International Journal of Control, Automation, and Systems (2011) 9(4):649-658 DOI 10.1007/s12555-011-0405-y
http://www.springer.com/12555
Robust Fuzzy Controller for Large-Scale Nonlinear Systems Using Decentralized Static Output-Feedback Geun Bum Koo, Jin Bae Park*, and Young Hoon Joo Abstract: This paper addresses a robust fuzzy control problem for an uncertain large-scale nonlinear system using decentralized static output-feedback scheme for both continuous-time and discrete-time cases. In both cases, sufficient design conditions are derived for robust asymptotic stabilization in terms of linear matrix inequalities (LMIs), and are therefore easily tractable by convex optimization. An illustrative example, a two-area power system with parametric uncertainties and the valve-position limit nonlinearity is provided to verify the effectiveness of the proposed technique. Keywords: Decentralized control, large-scale system, robust stability, static output-feedback, TakagiSugeno (T-S) fuzzy system.
1. INTRODUCTION High dimensionality, uncertainty, and nonlinearity are ubiquitous in practical control systems, such as power systems, communication networks, and economic models, which we categorize as an uncertain large-scale nonlinear system [1] that has gathered many research interests [2-9]. In the large-scale system, it has high dimensionality and information structure constraint problems to use traditionally centralized control technique. Thus the decentralized control framework is preferred to centralized control for stabilizing the largescale system. Similarly to the case of the centralized control, the state-feedback control method is not feasible in the largescale system. The large scale has been traditionally characterized by a large number of state variables. In the usual centralized state-feedback control framework, a control signal is to be manipulated in a single control unit, which requires every state information to be measured and transferred spaciously. However, due to the economical or physical constraints, measuring fullstate information may be costly or even impossible. To resolve the defect of the state-feedback technique, one may use an observer to estimate the unknown states. However, the observer-based output-feedback controller __________ Manuscript received March 31, 2010; revised November 11, 2010 and December 29, 2010; accepted March 6, 2011. Recommended by Editorial Board member Euntai Kim under the direction of Editor Young Il Lee. This work was supported by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 20104010100590). Geun Bum Koo and Jin Bae Park are with the School of Electrical and Electronic Engineering, Yonsei University, Seoul 120749, Korea (e-mails: {milbam, jbpark}@yonsei.ac.kr). Young Hoon Joo is with the Department of Control and Robotics Engineering, Kunsan National University, Kunsan, Chonbuk 573-701, Korea (e-mail:
[email protected]). * Corresponding author. © ICROS, KIEE and Springer 2011
not only requires more hardware resources, but also makes the dimension of the system increase greatly [9]. This provokes further dimensional complexity in design for large-scale systems. Differing from the schemes discussed above, the decentralized static output-feedback controller has lower dimension than that of the observerbased output-feedback and is also simpler to practically implement than the state-feedback, since it regards the large-scale system as a set of smaller subsystems interconnecting each other and uses only the subsystems' own output for feedback. Apart from the large scale issue, nonlinearity and uncertainties post additional difficulties to the control task. Various control approaches have been developed for a nonlinear system. Among them, the Takagi-Sugeno (T-S) fuzzy control technique is very popular today because it is highly regarded as a powerful resolution to bridge the gap between the fuzzy logic and the plentiful linear control theories [10]. A lot of achievements have been made for the T-S fuzzy-model-based control [3,10-17]. Furthermore, research endeavors are devoted to secure robustness against uncertainties in the T-S fuzzy system using statefeedback [11,12] and observer-based output-feedback [13]. However, they did not consider the static outputfeedback problem neither the decentralized control structure. In [10,14,15,18,21,22], a linear matrix inequality (LMI) approaches were presented for fuzzy static output-feedback. Decentralized fuzzy techniques were proposed in [6,8]. To the authors' best knowledge, the robust stabilization problem of uncertain large-scale T-S fuzzy systems using static output-feedback has not yet been studied in depth, thereby should be fully investigated. Motivated by the observations above, this paper aims at developing the robust decentralized fuzzy static output-feedback controller design technique for the uncertain large-scale nonlinear system for both continuous-time and discrete-time cases. Sufficient conditions for the existence of the controller to
Geun Bum Koo, Jin Bae Park, and Young Hoon Joo
650
asymptotically stabilize in the sense of Lyapunov in both cases are presented in terms of LMIs. The loadfrequency control (LFC) example of a two-area power system with parametric uncertainties and the valveposition limits nonlinearity is used for illustration of the proposed ideas, techniques, and procedures. This paper is organized as follows: Section 2. describes the large-scale T-S fuzzy system and the controller of interest. The robust decentralized fuzzy static output-feedback controller design conditions are proposed in Section 3. A numerical example is given for illustration in Section 4. Finally, the conclusions are given in Section 5. Throughout the paper, the subscripts i and j denote the fuzzy rule indices and the superscripts or subscripts k and l are used to indicate the subsystem. The
∑
notation
q l ≠k
means
∑
q l =1,l ≠ k
T
and (i)
denotes the
Cik and Aikl are nominal system matrices for the i th
rule in the k th subsystem, where Cik ∈ » l ×n is assumed to be full row rank and Aikl is the matrix representing the interconnection between the k th and the l th subsystems. ∆Aik , ∆Bik and ∆Aikl are the matrices expressing uncertainties with appropriate dimensions. Using the center-average defuzzification, product inference, and singleton fuzzifier, the input-output relation in the k th subsystem is represented as
+∑ l ≠k
r
(
Aikl
yk = ∑ θik Cik xk , i =1
where
(
)
xl ,
THEN uk = Kik yk .
The input-output form of the controller for the k th subsystem is then (3)
Substituting (3) into (2), the k th sub-closed-loop system is written as r
r
r
(2)
(( A + ∆A + ( B k i
k i
k i
)
)
+ ∆Bik K kj Chk xk
q + ∑ Aikl + ∆Aikl xl . l ≠k
(
)
(4)
The stabilization problem is formulated as follows: Problem 1: Find a gain matrix Kik stabilizing the closed-loop system (4) and satisfying the following design objectives. 1) The whole closed-loop system consisting of subclosed-loop system (4) satisfies robust asymptotic stability with the relaxed stable technique. 2) The sufficient condition for the stabilization contains not only continuous-time case, but also discrete-time case. 3. FUZZY DECENTRALIZED OUTPUT FEEDBACK CONTROLLER 3.1. Continuous-time controller design For solving Problem 1, hereafter, we assume that the uncertain matrices are norm-bounded and structured. Assumption 1: The uncertain matrices are represented as follow: ∆Aik
)
+ ∆Aik xk + Bik + ∆Bik uk
+ ∆Aikl
Rik : IF z1k is Γik1 and and z1k is Γik1 ,
(1)
r}, j ∈ P = {1, 2,… , p}, k ∈ Q = {1, 2,… , q}, Aik , Bik ,
q
The i th fuzzy rule for the decentralized static outputfeedback controller closing the k th subsystem (2) is formulated as follows:
i =1 j =1 h =1
where xk ∈ nk , uk ∈ » mk and yk ∈ »lk are the state, the control input, and the output for the k th subsystem, re-spectively. Γijk is a fuzzy set for z kj , i ∈ R = {1, 2,… ,
i =1
in Γijk .
xk = ∑∑∑θikθ kj θhk
xk = ( Aik + ∆Aik ) xk + ( Bik + ∆Bik )uk q THEN + ∑ ( Aikl + ∆Aikl ) xl l ≠k y = Ck x , i k k
)
in which Γijk ( z kj ) is the fuzzy membership grade of z kj
where Kik , (i, k ) ∈ R × Q is the feedback gain matrix.
Rik : IF z1k is Γik1 and and z1k is Γik1 ,
k i
j =1
i =1
Consider an uncertain large-scale T-S fuzzy system consisting of q subsystems, in which the i th IF--THEN rule in the k th subsystem is represented as follows:
(( A
i =1
n
ωik ( z k ) = ∏ Γijk ( z kj )
r
2. LARGE-SCALE T-S FUZZY SYSTEMS
r
r
∑ ωik ( z k ) ,
uk = ∑ θik K ik yk ,
transpose of the argument.
xk = ∑ θik
θik = ωik ( z k )
∆Bik
∆Aikl = Dik Fik Ei1k
Eik2
Eikl3 ,
where Dik , Eik1 , Eik2 , and Eikl3 are known real constant matrices of compatible dimensions, and Fi k is an unknown matrix function with Lebesgue-measurable elements and with ( Fi k )T Fi k ≤ I . Before proceeding, we recall the following matrix inequality which will be needed throughout the proofs.
Robust Fuzzy Controller for Large-Scale Nonlinear Systems Using Decentralized Static Output-Feedback
Lemma 1 [19]: Given constant matrices D and E, and a symmetric constant matrix S of appropriate dimensions, the following inequality holds: S + DFE + E T F T DT ≺ 0,
where F satisfies ( Fi k )T Fi k ≤ I , if and only if for some ε ∈ » >0 ε −1 E ε D ≺ 0. ε DT
S + ε −1 E T
cijkl , dijkl , Wi k , M k and somes scalars ε ik , such that the following LMIs are satisfied, then the whole continuoustime interconnected closed-loop system is robustly asymptotically stable in the presence of the parametric uncertainties: kl Ωiii + X iiikl Eiiikl k D i
(
)
(5)
X ikl11 X ikl1r 0, = ( X ikl )T = kl kl X ir1 X irr
Cik Q k
=M
k
Cik
∗ , −2Ql
kl dijh kl cijh
and εik = diag{ε ik I , ε ik I }, εijk = diag{ε ik I , ε kj I , ε ik I , ε kj I }, k εijh = diag{ε ik I , ε kj I , ε hk I , ε ik I , ε kj I , ε hk I }, (k,l) ∈ Q × ( Q \
{k}) and * denotes the transposed element in symmetric position. Proof: Consider the Lyapunov function candidate in the following form: q
V = ∑ Vk ( xk ), k =1
V is positive definite and radially unbounded. The time derivative of V along (4) is then q
(
k =1 q r
r
)
r
= ∑∑∑∑ θikθ kj θ hk k =1 i =1 j =1 h =1
T q k k k kl kl × Aijh + ∆Aijh xk + ∑ Ai + ∆Ai xl P xk l ≠k
(
(7)
)
(
(
)
(
∆Aijk = ∆Aik + ∆Bik K kj Chk .
We know that the following holds q
q
q
∑∑ xlT Pl xl = ( q − 1) ∑ xkT P k xk . k =1 l ≠ k
(11)
k =1
Let xkl = col{xk , xl } and P kl = diag{P k , P l }, and use (11), then (10) is reformulated as follows: q
q
r
r
r
V = ∑∑∑∑∑ θikθ kj θ hk
(
)
kl T kl kl × xklT (Φijh xkl ) P + P kl Φijh q
(9)
)
where Aijk = Aik + Bik K kj Chk ,
k =1 l ≠ k i =1 j =1 h =1
(8)
)
q k k + xkT P k Aijh + ∆Aijh xk + ∑ Aikl + ∆Aikl xl , l ≠k (10)
(6)
j = i + 1,… , r − 1,
h = j + 1,… , r , X ikl
kl aijh kl kl T = ( X ijh 0 , X ijh ) = kl bijh
V = ∑ xkT P k xk + xkT P k xk
∗ ≺ 0, −εijk
kl kl kl Ωijh + Ωihj + Ω kljih + Ω kljhi + Ω kl hij + Ω hji ∗ kl kl kl kl + X ijh + X ihj + X kljih + X kljhi + X hij + X hji kl kl E ijh + E ihj E kljih + E kljhi ≺ 0, kl kl E hij + Ehji k −εijh D ik D kj D hk
i = 1, 2,… , r − 2,
1 Aik Qk + BikW jk Chk + 2Qk kl Ωijh = q − 1 +Qk ( Aik )T + ( BikW jk Chk )T kl T ( Ai ) 1 kl E ijh = ( Eik1 Q k + Eik2 W jk Chk ) Eikl3 Ql , − q 1
where Vk(xk)= xkT P k xk , P k = ( P k )T 0, k ∈ Q . Clearly
∗ ≺ 0, i = 1, 2,… , r , k −εi
kl kl Ωiij + Ωiji + Ω kljii + X iijkl + X ijikl + X kljii Eiijkl + Eijikl E kljii D ik D kj i, j = 1, 2,… , r , j ≠ i,
where
D ikl = ε ik ( Dik )T
The stability condition of the closed-loop fuzzy interconnected (4) system is summarized in the following theorem. Theorem 1: If there exist some symmetric and positive definite matrices Qk, some matrices aijkl , bijkl ,
651
q
r
( )
= ∑∑∑ θik k =1 l ≠ k i =1
3
( )
kl xklT Φiii
T
kl P kl + P kl Φiii xkl
Geun Bum Koo, Jin Bae Park, and Young Hoon Joo
652 q
q
r
r
( )θ 2
+ ∑∑∑∑ θik k =1 l ≠ k i =1 j =1
(
k T kl jx
kl kl + P kl Φiij + Φiji + Φ kljii q
q r − 2 r −1
+ ∑∑ ∑
))
(
Φ kl + Φ kl + Φ kl iij iji jii
)
T
P kl
xkl
kl Ξiii
r
∑ ∑ θikθ kj θ hk
k =1 l ≠ k i =1 j = i +1 h = j +1
( (
kl kl kl × xklT Φijh + Φihj + Φ kljih + Φ kljhi + Φ kl hij + Φ hji kl kl kl + P kl Φijh + Φihj + Φ kljih + Φ kljhi + Φ kl hij + Φ hji
) P )) x , T
kl
kl
where kl Φijh =
1 k k k k k k q − 1 ( Ai + ∆Ai + ( Bi + ∆Bi ) K j Ch ) + I 0
Aikl + ∆Aikl . −I
If there exists X ikl = ( X ikl )T 0 such that the following inequalities are satisfied kl T kl kl (Φiii ) P + P kl Φiii + P kl X iiikl P kl ≺ 0, kl kl kl kl (Φiij + Φiji + Φ kljii )T P kl + P kl (Φiij + Φiji + Φ kljii )
+ P kl ( X iijkl + X ijikl + X kljii ) P kl ≺ 0,
(12) (13)
kl kl kl T kl + Φihj + Φ kljih + Φ kljhi + Φ kl (Φijh hij + Φ hji ) P kl kl kl + P kl (Φijh + Φihj + Φ kljih + Φ kljhi + Φ kl hij + Φ hji ) kl kl kl kl kl + P kl ( X ijh + X ihj + X kljih + X kljhi + X hij + X hji )P
≺ 0,
(14)
then (10) is majorized by q
q
ε ik > 0 such that
r
V ≤ −∑∑∑ xklT ΘTk Pˆ kl X ikl Pˆ kl Θ k xkl < 0, k =1 l ≠ k i =1
where
1 k k k k T q − 1 ( Ei1 + Ei2 Ki Ci ) + ( Eikl3 )T
1 ( Eik + Eik2 Kik Cik )T (ε ik )−1 I 0 q −1 1 × (ε ik )−1 I 0 ε ik ( Dik )T Pk ≺ 0.
i
Eikl3 0
Applying Schur complements results in kl Ξiii 1 ( E k + E k K k C k )T i2 i i q − 1 i1 ε ik ( Dik )T Pk ≺0
k − 1 (ε ) I 0 , i 0 (ε ik )−1 I ∗
Eikl3 0
(15)
which lacks the joint convexity in Pk and K ik . To recover the convexity, we take a congruence transformation with diag{( P k ) −1 , ( P l ) −1 , I , I } to (15), denote ( P k ) −1 =Qk, and import (9) with relation Kik M k = Wi k , to result in (5). Proving (6) and (7) is along a similar line above, so is omitted.
3.1. Paralleling to discrete-time case Similarly to the continuous-time case, sufficient conditions for robust asymptotic stabilizability of discrete-time T-S fuzzy interconnected systems are presented. The k th subsystem and its corresponding controller are described as follows: r
xk (t + 1) = ∑ θik i =1
Θ k = [θ1k , θ2k ,… , θrk ]T , θ k = diag{θ k ,… , θ k } ∈ 2 n×2 n , i
ε ik P k Dik 0
((G
k i
)
(
)
+ ∆Gik xk (t ) + H ik + ∆H ik uk (t )
q + ∑ Gikl + ∆Gikl xl (t ) , l ≠k
(
i
)
(16)
P kl = diag{P k , P l }, Pˆ kl = diag{P kl ,… , P kl } ∈ 2nr ×2 nr .
yk (t ) = ∑ θik Cik xk (t ),
(17)
First, applying Assumption 1 to (12) yields
uk (t ) = ∑ θik K ik yk (t ).
(18)
kl Ξiii
1 ( Aik + Bik K ik Cik ) + I = q −1 0
+ P kl X iiikl P kl
T
Aikl kl P − I
1 ( Aik + Bik K ik Cik ) + I + P q −1 0 kl
r
Aikl − I
which, based on Lemma 1, holds for all Fi k satisfying ( Fi k )T Fi k ≤ I if and only if there exists a constant
i =1 r
i =1
The sub-closed-loop system is then r
r
r
xk (t + 1) = ∑∑∑θikθ kj θhk i =1 j =1 h =1
((
(
)
)
× Gik + ∆Gik + Hik + ∆Hik K kj Chk xk (t ) (19) q +∑ Gikl + ∆Gikl xl (t ) , l ≠k
(
)
where t ∈ » ≥0 denotes the sequence index in discretetime.
Robust Fuzzy Controller for Large-Scale Nonlinear Systems Using Decentralized Static Output-Feedback
Theorem 2: If there exist some symmetric and positive definite matrices Qk, some matrices aijkl , bijkl , cijkl , dijkl , Wi k , M k and somes scalars ε ik , such that the
following LMIs are satisfied, then the whole discretetime interconnected closed-loop system is robustly asymptotically stable in the presence of the parametric uncertainties: Q kl + X iiikl ∗ ∗ Gˆ iiikl −Q k ≺ 0, i = 1, 2,… , r , Eˆiiikl −εik k ˆ Di Q kl + X iijkl + X ijikl + X kljii 1 ˆ kl ˆ kl ˆ kl 3 Giij + Giji + G jii Eˆ iijkl + Eˆ ijikl Eˆ kljii Dˆ ik Dˆ kj i, j = 1, 2,… , r , j ≠ i,
(
)
∗ k −Q
(20)
1 α − Qk − q − 1 36 Q kl = 0
and α is any scalar with 0 < α
