Decentralized Fuzzy Observer-Based Output-Feedback Control for

406

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

Decentralized Fuzzy Observer-Based Output-Feedback Control for Nonlinear Large-Scale Systems: An LMI Approach Geun Bum Koo, Jin Bae Park, Member, IEEE, and Young Hoon Joo

Abstract—This paper presents a decentralized fuzzy control problem for asymptotic stabilization of a class of nonlinear largescale systems that use an observer-based output-feedback scheme. A Takagi–Sugeno (T–S) fuzzy model is adopted for the nonlinear large-scale system, which has unknown interconnection terms, and a fuzzy controller is separately considered for measurable and nonmeasurable premise variable cases. Sufficient conditions are derived for both asymptotic stabilization and optimization of a maximum bound of interconnection and are formulated in terms of linear matrix inequalities. Finally, numerical examples are provided to verify the effectiveness of the proposed techniques. Index Terms—Decentralized fuzzy control, maximum bound of interconnection, nonlinear large-scale system, observer-based output-feedback, Takagi–Sugeno (T–S) fuzzy model.

I. INTRODUCTION ECENTLY, large-scale systems, which can be efficiently applied to many practical situations, such as power systems, transportation systems, industrial processes, and communication networks, have attracted much attention [2]–[4]. To control a large-scale system, it is essential to solve not only conventional problems, such as nonlinearity, but also additional problems such as: high dimensionality, structural constraints of the controller, and uncertain or unknown information about the interconnections. Because of these problems, a decentralized control technique is more feasible than the centralized control one, and the decentralized fuzzy control technique that uses a Takagi–Sugeno (T–S) fuzzy model is one of the most efficient control techniques for a large-scale system. In many previous studies [5]–[9], various control techniques have been proposed to conquer the high dimensionality and structural constraints of the controller problems. However, studies on uncertain or unknown interconnections are still insufficient.

R

Manuscript received September 22, 2012; revised January 23, 2013; accepted March 25, 2013. Date of publication April 23, 2013; date of current version March 27, 2014. This work was supported in part by the National Research Foundation of Korea grant funded by the Korea goverment under Grant 2012014088 and in part by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning grant funded by the Korea government Ministry of Knowledge Economy under Grant 20104010100590. G. B. Koo and J. B. Park are with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: milbam@ yonsei.ac.kr; [email protected]). Y. H. Joo is with the Department of Control and Robotics Engineering, Kunsan National University, Kunsan, Chonbuk 573-701, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TFUZZ.2013.2259497

Apart from the large-scale system issue, the observer-based output-feedback control, which conquers the drawback of the state-feedback technique that requires the measurement of fullstate information, is one of the traditional main topics. Especially, fuzzy observer-based output-feedback control techniques using the T–S fuzzy model have been proposed in many papers [10]–[22] and are broadly classified into two categories based on whether or not the premise variable is measurable. First, in the case of a measurable premise variable, by using the techniques of [23], relaxed stability conditions were proposed for T–S fuzzy systems with the fuzzy observer and controller [10], [11]. In [12], the fuzzy observer of the discrete nonlinear system was designed by using the fuzzy Lyapunov function. Also, fuzzy observer-based output-feedback controllers were presented for fuzzy systems with various problems, such as Markov stochastic process, input delay, and disturbance in [13]–[15]. Next, in the case of a nonmeasurable premise variable, it is hard to convert the stability condition to linear matrix inequality (LMI), because the closed-loop system has more complex form than the measurable premise variable case. For conversion as LMI form, new methods were variously proposed in [16]–[22]. In [16] and [17], the bilinear matrix inequality of stability condition was converted into LMI by substituting the solution of other LMI. To represent the LMI form, it used the assumption that the membership functions of fuzzy sets often satisfy a Lipschitz-like property in [18]. In [19], the stability condition was represented into LMI form with given scalars. In addition, by using the decoupled approach [20], [21] or Finsler’s lemma [22], stability conditions were successfully converted into LMIs. In [24], by using the decoupled approach, the stability condition was presented for the nonlinear large-scale system with the fuzzy observer-based output-feedback controller, which has the nonmeasurable premise variable. However, the unknown interconnection problem and the optimization problem of the maximum bound of the interconnection were not considered. To solve the unknown interconnection problem, decentralized output-feedback controllers were proposed into LMI formats in [25]–[28]. However, in these papers, all nonlinear terms were considered interconnection terms, and nonlinearity was not observed in subsystems. Thus, more studies are needed for nonlinear large-scale systems. In [29], the fuzzy controller was proposed for the nonlinear large-scale system with the uncertain interconnection term, but the maximum bound of the interconnection was not presented.

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KOO et al.: DECENTRALIZED FUZZY OBSERVER-BASED OUTPUT-FEEDBACK CONTROL FOR NONLINEAR LARGE-SCALE SYSTEMS

Motivated by the aforementioned analysis, this paper presents the decentralized fuzzy observer-based output-feedback control technique for the nonlinear large-scale system with an unknown interconnection term that is assumed to only satisfy the quadratic bound. Based on the T–S fuzzy model of the nonlinear largescale system, the decentralized fuzzy observer and controller are separately designed with two cases of premise variables: measurable and nonmeasurable. In the sense of Lyapunov, sufficient conditions to asymptotically stabilize and optimize the maximum bound of the interconnection are investigated for both closed-loop systems with each type of premise variable and presented in terms of LMIs. Finally, by simple examples, the validity of the proposed ideas, techniques, and procedures are shown. This paper is organized as follows. Section II describes the T–S fuzzy large-scale system and the decentralized observerbased output-feedback controller. The asymptotic stability condition for the closed-loop system with the measurable premise variable is presented in Section III. In Section IV, the stability condition for the nonmeasurable premise variable case is presented. Numerical examples are given for illustration in Section V. Finally, the conclusions are given in Section VI. Notation: The subscripts i, j, and h denote the fuzzy rule indices and the subscripts k and l denote the subsystem indices. (·)T denotes the transpose of the argument. The notation ∗ is used for the transposed element in symmetric positions. For simplicity, we will use xk in place of xk (t), and λm ax (A) (λm in (A)) is the maximum (minimum) eigenvalue of matrix A. II. TAKAGI–SUGENO FUZZY LARGE-SCALE SYSTEMS Consider a T–S fuzzy large-scale system which is composed of n subsystems. The kth subsystem can be described by the following rule: Rik

: IF z1k is Γki1 and · · · and zqk is Γkiq ,  x˙ k = Aki xk + Bik uk + hk (x) THEN yk = Cik xk

(1)

where xk ∈ Rn k , uk ∈ Rm k , and yk ∈ Rl k are the state, control input, and output for the kth subsystem, respectively, and xk and uk are within a compact set Bx k × Bu k := {xk : xk  ≤ Δx k } × {uk : uk  ≤ Δu k } for some Δx k ∈ R> 0 and Δu k ∈ R> 0 ; x = col{x1 , x2 , . . . , xn } is the whole state of the largescale system; Γkip , (i, p, k) ∈ {IR := {1, 2, . . . , r}} × {IQ := {1, 2, . . . , q}} × {IN := {1, 2, . . . , n}}, is a fuzzy set for zpk ; Aki , Bik , and Cik denote nominal system matrices with appropriate dimensions for the ith rule in the kth subsystem; hk (x) is a piecewise continuous vector function, which is assumed as follows: Assumption 1: The vector function hk (x) is unknown, but satisfies the quadratic inequality  k T k h (x) h (x) ≤ αk2 xT (H k )T H k x where αk > 0 is a bound scalar of the interconnection term, and H k is a constant matrix with appropriate dimension. Using the center-average defuzzification, product inference, and singleton fuzzifier to the fuzzy IF–THEN rule (1), the input–

407

output relation in the kth subsystem is represented as x˙ k =

r 

  μki (zk ) Aki xk + Bik uk + hk (x)

i=1

=: Ak (μk )xk + B k (μk )uk + hk (x) yk =

r 

μki (zk )Cik xk =: C k (μk )xk

(2)

i=1

where μki (zk )

=

 r

ωik (zk )

ωik (zk ),

ωik (zk ) =

q

Γkip (zpk )

p=1

i=1

in which Γkip (zpk ) is the fuzzy membership grade of zpk in Γkip . To design the fuzzy observer-based output-feedback controller, we assume the following. Assumption 2: The state variable xk is not measurable, but the premise variable zk and the output variable yk are measurable. Remark 1: The fuzzy observer-based output-feedback control techniques are classified into two cases. One is based on the measurable premise variable and the other is based on the nonmeasurable premise variable. In Assumption 2 and Section III, the premise variable zk is assumed as the measurable value [10]–[15]. On the other hand, in Section IV, the premise variable zk is assumed as the nonmeasurable value [20]–[22], [24]. Based on Assumption 2, we suppose the fuzzy observer-based output-feedback controller for the kth subsystem (2) x ˆ˙ k =

r 

   μki (zk ) Aki x ˆk + Bik uk + Lki yk − yˆk

i=1

  =: Ak (μk )ˆ xk + B k (μk )uk + Lk (μk ) yk − yˆk yˆk =

r 

μki (zk )Cik x ˆk =: C k (μk )ˆ xk

i=1

uk =

r 

μki (zk )Kik x ˆk =: K k (μk )ˆ xk

(3)

i=1

where Lki is the observer, and Kik is the controller gain matrices. ˆk . Then, substituting (2) and (3) into the Let ek := xk − x time derivative of ek , the kth sublosed-loop system is written as χ˙ k = Φk (μk )χk + Θk (x) where

χk =

Φk (μk ) =

k   h (x) , Θk (x) = ek hk (x)

(4)

xk

S1k (μk ) −B k (μk )K k (μk ) 0



S2k (μk )

S1k (μk ) = Ak (μk ) + B k (μk )K k (μk ) S2k (μk ) = Ak (μk ) − Lk (μk )C k (μk ). From the kth subclosed-loop system (4), the decentralized control problem can be stated as follows.

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

Problem 1: Find the gain matrices Lki and Kik stabilizing the closed-loop large-scale system, which is composed of subclosed-loop systems (4), such that the following design objectives are satisfied. 1) The equilibrium point of the closed-loop large-scale system consisting of subclosed-loop systems (4) satisfies an asymptotic stability. 2) A decentralized fuzzy controller maximizes the bound scalar αk of the interconnection term with guaranteeing stability of the closed-loop system. III. STABILIZATION OF THE CLOSED-LOOP SYSTEM WITH THE MEASURABLE PREMISE VARIABLE This section shows the stabilization condition of the closedloop T–S fuzzy large-scale system considering Problem 1. Before proceeding to our main results, the matrix inequality lemma has to be considered for the proof. Lemma 1 ( [30]): For any real matrices Xi , Yi for 1 ≤ i ≤ N , and S  0 with appropriate dimensions, we have 2

N  N N  N  

N  N 

  ˜ kij = Φkij T P k + P k Φkij , Φ

k  −Bik Kjk Ai + Bik Kjk k Φij = 0 Aki − Lki Cjk ˜ lk = [ H lk H

V =

V˙ =

···

.. .

k Z22 .. .

k T (Z1r )

k T (Z2r )

···

k ∈ IN

..

.

∗ .. .

k =1

k T

n  

χ˙ Tk P k χk + χTk P k χ˙ k



n   

T Φk (μk )χk + Θk (x) P k χk

k =1

  + χTk P k Φk (μk )χk + Θk (x)

where hi (1 ≤ i ≤ N ) are defined as hi ≥ 0, N i=1 hi = 1. The sufficient conditions for stabilization of the closed-loop system are summarized as follows. Proposition 1: If there exist some symmetric and positive k such definite matrix P k and some matrices Kik , Lki , and Zij that the optimization problem is satisfied, then the closed-loop large-scale system that is composed of subclosed-loop systems (34) is asymptotically stable, and αk2 = τk−1 is the maximum bound scalar of the interconnection term

⎢ ⎢ ⎣

χTk P k χk

k =1

i=1 j =1

11 ⎢ (Z k )T ⎢ 12

Vk =

n 

where P = (P )  0. Clearly, V is positive definite and radially unbounded. The time derivative of V evaluated along the closed-loop system (4) yields

hi hj (XijT SXij + YijT SYij )

(i, j, k, l) ∈ IJ × IR × IN × IN ⎡ Zk ∗ ··· ∗ ⎤

n  k =1

k

=

minτk subject to ⎡ ˜k ⎤ Φii − Ziik ∗ ∗ ⎢ ⎥ −I ∗ ⎢ Pk ⎥≺0 ⎣ ⎦ 1 ˜ lk H 0 − τl I 2n (i, k, l) ∈ IR × IN × IN   ⎡ k ˜ +Φ ˜ k − Zk − Zk T Φ ∗ ij ji ij ij ⎢ 1 ⎢ ⎢ − I Pk ⎢ 2 ⎣ lk ˜ H 0

0]

and H lk is the submatrix having nk columns of H l from the (n1 + n2 + · · · + nk −1 + 1)th column vector. Also, IJ × IR denotes all pairs (i, j) ∈ IR × IR such that 1 ≤ i < j ≤ r. Proof: Consider the Lyapunov function candidate as

hi hj hk hl XijT SYk l

i=1 j =1 k =1 l=1

≤

where

≤

n  

  χTk Φk (μk )T P k + P k Φk (μk ) + (P k )2 χk

k =1

  T + 2 hk (x) hk (x) ≤

n  

  χTk Φk (μk )T P k + P k Φk (μk ) + (P k )2 χk

k =1

 + 2αk2 xT (H k )T H k x .

(8)

Let H k = [ H k 1 H k 2 · · · H k n ] and H k l have nl columns; then, the following is satisfied: n 

αk2 xT (H k )T H k x =

k =1

∗ ∗ −

1 τl I 4n

⎤

αk2

n 

k =1

(5) =

⎥ ⎥ ⎥≺0 ⎥ ⎦

n 

xTl (H k l )T H k l xl

l=1

n  n 

αl2 xTk (H lk )T H lk xk . (9)

k =1 l=1

Applying (9) to (8) yields V˙ ≤

n 



Φk (μk )T P k + P k Φk (μk ) + (P k )2



k =1

(6)

+2

⎥ ⎥ ⎥≺0 ⎥ ⎦

χTk

n 

 ˜ lk χk ˜ lk )T H αl2 (H

l=1

=

Zrkr (7)

1  Φk (μk )T P k + P k Φk (μk ) + (P k )2 n k =1 l=1  ˜ lk χk . ˜ lk )T H + 2αl2 (H (10) n  n 

χTk

KOO et al.: DECENTRALIZED FUZZY OBSERVER-BASED OUTPUT-FEEDBACK CONTROL FOR NONLINEAR LARGE-SCALE SYSTEMS

From (10), if there exist a symmetric matrix Ziik and some k such that the following inequalities are satisfied: matrix Zij  k T k ˜ lk ≺ Z k (11) ˜ lk )T H Φii P + P k Φkii + (P k )2 + 2nαl2 (H ii  k    T Φij + Φkji P k + P k Φkij + Φkji ˜ lk )T H ˜ lk ≺ Z k + (Z k )T + 2(P k )2 + 4nαl2 (H ij ij

(12)

then time derivative of V is majorized by ⎡ μk I ⎤T ⎡ Z k 1 11 ⎢ μk I ⎥ ⎢ (Z k )T n n ⎥ ⎢  ⎢ ⎢ 2 ⎥ ⎢ 12 V˙ ≤ ⎢ . ⎥ ⎢ .. ⎢ . ⎥ ⎢ k =1 l=1 ⎣ . ⎦ ⎣ . μkr I

k )T (Z1r

∗

···

k Z22

···

.. .

..

k )T (Z2r

···

∗ ⎤⎡ μk1 I ⎤ ⎢ ⎥ ∗ ⎥ ⎥⎢ μk2 I ⎥ ⎥⎢ ⎥ ⎢ ⎥ .. ⎥ ⎥⎢ .. ⎥ . ⎦⎣ . ⎦

.

Zrkr

Remark 2: In inequalities (5) and (6) of Proposition 1, the number of subsystems n and the optimization value τk are in inverse proportion with each other. That means τk becomes more optimized when n has a small value. Thus, the smaller the number of subsystems, the more maximized is the bound of the interconnection. In Proposition 1, it is hard to directly solve the inequalities (5) and (6) and obtain the gain matrices Kik and Lki . Thus, the inequalities (5) and (6) need to be converted into the LMI problem, which is able to be easily solved using a convex optimization toolbox such as MATLAB. To convert into LMI format, withk as out the loss of generality, we define the matrices P k and Zij follows:

μkr I



(13)

< 0.

k

P =

By using the Schur complement, inequalities (11) and (12) yield the inequalities (5) and (6), respectively. Next, we need to show the existence of Δ0 ∈ R> 0 for any (Δx k , Δu k ) ∈ R> 0 × R> 0 such that (xk , uk ) from col{xk (0), ek (0)} on B0 stays in Bx × Bu for all t ∈ R≥0 . In the aforementioned proof procedure, we know if the inequalities (5), (6), and (7) are satisfied, then V˙ (x) < 0.

0

⇒ λm in (P k )col{xk (t), ek (t)}2 (15)

From xk (t) ≤ col{xk (t), ek (t)} and (15), we have  λm ax (P k ) xk (t) < col{xk (0), ek (0)}. (16) λm in (P k ) Next, the bound of uk is given by r     k  uk  ≤  μi (zk )Kik xk − ek  i=1

√ ≤ sup{Ki } 2col{xk (t), ek (t)} i∈Ir



≤ sup{Ki } i∈Ir

Ziik

=

k Zij

=

P1k

∗

0

P2k

P1k

∗

0

P1k

P1k

∗

0

P1k

 (19) 





akii

∗

bkii

ckii

akij

dkij

bkij

ckij

P1k

∗

0

P1k





P1k

∗

0

P1k

(20)  (21)

(14)

Integrating (14) from 0 to t, it can be shown that  t V˙ (τ )dτ = V (t) − V (0) < 0

< λm ax (P k )col{xk (0), ek (0)}2 .

409

2λm ax (P k ) col{xk (0), ek (0)}. (17) λm in (P k )

Therefore, from (16) and (17), if Δ0 is determined to following norm conditions:    λm in (P k ) 2λm ax (P k ) Δx , sup{Ki } Δu min λm ax (P k ) λm in (P k ) i∈Ir (18) then, the system trajectory of xk and uk starting from any initial condition col{xk (0), ek (0)} on B0 cannot leave the compact set  Bx × Bu .

where akii and ckii are symmetric matrices. Based on the aforementioned matrices, the main result is summarized as follows. Theorem 1: If there exist some symmetric and positive definite matrices Qk1 , P2k , some symmetric matrices akii , ckii , R1k , R2k and some matrices Mik , Nik , akij , bkij , ckij , dkij , such that the optimization problem is satisfied, then inequalities (5), (6), and (7) are satisfied with the maximum bound of the interconnection term

minτk ⎡

subject to

Ψkii − akii ⎢ −(M k B k )T − bk ⎢ i i ii ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

⎤

∗

∗

∗

∗

−ρkii Qk1 − ckii

∗

∗

∗

0

−I

∗

∗ ∗ 1 − τl I 2n

I 0

0

0

−R1k

H lk Qk1

0

0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

≺0 ⎡

(i, k, l) ∈ IR × IN × IN

Ωkii ⎢ Pk ⎢ 2 ⎢ ⎢ ⎢ I ⎣

0

∗

∗

−I

∗ 1 k − k Q1 ρii

0 I

0

(i, k) ∈ IR × IN

∗

(22) ⎤

∗ ⎥ ⎥ ⎥ ⎥≺0 ∗ ⎥ ⎦ −R1k (23)

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

⎡

˜k − a Ψ ˜kij ij

⎢ ˜ k )T − ˜bk ⎢ −(M ij ij ⎢ ⎢ ⎢ I ⎢ ⎢ ⎢ 0 ⎢ ⎣ H lk Qk1

∗

⎤

∗

∗

∗

∗

∗

∗

∗

∗

0

1 − I 2 0

−R2k

0

0

0

∗ 1 − τl I 4n

−ρkij Qk1

−

c˜kij

0

where

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

Λkij = (Fijk )T P1k + P1k Fijk − P1k akij P1k Ξkij = (Gkij )T P2k + P2k Gkij − P1k ckij P1k k Eij = −(Bik Kjk )T P1k − P1k bkij P1k

Fijk = Aki + Bik Kjk

≺0

Gkij = Aki + Lki Cjk .

(i, j, k, l) ∈ IJ × IR × IN × IN ⎤ ⎡ ˜k Ωij ∗ ∗ ∗ ⎥ ⎢ ⎢ Pk −1I ∗ ∗ ⎥ ⎥ ⎢ 2 2 ⎥ ⎢ ⎥≺0 ⎢ 1 k ⎢ I ∗ ⎥ 0 − k Q1 ⎥ ⎢ ρij ⎦ ⎣ 0 ⎡

I

(24)



⎢ bk ⎢ 11 ⎢ k T ⎢ (a12 ) ⎢ ⎢ k T ⎢ (d12 ) ⎢ ⎢ . ⎢ .. ⎢ ⎢ ⎢ (ak )T ⎣ 1r (dk1r )T

∗

∗

∗

···

∗

ck11

∗

∗

···

∗

(bk12 )T

ak22

∗

···

∗

(ck12 )T .. .

bk22 .. .

ck22 .. .

···

∗ .. .

(bk1r )T

(ak2r )T

(bk2r )T

···

akrr

(ck1r )T

(dk1r )T

(ck1r )T

···

bkrr

..

.

∗

⎤

(25)

∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥≺0 .. ⎥ . ⎥ ⎥ ⎥ ∗ ⎥ ⎦ k cr r

k ∈ IN where

(26)

T  Ψkij = Aki Qk1 + Bik Mjk + Aki Qk1 + Bik Mjk T  Ωkij = P2k Aki + Nik Cjk + P2k Aki + Nik Cjk ˜ kij = Ψkij + Ψkji Ψ ˜ kij = Ωkij + Ωkji Ω ˜ k = Bk M k + Bk M k M ij i j j i a ˜kij

=

akij

+

∗

P2k

−I



 ≺



−ρkii P1k

∗

0

−(R1k )−1

then the inequality (27) is majorized by

(i, j, k) ∈ IJ × IR × IN ak11

(Gkii )T P2k + P2k Gkii

(28)

−R2k

0

If there exists a symmetric matrix (R1k )−1 such that the following inequality is satisfied:

(akij )T

˜bk = bk + (dk )T ij ij ij c˜kij = ckij + (ckij )T and ρkij > 0 is a given constant scalar. Also, IJ × IR denotes all pairs (i, j) ∈ IR × IR such that 1 ≤ i < j ≤ r. Proof: By substituting (19) and (20) into (5), we obtain ⎡ k ⎤ Λii ∗ ∗ ∗ ∗ ⎢ E k Ξk ⎥ ∗ ∗ ∗ ⎢ ii ⎥ ii ⎢ k ⎥ ⎢ P1 ⎥ 0 −I ∗ ∗ (27) ⎢ ⎥≺0 ⎢ ⎥ 0 −I ∗ P2k ⎢ 0 ⎥ ⎣ ⎦ 1 lk H 0 0 0 − τl I 2n

⎡

∗

∗

∗

∗

−ρkii P1k − P1k ckii P1k

∗

∗

∗

0

−I

∗

∗

0

0

−(R1k )−1

∗

0

0

0

Λkii

⎢ k ⎢ Eii ⎢ ⎢ k ⎢ P1 ⎢ ⎢ ⎢ 0 ⎢ ⎣ H lk

−

1 τl I 2n

≺0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦ (29)

By using the congruences transformation with diag{(P1k )−1 , (P1k )−1 , I, R1k , I} and denoting (P1k )−1 = Qk1 and Kik (P1k )−1 = Mik for the inequality (29) and by using the Schur complement and denoting P2k Lki = Nik for the inequality (28), LMIs (22) and (23) can be obtained. We can again establish a similar argument to (6) in order to obtain (24) and (25) as follows: ⎡ ˜k Λij ⎢ k ˜ ⎢E ⎢ ij ⎢ ⎢ k ⎢ P1 (6) ⇒ ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎣ H lk

∗

∗

∗

∗

˜k Ξ ij

∗

∗

∗

0

1 − I 2

∗

∗

P2k

0

1 − I 2

∗

0

0

0

−

1 τl I 4n

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥≺0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(30)

where ˜ k = (F k + F k )T P k + P k (F k + F k ) Λ ij ij ji 1 1 ij ji − P1k (akij + (akij )T )P1k ˜ kij = (Gkij + Gkji )T P2k + P2k (Gkij + Gkji ) Ξ − P1k (ckij + (ckij )T )P1k ˜ k = −(B k K k + B k K k )T P k − P k (bk + (dk )T )P k . E ij i j j i 1 1 ij ij 1

KOO et al.: DECENTRALIZED FUZZY OBSERVER-BASED OUTPUT-FEEDBACK CONTROL FOR NONLINEAR LARGE-SCALE SYSTEMS

If there exists a symmetric matrix (R2k )−1 such that the following inequality is satisfied: ⎤ ⎡ k ∗ (Gij + Gkji )T P2k + P2k (Gkij + Gkji ) ⎣ 1 ⎦ − I P2k 2  k k  −ρij P1 ∗ ≺ . (31) 0 −(R2k )−1 then the inequality (30) is majorized by ⎡ ˜k Λij ∗ ∗ ∗ ⎢ ˜k k k k k k ⎢ Eij −ρij P1 − P1 c˜ij P1 ∗ ∗ ⎢ ⎢ k ⎢ P1 0 −I ∗ ⎢ ⎢ 0 k −1 0 0 −(R ⎢ 2) ⎣ H lk 0 0 0 ≺ 0.

∗ ∗ ∗

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

∗ 1 − τl I 4n (32)

From the same procedure that is used to obtain LMIs (22) and (23), LMIs (24) and (25) can be obtained.  Remark 3: In Theorem 1, LMIs can be guaranteed to have more relaxed conditions by using non-PDC methods [31], [32], fuzzy Lyapunov functions [33], [34], etc. However, in this paper, we omitted the discussion of the relaxed condition of the LMIs because it is out of the scope and purpose of this paper. IV. STABILIZATION FOR THE CLOSED-LOOP SYSTEM WITH THE NONMEASURABLE PREMISE VARIABLE In Section III, Proposition 1 and Theorem 1 are based on Assumption 2. However, Assumption 2 is a conservative condition because in the most fuzzy models, the premise variable zk consists of the state variable xk , and therefore zk is almost nonmeasurable, when xk is nonmeasurable. Thus, we now need to present a new stabilization condition for the T–S fuzzy largescale system with the following assumption. Assumption 3: State variable xk and premise variable zk are not measurable, but output variable yk is measurable. Based on Assumption 3, the fuzzy observer-based outputfeedback controller with the observed premise variable is supposed as follows: x ˆ˙ k =

r 

   μki (ˆ zk ) Aki x ˆk + Bik uk + Lki yk − yˆk

i=1

  μk )ˆ xk + B k (ˆ μk )uk + Lk (ˆ μk ) yk − yˆk =: Ak (ˆ yˆk =

r 

uk =



ˆk ) = Φk1 (μk , μ

S1k (μk , μ ˆk ) −B k (μk )K k (μk ) k k ˆk ) S3k (μk , μ ˆk ) S2 (μ , μ



ˆk ) = Ak (μk ) + B k (μk )K k (ˆ μk ) S1k (μk , μ   S2k (μk , μ ˆk ) = Ak (μk ) − Ak (ˆ μk ) + B k (μk ) − B k (ˆ μk )   μk ) − Lk (ˆ μk ) C k (μk ) − C k (ˆ μk ) × K k (ˆ ˆk ) = Ak (ˆ μk ) − Lk (ˆ μk )C k (ˆ μk ) S3k (μk , μ  k k  − B (μ ) − B k (ˆ μk ) K k (ˆ μk ). On the basis of the above, sufficient conditions for the stabilization of the closed-loop large-scale system composed of kthe subclosed-loop systems (34) are summarized as follows. Proposition 2: If there exist some symmetric and positive k definite matrix P k and some matrices Kik , Lki , Zij h such that the optimization problem is satisfied, then the closed-loop largescale system that is composed of subclosed-loop systems (4) is asymptotically stable, and αk2 = τk−1 is the maximum bound scalar of the interconnection term minτk subject to ⎡ ˜k k Φij j − Zij ∗ j ⎢ ⎢ Υkij j −I ⎢ ⎢ ⎢ 0 Pk ⎢ ⎢ ⎣ ˜ lk H 0

∗

∗

∗

∗

1 − I 2

∗

⎤ ⎥ ⎥ ⎥ ⎥ ⎥≺0 ⎥ ⎥ ⎦

1 τl I 2n (i, j, k, l) ∈ IR × IR × IN × IN ⎡ ˜k ˜ k − Zk − Zk Φij h + Φ ∗ ∗ ihj ij h ihj ⎢ k ⎢ Υij h −I ∗ ⎢ ⎢ ⎢ Υkihj 0 −I ⎢ ⎢ ⎢ 0 0 Pk ⎢ ⎢ ⎣ ˜ lk H 0 0 0

−

∗

∗

∗

∗

∗

∗

1 − I 4

∗

0

−

1 τl I 4n

(i, j, h, k, l) ∈ IR × IJ × IR ⎡ Zk ∗ ··· ∗ i11 ⎢ (Z k )T Zi22 ··· ∗ ⎢ i12 ⎢ .. .. .. ⎢ .. ⎣ . . . . k (Zi2r )T

···

× IN × IN ⎤ ⎥ ⎥ ⎥≺0 ⎥ ⎦

Zir r (37)

where μki (ˆ zk )Kik x ˆk =: K k (ˆ μk )ˆ xk

(33)

i=1

and the kth subclosed-loop system is rewritten as χ˙ k = Φk1 (μk , μ ˆk )χk + Θk (x)

(34)

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(36)

(i, k) ∈ IR × IN

μki (ˆ zk )Cik x ˆk =: C k (ˆ μk )ˆ xk

(35) ⎤

≺0

k )T (Zi1r

i=1 r 

where

411

  ˜ k = Φk T P k + P k Φk , Φ ij h ij h ihj

k k k Ai + Bi Kh Φkij h = k Ai − Akj − Lkj (Cik − Chk )

Υkij h = [ (Bik − Bjk )Khk

−Bik Khk Akj − Lkj Chk

−(Bik − Bjk )Khk ]



412

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

and IJ × IR denotes all pairs (j, h) ∈ IR × IR such that 1 ≤ j < h ≤ r. We consider the Lyapunov function candidate V = Proof: n n T k k =1 Vk = k =1 χk P χk as in Proposition 1; then, the time derivative of V yields n   T k  ˙ χ˙ k P χk + χTk P k χ˙ k V =

T  + Υkihj Υkihj + 4(P k )2 k k ˜ lk ≺ Zij ˜ lk )T H + 4nαl2 (H h + Zij h

then the time derivative of V is majorized by V˙ ≤

≤

 χTk (Φk (μk , μ ˆk ))T P k + P k Φk (μk , μ ˆk )

 + (Υk (μk , μ ˆk ))T Υk (μk , μ ˆk ) + 2(P k )2 χk   T + 2 hk (x) hk (x) ≤

μ ˆkr I

 χTk (Φk (μk , μ ˆk ))T P k + P k Φk (μk , μ ˆk )

 + (Υk (μk , μ ˆk ))T Υk (μk , μ ˆk ) + 2(P k )2 χk  + 2αk2 xT (H k )T H k x

ˆk ) = Φk (μk , μ

(38)

S1k (μk , μ ˆk ))

−B k (μk )K k (ˆ μk )

ˆk )) S4k (μk , μ

Ak (ˆ μk ) − Lk (ˆ μk )C k (ˆ μk )

Υk (μk , μ ˆk ) = [ S5k (μk , μ ˆk ))



−S5k (μk , μ ˆk )) ]

S4k (μk , μ ˆk ) = Ak (μk ) − Ak (ˆ μk )   − Lk (ˆ μk ) C k (μk ) − C k (ˆ μk )   S5k (μk , μ ˆk ) = B k (μk ) − B k (ˆ μk ) K k (ˆ μk ). Applying (9) to (38) yields V˙ ≤

n 

χTk



Φk (μk , μ ˆk )T P k + P k Φk (μk , μ ˆk )

k =1

+ Υk (μk , μ ˆk )T Υk (μk , μ ˆk ) + 2(P k )2 +2

n 



 ˜ lk χk ˜ lk )T H αl2 (H

l=1

=

1 Φk (μk , μ ˆk )T P k + P k Φk (μk , μ ˆk ) n k =1 l=1  + Υk (μk , μ ˆk )T Υk (μk , μ ˆk ) + 2(P k )2  ˜ lk χk . ˜ lk )T H + 2αl2 (H n  n 

k )T (Zi1r

∗

···

k Zi22 .. .

···

k (Zi2r )T

···

..

∗ ⎤⎡ μ ˆk1 I ⎤ ⎢ ˆk I ⎥ ∗ ⎥ 2 ⎥ ⎥⎢ μ ⎥ ⎥⎢ .. ⎥⎢ .. ⎥ . ⎦⎣ . ⎦

.

μ ˆkr I

k Zir r

< 0.

k =1

where

μki

⎡μ k ˆk1 I ⎤T ⎡ Zi11 k T k ⎥ ⎢ ⎢μ ⎢ ˆ2 I ⎥ ⎢ (Zi12 ) ⎥ ⎢ ×⎢ .. ⎢ .. ⎥ ⎢ ⎣ . ⎦ ⎣ .

k =1

n  

n  n  r  k =1 l=1 i=1

k =1 n  

(40)

χTk

k By using Lemma 1, if there exist symmetric matrix Zij j k and some matrix Zij h such that the following inequalities are satisfied:  k T k T  Φij j P + P k Φkij j + Υkij j Υkij j + 2(P k )2 k ˜ lk ≺ Ziii ˜ lk )T H + 2nαl2 (H (39)       k T T Φij h + Φkihj P k + P k Φkij h + Φkihj + Υkij h Υkij h

By using the Schur complement, inequalities (39) and (40) yield the inequalities (35) and (36), respectively. Next, the proof of the existence of Δ0 ∈ R> 0 directly follows the proof of Proposition 1.  Remark 4: In Proposition 2, if Bik = Bjk is satisfied for any i, j ∈ IR , LMIs (35) and (36) can be simplified as follows: ⎡ ˜k ⎤ k Φij j − Zij ∗ ∗ j ⎢ ⎥ 1 ⎢ ⎥ ∗ Pk − I ⎢ ⎥≺0 ⎢ ⎥ 2 ⎣ ⎦ 1 ˜ lk H 0 − τl I 2n ⎡ ˜k ⎤ ˜ k − Zk − Zk Φij h + Φ ∗ ∗ ihj ij h ihj ⎢ ⎥ 1 ⎢ ⎥ ∗ − I Pk ⎢ ⎥ ≺ 0. ⎢ ⎥ 4 ⎣ ⎦ 1 ˜ lk H 0 − τl I 4n To convert the inequalities (35) and (36) into LMI formats, the second main result is summarized as follows. Theorem 2: If there exist some symmetric and positive definite matrices Qk1 , P2k , some symmetric matrices akij j , ckij j , R3k , R4k and some matrices Mik , Nik , akij h , bkij h , ckij h , dkij h such that the optimization problem is satisfied, then inequalities (35), (36), and (37) are satisfied with the maximum bound of the interconnection term minτk ⎡ Ψ ˜k

ij j

subject to

⎢ ˜ ⎢ Mij j ⎢ ⎢ ˜k ⎢ Υij ⎢ ⎢ ⎢ I ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ H lk Qk 1 ⎢ ⎢ ⎣ Qk1 ≺0

∗

∗

∗

∗

∗

∗

˜k Q ij j

∗

∗

∗

∗

∗

˜k −Υ ij

−I

∗

∗

∗

∗

0

0

∗

∗

∗

0

0

1 − I 2 0

−R3k

∗

0

0

0

0

∗ 1 − τl I 2n

0

0

0

0

0

⎤

∗ −

1 k νij j

I

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

KOO et al.: DECENTRALIZED FUZZY OBSERVER-BASED OUTPUT-FEEDBACK CONTROL FOR NONLINEAR LARGE-SCALE SYSTEMS

⎡

(i, j, k, l) ∈ IR × IR × IN × IN k −νij jI

⎢ k ⎢ Sij j ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣

Ωkjj

∗

∗

(Ωkjj )T

∗

∗

P2k

1 − I 2

∗

I

0

0

I

+

0

−

1 ρkij j

∗

⎤

Qk1

ˆ ij h = −(Bik (Mjk + Mhk ))T − bkij h − (dkij h )T M ˆ k = −ρk Qk − ck − (ck )T Q ij h ij h 1 ij h ij h ˜ kij h = Ωkjh + Ωkhj Ω k k k k k k k k Sij h = P2 Ai − P2 Aj − Nj (Ci − Ch )

−R3k

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

−R4k

∗ 1 − τl I 4n

∗

0

ˆ kij h = Ψkij + Ψkih − akij h − (akij h )T Ψ

⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥≺0 ⎥ ⎥ ∗ ⎥ ⎥ ⎦

0

(i, j, k) ∈ IR × IR × IN ⎡ ˆk Ψij h ∗ ∗ ∗ ⎢ ˆ ij h ˆk ⎢ M Q ∗ ∗ ij h ⎢ ⎢ ⎢ Υ ˜k ˜k −Υ −I ∗ ij h ⎢ ij h ⎢ ⎢ 1 ⎢ I 0 0 − I ⎢ 4 ⎢ ⎢ 0 0 0 0 ⎢ ⎢ ⎢ lk k ⎢ H Q1 0 0 0 ⎢ ⎢ ⎣ 0 0 0 Qk1

0

∗ −

0

1 k νij h

(i, j, h, k, l) ∈ IR × IJ × IR × IN × IN k −νij hI

⎢ ⎢ Sk + Sk ihj ⎢ ij h ⎢ ⎢ ⎢ 0 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣

˜k Ω ij h

0

∗  k T ˜ + Ω ij h

∗

∗

∗

∗

P2k

1 − I 4

∗

I

0

0

I

−

∗

1 Qk ρkij h 1

I

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(43) ⎤

⎥ ∗ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎦

−R4k

0

≺0 (i, j, h, k) ∈ IR × IJ × IR × IN

⎡ ak i11 ⎢ bk ⎢ i11 ⎢ k T ⎢ (ai12 ) ⎢ ⎢ k T ⎢ (di12 ) ⎢ ⎢ .. ⎢ . ⎢ ⎢ k T ⎣ (ai1r ) (dki1r )T

∗

∗

∗

···

∗

cki11

∗

∗

···

∗

(bki12 )T

aki22

∗

···

∗

(cki12 )T

bki22

cki22

···

.. .

.. .

.. .

..

∗ .. .

(bki1r )T (cki1r )T

(aki2r )T (dki1r )T

(bki2r )T (cki1r )T

···

.

···

akir r bkir r

(44) ⎤ ∗ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ ⎥ ∗ ⎥ ⎥ .. ⎥ . ⎥ ⎥ ⎥ ∗ ⎦ ckir r

where

˜ kij h = col{(Bik − Bjk )Mhk , (Bik − Bhk )Mjk } Υ k and ρkij h > 0 and νij h > 0 are given constant scalars. Also, IJ × IR denotes all pairs (j, h) ∈ IR × IR such that 1 ≤ j < h ≤ r. Proof: By substituting (19) and (20) into (35), we obtain

⎡

∗

∗

∗

∗

∗

Ξkij j

∗

∗

∗

∗

ˆk −Υ ij

−I

∗

∗

∗

0

0

1 − I 2

∗

∗

P2k

0

0

1 − I 2

∗

0

0

0

0

Λkij j

⎢ k ⎢ Eij j ⎢ ⎢ ˆk ⎢ Υij ⎢ ⎢ ⎢ Pk ⎢ 1 ⎢ ⎢ ⎢ ⎢ 0 ⎢ ⎣ H lk

(45)

˜ ij h = −(B k M k )T − bk M i h ij h

1 τl I 2n

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥≺0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(46)

˜ k − P k ak P k ˜ k )T P k + P k Ψ Λkij h = (Ψ ih 1 1 ih 1 ij h 1 k k k T k k ˜k k k k Eij h = −(Bi Kh ) P1 + P2 Sij h − P1 bij h P1 k k k k k S˜ij h = Ai − Aj − Lj (Ci − Ch ) k k k ˜k ˜ k )T P k + P k Ω Ξkij h = (Ω jh 2 2 j h − P1 cij h P1

ˆ kij = (Bik − Bjk )Kjk . Υ If there exists a symmetric matrix R3k such that the following inequality is satisfied: ∗

k −νij jI

⎢ k ˜k ⎣ P2 Sij j ⎡

˜ k = Ψk − ak Ψ ij h ih ij h

−

⎤

where

⎡

≺0 (i, k) ∈ IR × IN

˜ k = (B k − B k )M k Υ ij i j j

(42) ⎤

≺0 ⎡

˜ k = −ρk Qk − ck Q ij h ij h 1 ij h

(41)

∗

413

˜k ˜ k )T P k + P k Ω (Ω 2 2 ii jj P2k

0

0 ⎢0 ≺⎣ 0

∗

∗

∗

−ρkij j P1k

∗

0

−(R3k )−1

⎤ ⎥ ⎦

⎤

⎥ ∗ ⎦

−I (47)

414

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

then the inequality (46) is majorized by ⎡ k k Λij j + νij jI ⎢ k ˜ ⎢ E ij j ⎢ ⎢ ˆ ⎢ Υkij ⎢ ⎢ ⎢ P1k ⎢ ⎢ 0 ⎢ ⎣ H lk

A. Example 1

∗

∗

∗

∗

∗

˜k Ξ ij j

∗

∗

∗

∗

ˆk −Υ ij

−I

∗

∗

∗

0

0

−I

∗

∗

0

0

0

−(R3k )−1

0

0

0

0

∗ 1 − τl I 2n

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥≺ 0 ⎥ ⎥ ⎥ ⎥ ⎦

Consider the fuzzy large-scale system which is composed of two subsystems as follows: r    μki (zk ) Aki xk + Bik uk + H k x x˙ k = i=1

yk = where

A11

˜ k = −(B k K k )T P k − P k bk P k E ij h i h 1 1 ij h 1 ˜ kij h = −ρkij h P1k − P1k ckij h P1k . Ξ By using the congruences transformation with diag{(P1k )−1 , (P1k )−1 , I, R3k , I}, using the Schur complement, and denoting (P1k )−1 = Qk and Kik (P1k )−1 = Mik for the inequality (48) and by using the Schur complement and denoting P2k Lki = Nik for the inequality (47), LMIs (41) and (42) can be obtained. The proofs of (43) and (44) are similar to the aforementioned process, and therefore are omitted here.  Remark 5: The optimal problems (22)–(25) of Theorem 1 and (41)–(44) of Theorem 2 are not strict LMI conditions, because k LMIs depend on the constant terms ρkij h and νij h . Thus, we k k assume that ρij h and νij h are given in advance and the method k to determine ρkij h and νij h is minutely described in [24]. Remark 6: In the most cases, because the premise variable zk depends on the state variable xk , if the state variable xk is not measurable, then the premise variable zk is also not measurable. In the nonmeasurable premise variable case, the stability condition is presented in Theorem 2. However, if the premise variable depends on the output variable or the state, which is directly transmitted into the output variable, the LMIs in Theorem 1 are efficient for use as the stability condition. The reason is because the LMI conditions in Theorem 1 are less conservative than the ones in Theorem 2, as will be shown in Section V. Remark 7: The major contributions of this paper are as follows. First, the observer-based decentralized fuzzy controller for the nonlinear large-scale system, which has unknown interconnection terms, is designed. Second, the observer design problems are considered not only for the measurable premise variable case, but also for the nonmeasurable one, and the controller design problems are solved for each case, respectively. Third, the optimization problem is studied for the maximum bound of the interconnection on which the closed-loop largescale system can be stabilized. V. NUMERICAL EXAMPLE In this section, three examples are given to verify the proposed techniques. The first example is the numerical fuzzy large-scale system and the second and third are, respectively, nonlinear systems of multiinverted pendulums.

μki (zk )Cik xk

i=1

(48)

where

r 

=

A21

=

2

0

1 −3 1

1



,

A12

=



A22

2



1

1 −3 1

1



, = 1 −2 −1 −2

 0.02 0.03 0.03 0.03 H 1 = α1 × 0.01 0.02 −0.01 0.04

 0.02 0.02 0.02 0.03 H 2 = α2 × 0.01 0.03 0.01 0



 1 xk 1 Bik = , Cik = [ 1 0 ], xk = xk 2 0 and a1 and a2 are unknown constant values. The purpose of this simulation is to obtain maximum bounds of a1 and a2 with satisfaction of the stability of the closed-loop large-scale system. First, to simulate the premise variable case, we suppose the function μki (zk ) as follows: μ11 (z1 ) = μ11 (x1 1 ) = exp (−2x21 1 ) μ12 (z1 ) = μ12 (x1 1 ) = 1 − μ11 (x1 1 ) μ21 (z2 ) = μ21 (x2 1 ) =

1 1 + exp (−x2 1 )

μ22 (z2 ) = μ22 (x2 1 ) = 1 − μ21 (x2 1 ). Then, because the premise variables can be directly measured by output variables, Assumption 2 is satisfied and Theorem 1 can be used. By assuming ρk1j = 7 and ρk2j = 7.1, using Theorem 1 and solving the corresponding LMIs, we obtain observer and controller gains as follows: K11 = [ −5.1570

−0.7281 ]

K21 = [ −5.3930

−1.4717 ]

= [ −5.7017

−3.7703 ]

K12

K22 = [ −5.1133 −0.4901 ]



 281.3359 281.4767 1 1 L1 = , L2 = 29.0927 52.7231



 281.2414 281.2415 L21 = , L22 = 125.7921 123.7942 and we also obtain the maximum bounds of the interconnection as α1 = 9.3250 and α2 = 9.5433. This means that, even if the values α1 and α2 are any values below 9.3250

KOO et al.: DECENTRALIZED FUZZY OBSERVER-BASED OUTPUT-FEEDBACK CONTROL FOR NONLINEAR LARGE-SCALE SYSTEMS

Fig. 1. Time responses of states with measurable premise variables: x1 1 (solid), x1 2 (dotted), x2 1 (dashed), and x2 2 (dash–dotted).

415

Fig. 3. Time responses of states with nonmeasurable premise variables: x1 1 (solid), x1 2 (dotted), x2 1 (dashed), and x2 2 (dash–dotted).

Then, because the premise variables cannot be directly measured by output variables, the nonmeasurable premise variable case has to be applied. By assuming ρk1j h = 7, ρk2j h = 7.1, and k νij h = 0.12, using Theorem 2 and solving the corresponding LMIs, we obtain observer and controller gains as follows: K11 = [ −5.2601

−1.0543 ]

K21 = [ −5.2896

−1.0555 ]

= [ −5.5089

−1.0926 ]

K12

Fig. 2. Time responses of errors with measurable premise variables: e1 1 (solid), e1 2 (dotted), e2 1 (dashed), and e2 2 (dash–dotted).

and 9.5433, respectively, the stability of the closed-loop largescale system with the aforementioned gains is always satisfied. Time responses of the states and errors for each subsystem with the initial condition x1 (0) = e1 (0) = [1, −1]T and x2 (0) = e2 (0) = [2, −2]T are shown in Figs. 1 and 2. As shown in the figures, all states and errors of each subsystem are converged into zero, and we can know the proposed decentralized fuzzy observer-based output-feedback control technique satisfies an asymptotic stability of the closed-loop fuzzy large-scale system with the measurable premise variable. Next, we again suppose the function μki (zk ) as follows: μ11 (z1 ) = μ11 (x1 2 ) = exp (−2x21 2 ) μ12 (z1 ) = μ12 (x1 2 ) = 1 − μ11 (x1 2 ) μ21 (z2 ) = μ21 (x2 2 ) =

1 1 + exp (−x2 2 )

μ22 (z2 ) = μ22 (x2 2 ) = 1 − μ21 (x2 2 ).

K22 = [ −5.5413 −1.0963 ]



 0.6239 0.6239 1 3 1 3 L1 = 10 × , L2 = 10 × 0.0266 0.0360



 0.3704 0.3704 L21 = 103 × , L22 = 103 × 1.9138 1.9118 and we also obtain the maximum bounds of the interconnection as α1 = 1.6400 and α2 = 1.6784. Comparing the maximum bounds αk of the nonmeasurable premise variable case with the maximum bounds αk of the measurable one, Theorem 1 tenders the more extensive range of the bound of interconnection than Theorem 2. Time responses of states and errors for each subsystem with the same initial condition as the measurable premise variable case are shown in Figs. 3 and 4. As shown in the figures, we can also know the effectiveness of the proposed technique in the nonmeasurable premise variable case. B. Example 2 For simulating the measurable premise variable case, we consider a double-inverted pendulum system connected by a spring [35], and composed of two subsystems as follows: x˙ k 1 = xk 2 x˙ k 2 =

mk gl κ 1 κ sin (xk 1 ) − xk 1 + uk + xl (49) Jk Jk Jk Jk 1

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

Fig. 4. Time responses of errors with nonmeasurable premise variables: e1 1 (solid), e1 2 (dotted), e2 1 (dashed), and e2 2 (dash–dotted).

where {(k, l) ∈ I2 |k = l}; xk 1 is the angular displacement of the kth pendulum; m1 = 2 kg and m2 = 2.5 kg are the masses of each pendulum; J1 = 2 kg and J2 = 2.5 kg are the moments of inertia; g = 9.8 m/s2 is the gravity constant; l = 1m is the length of the pendulum; and κ is the unknown spring constant, but assumed to be 1 N/m. The purpose of this simulation is to determine the maximum bound of the spring constant with stabilization of the fuzzy large-scale system. To construct the T–S fuzzy large-scale system of the double-inverted pendulum system (49), we establish the fuzzy IF–THEN rules as follows:  x˙ k = Aki xk + Bik uk + H k x k k Ri : IF xk 1 is Γi , THEN yk = Cik xk where

Ak1 =

H1 =

2

H =

0

1

m1 gl/Jk

0

0

κ/Jk

 0 k Bi = , 1/Jk Γk1

Ak2 =

,

0

−κ/Jk 0



0

0

0 κ/Jk 0

0

0

0 −κ/Jk Cik = [ 1

xk + sin (xk 1 ) = 1 , 2xk 1

0



0

1

−m1 gl/Jk

0





0

Fig. 5. Time responses of states of the double-inverted pendulum systems with measurable premise variables: x1 1 (solid), x1 2 (dotted), x2 1 (dashed), and x2 2 (dash–dotted).

obtained as follows: K11 = [ −797.8108

−53.0191 ]

K21 = [ −748.2240

−51.9798 ]

= [ −997.2635

−66.2738 ]

K12

K22 = [ −935.2800 −64.9747 ]



 0.0235 0.0235 L11 = 105 × , L12 = 105 × 4.2066 4.2064



 0.0235 0.0235 L21 = 105 × , L22 = 105 × . 4.2066 4.2064 and the maximum bound of the interconnection α1 = α2 = 8.6451. This means that, even if the spring constant κ has any value below 8.6451N/m, the stability of the closed-loop fuzzy large-scale system with above gains is always satisfied. Time responses of states and errors for each subsystem with the initial conditions x1 (0) = e1 (0) = [π/3, 0]T and x2 (0) = e2 (0) = [π/4, 0]T are shown in Figs. 5 and 6. From the results of the simulation, we can know that the proposed decentralized fuzzy observer-based output-feedback control technique is effective in the nonlinear large-scale system with the measurable premise variable.

0] C. Example 3

Γk2

=1−

Γk1

for (i, k) ∈ I2 × I2 . Because the premise variable is xk 1 , we can directly measure the premise variable by the output one. Thus, the decentralized fuzzy observer-based output-feedback controller can be designed by Theorem 1 that tenders the stabilization algorithm in the measurable premise variable case. By assuming ρkij = 35, the observer and controller gains are

We also consider the system of the two-inverted pendulums connected by a spring that is mounted on two carts [36] to validate the proposed method in the nonmeasurable premise variable case. The dynamic equation of the nonlinear large-scale system is described by θ¨k − {(g/cl) − (κa(a − cl)/cmk l2 )}θk + (mk /Mk )sin(θk )θ˙k2 = (κa(a − cl)/cmk l2 )θl + (1/cmk l2 )uk

(50)

KOO et al.: DECENTRALIZED FUZZY OBSERVER-BASED OUTPUT-FEEDBACK CONTROL FOR NONLINEAR LARGE-SCALE SYSTEMS

417

for (i, k) ∈ I4 × I2 . Using the center-average defuzzification, product inference, and singleton fuzzifier to the fuzzy IF–THEN rule (51), the two-inverted pendulum system (50) is inferred as x˙ k =

4 

  μki (xk 1 , xk 2 ) Aki xk + Bik uk + H k x

i=1

yk =

4 

μki (xk 1 , xk 2 )Cik xk

i=1

where

Fig. 6. Time responses of errors of the double-inverted pendulum systems with measurable premise variables: e1 1 (solid), e1 2 (dotted), e2 1 (dashed), and e2 2 (dash–dotted).

where {(k, l) ∈ I2 |k = l}, θk (t) is the angle of the kth pendulum, ck = mk /(mk + Mk ); M1 = 4.8 kg and M2 = 5 kg are the masses of each cart; a = 0.2 m is the length from the cart to the spring; m1 = 0.9 kg, m2 = 1 kg, l = 1 m, and g = 9.8 m/s2 are defined in Example 2; and, as same with Example 2, κ is the unknown spring constant, but assumed to be 1N/m. Assuming θ˙k (t) ∈ [−Ωkm in , Ωkm ax ] and choosing xk = [xk 1 , xk 2 ]T = [θk , θ˙k ]T , the fuzzy IF–THEN rule of the twoinverted pendulum system (50) can be constructed as follows: Rik : IF xk 1 is Γki1 , and xk 2 is Γki2  x˙ k = Aki xk + Bik uk + H k x THEN yk = Cik xk where



Ak1 =  Ak3 = ⎡ H =⎣ 1

⎡

0 g cl 0 g cl

1

0 κa(a − cl) − cm1 l2

1 + sin (xk 1 ) , = 2

Γk12 =



mk k , Ak2 = Ωm in Mk  1 mk k , Ak4 = − Ωm in Mk

0 H 2 = ⎣ κa(a − cl) cm2 l2 ⎤ ⎡ 0 Bik = ⎣ 1 ⎦, cmk l2 Γk11



Ωkm ax − xk 2 , Ωkm ax − Ωkm in

0

0

mk k Ω Mk m ax 0 g cl

0 κa(a − cl) − cm2 l2 Cik = [ 0 1 ] Γk21

=1−

Γk11

Γk22 = 1 − Γk12

1 mk k − Ω Mk m ax ⎤ 0 ⎦ 0 0 0

⎤ ⎦

Ωk − xk 2 1 + sin (xk 1 ) × km ax 2 Ωm ax − Ωkm in

μk2 (xk 1 , xk 2 ) =

xk − Ωkm in 1 + sin (xk 1 ) × k2 2 Ωm ax − Ωkm in

μk3 (xk 1 , xk 2 ) =

Ωk − xk 2 1 − sin (xk 1 ) × km ax 2 Ωm ax − Ωkm in

μk4 (xk 1 , xk 2 ) =

xk − Ωkm in 1 − sin (xk 1 ) × k2 . 2 Ωm ax − Ωkm in

While the premise variables are xk 1 and xk 2 , the state variable xk 2 can be only directly measured by the output variable. Thus, this simulation is affiliated with the nonmeasurable premise variable and Theorem 2 is applied for the controller design. We assume Ω1m in = −5, Ω1m ax = 2, Ω2m in = −4, Ω2m ax = 3, ρk1j h = k 35, ρk2j h = 35.1, ρk2j h = 35.2, ρk4j h = 35.3, and νij h = 0.2. By using Theorem 2 and solving the corresponding LMIs, we obtain observer and controller gains as follows: K11 = [ −24.1316

−3.1196 ]

= [ −24.1457

−3.1222 ]

K31 = [ −24.1597

−3.1249 ]

= [ −24.1735

−3.1275 ]

K12 = [ −25.8324

−3.6306 ]

= [ −25.8597

−3.6359 ]

K32 = [ −25.8806

−3.6398 ]

K21 K41



1

0 κa(a − cl) cm1 l2

0 0

0 g cl 

(51)

μk1 (xk 1 , xk 2 ) =



K22 K42 L11 L13 L21 L23

= [ −25.8951 −3.6424 ]

 3.1357 = 103 × , L12 = 103 2.8154

 3.1357 3 = 10 × , L14 = 103 2.8173

 2.9473 3 = 10 × , L22 = 103 2.6182

 2.9473 = 103 × , L24 = 103 2.6199

×

×

×

×

3.1878 2.8485 3.1884 2.8483 2.9675 2.6319 2.9678

   

2.6310

and we also obtain the maximum bound of the interconnection as α1 = α2 = 17.8033. This means that, even if the value of the spring constant is any value below 17.8033 N/m, the stability of the closed-loop large-scale system with aforementioned

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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 22, NO. 2, APRIL 2014

solving the corresponding LMIs for the aforementioned system, we obtain the maximum bound of the interconnection as α1 = α2 = 43.6145. Thus, comparing the values of the maximum bound of the interconnection, we show that Theorem 1 tenders a less conservative condition than Theorem 2. VI. CONCLUSION In this paper, a decentralized observer-based output-feedback fuzzy controller has been proposed for asymptotic stabilization of the nonlinear large-scale system. It is assumed that the largescale system has unknown interconnection terms, and showed the maximum bound of the interconnection with asymptotic stability of the closed-loop system. By using the Lyapunov functional, sufficient conditions were derived and formulated in the LMI format. Finally, the numerical examples convincingly demonstrated the advantages of the developed method. Fig. 7. Time responses of states of the two inverted pendulum systems with nonmeasurable premise variables: x1 1 (solid), x1 2 (dotted), x2 1 (dashed), and x2 2 (dash–dotted).

Fig. 8. Time responses of errors of the two inverted pendulum systems with nonmeasurable premise variables: e1 1 (solid), e1 2 (dotted), e2 1 (dashed), and e2 2 (dash-dotted).

gains is always satisfied. Time responses of states and errors for each subsystem with the initial conditions xk (0) = ek (0) = [π/3, 0]T are shown in Figs. 7 and 8. From the simulation results, we can know that the proposed technique satisfies an asymptotic stability of the closed-loop large-scale system with the nonmeasurable premise variable and presents the maximum bound of the interconnection. To emphasize the superiority of the proposed algorithm, we compare it with the fuzzy observer-based fuzzy control technique [21] not considering the persistent bounded disturbances. To simulate the control algorithm of [21], all subsystems are united as one system. However, the compared method does not even provide a feasible solution in not only κ = 17.8033, but also κ = 1. From this comparison, we can know that the decentralized control technique is more effective than the centralized one in the large-scale system. In addition by using Theorem 1 and

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Geun Bum Koo received the B.S. degree in electrical and electronic engineering from Yonsei University, Seoul, Korea, in 2007. Since 2007, he has been pursuing the Ph.D. degree with Yonsei University. His current research interests include large-scale systems, decentralized control, sampled-data control, digital redesign, nonlinear control, and fuzzy systems.

Jin Bae Park (M’12) received the B.S. degree in electrical engineering from Yonsei University, Seoul, Korea, and the M.S. and Ph.D. degrees in electrical engineering from Kansas State University, Manhattan, KS, USA, in 1977, 1985, and 1990, respectively. Since 1992, he has been with the Department of Electrical and Electronic Engineering, Yonsei University, where he is currently a Professor. His major research interests include robust control and filtering, nonlinear control, intelligent mobile robot, fuzzy logic control, neural networks, Hadamard transforms, chaos theory, and genetic algorithms. Dr. Park served as the Editor-in- Chief (2006–2010) for the International Journal of Control, Automation, and Systems and the Vice-President (20092011) for the Institute of Control, Robot, and Systems Engineers (ICROS). He is currently serving as the President for the ICROS.

Young Hoon Joo received the B.S., M.S., and Ph.D. degrees in electrical engineering from Yonsei University, Seoul, Korea, in 1982, 1984, and 1995, respectively. He was with Samsung Electronics Company, Seoul, Korea, from 1986 to 1995, as a Project Manager. He was with the University of Houston, Houston, TX, USA, from 1998 to 1999, as a Visiting Professor with the Department of Electrical and Computer Engineering. He is currently a Professor with the Department of Control and Robotics Engineering, Kunsan National University, Kunsan, Korea. His research interests include intelligent robots, intelligent control, human–robot interaction, and intelligent surveillance systems. He served as the President of the Korea Institute of Intelligent Systems from 2008–2009. He has been serving as an Editor for the Intelligent Journal of Control, Automation, and Systems since 2008 and is also currently serving as the Vice President for the Korean Institute of Electrical Engineers.