lmibasedstabilityandstabilizationofsecond ...math.niu.edu/~dattab/psfiles/galkowski.jac.pdf_ kowski,...

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PROD. TYPE: COMED: Selva

PAGN: Babu -- SCAN:

Asian Journal of Control, Vol. 12, No. 2, pp. 1 12,Published online in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/asjc.171

LMI BASED STABILITY AND STABILIZATION OF SECOND-ORDER

LINEAR REPETITIVE PROCESSES

Pawel Dabkowski, Krzysztof Gal_kowski, Biswa Datta and Eric Rogers

ABSTRACT

This paper develops new results on the stability and control of a classof linear repetitive processes described by a second-order matrix discrete ordifferential equation. These are developed by transformation of the second-order dynamics to those of an equivalent first-order descriptor state-spacemodel, thus avoiding the need to invert a possibly ill-conditioned leadingcoefficient matrix in the original model.

Key Words: LMI, discrete and differential second-order linear repetitiveprocesses, ill-conditioning, descriptor systems.

I. INTRODUCTION

Second order linear control systems arise in awide variety of practical applications involving, for ex-3ample, vibrating structures, power systems, economics,and computer networks. One obvious way to solve a5control problem for a linear second-order system is totransform the model to first-order state-space form and7then use any of the well known and tested computa-tional methods. Unfortunately, such reduction requires9explicit computation of the inverse of the leading

11

Manuscript received December 24, 2008; accepted April 14,2009.P. Dabkowski and K. Galkowski are with the Institute

of Physics, Nicolaus Copernicus University Torun, Poland(e-mail: [email protected]).K. Galkowski is also with the Institute of Control and

Computation Engineering, University of Zielona Gora, ZielonaGora, Poland (e-mail: [email protected]).B. Datta is with Departmemt of Mathematical

Sciences and Vibration and Acoustic Center of theCollege of Engineering and Engineering Technology,Northern Illinois University, Dekalb, Illinois, 600115, U.S.A.(e-mail: [email protected]).E. Rogers is with School of Electronics and Com-

puter Science, University of Southampton, SO17 1BJ, U.K.(e-mail: [email protected]).This work is partially supported by the Nicolaus Copernicus

University Torun grant 351−F and the Ministry of Science andHigher Education in Poland under the project N N514 293235.

coefficient matrix, which could be numerically prob-lematic due, for example, to possible ill-conditioning of 13this matrix or the computational cost involved. For ex-ample, in vibration control analysis, this matrix, termed 15the mass matrix, is often diagonal and therefore can beill-conditioned whenever some (or all) of the diagonal 17entries are small (see [1]).

Another area where such problems can arise is in 19the application of the Crank-Nicholson discretizationscheme to partial differential equations (PDEs) [2, 3]. 21Here the resulting model coefficient matrices are oftentri-diagonal but the inverse of the leading coefficient one 23may not have this computationally attractive property.A similar situation arises for first-order descriptor sys- 25tems, where the coefficient matrix on the left-hand sidemay be very close to singular or left-multiplying the 27model by the inverse of this matrix involves the loss ofother essential problem features. To overcome such dif- 29ficulties research has been focussed in recent years ondeveloping methods for second-order state-space mod- 31els that do not require explicit computation of a matrixinverse. 33

As a result of such research, there as been muchprogress on the solution of control related problems 35for systems described by second-order state-space mod-els. Examples here include including stability, feedback 37stabilization, partial pole placement, robust pole place-ment, and model order reduction. These solutions have 39been developed, in the main, by either first convert-ing to an equivalent descriptor system or proceeding 41

q 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society

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directly with the coefficient matrices of the second-1order state-space model [4–12]. The latter approach hasthe further advantage that any special structure, such as3sparsity, in the coefficient matrices, which often arisesin practical applications, can be preserved and exploited5in the computations associated with numerical exam-ples. Such problems can also arise in 2D linear systems7where, for example, repetitive processes have found ap-plication in modeling spatio-temporal dynamics such as9large flexible structures [13].

The unique characteristic of a repetitive, or multi-11pass [14], process is a series of sweeps, termed passes,through a set of dynamics defined over a fixed finite du-13ration known as the pass length. On each pass an out-put, termed the pass profile, is produced which acts as15a forcing function on, and hence contributes to, the dy-namics of the next pass profile. This, in turn, leads to17the unique control problem in that the output sequenceof pass profiles generated can contain oscillations that19increase in amplitude in the pass-to-pass direction.

Physical examples of repetitive processes include21long-wall coal cutting and metal rolling operations [15].Also in recent years applications have arisen where23adopting a repetitive process setting for analysis hasdistinct advantages over alternatives. Examples of these25so-called algorithmic applications include classes of it-erative learning control schemes [16] and iterative al-27gorithms for solving nonlinear dynamic optimal controlproblems based on the maximum principle [17]. In this29last case, for example, use of the repetitive process set-ting provides the basis for the development of highly31reliable and efficient solution algorithms and in the for-mer it provides a stability theory which, unlike alter-33natives, provides information concerning an absolutelycritical problem in this application area, i.e. the trade-35off between convergence and the learnt dynamics.

Attempts to control these processes using standard37(or 1D) systems theory/algorithms fail (except in a fewvery restrictive special cases) precisely because such an39approach ignores their inherent 2D systems structure,i.e. information propagation occurs from pass-to-pass41and along a given pass and also the initial conditions arereset before the start of each new pass. To remove these43deficiencies, a rigorous stability theory has been devel-oped [15] based on an abstract model of the dynamics45in a Banach space setting which includes a very largeclass processes with linear dynamics and a constant pass47length as special cases. Also the results of applying thistheory to a range of sub-classes, including those consid-49ered here, have been reported [15]. This stability theoryconsists of the distinct concepts of asymptotic stability51and stability along the pass respectively where the for-mer is a necessary condition for the latter.53

Recognizing the unique control problem, this sta-bility theory is of the bounded input bounded output 55(BIBO) form, i.e. bounded previous pass profiles are re-quired to produce bounded sequences of pass profiles 57(where boundedness is defined in terms of the normon the underlying Banach space). Asymptotic stability 59guarantees this property over the finite and fixed passlength whereas stability along the pass is stronger in that 61it requires this property uniformly, i.e. for all possiblevalues of the pass length (and hence it is not surpris- 63ing that asymptotic stability is a necessary condition forstability along the pass). 65

In this paper we develop new results on the sta-bility and control of linear repetitive processes where 67the pass-to-pass updating is governed by a matrix linearsecond-order discrete equation with possible numerical 69ill-conditioning. The major outcome is Linear MatrixInequality (LMI)-based algorithms for stability testing 71and control law design, including the case when thereis uncertainty associated with the process dynamics. 73

Throughout this paper, the null matrix and theidentity matrix with the required dimensions are de- 75noted by 0 and I , respectively. Moreover, M>0(<0)denotes a real symmetric positive (respectively nega- 77tive) definite matrix, and � denotes a block matrix entryin a symmetric matrix. 79

II. BACKGROUND

Let �<∞ denote the pass length and use an integer 81subscript k ≥ 0 to denote the pass number or index.Then the most basic discrete linear repetitive process 83state-space model [15] has the following form over 0 ≤p ≤ � − 1, k ≥ 0, 85

xk+1(p + 1) = Axk+1(p)+Buk+1(p)+B0yk(p)

yk+1(p) = Cxk+1(p)+Duk+1(p)+D0yk(p).(1)

Here on pass k, xk(p)∈ Rn is the state vector, 87yk(p)∈ Rm is the pass profile vector, and uk(p)∈ Rr isthe vector of control inputs. The boundary conditions 89(i.e. the pass state initial vector sequence and the initialpass profile) are 91

xk+1(0) = dk+1, k ≥ 0

y0(p) = f (p), 0 ≤ p ≤ � − 1(2)

where the n× 1 vector dk+1 has known constant entries 93and f (p) is an m × 1 vector whose entries are knownfunctions of p. 95

In a differential linear repetitive process [15] thealong the pass dynamics are governed by a linear ma- 97trix differential equation and, with the along the pass


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variable denoted by t, the most basic state-space model1has the following form over 0 ≤ t ≤ �, k ≥ 0,

xk+1(t) = Axk+1(t) + Buk+1(t) + B0yk(t)

yk+1(t) = Cxk+1(t) + Duk+1(t) + D0yk(t)(3)

3

where all notation is the same as the discrete case,except that the initial pass profile is now taken as5y0(t)= f (t), where the entries in f (t) are knownfunctions over 0 ≤ t ≤ �.7

The stability theory [15] for linear repetitive pro-cesses is based on an abstract model in a Banach space9setting which includes a wide range of such processesas special cases, including both cases considered in this11work. In terms of their dynamics it is the pass-to-passcoupling (noting again their unique feature) which is13critical in the analysis of linear repetitive processes.This is of the form yk+1 = L�yk , where yk ∈ E� (E� a15Banach space with norm ‖ · ‖) and L� is a boundedlinear operator mapping E� into itself. (In the cases17considered here L� are discrete and differential linearsystems convolution operators respectively.)19

Recognizing the unique control problem, this sta-bility theory is of the bounded input bounded output21(BIBO) form, i.e. bounded previous pass profiles are re-quired to produce bounded sequences of pass profiles23(where boundedness is defined in terms of the normon the underlying Banach space). Asymptotic stability25guarantees this property over the finite and fixed passlength whereas stability along the pass is stronger in that27it requires this property uniformly, i.e. for all possiblevalues of the pass length (and hence it is not surpris-29ing that asymptotic stability is a necessary condition forstability along the pass).31

Asymptotic stability, i.e. BIBO stability over thefixed finite pass length �>0, requires the existence of fi-33nite real scalars M�>0 and �� ∈ (0, 1) such that ‖Lk

�‖ ≤M��

k�, k ≥ 0, (where || · || denotes the induced operator35

norm). For the discrete and differential linear repetitiveprocesses considered in this work it has been shown37elsewhere (see, for example, Chapter 3 of [15]) that thisproperty holds if, and only if, all eigenvalues of the ma-39trix D0 have modulus strictly less than unity, writtenhere as r(D0)<1 where r(·) denotes the spectral radius41of its matrix argument.

Suppose that a process described by either of the43state-space models considered here is asymptoticallystable and also that the input sequence applied {uk+1}k45converges strongly as k → ∞ (i.e. in the sense of thenorm on the underlying function space) to u∞. Then the47strong limit y∞ := limk→∞yk is termed the limit profilecorresponding to this input sequence. For the discrete

process, it can be shown that the limit profile is given by 49

x∞(p + 1) = (A + B0(I − D0)−1C)x∞(p)

+(B+B0(I−D0)−1D)u∞(p)

y∞(p) = (I − D0)−1Cx∞(p)

+(I − D0)−1Du∞(p)

x∞(0) = d∞

(4)

where d∞ is the strong limit of the sequence {uk}. 51In physical terms, this result states that under

asymptotic stability the repetitive dynamics can, after 53a ’sufficiently large’ number of passes have elapsed,be replaced by those of a 1D discrete linear system. 55In particular, this property demands that the amplify-ing properties of the coupling between successive pass 57profiles are completely damped out after a sufficientlylarge number of passes have elapsed. This fact has clear 59implications in terms of the control of these processes.

The fact that the pass length is finite means that the 61limit profile may have unacceptable along the pass dy-namics. For example, consider the case when A=−0.5, 63B = 1, B0 = 0.5+�, C = 1, D = 0, D0 = 0 where � is areal scalar. This example is asymptotically stable since 65D0 = 0 and the state matrix of the resulting limit profilestate-space model is �. Hence the limit profile is unsta- 67ble unless |�|<1. Clearly this is not acceptable in manycases. 69

The limit profile for the differential case is

x∞(t) = (A + B0(I − D0)−1C)x∞(t)

+(B + B0(I − D0)−1D)u∞(t)

y∞(t) = (I − D0)−1Cx∞(t)

+(I − D0)−1Du∞(t)

x∞(0) = d∞

(5)

71

In order to avoid cases where asymptotic stabil-ity results in an unstable limit profile, the obvious route 73is to demand the BIBO property for all possible valuesof the pass length (mathematically this can be analyzed 75by letting � → ∞). This is the stability along the passproperty which (in abstract model terms) requires the 77existence of finite real scalars M∞>0 and �∞ ∈ (0, 1)such that ||Lk

�|| ≤ M∞�k∞, k ≥ 0. For discrete pro- 79cesses described by (1) and (2), it has been shown else-where that this requires 81

• r(D0)<1 (asymptotic stability),• r(A)<1, and 83• r(G(z))<1, ∀|z|= 1, where G(z)=C(z I−A)−1

B0 + D0. 85


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In the case of processes described by (3) and the corre-1sponding boundary conditions, the corresponding con-ditions are3

• r(D0)<1 (asymptotic stability),• all eigenvalues of the matrix A have strictly nega-5

tive real parts, and• r(G(s))<1, ∀s : Res ≥ 0,7

where G(s) =C(s I − A)−1B0 + D0.

Note here that (1D) stability of the state matrix A is also9only necessary for stability along the pass, as the simpleexample above for the discrete case demonstrates.11

For the processes considered here stability alongthe pass is independent of the boundary conditions as-13sumed in this paper and hence they will not be explic-itly stated in the theorems to follow that give the main15results. Note, however, that the form of the boundaryconditions is critical to the stability properties of linear17repetitive processes. In particular, it can be shown [15]that if the state initial vector on each pass is a function19of points along the previous pass then this alone cancause instability.21

In terms of stability analysis and control law de-sign, the most productive route is via a 2D Lyapunov23equation [15] characterization of stability along the passwhich, in turn, arises from a Lyapunov function inter-25pretation of stability along the pass. The starting pointis to note that any candidate Lyapunov function needs to27capture the ‘energy’ associated with information prop-agation both along the pass and from pas-to-pass. The29function used here for the discrete case is

V (k, p)= xTk+1(p)W1xk+1(p)+yTk (p)W2yk(p) (6)31

where W1>0 and W2>0, with associated increment

�V (k, p) = xTk+1(p + 1)W1xk+1(p + 1)

+yTk+1(p)W2yk+1(p)

−xTk+1(p)W1xk+1(p)

−yTk (p)W2yk(p) (7)

Then we have the following result via the 2D Lyapunov33equation.

Theorem 1 ([15]). A discrete linear repetitive process35described by (1) with Lyapunov function (6) is stablealong the pass if37

�V (k, p)<0 (8)

for all 0 ≤ p ≤ � − 1, k ≥ 0.39

Now we have the following results which are cen-tral to the analysis in this paper. 41

Theorem 2 ([15]). A discrete linear repetitive processdescribed by (1) is stable along the pass if ∃ matrices 43P>0 and Q>0 such that⎡⎢⎢⎢⎣

−P + Q 0 AT1 P

0 −Q AT2 P

P A1 P A2 −P

⎤⎥⎥⎥⎦ <0 (9)

45

where

A1 =[A B0

0 0

], A2 =

[0 0

C D0

].

47

Theorem 3 ([18]). A differential linear repetitive pro-cess described by (3) is stable along the pass if ∃ ma- 49trices P1>0 and P2>0 such that

⎡⎢⎢⎣−P2 P2C P2D0

CT P2 AT P1 + P1A P1B0

DT0 P2 BT

0 P1 −P2

⎤⎥⎥⎦<0 (10)

The discrete processes considered in the remainder 51of this paper are, with the notation as above, describedby the following state-space model which is second- 53order in the pass-to-pass direction

xk+1(p + 1) = Axk+1(p) + Buk+1(p)

+B10yk+1(p) + B00yk(p)

D2yk+2(p) = C1xk+1(p) + D1uk+1(p)

+D10yk+1(p) + D00yk(p)

(11)

55

It is also necessity to extend the boundary conditions of(2) by adding 57

y1(p)= f1(p), p= 0, 1, . . . , (� − 1) (12)

where f1(p) is anm × 1 vector whose entries are known 59functions of p. This model (11) can be transformed tofirst-order form by introducing 61

Yk(p)=[

yk(p)

yk+1(p)

]


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to obtain1

xk+1(p + 1) = Axk+1(p) + Buk+1(p)

+B0Yk(p)

�0Yk+1(p) = Cxk+1(p) + Duk+1(p)

+D0Yk(p)

(13)

where3

B0 = [B00 B10] C =[

0

C1

]

D =[

0

D1

], D0 =

[0 Im

D00 D10

]

�0 =[Im 0

0 D2

].

The corresponding model in the differential case5is

xk+1(t) = Axk+1(t) + Buk+1(t)

+B10yk+1(t) + B00yk(t)

D2yk+2(t) = C1xk+1(t) + D1uk+1(t)

+D10yk+1(t) + D00yk(t)

(14)

7

and it is necessary to add

y1(t)= f1(t), 0 ≤ t ≤ �9

to the boundary conditions where f1(t) is anm × 1 vec-tor whose entries are known functions of t . In first-order11form we have for this case we introduce

Yk(t)=[

yk(t)

yk+1(t)

]13

to obtain

xk+1(t) = Axk+1(t)+Buk+1(t)+B0Yk(t)

�0Yk+1(t) = Cxk+1(t)+Duk+1(t)+D0Yk(t)(15)

15

Obviously, left-multiplying the second equationof (13) or (15) as appropriate by the matrix �−1

017yields the repetitive process model of (1) or (3) re-spectively and then existing results can be applied. In19terms of applications, however, problems will arise ifthis matrix is ill-conditioned. In this paper we develop21methods that do not require this inversion and the pos-sible ill-conditioning associated with constructing the

inverse. Note, however that the methods developed in 23this paper do not extend to the case where the matrixD2 is singular. To deal with this case it is necessary to 25use further results for singular linear systems, see, forexample, [19]. 27

III. ANALYSIS

First note that stability along the pass is indepen- 29dent of the boundary conditions assumed in this paperand hence theywill not be referred to again in this paper. 31Note, however, that the form of the boundary conditionsis critical to the stability properties of linear repetitive 33processes. In particular, it can be shown [15] that if thestate initial vector on each pass is a function of points 35along the previous pass then this alone can cause insta-bility. 37

Consider the discrete case. Then we cannot di-rectly apply Theorem 2 to obtain a condition for stabil- 39ity along the pass of a process described by (11) sincethis requires the numerical inversion of the matrix D2. 41Instead, we have the following result.

Theorem 4. A discrete linear repetitive process de- 43scribed by (13) is stable along the pass if ∃ matricesY>0 and Z>0 such that 45⎡⎢⎢⎣

−Y + Z 0 Y AT1

0 −Z Y AT2

A1Y A2Y −�Y�T

⎤⎥⎥⎦< 0 (16)

where 47

A1 =[A B0

0 0

], A2 =

[0 0

C D0

]

and 49

�=[In 0

0 �0

].

Proof. First left-multiply the second equation in (13) 51by �−1

0 and apply Theorem 2 to the result. The proof isthen completed by application of obvious congruence 53transforms and change of variables. �

Theorem 5. A differential linear repetitive process de- 55scribed by (15) is stable along the pass if ∃ matrices


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Y1>0 and Z1>0 such that1 ⎡⎢⎢⎣−�0Z1�

T0 CY1 D0Z1

Y1CT Y1A

T + AY1 B0Z1

Z1 DT0 Z1 B

T0 −Z1

⎤⎥⎥⎦ <0 (17)

Proof. First Left-multiply the second equation of (15)3by �−1

0 and apply Theorem 3 to the result. Next,left and right-multiply the result of this last step by5diag(�0P

−12 , P−1

1 , P−12 ) to obtain (17). �

If the along the pass stability property is not7present for a given example then it will clearly be nec-essary to introduce regulation action to guarantee this9property. Moreover, given the critical role of the pass-to-pass updating, it follows that any control law must11have a contribution from the previous two passes hereplus current pass state or pass profile activated action.13Here we consider a law of the form

uk+1(p) = K1xk+1(p) + K2yk(p) + K3yk+1(p)

= K

[xk+1(p)

Yk(p)

](18)

for the discrete case with differential counterpart15

uk+1(t) = K1xk+1(t) + K2yk(t) + K3yk+1(t)

= K

[xk+1(t)

Yk(t)

](19)

where

K = [K1 K2 K3] = [K1 K2].17

Note here that the pass profile vector is the pro-cess output and here it assumed that noise corruption19and other disturbances are negligible. Moreover, ill-conditioning of the matrix D2 is not a problem here in21practical implementation of the control law. Also thecurrent pass state vector in this stabilization law will, in23general, require an observer.

The controlled process in the discrete case is de-25scribed by

xk+1(p + 1) = Anewxk+1(p) + Bn1yk+1(p)

+Bn2yk(p)

D2yk+2(p) = Cnxk+1(p) + Dn1yk+1(p)

+Dn2yk(p)

(20)

27

and in the differential case by

xk+1(t) = Anewxk+1(t) + Bn1yk+1(t)

+Bn2yk(t)

D2yk+2(t) = Cnxk+1(t) + Dn1yk+1(t)

+Dn2yk(t)

(21)

29

where

Anew = A + BK1, Bn1 = B10 + BK3

Bn2 = B00 + BK2, Cn =C1 + D1K1

Dn1 = D10 + D1 K3, Dn2 = D00 + D1 K2

or, more compactly, 31

xk+1(p + 1) = Anewxk+1(p) + BnewYk(p)

�0Yk+1(p) = Cnewxk+1(p) + DnewYk(p)(22)

and 33

xk+1(t) = Anewxk+1(t) + BnewYk(t)

�0Yk+1(t) = Cnewxk+1(t) + DnewYk(t)(23)

respectively, where 35

Bnew = [Bn2 Bn1] = B0 + BK2

Cnew =[

0

Cn

]= C + DK1

Dnew =[

0 Im

Dn2 Dn1

]= D0 + DK2.

In the discrete case we now have the following re- 37sult for stability along the pass of the controlled processtogether with a formula for computing the control law 39matrix.

Theorem 6. Suppose that a control law of the form 41(18) is applied to a discrete linear repetitive processdescribed by (13). Then the resulting controlled process 43is stable along the pass if ∃ matrices Y>0, Z>0, andN = [N1 N2] such that 45

⎡⎢⎢⎣−Y + Z � �

0 −Z �

A1Y + B1 N A2Y + B2 N −�Y�T

⎤⎥⎥⎦< 0 (24)


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where1

B1 =[B

0

], B2 =

[0

D

]

If this condition holds, a stabilizing control law matrix3K is given by

K = NY−1 (25)5

Proof. The proof is a direct consequence of interpret-ing the LMI of Theorem 4 in terms of the case con-7sidered here with KY = N and application of routinemanipulations. �9

For the differential case we have the followingresult.11

Theorem 7. Suppose that a control law of the form (19)is applied to a differential repetitive process described13by (15). Then the resulting controlled process is stablealong the pass if ∃ matrices Y>0, Z>0, N , and M such15that

⎡⎢⎢⎢⎢⎢⎢⎣Y AT + NT BT

+AY + BN� �

(B0Z + BM)T −Z �

CY + DN D0Z + DM −�0Z�T0

⎤⎥⎥⎥⎥⎥⎥⎦<0 (26)

If this condition holds, a stabilizing control law matrix17K is given by

K1 = NY−1

K2 = MZ−1(27)

19

Proof. This follows analogous steps to the proof of thelast result and hence the details are omitted here. �21

IV. STABILITY AND STABILIZATIONOF UNCERTAIN PROCESSES

In the most cases the model matrices are subjectto uncertainty and only the nominal model is known.25The standard route in robust control to deal with thiscase is to assume an uncertainty model and here we use27the norm bounded type of uncertainty under which the

discrete linear repetitive process of (11) takes the form

xk+1(p + 1) = (A + �A)xk+1(p)

+(B + �B)uk+1(p)

+(B10 + �B10)yk+1(p)

+(B00 + �B00)yk(p)

D2yk+2(p) = (C1 + �C1)xk+1(p)

+(D1 + �D1)uk+1(p)

+(D10 + �D10)yk+1(p)

+(D00 + �D00)yk(p)

(28)

29

and in the differential case

xk+1(t) = (A + �A)xk+1(t)

+(B + �B)uk+1(t)

+(B10 + �B10)yk+1(t)

+(B00 + �B00)yk(t)

D2yk+2(t) = (C1 + �C1)xk+1(t)

+(D1 + �D1)uk+1(t)

+(D10 + �D10)yk+1(t)

+(D00 + �D00)yk(t)

(29)

31

For analysis purposes these models can be rewritten as

xk+1(p + 1) = (A + �A)xk+1(p)

+(B + �B)uk+1(p)

+(B0 + �B0)Yk(p)

�0Yk+1(p) = (C + �C)xk+1(p)

+(D + �D)uk+1(p)

+(D0 + �D0)Yk(p)

(30)

33

and

xk+1(t) = (A + �A)xk+1(t)

+(B + �B)uk+1(t)

+(B0 + �B0)Yk(t)

�0Yk+1(t) = (C + �C)xk+1(t)

+(D + �D)uk+1(t)

+(D0 + �D0)Yk(t)

(31)

35


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respectively, where1

�B0 = [�B00 �B10], �C =[

0

�C1

]�D =

[0

�D1

],

�D0 =[

0 Im

�D00 �D10

]

and the rest of notation follows that of the previoussection.3

Now introduce the following notation.

⎡⎣ A + �A B0 + �B0 B + �B

C + �C D0 + �D0 D + �D

⎤⎦= � + ��.

Also it assumed that we can write5

��=HFE=[H1

H2

]F[E1 E2] (32)

where H , E1, E2 are given matrices of compatible di-7mensions, and F is unknown matrix which satisfies‖F‖<1, or FT F<I .9

4.1 Stability

4.1.1 Discrete processes.11

Introduce the following notation

� A1 =[

�A �B0

0 0

]� A2 =

[0 0

�C �D0

]�B1 =

[�B0

]�B2 =

[0

�D

]and13

H1 =[H1

0

], H2 =

[0H2

].

Then we have the following result by direct application15of an existing LMI based stability condition.

Theorem 8. A discrete linear repetitive process de-scribed by (30) with uncertainty modeled as (32) is 17stable along the pass if, ∃ matrices P>0, and Q>0,such that 19

( A + � A)T P( A + � A) + Q<0 (33)

where 21

Q =[P − Q 0

0 Q

], A= [A1 A2]

and 23

� A=[� A1 � A2]To remove the uncertain term F in this last result 25

(which means that it is numerically intractable) we ap-ply the elimination lemma [20] to obtain the following 27result.

Theorem 9. The condition of Theorem 8 holds if, and 29only if, ∃ a scalar �>0 and matrices P>0, and Q>0,such that

31

[ −�P−1�T + �H H T A

AT Q + �−1 ET1 E1

]<0 (34)

where

H = [H1 H2], E1 = diag(E1, E1) 33

Now we have the following result whose proof followsafter standard algebraic manipulations that are omitted 35here.

Theorem 10. A discrete linear repetitive process de- 37scribed by (30) with uncertainty modeled as (32) is sta-ble along the pass if ∃ a scalar �>0, and matrices Y>0, 39and Z>0, such that⎡⎢⎢⎢⎣

−�Y�T AY �H 0

Y AT Z 0 Y ET1

�H T 0 −�I 0

0 E1Y 0 −�I

⎤⎥⎥⎥⎦ <0 (35)

41

where

Y =[Y 00 Y

], Z =

[ −Y + Z 00 −Z

]43

4.1.2 Differential processes.

We require the following well known result.


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Lemma 11 ([21]). Let �1, �2 be real matrices of1appropriate dimensions. Then for any matrix F satisfy-ing FTF ≤ I and a scalar �>0 the following inequality3holds

�1F�2 + �T2 FT�T

1 �−1�1�T1 + ��T

2 �2 (36)5

Applying Lemma 11 to the result of Theorem 5interpreted in terms of the uncertain process (31) gives,7after routine algebraic manipulations, the following re-sult.9

Theorem 12. A differential linear repetitive processdescribed by (31) is stable along the pass if ∃ a scalar11�>0, and matrices Y1>0, and Z1>0, such that

⎡⎢⎢⎢⎢⎢⎢⎣�1 CY1 D0Z1 0 0

Y1CT �2 B0Z1 Y T

1 ET1 Y T

1 ET1

Z1 DT0 Z1 B

T0 −Z1 ZT

1 ET2 ZT

1 ET2

0 E1Y1 E2Z1 −�I 00 E1Y1 E2Z1 0 −�I

⎤⎥⎥⎥⎥⎥⎥⎦<0 (37)

where13

�1 = − �0Z1�T0 + �H2H

T2

and15

�2 = Y1AT + AY1 + �H1H

T1

4.2 Robust control

Using the analysis so far in this paper we can nowestablish the following results.19

4.2.1 Discrete processes.

Theorem 13. Suppose that a control law of the form21(18) is applied to a discrete linear repetitive processdescribed by (30) with uncertainty modeled by (32).23Then the resulting controlled process is stable along thepass if ∃ matrices P>0, and Q>0, such that

25

Q + ( A + B K + � A + �B K )T

×P( A + B K + � A + �B K )<0

Theorem 14. Suppose that a control law of the form(18) is applied to a discrete linear repetitive process 27described by (30) with uncertainty modeled by (32).Then the resulting controlled process is stable along the 29pass if ∃ a scalar �>0, and matrices Y>0, Z>0, and N ,such that 31

⎡⎢⎢⎢⎢⎢⎣−�Y�T ∗ ∗ ∗

Y AT + N T BT Z ∗ ∗�H T 0 −�I ∗0 E1Y + E2 N 0 −�I

⎤⎥⎥⎥⎥⎥⎦<0 (38)

where

H = [H1 H2], E1 = diag(E1, E1)

E2 = diag(E2, E2), A= [A1 A2]B = [B1 B2]

Z =[ −Y + Z 0

0 −Z

]

Y =[Y 0

0 Y

], N =

[N 0

0 N

]

If this condition holds, a stabilizing control law matrix 33is given by

K = NY−1. (39) 35

4.2.2 Differential processes.

Using Theorem 12 we have the following result. 37

Theorem 15. Suppose that a control law of the form(19) is applied to a differential linear repetitive process 39described by (31) with uncertainty modeled by (32).Then the resulting controlled process is stable along the 41pass if, ∃ a scalar �>0, and matrices Y1>0, Z1>0, N1,and N2, such that

43⎡⎢⎢⎢⎢⎢⎢⎣�11 � � � �

�21 �22 � � �

Z1 DT0 + NT

2 D Z1 BT0 + NT

2 BT −Z1 � �

0 E1Y1 + E3N1 0 −�I �

0 0 E2Z1 + E3N2 0 −�I

⎤⎥⎥⎥⎥⎥⎥⎦<0 (40)


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where

1

�11 = −�0Z1�T0 + 2�H2H

T2

�21 = Y1CT + NT

1 D + 2�H1HT2

�22 = Y1AT + AY1 + NT

1 BT + BN1 + 2�H1HT1

If this condition holds, a stabilizing control law matrixis given by3

K1 = N1Y−11

K2 = N2Z−11

(41)

V. CONCLUSIONS

This paper has developed new results on stabil-ity and stabilization of discrete and differential linear7repetitive processes whose dynamics second-order inthe pass-to-pass direction, with particular attention to9avoiding numerical ill-conditioning. The resulting sta-bility conditions and control law design algorithms are11LMI based. The core feature is that the algorithms de-veloped do not require the inversion of a possibly ill-13conditionedmatrix. Also the analysis has been extendedto the case when there is uncertainty associated with the15process model. Further work consists amongst others, ofattempting to use these results to design iterative learn-17ing control schemes for second-order ill-conditioned 1Dlinear systems, as frequently arise in electro-mechanical19systems. Note also that all results here can be gener-alized to higher order processes which are related to21so-called non-unit memory linear repetitive processes,which find application in modeling coal mining sys-23tems.

The results in this paper, and the methods used25to derive them, can also be extended to the case whenthe along the pass dynamics are second-order as, for27example, in the following discrete state-space model

A0xk+1(p + 2) = A1xk+1(p + 1) + A1xk+1(p)

+Buk+1(p) + B00yk(p)

yk+1(p) = C1xk+1(p) + D1uk+1(p)

+D00yk(p)

(42)

29

where the matrix A0 is nonsingular but possibly ill-conditioned. Such models open up other application ar-31eas, such as the development of iterative learning con-trol schemes for descriptor first or the second-order sys-33tems. This would, however, require the use of only out-put feedback control as the state vector here is much35harder to recover and the special singular observer mustbe used [22].37

REFERENCES

1. Inman, D., “Vibration with Control Measurement 39and Stability,” Prentice Hall, New Jersey,Englewood Cliffs, (1989). 41

2. Garcia, A., “Numerical Methods for Physics,”Prentice Hall, Engelwood Cliffs, (1994). 43

3. Steffen, P. and R. Rabenstein, “Implicitdiscretization of linear partial differential equations 45and repetitive processes,” NDS09 6th Int. WorkshopMultidimens. (nD) Syst., Thessaloniki, Greece, June 4729–July 1 (2009).

4. Chu, E. K. and B. N. Datta, “Numerically robust 49pole assignment for second-order systems,” Int. J.Control, Vol. 64, No. 6, pp. 1113–1127 (1996). 51

5. Datta, B. and D. Sarkissian, “Feedback controlin distributed parameter gyroscopic systems A 53solution of the partial eigenvalue assignmentproblem,” Mechanical Systems and Signal 55Processing, Vol. 16, No. 1, pp. 3–17 (2001). (15),January, (Invited paper) , special issue on Vibration 57Control.

6. Datta, B. and D. Sarkissian, “Theory and 59computations of some inverse eigenvalue problemsfor the quadratic pencil,” Contemp. Math., Vol. 280, 61pp. 221–240 (2001).

7. Datta, B. and D. Sarkissian, “Computational 63methods for feedback control in damped gyroscopicsecond-order systems,” Proc. 41st IEEE Int. Conf. 65Decis. Control, (2002).

8. Chahlaoui, Y., D. Lemonnier, A. Vandendorpe, 67and P. V. Dooren, “Second-order balancedtruncation,” Linear Algebra Applic., Vol. 415, 69No. 2–3, pp. 373–384 (2006).

9. Henrion, D., M. Sebek, and V. Kucera, “Robust 71pole placement for second-order systems: an lmiapproach,” Proc. IFAC Symp. Rob. Control Design, 73L.-C. R. Report, Ed., July (2002).

10. Datta, B. and F. Rincon, “Feedback stabilization 75of the second-order model: A nonmodal approach,”Lin. Algebra. Applicat., Vol. 188, pp. 138–161 77(1993).

11. Ram, Y. M. and S. Elhay, “An inverse eigenvalue 79problem for the symmetric tridiagonal quadraticpencil with applications to damped oscillatory 81systems,” SIAM J. Appl. Math., Vol. 56, pp. 232–224 (1996). 83

12. Ram, Y. M. and S. Elhay, “Pole assignmnetin vibrating systems with multi-input control,” 85J. Sound Vibrat., Vol. 230, pp. 101–119, (2001).

13. Cichy, B. P., Augusta, E. Rogers, K. Gałkowski, and 87Z. Hurak, “On the control of distributed parametersystems using a multidimensional systems setting,”


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Mech. Syst. Sig. Proc., Vol. 22, pp. 1566–15811(2008).

14. Rogers, E. and D. H. Owens, Stability Analysis3for Linear Repetitive Processes, ser. Lecture Notesin Control and Information Sciences. Vol. 175,5Springer, (1992).

15. Rogers, E., K. Gałkowski, and D. H. Owens,7Control Systems Theory and Applications for LinearRepetitive Processes, Ser. Lecture Notes in Control9and Information Sciences. Vol. 349, Springer,Berlin, Germany, (2007).11

16. Amann, N., D. H. Owens, and E. Rogers,“Predictive optimal iterative learning control,” Int.13J. Control, Vol. 69, No. 2, pp. 203–226 (1998).

17. Roberts, P. D., “Numerical investigation of a15stability theorem arising from 2-dimensionalanalysis of an iterative optimal control algorithm,”17Multidimens. Syst. Signal Process., Vol. 11,No. (1–2), pp. 109–124 (2000).19

18. Paszke, W., Analysis and Synthesis ofMultidimensional System Classes Using Linear21Matrix Inequality Methods, ser. Lecture Notes inControl and Computer Science. Vol. 8, University of23Zielona Gora Press, Zielona Gora, Poland, (2005).

19. Xu, S. and J. Lam, Robust Control and Filtering25of Singular Systems, (Lecture Notes in Control andInformation Sciences). Vol. 332, Springer: (2008).27

20. Boyd, S., L. E. Ghaoui, E. Feron, and V.Balakrishnan, Linear Matrix Inequalities in System29and Control Theory, Ser. SIAM Studies in Appliedand Numerical Mathematics. Vol. 15, SIAM:31Philadelphia, U.S.A., (1994).

21. Khargonekar, P. P., I. R. Petersen, and K. Zhou,33“Robust stabilization of uncertain linear systemsQuadratic stabilizability and H∞ control theory,”35IEEE Trans. Autom. Control, Vol. 35, No. 3,pp. 356–361 (1990).37

22. Lam, J., Z. Shu, S. Xu, and E. Boukas,“Robust h-infinity control of descriptor discrete-39time markovian jump systems,” Int. J. Control,Vol. 80, No. 3, pp. 374–385 (2007).41

Pawel Dabkowski was born in43Torun, in 1982. He received theM.Sc. degree from the Nicolaus45Copernicus University of Torun,Poland in 2006. Currently he is47with the Institute of Physics inthe Nicolaus Copernicus Univer-49sity of Torun. His research inter-ests include the control theory and

multidimensional systems, especially the Linear Repet- 51itive Processes.

Krzysztof Gakowski received 53the M.S., Ph.D. and Habilita-tion (D.Sc.) degrees in elec- 55tronics/automatic control fromTechnical University of Wrocaw, 57Poland in 1972, 1977 and 1994respectively. In October 1996 he 59joined the Technical University ofZielona Gra (now the University 61of Zielona Gra), Poland where

he holds the professor position, and he is a visiting 63professor in the School of Electronics and ComputerScience, University of Southampton, U.K. In 2002, he 65was awarded the degree “Professor of Technical Sci-ence” the highest scientific degree in Poland. He spent 67academic year 2004-2005 and 2006-2007 in The Uni-versity of Wuppertal, Germany as and awardee of The 69Gerhard Mercator Guest Professor funded by DFG.He holds also the Professor position at The Nicolaus 71Copernicus University in Torun, Poland at The Depart-ment of Physics, Astarnomy and Computer Science. 73He is an associate editor of Int. J. of MultidimensionalSystems and Signal Processing, International Journal 75of Control and Int. J. of Applied Mathematics andComputer Science. He has co-organised four Inter- 77national Workshops NDS 1998, Lagov, Poland, NDS2000, Czocha Castle, Poland, the third as a Symposium 79within MTNS 2002 in Notre Dame IN, US, and thelast one NDS 2005 in Wuppertal, Germany. In 2007 81NDS 2007 has been organized in Aveiro, Portugal andin 2009 will be at Thessaloniki. In 2004, he obtained 83a Siemens Award for his research contributions. Hisresearch interests include multidimensional (nD) sys- 85tems and repetitive processes theory and applications,control and related numerical and symbolic algebra 87methods.

Biswa Nath Datta is a Distin- 89guished Research Professor atNorthern Illinois University. He 91is a professor of MathematicalSciences Department and an ad- 93junct professor of Electrical andMechanical Engineering Depart- 95ments at this University. He alsoheld visiting professorship at Uni- 97versity of Illinois at Urbana-

Champaign, Pennsylvania State University, Southern 99Illinois University, University of California at San


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Diego, State University of Campinas, Campinas, Brazil,1as well as many other universities and research or-ganizations around the world, including the Boeing3Company.

He is the author of more than 110 interdisciplinary5research papers blending linear and numerical linearalgebra with control theory and vibration engineering.7He also authored the books, Numerical Methods forLinear Control Systems-Design and Analysis, and the9associated software packages, MATLAB-based MAT-CONTROL, andMATCOM, andMATHEMTICA-based11Control Systems Professional- Advanced NumericalMethods. He was elected to a Fellow of IEEE in132000 for his interdisciplinary contributions. He hasserved on the editorial board of premier journals of15linear algebra, such as SIAM J. Matrix Analysis andApplications and Linear Algebra and its Applications17(Special Editor) and is currently serving on the ed-itorial board of about a dozen of mathematics and19engineering journals, including Numerical LinearAlgebra with Applications andMechanical Systems and21Signal Processing. Datta has served as the vice-chairof SIAM Linear Algebra Activity Group and organized23several successful interdisciplinary conferences spon-sored by American Mathematical Society and SIAM,25and a MTNS(Mathematical Theory of Networks andSystems) conference. He also co-edited several Pro-27ceedings books of these conferences.

Eric Rogers was born in 1956 29near Dungannon in Northern Ire-land He read Mechanical Engi- 31neering as an undergraduate inQueens University, Belfast, UK 33and was awarded his Ph.D. degreeby The University of Sheffield UK 35for a thesis in the area of mul-tidimensional systems theory. Re- 37cently he has been awarded the

D.Sc. degree by Queens University Belfast for research 39in nD systems theory and applications. He has been withThe University of Southampton UK since 1990 where 41he is currently Professor of Control Systems Theoryand Design in The School of Electronics and Computer 43Science. His current major research interests includemultidimensional systems theory and applications, with 45particular emphasis on behavioral systems theory ap-proaches and systems with repetitive dynamics, itera- 47tive learning control, flow control, and active controlof microvibrations. He is currently the editor of The 49International Journal of Control, an associate editor ofMultidimensional Systems and Signal Processing, and 51a member of the editorial board of Applied Mathemat-ics and Computer Science. In addition, he has served 53extensively on IEEE, IFAC and IEE technical commit-tees and acted as a consultant to numerous companies 55and government agencies in the UK and abroad.


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