Decentralized Control of Nonlinear Large-Scale ... - Semantic Scholar

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 104, No. 2, pp. 459–475, FEBRUARY 2000

Decentralized Control of Nonlinear Large-Scale Systems Using Dynamic Output Feedback1 X. G. YAN,2 J. LAM,3 H. S. LI,4

AND

I. M. CHEN4

Communicated by F. L. Chernousko

Abstract. In this paper, a class of nonlinear large-scale systems with similar subsystems is studied. Both matched and unmatched uncertainties are considered by utilizing their bounding functions, and the interconnections take a more general form than considered previously. Based on a constrained Lyapunov equation, a nonlinear dynamic output feedback decentralized controller is presented. Unlike existing results, matched uncertainties are considered in the control design; by using a decomposition of the interconnections, the known and uncertain interconnections are treated separately; thus, the robustness is improved and conservativeness is reduced significantly. The computation effort for solving the Lyapunov equation is greatly reduced by taking into account the similar subsystem structure. Finally, simulation is used to illustrate the effectiveness of our results. Key Words. Stabilization, nonlinear large-scale systems, decentralized control, dynamic output feedback.

1. Introduction Nonlinear large-scale systems are difficult to control due to various reasons, such as lack of centralized computing capability, system nonlinearity, interconnection of subsystems, and system uncertainty. Some of 1

This project was partially supported by Hong Kong University Grant CRCG 337y064y0063 and was completed while the first author was with the University of Hong Kong. 2 Research Fellow, School of Mechanical and Production Engineering, Nanyang Technological University, Republic of Singapore. 3 Associate Professor, Department of Mechanical Engineering, University of Hong Kong, Pokfulam Road, Hong Kong, PRC. 4 Assistant Professor, School of Mechanical and Production Engineering, Nanyang Technological University, Republic of Singapore.

459 0022-3239y00y0200-0459$18.00y0  2000 Plenum Publishing Corporation

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the difficulties associated with a centralized control scheme can be alleviated via a decentralized control structure in which information transfer between subsystems is avoided. Therefore, decentralized control is considered as an effective method to deal with large-scale interconnected systems. In addition, it is often useful to utilize the system structural characteristics, such as symmetric structure (Ref. 1), cascaded structure (Ref. 2), or similar structure (Ref. 3) to study special large-scale systems as a first step toward general large-scale systems. In recent years, attention has been devoted to the stabilization of nonlinear large-scale systems, and fruitful results have been obtained; see e.g. Refs. 4 and 5. However, these works are all based on the assumption that full state information of each subsystem is available. Although such assumption is not required for output feedback control, stringent conditions may have to be imposed on large-scale systems (Refs. 3 and 5–7). Moreover, there are sometimes great differences in the control of nonlinear systems using true state feedback and estimated state feedback, even with the same control scheme. For example, consider the nonlinear system x˙ G−2(tC2)−2Cx2Cxu,

x(0)G−1.

(1)

It is obvious that xˆ Gx(t)A1y(tC2) may be considered as an estimated state of the system (1) which asymptotically tracks the true state. Then, it is easy to observe that the closed-loop system is stable using the state feedback control uG−x, but unstable using the estimated state uG−xˆ. Indeed, the solution of the former is xG2(tC2)−1, while that of the latter is xG(tC2)[1y(tC2)2A3y4]. Consequently, it becomes important and valuable to establish an observer to estimate the system state which can be used reliably to stabilize the system. It is well known that many results related to observer-based stabilization for nonlinear systems and linear large-scale systems have been obtained. However, the corresponding results for nonlinear large-scale systems are very scarce, and relatively few results are available in the literature (Refs. 4, 8, 9). Chen et al. (Ref. 4) and Wang et al. (Ref. 9) have studied a class of nonlinear time-delay large-scale systems with linear interconnections and linear isolated subsystems subject to nonlinear disturbances bounded by the norm of the state vector. A severe restriction on the nominal subsystem is that it is output feedback stabilizable; specifically, practical stability

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instead of asymptotic stability is considered by Chen in Ref. 8. Moreover, the interconnection is considered as a disturbance bounded by the norm of the state variables. In this paper, nonlinear large-scale interconnected systems are considered. Both matched and unmatched uncertainties are considered. No statistical information about the uncertainties is imposed; only their bounds are assumed to be known. The bounding functions of the uncertain interconnections take more general forms than considered previously. Under certain conditions, the system considered is transformed first into a largescale interconnected system with linear nominal subsystems by exploiting the characteristics of the nonlinear system. Then, a dynamic output feedback control scheme is proposed in which the bounding functions of the matched uncertainties are exploited. Using a decomposition technique, the general known interconnections are treated separately from the uncertain interconnections. This is in contrast with existing results where all interconnections are treated as disturbances. Moreover, nonlinearities in the system are also used in analysis and synthesis. For these reasons, the robustness of the associated closed-loop systems is greatly enhanced. This paper is organized as follows. In Section 2, we provide notation and preliminaries for analyzing nonlinear systems. In Section 3, large-scale systems with similar structure are introduced and certain basic assumptions are imposed. In Section 4, a dynamic output feedback control is presented. In Section 5, a numerical example is used to illustrate our method. Finally, Section 6 concludes the paper.

2. Notation and Preliminaries For a given matrix A, let AT denote its transpose; let σ¯ (A) denote the maximum singular value of A ; when A is real symmetric, let λq (A) and λr (A) denote its minimum and maximum eigenvalues, respectively; let AH0 denote that A is positive definite. Let L F (x) denote the Lipschitz constant of the function F(x) in its domain of definition; also, let uu·uu denote the Euclidean norm or its induced norm. Consider the following two systems: (ΣI)

x˙ GfI (x)CgI (x)u, yI GhI (x),

(ΣII) ˙xˆGfII (xˆ)CgII (xˆ)u, yII GhII (xˆ),

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where x, xˆ ∈R n, yI , yII ∈R l, and u∈R m are the state vectors, outputs, and inputs of the systems ΣI and ΣII , respectively. Definition 2.1. ΣI is said to be similar to ΣII in the domain E if there exists a diffeomorphism T: x > xˆ defined in E such that in the coordinate xˆ defined by T, the system ΣI possesses the same form as ΣII . In this case, T(x) is called a similarity transformation from ΣI to ΣII . Remark 2.1. Similarity between systems is an equivalence relationship. That is, similarity possesses the properties of reflectivity, symmetry, and transitivity; it is an extension of equivalence between linear systems. Definition 2.2. The system ΣI is said to be output feedback linearizable in the domain E if there exist a diffeomorphism T: x > z, α ( y)∈R m, and a nonsingular matrix β ( y)∈R mBm such that in the coordinate z defined by T, the closed-loop system resulting from the input uGα ( y)Cβ ( y)v to ΣI is described by z˙ GAzCBv, yGCz, with the realization (A, B, C) both controllable and observable. Remark 2.2. It should be noticed that output feedback linearizability defined above does not imply static output feedback stabilizability. However, the fact that a system is output feedback linearizable implies that it is state feedback stabilizable. It should be mentioned that static output feedback stabilizability is a very strong condition (see e.g. Refs. 3, 7, 10) and it remains an open problem even for linear systems (Ref. 10). Lemma 2.1. Suppose that ΣI is similar to ΣII in the domain E. Then, ΣI is output feedback linearizable if and only if ΣII is output feedback linearizable. Proof. Necessity. By output feedback linearization of ΣI , it follows that there exist a similarity transformation T1 : x > z, α ( y)∈R m, and a nonsingular matrix β ( y)∈R mBm such that in the coordinate z defined by T1 , the closed-loop system x˙ GfI (x)CgI (x)[α ( yI )Cβ ( yI )v],

(2)

yI GhI (x),

(3)

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has the following form: z˙ GAzCBv,

(4)

yGCz.

(5)

It is obvious that the system (2)–(3) is similar to (4)–(5) with similarity transformation T1 . Now, suppose that ΣI is similar to ΣII with similarity transformation T2 . Then, it is observed that the system ˙xˆGfII (xˆ)CgII (xˆ)[α ( yÎI)Cβ ( yÎI)v],

(6)

yII GhII(xˆ),

(7)

is similar to (2)–(3) with similarity transformation T −1 2 . By the properties of similarity, it follows that (6)–(7) is similar to (4)–(5) with similarity transformation T −1 2 ° T1 . Therefore, for ΣII , there exists a composition of transformations given by T −1 2 ° T1 and output feedback uGα ( yII)Cβ ( yII)v such that the resulting closed-loop system is linearizable. Sufficiency. It may be obtained directly from the necessity proof and the symmetry property of the similarity transformation. Hence, the result follows. h

3. System Description Consider a nonlinear large-scale interconnected system described by N

x˙i Gfi (xi )Cgi (xi )[uiC∆Ψi (xi )]C ∑ Hij (xj )C∆Hi (x),

(8)

yi Ghi (xi ),

(9)

j G1 j≠i

iG1, 2, . . . , N.

Here xi ∈Ωi ⊂R n (Ωi is a neighborhood of xi G0), ui , and yi ∈R m are the state vector, input, and output vector of the ith subsystem, respectively; fi (xi ), gi (xi ) are both smooth vectors, hi is a smooth function in Ωi , ∆Ψi (xi ) N is the matched uncertainty of the ith isolated subsystem; ∑j G1 Hij (xj ), jGi, is the known interconnection, the uncertain interconnection ∆Hi (x) includes all unmatched uncertainties, and they are all continuous in their arguments. Without loss of generality, it is supposed that fi (0)G0 and hi (0)G0. Also,

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we write xGcol (x1 , x2 , . . . , xN )∈Ω1BΩ2B· · ·BΩN GΩ. Definition 3.1. Consider the system (8)–(9). The systems x˙i Gfi (xi )Cgi (xi )ui

(10)

yi Ghi (xi ),

(11)

iG1, 2, . . . , N,

are called nominal subsystems of the system (8)–(9); the systems x˙i Gfi (xi )Cgi (xi )[uiC∆Ψi (xi )],

(12)

yi Ghi (xi ),

(13)

iG1, 2, . . . , N,

are called isolated subsystems of the system (8)–(9). Definition 3.2. The system (8)–(9) is said to be a similar interconnected large-scale system or to possess a similar structure if all its nominal subsystems are similar to one another. Remark 3.1. It should be noticed from Remark 2.1 that there exists one system such that each nominal subsystem of the system (8)–(9) is similar to the system, if (8)–(9) is a similar interconnected large-scale system. The similar large-scale system introduced here is an extension of the systems dealt with in Refs. 1, 3, 11, 12. Assumption A1. The system (8)–(9) possesses a similar structure, and there exists one output feedback linearizable nominal subsystem. It should be noticed from Definition 2.2 and Remark 2.2 that Assumption A1 does not imply output feedback linearizability of the nominal subsystem (10)–(11) of the system (8)–(9). In fact, we do not require that the nominal subsystem (10)–(11) of the system (8)–(9) is output feedback stabilizable in this paper; on the other hand, the nominal subsystem is required to be linear and output feedback stabilizable in Ref. 8. By Lemma 2.1, it is observed that, under Assumption A1, all nominal subsystems of the system (8)–(9) are output feedback linearizable, and there exists a controllable and observable linear system to which all nominal subsystems of (8)–(9) are similar. Therefore, there exist diffeomorphisms Ti : xi > zi and output feedback ui Gα i ( yi )Cβ i ( yi )vi

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in Ωi , for iG1, 2, . . . , N, such that, in the new coordinate zG col (z1 , z2 , . . . , zN ), the system (8)–(9) is described by N

z˙i GAziCB[viC∆Φi (zi )]C ∑ Mij (zi , zj )C∆Mi (z),

(15)

yi GCzi ,

(16)

j G1 j≠i

iG1, 2, . . . , N,

where the realization (A, B, C) is controllable and observable, and where ∆Φi (zi )G[ β −1 i ( yi )∆Ψi (xi )]xi GT −1 i (zi ) ,

(17)

−1 (z)), ∆Mi (z)G[∂Ti (xi )y∂xi ]xi GT −1 i (zi) · ∆Hi (T

(18)

−1 j

Mij (zi , zj )G[∂Ti (xi )y∂xi ]xi GT −1 (zj )), i (zi ) · Hij (T

(19)

j≠i,

with i, jG1, 2, . . . , N and zGT(x)Gcol (T1(x1), T2(x2), . . . , TN (xN )). Remark 3.2. It should be noted that nonlinear geometric theory (see e.g. Refs. 13–14) may be useful for obtaining Ti , α i (·), β i (·), although there is no general method available. Specifically, they may be derived from a single ui and single yi system with minimum phase by the method proposed in Ref. 14. From fi (0)G0 and hi (0)G0, it is observed that Ti (·) may be chosen such that Ti (0)G0 for iG1, 2, . . . , N. In view of Remark 3.2, all diffeomorphisms dealt with in this paper are those which transform origin to origin. Assumption A2. Mij (zi , zj ), j≠i, is Lipschitz in Ti (Ωi )BTj (Ωj ) with Lipschitz constants L iMij and L jMij . That is, for any zi , z˜i ∈Ti (Ωi ) and zj , z˜j ∈ Tj (Ωj ), uuMij (zi , zj )AMij (z˜i , z˜j )uuYL

i Mij

uuziAz˜i uuCL

j Mij

uuzjAz˜j uu,

j≠i,

and Mij (zi , zj ) has the following decomposition: Mij (zi , zj )GΠij ( yi , yj )zj , with Πij ∈R

nBn

j≠i,

(20)

for all i, jG1, 2, . . . , N.

Assumption A3. There exist known continuous functions ρi (·) and γ i (·), defined in their domains of definition, such that, for iG1, 2, . . . , N, (i)

uu∆Φi (zi )uuYρi (uuyi uu)uuyi uu,

(ii) uu∆Mi (z)uuYγ i ( y)uuzuu.

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4. Dynamic Output Feedback Controller Design Consider the system (15)–(16). From the controllability and observability of the realization (A, B, C), it follows that there exist K, L such that, for any QH0 and SH0, the following Lyapunov equations: (AABK)TPCP(AABK)G−Q,

(21)

T

(22)

(AALC) RCR(AALC)G−S

have unique solutions PH0 and RH0, respectively. Before the main results are shown, the following assumption is imposed as in Refs. 3, 15, 16.

Assumption A4. There exists matrix F such that B TPGFC, with P defined by (21).

Consider the system (8)–(9). Construct the controller described by ˙xî Gfi (xî )Cgi (xî )ui N

C[∂Ti (xî )y∂xî ]−1L[ yiAhi (xî )]C ∑ Hij (xˆj ),

(23)

j G1 j≠i

yi Ghi (xi ),

(24)

ui Gα i ( yi )Cβ i ( yi )[−KTi (xî )Cη i ( yi )],

iG1, 2, . . . , N,

(25)

where K satisfies (21) and η i (·) is defined by, for iG1, 2, . . . , N,

η i ( yi )G

5

A[FyiyuuFyi uu] ρi (uuyi uu)uuyi uu, 0,

Fyi ≠0, Fyi G0,

(26)

with ρi (uuyi uu), for iG1, 2, . . . , N, and F defined respectively by Assumptions A3 and A4. Now, we have the following result.

Theorem 4.1. Under Assumptions A1–A4, the system (8)–(9) is stabilized by the controller (23)–(25) if there exists a neighborhood about the origin Ω ′⊆Ω such that W TCWH0 in Ω′\{0}, where WG[wij ]2NB2N is

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defined by wij Gλq (Q)A2λr (P)γ i ( y), wij Gλq (S )A2λr (R)

1YiYN, iGj,

2N

∑

k GNC1,k ≠ i

L

iAN M (iAN)(kAN)

,

NC1YiY2N, iGj,

wij GA2(σ¯ (PΠij ( yi , yj ))Cλr (P)γ i ( y)),

1Yi, jYN, j≠i,

wij GA2λr (R)L

NC1Yi, jY2N, j≠i,

jAN M (iAN)( jAN)

,

wij GAσ¯ (RB)σ¯ (C)ρi (uuyi uu) Aσ¯ (PBK)Aλr (R)γ i ( y),

jAiGN, 1YiYN,

wij GAσ¯ (RB)σ¯ (C)ρj (uuyj uu) Aσ¯ (PBK)Aλr (R)γ j ( y),

iAjGN, 1YjYN,

wij GAλr (R)γ jAN ( y), wij GAλr (R)γ iAN ( y),

jAi≠N, 1YiYN, NYjY2N, jAi≠N, 1YjYN, NYiY2N,

where P, Q, R, S are defined by (21)–(22). Proof. It is obvious that the closed-loop system obtained by applying the controller (23)–(25) to the system (8)–(9) is described by x˙i Gfi (xi )Cgi (xi ){α i ( yi )Cβ i ( yi )[−KTi (xî )Cη i ( yi )]C∆ψi (xi )} N

C ∑ Hij (xj )C∆Hi (x),

(27)

j G1 j≠i

˙xî Gfi (xî )Cgi (xî ){α i ( yi )Cβ i ( yi )[−KTi (xî )Cη i ( yi )]} N

C[∂Ti (xî )y∂xî ]−1L[ yiAhi (xî )]C ∑ Hij (xˆj ),

(28)

j G1 j≠i

yi Ghi (xi ),

iG1, 2, . . . , N,

(29)

where K, L are defined by (21)–(22) and η i (·) is defined by (26). For the system (27)–(29), we construct a Lyapunov function candidate as N

VG ∑ {(Ti (xi ))TPTi (xi )C[Ti (xi )ATi (xî )] TR[Ti (xi )ATi (xî )]},

(30)

i G1

where P, S are defined by (21)–(22). It follows from Assumption A1 that, in the new coordinates z, zˆ, system (27)–(29) is described by z˙i GAziABKzîCB[η i ( yi )C∆Φi (zi )] N

C ∑ Mij (zi , zj )C∆Mi (z), j G1 j≠i

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JOTA : VOL. 104, NO. 2, FEBRUARY 2000 N

˙zî G(AABK)zîCBη i ( yi )CL( yiACzî )C ∑ Mij (zî , zˆj ),

(32)

yi GCzi ,

(33)

j G1 j≠i

iG1, 2, . . . , N.

where the realization (A, B, C) is the same as (15)–(16), and where ∆Φi , ∆Mi , Mij are defined by (17)–(19). Let ei _Ti (xi )ATi (xî )GziAzî ,

for iG1, 2, . . . , N.

We have, for iG1, 2, . . . , N,

3e˙ 4 G30 z˙i

AABK

i

43 4 3

4

∑j G1, j ≠ iMij (zi , zi )C∆Mi (z) . N ∑j G1, j ≠ i [Mij (zi , zi )AMij (zî , zˆj )]C∆Mi (z) N

3

C

BK zi B[∆Φi (zi )Cη ( yi )] C AALC ei B∆Φi (zi )

4

(34)

The time derivative of V along the trajectories of the system (27)–(29) is given by N

V˙ u(27)–(29) GA ∑ [ziT , eiT ] diag(Q, S ) i G1

B

3 e4 zi

N

C2 ∑ (ziT , eiT) diag(P, R) i G1

i

B[∆Φi (zi )Cη i ( yi )CKei ]

53 B∆Φ (z ) i

i

4

Mij (zi , zi )C∆Mi (z) ∑ C Nj G1, j ≠ i . ∑j G1, j ≠ i [Mij (zi , zi )AMij (zî , zˆj )]C∆Mi (z)

3

N

46

By the structure (26) of η i ( yi ) and Assumption A4, it is observed that: (i)

if Fyi G0, then for iG1, 2, . . . , N, ziTPB[∆Φi (zi )Cη i ( yi )]G(FCzi ) T∆Φi (zi )G(Fyi ) T∆Φi ( yi )G0;

(ii)

if Fyi ≠0, then for iG1, 2, . . . , N, ziTPB[∆Φi (zi )Cη i ( yi )] YuuziT (FC) T uu ρi (uuyi uu)uuyi uuA[ziT (FC)TFyi yuuFyi uu] ρi (uuyi uu)uuyi uu YuuFCzi uu ρi (uuyi uu)uuyi uuA[(FCzi )TFCzi yuuFCzi uu] ρi (uuyi uu)uuyi uu G0.

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Therefore, for iG1, 2, . . . , N, ziTPB[∆Φi (zi )Cη i ( yi )]Y0.

(36)

Then, by Assumption A3, it follows that, for iG1, 2, . . . , N, eiTRB∆Φi (zi )CziTPBKei Yuuei uuuuRBuu ρi (uuyi uu)uuyi uuCuuziTPBKei uu Y[ ρi (uuyi uu)σ¯ (RB)σ¯ (C)Cσ¯ (PBK)]uuzi uuuuei uu.

(37)

From Assumptions A2–A3 and the fact that uuzuuYuuz1uuCuuz2 uuC· · ·CuuzN uu, it is observed that

3

4

N

ziT P ∑ Mij (zi , zj )C∆Mi (z) j G1 j≠i

N

G ∑ ziT PΠij ( yi , yj )zjCziTP∆Mi (z) j G1 j≠i N

Y ∑ σ¯ (PΠij ( yi , yj ))uuzi uuuuzj uuCλr (P)γ i ( y)uuzi uuuuzuu j G1 j≠i N

N

j G1 j≠i

j G1

Y ∑ σ¯ (PΠij )uuzi uuuuzj uuC ∑ λr (P)γ i ( y)uuzi uuuuzj uu. Also, Assumptions A2–A3 give

3

4

N

eiTR ∑ [Mij (zi , zi )AMij (zî , zˆj )]C∆Mi (z) j G1 j≠i

N

Y ∑ λr (R)uuei uuuuMij (zi , zj )AMij (zî , zˆj )uuCλr (R)γ i ( y)uuei uuuuzuu j G1 j≠i N

Y ∑ λr (R)uuei uu[LMi ij uuziAzî uu j G1 j≠i

N

CLMj ij uuzjAzˆj uu]C ∑ λr (R)γ i ( y)uuei uuuuzj uu j G1

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JOTA : VOL. 104, NO. 2, FEBRUARY 2000 N

G ∑ λr (R)[LMi ij uuei uu2CLMj ij uuei uuuuej uu] j G1 j≠i

N

C ∑ λr (R)γ i ( y)uuei uuuuzj uu j G1

3

N

Gλr (R) C ∑

LMi

k G1 k≠i

ik

4

N

uuei uu2Cλr (R) ∑

LMj

ij

uuei uuuuej uu

j G1 j≠i

N

C ∑ λr (R)γ i ( y)uuei uuuuzj uu.

(39)

j G1

Substituting (36)–(39) into (35) yields V˙ u(27)–(29) N

YA ∑ [λq (Q)uuzi uu2Cλq (S )uuei uu2] i G1

N

C2 ∑ [ ρi (uuyi uu)σ¯ (RB)σ¯ (C)Cσ¯ (PBK)]uuzi uuuuei uu i G1 N

C2 ∑

5

6

N

2 ∑ [σ¯ (PΠij ( yi , yj ))Cλr (P)γ i ( y)]uuzi uuuuzj uuCλr (P)γ i ( y)uuzi uu

i G1 j G1 j≠i

3 4

N

N

C2 ∑ λr (R) ∑ i G1 N

LMi

ik

k G1 k≠i

N

N

uuei uu2C2 ∑ ∑ λr (R)LMj ij uuei uuuuej uu i G1 j G1 j≠i

N

C2 ∑ ∑ λr (R)γ j ( y)uuzi uuuuej uu i G1 j G1

GA(1y2)Y T (W TCW)Y,

(40)

where YG(uuz1uu, uuz2uu, . . . , uuzN uu, uue1uu, uue2uu, . . . , uueN uu) T. It is obvious that YG0 if and only if xi Gxî G0 for iG1, 2, . . . , N. Then, from the positive-definiteness of W TCW, it follows that the system (27)– (29) is asymptotically stable at origin. Hence, the result follows. h It should be emphasized that the proof of Theorem 4.1 is constructive. The robustness is enhanced greatly compared with existing results in which all the uncertainties are estimated or not considered in the control design.

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In addition, from the continuity of ρi (·), it is observed that ui is continuous if F is nonsingular. From the proof above, it should be emphasized that Assumption A4 is unnecessary if the matched uncertainties ∆Ψi , with iG 1, 2, . . . , N, do not appear in (8). Corollary 4.1. Under the conditions of Theorem 4.1, (23) is an asymptotic observer of the system (8)–(9). In other words, under certain conditions, the system (8)–(9) is stabilized by the controller (25), based on the system output and estimated states given by (23).

5. Example In this section, an example is used to illustrate the proposed control scheme. Consider the nonlinear interconnected system given by

3

x˙1 G

A2x11 x11A6x311C[x11C4(x12Ax311)]2y100

31C[x

C

0

C4(x12Ax311)]2y100

11

C(1y4)

3 A2x

x˙2 G

3

4[u C∆Ψ (x )] 1

1

x22Cx321 C∆H1(x), 3x211 (x22Cx321)

4

A4x21

C

4 1

(41)

4C3 A3x 4[u C∆Ψ (x )] 1

C10x

22

3 21

2 21

3(1y4)x 4C∆H (x), 0

2

2

2

2

(42)

11

y1 G0.1(x11C4x12A4x311),

(43)

y2 G0.1(x321C4x21Cx22),

(44)

where xi Gcol (xi1 , xi2),

for iG1, 2,

xGcol (x1 , x2).

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472


Suppose that the uncertainty bounds are known with u∆Ψ1(x1)uYy21y(1Cy21), u∆Ψ2(x2)uYuy2 u, uu∆H1(x)uuY(1y4)uy1u[(x211Cx221)y(9x411C2)]1/2, uu∆H2(x)uuY(1y4)uy1u[(x211Cx221)y(9x421C2)]1/2. Introduce the following global transformation of the system (41)–(44): (T1) z11 Gx11 , z12 Gx12Ax311 , (T2) z21 Gx22Cx321 , z22 Gx21 , and output feedback controls u1 G−y21y(1Cy21)C[1y(1Cy21)]v1 , u2 G10y2Cv2 . It follows that, in the new coordinate z, the closed-loop system is described by

3

A2 0 0 1y4 0 z1C [v1C∆Φ1(z1)]C z2C∆M1(z), 1 0 1 0 0

3

A2 0 0 1y4 0 z2C [v2C∆Φ2(z2)]C z1C∆M2(z), 1 0 1 0 0

z˙1 G z˙2 G

4 34

3

4

(45)

4 34

3

4

(46)

y1 G[0.1, 0.4]z1 ,

(47)

y2 G[0.1, 0.4]z2 ,

(48)

where ∆Φ1(z1)G(1Cy21)∆Ψ1(x1), ∆Φ2(z2)G∆Ψ2(z2), ∆M1(z)G ∆M2(z)G

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3 A3x 1

2 11

3

4

0 ∆H1(x), 1

3x221 1 ∆H2(x), 1 0

4

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473


and their bounds are estimated respectively as follows: uΦ1(z1)uYuy1u2, uΦ2(z2)uYuy2u, uu∆M1(z)uuY(1y4)uy1uuuzuu, uu∆M2(z)uuY(1y4)uy1uuuzuu. Next, choose

3 54 ,

KG[0, 2],

LG

0

QGSG8I.

Computing directly, it follows that PG

3

3 0.25

4

2.25 0.5 , 0.5 2

30

Π12 GΠ21 G

4

1y4 0 , 0

i Mij

G0, i, jG1, 2, i≠j, ρ2(uuy2uu)G1, FG5,

L

4

RG

2.0625 0.25 , 2

LMj

G1y4,

ij

ρ1(uuy1uu)Guy1u, γ 1( y)Gγ 2( y)G(1y4)uy1u.

It is shown that Theorem 4.1 is satisfied in Ω′G{(x11 , x12 , x21 , x22)uux11C4x12A4x311 uY7.7, (x21 , x22)∈R2}. Therefore, the control to stabilize the system (41)–(44) is described by

3

˙xˆ1 G

A2xˆ11 xˆ11A6xˆ311C[xˆ11C4(xˆ12Axˆ311)]2y100

31C[xˆ 0

C

C4(xˆ12Axˆ311)]2y100

11

4u

4

1

0 xˆ22Cxˆ321 [y1A0.1(xˆ11C4xˆ12A4xˆ311)]C(1y4) , 3xˆ211 (xˆ22Cxˆ321) 5

34

C

3 A2xˆ

˙xˆ2 G

3

A4xˆ21

4C3 A3xˆ 4 u

(49)

1

C10xˆ

22

3 21

2 21

3A15xˆ 4[ y A0.1(xˆ

C

4

5

2 21

2

22

2

Cxˆ321C4xˆ21)]C(1y4)

3 xˆ 4 , 0

(50)

11

u1 G−[1y(1Cy21)][ y21C2xˆ12A2xˆ311Cη 1( y1)],

(51)

u2 G10y2A2xˆ21Cη 2( y2),

(52)

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Fig 1. Evolution of the state variables of the closed-loop system.

where

η 1( y1)G−y1 uy1u,

η 2( y2)G−y2 .

The simulation with initial state x0 G(2, 6, A1, 4) is shown by Fig. 1.

6. Conclusions We have presented a dynamic output feedback control scheme to stabilize a class of nonlinear interconnected systems. The systems considered correspond to a generalization of those studied previously on decentralized stabilization based on estimated states. Moreover, the conservativeness of the results is also reduced. The role of coordinate transformation is clearly shown in our analyses. We mention that our method may be extended to the case where the dimensions of each subsystem are different by introducing a new similar structure. Last, but not least, it is demonstrated that the system structure plays an important role in reducing the computation effect for the Lyapunov equation.

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475

References 1. LAM, J., and YANG, G. H., Balanced Model Reduction of Symmetric Composite Systems, International Journal of Control, Vol. 65, pp. 1031–1043,1996. 2. QU, Z., and DAWSON, D. M., Robust Control of Cascaded and Individually Feedback Linearizable Nonlinear Systems, Automatica, Vol. 30, pp. 1057–1064, 1994. 3. YAN, X. G., WANG, J. J., LU, X. Y., and ZHANG, S. Y., Decentralized Output Feedback Robust Stabilization for a Class of Nonlinear Interconnected Systems with Similarity, IEEE Transactions on Automatic Control, Vol. 43, pp. 294– 299, 1998. 4. CHEN, Y. H., WANG, W. J., and MAU, L. H., Robust Stabilization of LargeScale Time-Delay Systems with Estimated State Feedback, Journal of Optimum Theory and Application, Vol. 89, pp. 543–559, 1996. 5. JAIN, S., and KHORRAMI, F., Decentralized Adaptive Output Feedback Design for Large-Scale Nonlinear Systems, IEEE Transactions on Automatic Control, Vol. 42, pp. 729–735, 1997. 6. MAHMOUD, M. S., Stabilizing Control for a Class of Uncertain Interconnected Systems, IEEE Transactions on Automatic Control, Vol. 39, pp. 2484–2488, 1994. 7. YAN, X. G., and DAI, G. Z., Decentralized Output Feedback Robust Control for Nonlinear Large-Scale Systems, Automatica, Vol. 34, pp. 1469–1472, 1998. 8. CHEN, Y. H., Decentralized Robust Output and Estimated State Feedback Controls for Large-Scale Systems, International Journal of Control, Vol. 46, pp. 1979–1992, 1987. 9. WANG, W. J., and MAU, L. G., Stabilization and Estimation for Perturbed Discrete Time-Delay Large-Scale Systems, IEEE Transactions on Automatic Control, Vol. 42, pp. 1277–1282, 1997. 10. SYRMOS, V. L., ABDALLAH, C. T., DORATO, P., and GRIGORIADIS, K., Static Output Feedback: A Survey, Automatica, Vol. 33, pp. 125–137, 1997. 11. SUNDARESHAN, M. K., and ELBANNA, R. M., Qualitative Analysis and Decentralized Controller Synthesis for a Class of Large-Scale Systems with Symmetrically Interconnected Subsystems, Automatica, Vol. 27, pp. 383–388, 1991. 12. YAN, X. G., LAM, J., and DAI, G. Z., Decentralized Robust Control for Nonlinear Similar Large-Scale Systems, Computer and Electrical Engineering, Vol. 25, pp. 169–179, 1999. 13. ISIDORI, A., Nonlinear Control Systems, 3rd Edition, Springer Verlag, London, Great Britain, 1995. 14. MARINO, R., and TOMEI, P., Nonlinear Control Design, Prentice Hall International, Englewood Cliffs, New Jersey, 1995. 15. ZAK, S. H., On the Stabilization and Observation of NonlinearyUncertain Dynamic Systems, IEEE Transactions on Automatic Control, Vol. 35, pp. 604– 607, 1990. 16. CHENG, C. F., Output Feedback Stabilization for Uncertain Systems: Constrained Riccati Approach. IEEE Transactions on Automatic Control, Vol. 43, pp. 81–84, 1998.

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