--~H

There are two distinct eigen responds to the last row of is zero; therefore, the equati( corresponding to the first coli therefore, the equation is obs

Example 3

Consider the following two J

[~1-X2_ [~1-and X2_

That the state equation (5< Equation (5-77) is a time-va matrix is in the Jordan form

---------

MICAL EQUA nONS

em. In order not to be I be of the form

bI lll : b 211 I bill

: b 112

-------------~-~~~ -A2 -1: bl2l

s -A2 : b /21

(5-73) There are two Jordan

; =Ab (5-73) becomes b lll

b2ll

bill

(5-74)b112 I bll2

--T-----: b l2l

'2 : b'21 matrix in (5-74) can be

o o

(5-75) : bll2 : 0 I

'2 : 0 1 (5-75), or equivalently, only if bill and bll2 are

CONTROLLABIUTY AND OBSERVABILITY OF JORDAN-FORM DYNAMICAL EQ\JATlONS 213

linearly independent. By proceeding similarly for each distinct eigenvalue, the theorem can be established. Q.E.D.

Observe that in order for the rows of an r(i) x p matrix B~ to be linearly t

independent, it is necessary that r(i)sp. Hence in the case of single input-that is, p = I-it is necessary to have r(i) = 1 in order for the rows of B~ to be linearly independent. In words, a necessary condition for a single-input Jordanform dynamical equation to be controllable is that there is only one Jordan block associated with each distinct eigenvalue. For p = 1, the matrix B~ reduces to a vector. Thus we have the following corollary.

Corollary 5-21

A single-input linear time-invariant Jordan-form dynamical equation is controllable if and only if there is only one Jordan block associated with each distinct eigenvalue and all the components of the column vector B that correspond to the last row of each Jordan block are different from zero.

A single-output linear time-invariant Jordan-form dynamical equation is observable if and only if there is only one Jordan block associated with each distinct eigenvalue and all the components of the row vector C that correspond to the first column of each Jordan block are different from zero.

Example 2

Consider the single-variable Jordan-form dynamical equation

~ ~ ~ i~]. [1~l OO:I]xy=[1 X= [~__~ __~_!~ x + +u

There are two distinct eigenvalues 0 and 1. The component of B which corresponds to the last row of the Jordan block associated with eigenvalue 0 is zero; therefore, the equation is not controllable. The two components of C corresponding to the first column of both Jordan blocks are different from zero; therefore, the equation is observable.

Example 3 Consider the following two Jordan-form state equations:

(5-76)[~:J=[ -~ -~Jx +[:}

[XlJ [-1 OJ [e- t Jand (5-77)x2 = 0 - 2 x + e - 2t u

That the state equation (5-76) is controllable follows from Corollary 5-21. Equation (5-77) is a time-varying dynamical equation; however, since its A matrix is in the Jordan form and since the components of B are different from

I


zero for all t, one might be tempted to conclude that (5-1'1) is controllable. Let u~ .;heck this by using Theorem 5-4. For any fixed to, we have

-(to-t) 0 J[ -t ] [-to]c)(to t)B(t)= [ eO e-2(to-t) :-2t = :-2to

It is clear that the rows of c)(to - t)B(t) are linearly dependent in t. Hence the state equation (5-77) is not controllable at any to.

From this example we see that, in applying a theorem, all the conditions should be carefully checked; otherwise, we might obtain an erroneous conclusion.

*5-7 Output Controllability and Output Function Controllability

Similar to the (state) controllability of a dynamical equation, we may define controllability for the output vector of a system. Although these two concepts are the same except that one is defined for the state and the other for the output, the state controllability is a property of a dynamical equation, whereas the output controllability is a property of the impulse-response matrix of a system.

Consider a system with the lilput-output description

y(t) ~ G(t,r)u(r) drr where u is the p x 1 input vector, y is the q x 1 output vector, G(t, T) is the q x p impulse-response matrix of the system. We assume for simplicity that G(t, T) does not contain t>-functions and is continuous in t and T for t > T.

Definition 5-10 A system with a continuous impulse-response matrix G(t, T) is said to be output controllable at time to if, for any Y l' there exist a finite t 1 > to and an inputu[to,td that transfers the output from y(to) = 0 to y(td = Y1 Theorem 5-22

A system with a continuous G(t, T) is output controllable at to if and only if there exists a finite t 1 > to such that all rows of G(t 1, T) are linearly independent in r on [to, t 1] over the field of complex number.

The proof of this theorem is exactly the same as the one of Theorem 5-4 and is therefore omitted.

We study in the following the class of systems tha~ also have linear timeinvariant dynamical-equation descriptions. Consider the system that is describable by

FE: x=Ax +Bu y=Cx

OUTPUT CONTROLl

where A, B, and Care n x n response matrix of the systel

The transfer-function matrix

It is clear that G(s) is a strict

Corollary 5-22

A system whose transfer fun< output controllable if and OJ over the field of complex nUll

[CD has rank q.

The proof of Theorem 5-~ trivial consequence of this co controllable. We see that 4 can also be stated in terms a independence of G(s), the COil

The state controllability the output controllability is c these two concepts are not nc

Example 1 Consider the network shown observable, but it is output a

tn

1 yu

tn

Figure 5-14 A network which i observable.

OUTPUT CONTROLLABILITY AND OUTPUT FUNCTION CONTROLLABILITY 215MICAL EQUATIONS

Lt (5-77) is controllable. 1 to, we have

-to] = [:-2to )endent in t. Hence the

orem, all the conditions ,tain an erroneous con-

Function

:quation, we may define ough these two concepts l the other for the output, 1 equation, whereas the lonse matrix of a system. In

{ector, G(t, r) is the q x p or simplicity that G(t, r) drfort>r.

;(t, r) is said to be output lite t1 > to and an input

=Yl'

le at to if and only if there linearly independent in r

the one of Theorem 5-4

at also have linear timeier the system that is

where A, B, and Care n x n, n x p, q x n real constant matrices. The impulseresponse matrix of the system is

i

The transfer-function matrix of the system is

(5-78)

It is clear that G(s) is a strictly proper rational function matrix.

Corollary 5-22

A system whose transfer function is a strictly proper rational-function matrix is output controllable if and only if all the rows of G(s) are linearly independent over the field of complex numbers or if and only if the q x np matrix

[CB: CAB: ... : CAn-1BJ (5-79)

has rank q.

The proof of Theorem 5-7 can be applied here with slight modification. A trivial consequence of this corollary is that every single-output system is output controllable. We see that although the condition of output controllability can also be stated in terms of A, B, and C, compared with checking the linear independence of G(s), the condition (5-79) seems more complicated.

The state controllability is defined for the dynamical equation, whereas the output controllability is defined' for the input-output description; therefore, these two concepts are not necessarily related.

Example 1

Consider the network shown in Figure 5-14. It is neither state controllable nor observable, but it is output controllable.

1+In I+

1 Yu ~

In L--_____------'

Figure 5-14 A network which is output controllable but neither (state) controllable nor observable.

216 CONTROLLABILITY AND OBSERVABILITY OF LINEAR DYl'T:\MICAL EQUATIONS

. YI-----1

~Y2

Figure 5-15 A system which is controllable and observable but not output controllable.

Example 2 Consider the system shown in Figure 5-15. The transfer-function matrix of the system is

s ;1]. [ s +1

the rows of which are linearly dependent. Hence the system is not output controllable. The dynamical equation of the system is

x -x +u Y~G} which is controllable and observable.

If a system is output controllable, its output can be transferred to any desired value at certain instant of time. A related problem is whether it is possible to steer the output following a preassigned curve over any interval of time. A system whose output can be steered over any interval of time is said to be output function controllable or functional reproducible.

Theorem 5-23

A system with a q x p proper rational-function matrix G(s) is output function controllable if and only if pG(s) = q in ~(s), the field of rational functions with real coefficients. Proof

If the system is initially relaxed, then we have

y(s) = G(s)u(s) (5-80) If pG(s) q-that is, all the rows of G(s) are linearly independent over the field of rational functions-then the q x q matrix (;{s)G*(s) is nonsingular (Theorem 2-8). Consequently, for any y(s), if we ,choose

u(s) = G*(s)(G(s)G*(S-l y(s) (5-81) then Equation (5-80) is satisfied. Consequently, if pG{s) = q, then the system is output function controllable. If pG(s) < q, we can always find a y(s), not in the

range of G(s), for which th

If the input is restricted then the given output func computed from (5-81) will] given output has some diSCI needed to generate the disc(

The condition for outp terms of the matrices A, B, a interested reader is referred

Dual to the output functil These problems are intimat, q x p proper rational matri: exists a p x q rational matriJ

G(s)G. A system is said to have an ir A necessary and sufficient ( pG(s) q in [~(s). This cor controllability. Many quest unique? Is it a proper rati stable? What are its equiv problems will not be stuQie( References S172, S185, S218,

* 5-8 Computational I In this section, we discuss s( chapter. As discussed in Sec conditioned; a computationa unstable. If we use a nume problem, the result will gener if we use a numerically stable res ult will be correct. If WI problem, well or ill condition a problem, if we must use an stable method, the unstable possible, in the computation.

As discussed in Theorem trollability of a state equatioJ more suitable for computer a computational problems, as,

The computation of the c is straightforward. Let Ko ~

IAMICAL EQUATIONS

Ie but not output controllable.

:ansfer-function matrix of

the system is not output is

:an be transferred to any problem is whether it is curve over any interval of .ny interval of time is said 'ucible.

'ix (;(s) is output function of rational functions with

(5--80) .fly independent over the (;(s )(;*(s) is nonsingular ose

(5--81 ) ~(s) = q, then the system is ways find a y(s), not in the

COMPUTATIONAL PROBLEMS 217

range of (;(s), for which there exists no solution o(s) in (5-80) (Theorem 2-4). Q.E.D.

If the input is restricted#, to the class of piecewise continuous functions of t, then the given output function should be very smooth; otherwise, the input computed from (5-81) will not be piecewise continuous. For example, if the given output has some discontinuity, an input containing

j


At the end, we have U = [Ko Kl Kn - l ]. The rank of U can then be computed by using the singular value decomposition (Appendix E) which is a numerically stable method. If the dimension n of the equation is large, this process requires the computation of AkB for large k and may transform the problem into a less well ,conditioned problem. For convenience in discussion, we assume that all eigenvalues Ai of A are distinct and B is an n x 1 vector. We also arrange Ai so that 1..1,11 ;;::::IA21 ;;:::: ... ;;::::IAnl. Clearly, we can write B as

B=(XI VI + (X2V2 + ... + (Xnvn where Vi is an eigenvector associated with eigenvalue Ai; that is, AVi = AiVi, It is straightforward to verify

AkB =(Xl"~.1Vl +(X2A~v2 + ... +(XnA~n

If 1..1,11 is much larger than all other eigenvalues, then we have AkB-+(XIA~Vl for k large

In other words, AkB tends to approach the same vector, VI' as k increases. Hence, it will be difficult to check the rank of U if n is large.

The same conclusion can also be reached by using a different argument. The condition number of a matrix Amay be defined as cond A~ IIAI1211A -1112 = (It/(ls, where (I, and (Is are the largest and smallest singular values of A. It

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