Int. J. Fuzzy Syst. https://doi.org/10.1007/s40815-022-01296-x Nonlinear Pseudo State-Feedback Controller Design for Affine Fuzzy Large-Scale Systems with H1 Performance Iman Zamani1 • Mohsen Shafieirad2 • Mohammad Manthouri1 • Mohammad Sarbaz1 • Asier Ibeas3 Received: 26 June 2019 / Revised: 11 December 2021 / Accepted: 19 January 2022  The Author(s) 2022 Abstract This paper treats robust controller design for 1 Introduction Affine Fuzzy Large-Scale Systems (AFLSS) composed of Takagi–Sugeno-Kang type fuzzy subsystems with offset Large-scale systems (LSS) have been widely used to terms, disturbances, uncertainties, and interconnections. describe real-world problems, including the internet, eco- Instead of fuzzy parallel distributed compensation, a nomic systems, mobile networks, chemical processes, and decentralized nonlinear pseudo state-feedback is developed electronic power grids [1, 2]. Because of the complexity of for each subsystem to stabilize the overall AFLSS. Using dynamical behaviors in these systems, it is necessary to Lyapunov stability, sufficient conditions with low codem- seek techniques that reduce the complexity of the mathe- putational effort and free gains are derived in terms of matical models and computational effort. Hence, there matrix inequalities. The proposed controller guarantees have been considerable efforts in modeling, analysis, asymptotic stability, robust stabilization, and H 1 control optimization and control of LSS [2, 3], adaptive decen- performance of the AFLSS. A numerical example is given tralized stabilization [4, 5], decentralized H 1 filtering [6], to illustrate the feasibility and effectiveness of the proposed observer-based output feedback control [7], state estima- approach. tion [8, 9], and many approaches have also been presented to investigate their stability, stabilization, and optimization Keywords Affine fuzzy large-scale system (AFLSS)  [10–13]. Decentralized nonlinear controller  Parametric Using fuzzy systems, qualitative knowledge can be uncertainty  H? control performance represented in nonlinear functional forms. Among fuzzy models, the Takagi–Sugeno-Kang (TSK) model can pro- vide a fuzzy representation of complex nonlinear systems. Stability analysis and controller design of fuzzy systems have also been extensively treated [14–19]. Some approa- ches based on the parallel distributed compensation (PDC) design have been reported [19, 20]. In addition, nonlinear & Asier Ibeas state feedback controllers for fuzzy systems [16–22], and asier.ibeas@uab.cat strategies based on fuzzy Lyapunov functions have been Iman Zamani developed [13–16]. Moreover, stability analysis and sta- zamaniiman@shahed.ac.ir bilization of fuzzy large-scale systems (FLSS) have been 1 Electrical and Electronic Engineering Department, Shahed studied for the discrete and continuous time [23–28]. One University, Tehran, Iran can also study the stabilization of the FLSS based on 2 adaptive and observer design methods [29–32]. One of the Department of Electrical and Computer Engineering, University of Kashan, Kashan, Iran most important strategies for controller design is to stabi- 3 lize the system robustly while satisfying H 1 -norm boun- Departament de Telecomunicació i Enginyeria de Sistemes, Escola d’Enginyeria, Universitat Autònoma de Barcelona, ded constraints for fuzzy large-scale systems [33–35]. For Barcelona, Spain continuous-time FLSS with parametric uncertainties, only 123 International Journal of Fuzzy Systems 8 a few results are available for stability analysis and robust > > l IFni1 ðtÞ is Mi1 l and    nini ðtÞ is Min < i stabilization. This could be a result of the complexity of Sli ¼ X J > > THEN x_i ðtÞ ¼ Ali xi ðtÞ þ Bli ui ðtÞ þ Dli di ðtÞ þ ali þ Cijl xj ðtÞ such systems. However, robust stabilization and H 1 con- : j ¼ 1j 6¼ i troller design for FLSS with affine terms have not yet been fully investigated, due to the difficulty of extending exist- ð1Þ ing stability results. where Sli is l th rule of Si , ui ðtÞ 2 Rmi is control input of Si In this paper, we will concentrate our efforts on at time t, xi ðtÞ 2 Rni is state vector of the i th subsystem; asymptotic stability, robust stabilization, and H 1 controller xi ðtÞ ¼ ½xi1 ðtÞ; xi2 ðtÞ;    ; xini ðtÞT , Clij is interconnection design of an affine fuzzy large-scale system (AFLSS) consisting of J interconnected subsystems. To investigate matrix between the i th and j th subsystem of the l th rule of the stabilization of the overall system, each subsystem is Si , r i is number of rules in subsystem Si , ni is number of   decomposed into a set of fuzzy regions, for which a Tak- states in subsystem Si , Ali ; Bli is controllable system agi–Sugeno fuzzy model expresses the dynamical behavior matrices of rule l in subsystem Si , M lik is grade of mem- of the subsystem. The whole large-scale system model is bership of nik ðtÞ; k ¼ 1; 2;    ; ni , nik ðtÞ is known premise obtained by smoothly connecting all subsystems. Sufficient variable. ni ðtÞ is used to denote the vector containing all conditions for asymptotic stability of the overall system are individual elements ni1 ðtÞ  nini ðtÞ; k ¼ 1; 2;    ; ni , ali is   derived using a new decentralized nonlinear pseudo state- constant and deterministic offset term, di ðtÞ; Dli is dis- feedback controller. A positive definite matrix ðPi Þ is turbance and its coefficient matrix of rule l in subsystem Si , shown to satisfy linear matrix inequalities corresponding to where kdi ðtÞk  bi 2 and bi is scalar. The counters varying each sub-system. as i ¼ 1; 2;    ; J; j ¼ 1; 2;    ; J; l ¼ 1; 2;    ; ri . Using a Motivated from the previous work and their shortages in standard fuzzy inference method (product fuzzy inference) fuzzy large scale systems analysis as stated before, a new and also a central-average deffuzzifier, Eq. (1) can be approach has been given for affine fuzzy large-scale sys- obtained as tems with H_? performance in this paper. In comparison with the previous works which all gains for subsystems X ri   x_i ðtÞ ¼ lli ðni ðtÞÞ Ali xi ðtÞ þ Bli ui ðtÞ þ Dli di ðtÞ þ ali must be computed exactly according to all states of the l¼1 system, here it is not required to have all gains and some of X J X ri them can be selected optionally. In addition, the proposed þ lli ðni ðtÞÞCijl xj ðtÞ ð2Þ j¼1 l¼1 method in this paper is applicable with much less compu- tation and does not include restrictive conditions such as j 6¼ i bounded norm. In contrast to the PDC method in which the Qi control law is based on Lyapunov stability, the approach where wi l ðni ðtÞÞ ¼ nk¼1 M lik ðni ðtÞÞ  0 and lli ðni ðtÞÞ ¼ presented in this paper is simpler. There are some other  merits for the proposed method will stated in the wli ðni ðtÞÞ Pri l represents the firing strength of l¼1 wi ðni ðt ÞÞ continuation. The structure of this paper is as follows. Preliminaries the l th rule of the i th subsystem. In this paper, we assume Pr i l and the problem formulation are presented in Sect. 2. In that l¼1 wi ðni ðtÞP Þ [ 0; 8t. Therefore, we have l ri l Sect. 3, decentralized nonlinear pseudo state-feedback li ðni ðtÞÞ  0 and l¼1 li ðni ðtÞÞ ¼ 1; 8t. For stability controller is introduced, and the main results are obtained. purposes, decentralized nonlinear pseudo state-feedback Robust stabilization and H 1 controller design are investi- controller [17, 26] is represented for each subsystem as gated in Sect. 4. A numerical example is presented in Sect. follows: 5. Finally, concluding remarks are presented in Section 6. X ci ui ð t Þ ¼  mki ðtÞKik xi ðtÞ ð3Þ k¼1 2 Preliminaries Pci k where k¼1 mi ðtÞ ¼ 1; 0  mi ðtÞ  1ði ¼ 1; 2;    ; J; k ¼ 1; 2;    ; ci Þ and K ki ’s k Consider the AFLSS consisting of J interconnected sub- are state feedback gains with appropriate dimensions. The systems Si ði ¼ 1; 2;    ; JÞ, each described as follows: mki ðtÞ’s are also nonlinear functions defined as follows; P i PJ   IF rl¼1 j¼1 l l i ð n i ð t Þ ÞH lk ij  0; Then j 6¼ i 123 I. Zamani et al.: Nonlinear Pseudo State-Feedback Controller Design… mki ðtÞð2  k  ci Þ The rule set of the i th subsystem is divided into I i0 and 8 P P   > ri J l lk I i1 . I i0 represents rules that contain the origin and I i1 are the > > > > l¼1 j ¼ 1 li ðni ðtÞÞHij > > remaining rules that do not contain the origin. As a result, if > > Xci X ri XJ   > > j 6¼ i  l  > > Pci Pri PJ    IF  li ðni ðtÞÞHijlh  6¼ 0 l 2 I i0 , then Dli di ðtÞ ¼ 0; ali ¼ 0 which guarantees the triv- > > l l ð n ð t Þ ÞH lh  > > h¼1 l¼1 j ¼ 1 i i ij h¼1 l¼1 j¼1 < ial solution x_i  0 is the origin (i.e. xi  0). In this paper, ¼ j ¼ 6 i j 6¼ i > > > we assume that there is no perturbation for the rule l 2 I i0 . > >   > > 1 Xci X r i X J  l  We remark due to the problem formulation, Dli might be > > IF li ðni ðtÞÞHijlh  ¼ 0 > > c > > i h¼1 l¼1 j ¼ 1 scalar or considered to be a matrix. Now, assume that Z li is > > > > : a bounded region in Rn , where the l th rule of the i th j 6¼ i ð4Þ subsystem fires. For I i1 , one can obtain a hyper-ellipsoid T P containing Z li with definition 1  xli Hc xli þ xT Hc xli þ and m1i ðtÞ ¼ 1  ck¼2 i mk ðtÞ. T   Pr i PJ  i  xli Hc x  xT Hc x  0 such that its parameters xli ; Hli sat- l lk k IF l j¼1 i i ð n ð t Þ ÞH ij \0, mi ðtÞðk 6¼ 1Þ ¼ 0 T l¼1 isfy }1  xli Hli xli \0}, xli represents its center and Hli is a j 6¼ i positive definite matrix that characterizes the hyper-ellip- and m1i ðtÞ ¼ 1. soid [15]. T lh Note that H lh 1 l ij ¼ J1 xi ðtÞ Qi;cont xi ðtÞ  F ij ðtÞ. Also, Definition 1 A Euclidean hyper-ellipsoid with center xc Qlh   i;cont is defined in the next sections according to Theorems and radius r can be defined as E xc ; P1 c ¼ xjð x  xc ÞT P1 c T 1 and 2, and F lij ðtÞ ¼ xTj ðtÞClij Pi xi ðtÞ þ xi ðtÞT Pi Clij xj ðtÞ. In ð x  xc Þ  1g ¼ xj1  xc Hc xc þ x Hc xc þ xc T T T Hc x  here, the ci ’s are parameters for designing the controller of xT Hc x  0g where P1 c ð¼ Hc Þ is a positive definite matrix. each subsystem. So, we can define I i0 and I i1 as I i1 ¼ For readability, arguments ‘‘t’’ and ni ðtÞ are omitted n o T from xðtÞ; mki ðtÞ; F lij ðtÞ; lli ðni ðtÞÞ, and ui ðtÞ. Consequently, l : 1  xli Hli xli \0; 1\l\r i and I i0 ¼ f1; 2; :::; r i gnI i1 . these terms are abbreviated as x; mki ; F lij ; lli , and ui , T The term ‘‘1  xli Hli xli ’’ has been used to analyze the sta- respectively. Using Eqs. (2)-(4), the closed-loop fuzzy bility of an affine fuzzy model as stated later in the proof of subsystem now becomes the theorems in this paper. 0 1 B C Example a: In Fig. 1, it is assumed that Z 3i (the region Xri B l Xci X J C that rule A3 B3 fires) does not contain the origin x_i ¼ lli B A B i x i  Bl i m k k K i i x i þ D l i di ð t Þ þ a l i þ C l C x ij Cj l¼1 @ k¼1 j¼1 A ðð X; Y Þ ¼ ð0; 0ÞÞ. As a result, for a typical hyper-ellipsoid T j 6¼ i that contains Z 3i , }1  xli Hli xli \0}. For example, a typical ð5Þ hyper-ellipsoid can be described as    5 3=81 0 where, by considering Y lk l l k x3i ¼ ; H3i ¼ . i ¼ Ai  Bi K i , we obtain 0:5 0 2 X J Xri Xci  lk l k Yi xi þ Dli di ðtÞ þ ali In summary, the stability conditions for AFLSS (6) can x_i ¼ li mi þ Cijl xj l¼1 k¼1 J1 be presented as follows. j¼1 j 6¼ i ð6Þ Our task is to design the K ki ’s such that the overall AFLSS is asymptotically stable. The stability conditions 1.5 for the AFLSS described by Eq. (6) can be summarized by theorems stated in the following sections. 0 -0.5 3 Stability Problem 0 0.5 5.5 In this section, we consider decentralized controllers for the AFLSS described in Eq. (6). For stabilization, the follow- ing description is required. Fig. 1 Fuzzy rules and hyper-ellipsoids 123 International Journal of Fuzzy Systems   Theorem 1 AFLSS (2) can be made asymptotically T l 1 as Y l1 i P i þ P i Y l1 i þ - l i s i þ 1 Pi -li qli Hli ¼ Ql1 i  stable by nonlinear pseudo state-feedback controllers in (3) if there exist symmetric positive definite matrices’s, Pi’s gli I i , consequently.  l T     positive scalars and gli ; qli ; sli state feedback gains, such Ai  Bli Ki1 Pi þ Pi Ali  Bli Ki1 þ -li sl1 þ 1 Pi i that the following conditions are met:  -li qli Hli   gli Ii ð8Þ  M\0; i ¼ 1; 2;    ; J J[1 ð7aÞ 1 for l 2 I i1 where -li ¼ . Then, by pre-multiplying and 0 for l 2 I i0 gli Ii  Ql1 i ; l ¼ 1; 2;    ; ri ; i ¼ 1; 2;    ; J ð7bÞ post-multiplying both sides of Eq. (8) by P1 i ,letting T i ¼ For l 2 I i1 ; Pi 1 and vi ¼ K i T i , we have Ui þ T i -li qli Hli þ 1 1 l1 Pi T T T  0; gli I i ÞT i  0 in which Ul1 l 1 l l l 1 i ¼ T i Ai  v i B i þ Ai T i  B i v i þ alT þ qli xlT l sli b2i DlT l l l   i P i i Hi i Pi Di þ qi Oi ð7cÞ 1 -li sli þ 1 T i . Now, using the Schur complement we i ¼ 1; 2;    ; J obtain the following where I i is the identity matrix in Rni Rni . Also "  1 # (  l1   -li qli Hli þ gli Ii Qlk lkT lk i ¼ Yi Pi þ Pi Yi þ si þ 1 Pi  qli Hli ; l 2 Ii1 0 ð9Þ Ti Ul1 i Qlk lkT lk i ¼ Y i P i þ P i Yi ; l 2 Ii0 ( then K 1i ¼ v1i T 1 i . Equation (7c) can also be simplified. By   m1i gi ; i¼j pre-multiplying and post-multiplying both sides of (7c) by   M ¼ mij ¼ diag P1 m1i ri kmax ðPi Þdij ; i 6¼ j i ; I i , and using the Schur complement, we obtain 2 3 Ti where for l 2 I i1 , Hli is a positive definite matrix of the 4 alT l l l qli Oli 50 i þ qi Ti xi Hi ð10Þ hyper-ellipsoid that encircles the l th rule region. More- 0 sli bi Dli sli Ti T over, Oli ¼ 1  xli Hli xli , which is negative for l 2 I i1 ,      where T i ¼ P1i . Because the above inequality is now in gi ¼ min gli , dij ¼ max kClij k ni þ nj and l l terms of T i and v1i , we can rewrite Theorem 1 as follows, Qlk lk lk i;cont ¼ Qi . We remark that Qi;cont is used to define the Corollary 1 AFLSS (2) can be stabilized asymptotically parameters of the controllers in Eq. (4). In here, ‘‘ ’’ by nonlinear pseudo state-feedback controllers (3), if there denotes the matrix entries implied by symmetry, and are symmetric positive definite matrices T i ’s, a positive kmax ðPÞ denote the maximum eigen values of matrix P. T scalar set gli ; qli ; sli and vectors v1i ’s such that (7a), (9), Proof: See Appendix. and for l 2 I i1 (10) are satisfied. We remark that the Remark 1: Conditions (7b) and (7c) represent bilinear notations used here are similar to Theorem 1. matrix inequalities (BMIs) that are. By pre-defining scalar Due to Corollary 1, the control synthesis procedure can variables, (7b) and (7c) can be rewritten in terms of LMIs be summarized as the following algorithm. 123 I. Zamani et al.: Nonlinear Pseudo State-Feedback Controller Design… ALGORITHM A Step 1: Divided the rules into 0 and 1 for each subsystems. For 1 , ind a hyper-ellipsoid containing the iring region by its parameters ( ̅ , ). Step 2: Choose an optional number for each subsystems and ind symmetric positive deinite matrices ’s, positive scalars { , , } such (7a), (9) and (10) are held. Step 3: Extract 1 from Step 2 as follows −1 =− − −( + 1) + , ∈ 1 { =− − , ∈ 0 Step 4: Extract ℎ as follows ℎ 1 −1 = ( ) ( + +( + 1) − ) ( )− ( ), ∈ 1 −1 ℎ 1 = ( ) ( + )− ( ), . ∈ 0 { −1 wh ( )= ( ) ( )+ ( ) ( ) Step 5: Extract from Step 2 for = 1, and the remaining gains optional (k=1). Now, construct the ( )(2 ≤ ≤ ) as (4) and the decentralized controller as follows ( ) = −∑ ( ) ( ) =1 3.1 H1 Controller Design for the AFLSS nonlinear pseudo state-feedback controllers (3), the closed- loop system can be described as follows: The H 1 problem is concerned with the design of a con- X J X ri Xci  lk troller that stabilizes a system for which an H 1 -norm Y^i xi þ Dli di ðtÞ þ a^li x_i ¼ lli mki þ Cijl xj bounded constraint on the disturbance attenuation is satis- J  1 j ¼ 1 l¼1 k¼1 fied. Consider the subsystems Si described by the following j 6¼ i rule-based equations. 8 ð12Þ l l > > IFni1 ðtÞ 1mu is Mi1 and    nini ðtÞ is Min > > 8 i lk l l > > > > > > > > x_i ¼ Ai xi þ Bi ui þ Di di ðtÞ þ a^i þ ^l ^l l l XJ where Ybi ¼ Abi  Bbi K ki . In this section, we consider H 1 < > > Cijl xj > < controller design for the AFLSS (11), with nonlinear con- Sli ¼ j¼1 ð11Þ > > THEN > > > > > j¼ 6 i trollers presented in (3). The objective is to design suit- > > > > > > > > able controllers for the AFLSS (11) that guarantee the : : l l yi ðtÞ ¼ Ci xi þ Ei ui performance in the H 1 sense. By specifying a prescribed where di ðtÞ is the square integrable disturbance, dT ðtÞ ¼ level of disturbance attenuation, we determine the decen-  T T tralized fuzzy control law ui ðtÞ such that the induced L2 - d1 ðtÞ; dT2 ðtÞ; . . .; dTJ ðtÞ , yi ðtÞ is the controlled output and  T l norm of the operator from dðtÞ to the controlled output yðtÞ yT ðtÞ ¼ yT1 ðtÞ; yT2 ðtÞ; . . .; yTJ ðtÞ , and Abi ¼ Ali þ DAli ðtÞ; D is less than c, under zero initial conditions, i.e., T l R1 R1 2 Ali ðtÞ ¼ H lai F lai ðtÞLlai ; F lai ðtÞF lai ðtÞ  Rlai , Bbi ¼ Bli þ DBli ðtÞ; 2 2 0 jyðtÞj dt  0 c jdðtÞj dt: We remark that the closed- T DBli ðtÞ ¼ H lbi F lbi ðtÞLlbi ; F lbi a li ¼ ali þ D ðtÞF lbi ðtÞ  Rlbi ,b loop system must be asymptotically stable when dðtÞ ¼ 0. T If such a control law exists, then the system is said to be al ðtÞ; Dali ðtÞ ¼ H lai F lai ðtÞLlai ; F lai ðtÞF lai ðtÞ  Rlai in which ni o stabilizable with H 1 -norm bound c. Now, by removing the Rlai ; Rlbi ; Rlai are symmetric positive matices, argument ‘‘t’’ from yi ðtÞ, and employing the proposed DAli ðtÞ; DBli ðtÞ; Dali ðtÞ represents the system uncertainties controllers, we obtain ! satisfying the norm bounded condition. Hil ¼ ½Hal i ; Hbl i ; Hal i  Xri X ci l l l k k yi ¼ li Ci xi  Ei mi Ki xi and Lli ¼ ½Llai ; Llbi ; Llai  are known constant matrices, l¼1 k¼1 andFlai ðtÞ, F lbi ðtÞ, and F lai ðtÞ belong to Xi set as Xi ¼ X ci X ri    ¼ lli Cil  Eil Kik xi ð13Þ Fi ðtÞjFiT ðtÞFi ðtÞ  Ii ; where elements of Fi ðtÞare Lebe   k¼1 l¼1 sgue measurableg. C li ; Eli are output matrices. Using 123 International Journal of Fuzzy Systems 8 T 1 > >  Qlk lk i;cont ¼ Qi þ D0 ð4; 1Þ ðDiag0 ðÞÞ D0 ð4; 1Þ > >     > < T þ ci ri Cil  Eil Kik Cil  Eil Kik ; for l 2 Ii0 >  Qlk ¼ Qlk þ D1 ð5; 1ÞT Diag1 ðÞ1 D1 ð5; 1Þ þ c2 Pi Dl DlT Pi > 2 > > i;cont i 1 i i 3 > :  T   1 þ ci ri Cil  Eil Kik Cil  Eil Kik  sil1 Pi ; for l 2 Ii1 and D0 ð4; 2Þ; D1 ð5; 3Þ are matrices that are defined as follows: 8 > D0 is a 4 2 block matrix where the first column > > h i > > > > is HalTi Pi ; HblTi Pi ; 2lai Llai ; 2lbi Llbi Kik and the other is zero > > > > 1 > < D1 is a 5 3 block matrix where the first column 3 h i 3 > > is HalTi Pi ; HalTi Pi ; HblTi Pi ; 2lai Llai ; 2lbi Llbi Kik and the others are zero > > > > > > r is a 4 1 block matrix where the first column > > h i Fig. 2 Block diagram of the controller > > : is sli bi Dli ; ali ; qli xlT l 1 l l i Hi Pi ; 2ai Lai The block diagram in Fig. 2 shows the details of the control procedure: Diag11 ðÞ; Diag21 ðÞ and Diag0 ðÞ are diagonal matrices 8 that are defined as > Xri   > > S1 : x_i ðtÞ ¼ lli ðni ðtÞÞ Ali xi ðtÞ þ Bli ui ðtÞ þ Dli di ðtÞ þ ali 8   > > > > l¼1 > > Diag 1 ð Þ ¼ diag 2l R l1 ; 2l R l1 ; 2l R l1 ; 2 l I i ; 2l Ii > > > > 1 a i ai a i bi b i bi a i b i > > X J Xri <   > > < þ lli ðni ðtÞÞCijl xj ðtÞ Diag21 ðÞ ¼ diag sli P1 1 1 i ; Pi ; Pi ; 2ai Ii l > >   j¼1 l¼1 > > > > : Diag ðÞ ¼ diag 2l Rl1 ; 2l Rl1 ; 2l I ; 2l I > > 0 ai ai bi bi ai i bi i > > j 6¼ i > > > > Xci > > and diagðM 1 ;    ; M n Þ is a diagonal matrix such that its > : u1 : u i ð t Þ ¼  mki ðtÞKik xi ðtÞ k¼1 ði; iÞ th entry is M i ði ¼ 1; 2;    ; nÞ. Also, semicolons ð;Þ ð14aÞ are used to separate the rows of a matrix. All notations are similar to Theorem 1. Theorem 2 The AFLSS defined in (12) and (13), is sta- Proof: See Appendix. bilizable with H 1  norm bound c, using nonlinear pseudo state-feedback controllers (3), if there exist symmetric Remark 2: Equations (14b) and (14c) represent bilinear positive definite matrices Pi , positive scalars matrix inequalities (BMIs). For simplicity analogous to n o   Remark 1 and Corollary 1, we can rewrite (14b) and (14c) gli ; lai ; lai ; lbi ; qli , non-negative scalars sli sli  0 , inte- in terms of LMIs with pre-defined scalar variables. We   gers ci and state gains K 1i such that (7a) and the following further remark that, forl 2 Ii1 , because Qlk l 1 i  si Pi is conditions are satisfied: For l 2 I i0 ; independent of sli ; Qlk l i;cont is also independent of si . Thus, for 2 3 (14d), we can set sli ¼ 0 and delete the column and row that gli Ii  Ql1 i 6   7 includes sli ; resulting in a less restrictive condition. 4 ci ri Cil  Eil Ki1 ci ri Ii 5 ð14bÞ Remark 3: For l 2 I i1 , if Dli di ðtÞ 6¼ 0 and di ð0Þ ¼ 0, then D0 Diag0 ðÞ  l   l  Qlk lk T 1 2 l lT l k T i;cont ¼ Qi þ D0 ð4; 1Þ ðDiag0 ð ÞÞ D0 ð4; 1Þ þ c Pi Di Di Pi þ ci r i C i  E i K i C i  Eli K ki ,  0 i ¼ 1; 2;    ; J; k ¼ 1 and (14b) can be rewritten as the following inequality. For l 2 I i1 ; 2 3 gli Ii  Ql1i 2 3 6 T 7 gli Ii  Ql1 6 Dli Pi c2 I 70 4 ci ri  C l  E l K 1  i 6 lT 2 7 5 6  Dl i Pi l 1  c I 7  0; i ¼ 1; 2;    ; J; k ¼ 1 0 ci ri Ii 4 ci r i C  E K 0 ci ri Ii 5 i i i i i i D1 Diag11 ðÞ D0 ð4; 3Þ Diag0 ðÞ ð14cÞ If ð14aÞ  ð14dÞ hold for c, then the latter inequality !li holds is feasible for all attenuation levels bc [ c. The 0; i ¼ 1; 2;    ; J ð14dÞ following theorem denotes the suboptimal solution for the r Diag21 ðÞ H 1 optimal control problem. where 123 I. Zamani et al.: Nonlinear Pseudo State-Feedback Controller Design… Lyapunov–Krasovskii functional method, stability condi- tions, which are less conservative and more applicable than the existing results, have been derived in terms of linear matrix inequalities (LMIs). In the mentioned papers, affine systems have not been considered in fuzzy large-scale systems. Recently, some works have been considered in the field of AFLSS but robust stability, but H 1 controller design, nonlinear controller with more flexibility and low computation as will explain in the continuation, have not been considered. Fig. 3 Membership functions of subsystem S1 In summary, we observe the following: Theorem 3 The suboptimal solution for the H 1 optimal (i). The conditions of Theorems 1, 2 and 3 and control problem can be obtained by solving the following Corollary 1 are satisfied only for Ql1 1 i;cont and K i , minimization problem. and there is no need for Qlk i;cont ð2  k  ci Þ to be 8 > < minimize c positive. Therefore, the K ki ’s are optional 1 1 ð15Þ ðfork 6¼ 1Þ and do not affect stability. As a result, 2 2 2 > : subject to r jyðtÞj dt  r c jdðtÞj dt the number of state feedback gains is greatly 0 0 decreased. We remark that K ki ðk 6¼ 1Þ modify the which can also be stated as follows type of response, i.e., the amount of oscillation, ( damping, settling time, etc. Consequently, there is minimize c ð16Þ greater flexibility in the control design. So, giving subject to ð14aÞ; ð14bÞ; ð14cÞ; and ð14dÞ: a new controller with more optional gains, relaxed conditions are one of the majority of the merits of the paper. In the previous papers, the Remark 4: For comparison, in [35], stability analysis and H 1 performance for AFLSS, all gains for subsys- H 1 controller design of continuous-time non-affine fuzzy tems must be computed exactly according to all large-scale systems by using piecewise Lyapunov functions states of the system. But here it is not required to is considered. Extending the piecewise Lyapunov function have all gains and some of them can be selected approach to the fuzzy large-scale system is the main optionally. advantage of this paper. Reference [25] which was one of (ii). The theorems presented in this paper, and the the pioneer works in this field, has considered a particular proposed method is applicable with much less class of interactions and feedback gains, as the main con- computation. This method does not include tribution of the manuscript and also state feedback con- restrictive conditions such as bounded norm. In troller has been used instead of PDC, in this paper. In addition, the proposed method is not a pre- another work, stability, and stabilization of standard fuzzy designed scheme. That is, it is not necessary to large-scale systems, as the main contribution, based on an check the stability of the predesigned system by existing method (PDC), has been the main motivation for trial and error. considering that article to be published [24]. Stability (iii). In contrast to the PDC method presented [24] in analysis and H 1 controller design of discrete-time standard which the control law is based on Lyapunov fuzzy large-scale systems, based on a commonly used stability for which ensuring V_ð x; tÞ\0, is difficult, method, namely piecewise Lyapunov functions, has been the approach presented in this paper is simpler. the reason for the publication of [36]. Even, a new stabi- (iv). The number of controllers (ci ) is also optional for lization criterion for large-scale T–S fuzzy systems, based Theorem 1 and Corollary 1. However, for Theo- on a commonly used method (PDC), as its main contri- rems and 3, it is obtained via matrix inequalities. bution, has received attention in [37]. In [37], extending By choosing a proper ci , the amount of compu- some widely used methods to uncertain fuzzy large-scale tation including the number of inequalities and the systems have been considered as the main contribution of number of controller gains which have to be the manuscript. In [33], by using some changes in designed is reduced. (v). Considering (7a), we can use show matrix M as follows: 123 International Journal of Fuzzy Systems 30 X11 the uncertainties term in Qlk i;cont , to a reach negative X12 20 amount for the derivative of the Lyapunov function 10 as follows: 8 T 1 > > Qlk lk i;cont ¼ Qi þ D0 ð4; 1Þ ðDiag0 ðÞÞ D0 ð4; 1Þ 0 > >     > > T > > þ ci ri Cil  Eil Kik Cil  Eil Kik ; for l 2 Ii0 < -10 T 1  Qlk lk i;cont ¼ Qi þ D1 ð5; 1Þ Diag1 ðÞ 1 D1 ð5; 1Þ > > -20 > >  l k T   l  > > 2 l lT l þ c Pi Di Di Pi þ ci ri Ci  Ei Ki Ci  Eil Kik > > : -30  sl1 i Pi ; for l 2 Ii1 0 10 20 30 40 50 60 70 80 90 100 time Remark 5: To comparison with some works, by referring Fig. 4 Trajectories of subsystem 1 to some previous studies like [24, 36], and [38] in fuzzy 0 10 1 large-scale systems, we will reach some strong points in m11 0    0 g1 r1 kmax ðP1 Þd12    r1 kmax ðP1 Þd1J B 0 m12    0 C B r2 kmax ðP2 Þd21 B CB g 2    r 2 k max ð P 2 Þd 2J C C this paper. In [24], a Decentralized PDC controller is M¼B . .. . . . CB .. .. .. .. C @ .. . . .. A @ . . . . A designed for a T-S fuzzy large-scale system. It is evident 1 rJ kmax ðPJ ÞdJ1 rJ kmax ðPJ ÞdJ2    0 0    mJ gJ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Controlerparameters Interconnectioneffectmatrix that some essential assumptions are not considered in this paper. Disturbances are not applied and evaluated for these systems. It is obvious disturbance rejection is one of the By using Sylvester’s criterion to check for positive most critical part of designing as an effective controller. definite matrices [21], It is easy to show that these condi- Besides, the computational burden of the paper is not tions are independent of m1i ði ¼ 1; 2;    ; JÞ, and we can suitable for today’s approaches. By noticing [36], we can obtain gi ði ¼ 1; 2;    ; JÞ. All leading minors have to be see that disturbances and uncertainties are not considered positive to guarantee positive-ness of M. For kth leading again in this paper. On the other hand, the computed gains minor ðM k Þ, we have for stabilizing are so big and it will be a negative point  1    m1  0   0 g1 r1 kmax ðP1 Þd12  r1 kmax ðP1 Þd1k  causes more costs. In [38], the author designed an output-  0 m12   r2 kmax ðP2 Þd21 0 g2  r2 kmax ðP2 Þd2k    feedback control problem for a class of switched Takagi– jMk j ¼  . .. .. .. .. .. .. ..   .. . .  . . . . .     Sugeno. Not only did the author not consider disturbances  0 0    m1k  rk kmax ðPk Þdk1 rk kmax ðPk Þdk2  g  k and uncertainties for the system, but also the proposed    g1 r1 kmax ðP1 Þd12    r1 kmax ðP1 Þd1k  algorithm is just applicable for switched systems. It is not   r2 kmax ðP2 Þd21 g2    r2 kmax ðP2 Þd2k   possible to apply this method to affine systems. Here, we ¼ m11 m12    m1k  .. .. ..   . . . ...    proposed a nonlinear Pseudo state-feedback controller for  rk kmax ðPk Þdk1 rk kmax ðPk Þdk2  g  k affine fuzzy large-scale systems with H 1 performance. Using Lyapunov stability, sufficient conditions with low since m11 m12    m1k  0, the only thins has to be checked computational effort and free gains are derived in terms of is   matrix inequalities are a strength part of this work. As it is  g1 r1 kmax ðP1 Þd12    r1 kmax ðP1 Þd1k   mentioned, a prominent part of this paper is that the  r2 kmax ðP2 Þd21 g2    r2 kmax ðP2 Þd2k    .. .. .. ..  [ 08k ¼ 1; 2; 3;    ; J algorithm does not need to compute gains for each sub-  . . . .     rk kmax ðPk Þdk1 rk kmax ðPk Þdk2  gk  system, and due to the proposed example, by computing three gains for six subsystems, the overall system will be stabilized. (vi). In the above method, the hyper-ellipsoid technique Remark 6: To compare with state feedback method, in has been extended to overcome the complexity of this paper a nonlinear pseudo state-feedback controller for AFLSS for designing a new nonlinear controller to affine fuzzy large-scale systems is studied, in which, by guarantee the robust stability and H 1 performance this algorithm the computational burden is declined dra- of the AFLSS. matically. Besides, in state feedback controller, to stabilize (vii). Qlk i;cont is a key parameter in controller design and outputs of the system, the exact states and parameters of also in the stability analysis. But according to the the system must be existed, but in this paper, it is not controller structure and also the firing regions of required to compute all gains in each subsystem and we can rules, they are different for the main system with avoid some of them so that they are considered optional. It and without uncertainties. Consider the (A.6) and is one of the main novelties of the proposed method that by (A.7),we have to consider Qlk lk i;cont ¼ Qi and when having just some of gains the overall system will be the systems has uncertainties, we have to consider 123 I. Zamani et al.: Nonlinear Pseudo State-Feedback Controller Design… 30 X21 4 Numerical Example X22 20 In this section, an example is presented to demonstrate the 10 results of the proposed stabilization procedure for affine 0 fuzzy large-scale systems. Consider the following AFLSS, -10 composed of three subsystems described as. -20 • Subsystem S1 : Rule1: -30 8 0 10 20 30 40 50 60 70 80 90 100 > > X3 > > x_1 ¼ A^11 x1 þ B11 u1 þ D11 d1 ðtÞ þ a11 þ 1 C1j xj time > < 1 3 IF x11 is M1 and x12 is M1 THEN j ¼ 1 > > j 6¼ 1 > > Fig. 5 Trajectories of subsystem 1 > : y1 ¼ C11 x1 þ E11 u1 Rule2: stabilized according to the example. In addition, By con- 8 Pi > > > > x_1 ¼ A^1 x1 þ B1 u1 þ D1 d1 ðtÞ þ a1 þ 2 2 2 2 X 3 2 sidering ui ðtÞ ¼  ck¼1 mki ðtÞK ki xi ðtÞ in the proposed > < j¼1 C1j xj IF x11 is M12 and x12 is M12 THEN > method, we are using a nonlinear averaging method by > > > > j 6¼ 1 : y1 ¼ C12 x1 þ E12 u1 computing mki ðtÞ at instant and we do averaging among Rule3: some state feedback controllers. So mki ðtÞ can be consid- 8 > X 3 > > > x_1 ¼ A^1 x1 þ B1 u1 þ D1 d1 ðtÞ þ a1 þ 3 3 3 3 3 C1j xj ered as varying average coefficients according to the con- > < j¼1 IF x11 is M13 and x12 is M11 THEN ditions of all subsystems. > > j 6¼ 1 > > > : y1 ¼ C13 x1 þ E13 u1 Due to Theorem 3, the control synthesis procedure can be summarized as the following algorithm. ALGORITHM B Step 1: Divided the rules into 0 and 1 for each subsystems. For 1 , ind a hyper-ellipsoid containing the iring region by its parameters ( ̅ , ). Step 2: Extract matrices { 0 , 1 ¸∇, 1 1 (∙), 2 1 (∙), 0 (∙)} according to uncertainties of each subsystems.. Step 3: Solve the following minimization problem. { (14 ), (14 ), (14 ), (14 ). and extract symmetric positive deinite matrices ’s, positive scalars }¸ and gains . Step 4: Extract from Step 3 as follows −1 =− − −( + 1) + , ∈ 1 { =− − , ∈ 0 Step 5: Extract , from Step 3 as follows −1 , = − ∆0 ( 0 (∙)) ∆0 − ( − ) ( − ); ∈ 0 { −1 −1 1 −2 , = − ∆1 ( 1 (∙)) ∆1 − − ( − ) ( − )+ ; ∈ 1 1 Step 6: Extract ℎ as ℎ = −1 ( ) , ( )−( ( ) ( )+ ( ) ( )). Step 7: Extract from Step 3. Now, construct the ( )(2 ≤ ≤ ) as (4) and the decentralized controller as follows ( ) = −∑ ( ) ( ) =1 123 International Journal of Fuzzy Systems where 1 0 1 1:5 0 Ab3 ¼ ; B13 ¼ ; D13 ¼ ; a13 d1 d1 d3 d1 3 d2 1 d2 0:25 1 A^11 ¼ ; A^21 ¼ ; ; A^31 ¼ d2 4 1 d3 0 2 0 1 0 0 1 ¼ ; C131 ¼ ; C 132 ¼ ; C 13 0:5 0:5 0 1 0 0 1 0 4 0 B11 ¼ ; B21 ¼ ; B31 ¼ 1 ; D11 ¼ ; D21 ¼ T 0 0:5 =3 0 1 1 ¼ ; E13 ¼ 2: 0 1 0 0 1 D31 ¼ ; a11 ¼ ; a21 ¼ ; a31 ¼ ; The normalized membership functions of subsystems 2 1 1 0 0 and 3 are as follows: 1 1 0 1 2 0 1 C12 ¼ 1 ; C13 ¼ ; C122 ¼ ; M 12 ð xÞ ¼ M 13 ð xÞ ¼ 17 ðx þ 4Þ; M 22 ð xÞ ¼ M 23 ð xÞ ¼ 17 ð x þ 3Þ. 0 0 2 0 0 1 Here we considered d1 ¼ 0:5; d2 ¼ 0:4; d3 ¼ 0:3 2 0 1 The uncertainties are considered as follows. It is also C13 ¼ l 1 0 assumed that Ab ¼ Al þ H l F l Ll , where i i ai ai ai 2 1 1 0 1 T d1 ¼ h½ð1  0:25%Þð1þ 0:25%Þd 2i 3 3 C12 ¼ ; C13 ¼ ; C11 ¼ ; 2 2 0 0 0 2 1 ¼ =3  10% =3  10% d3 1 T 1 T ¼ ½ð0  0:5%Þð0 þ 0:5%Þ C12 ¼ ; C13 ¼ ; E11 ¼ E12 ¼ E13 ¼ 1 1 0 d4 ¼ ½ð2  40%Þð2 þ 40%Þd5 ¼ ½ð1  15%Þð1 þ 15%Þ The normalized membership functions of subsystem 1 are shown in Fig. 3. and • Subsystem S2 : A11 ¼ 1 2=3 ; A2 ¼ 2 2=3 ; A3 ¼ 1 3 ; A1 Rule1: 1 1 1 2 2 8 1 4 0 2 > b 1 x1 þ B1 u2 þ D1 d2 ðtÞ þ a1 þ P3 1 1 4 1 0 1 ¼ =3 1 ; A22 ¼ < x_2 ¼ A j ¼ 1 C2j xj ; A13 ¼ 2 2 2 2 M12 M22 1 2=3 IF x21 is and x22 is THEN j 6¼ 2 > : y2 ¼ C12 x2 þ E12 u2 2 1 0 1 Rule2: 0 n1 8 > b 2 x2 þ B2 u2 þ D2 d2 ðtÞ þ a2 þ P3 < x_2 ¼ A 2 F 1a1 ¼ diagðn1 ; n2 Þ; F 2a1 ¼ ; F 3a1 2 2 2 2 j ¼ 1 C2j xj n2 0 IF x21 is M22 and x22 is M12 THEN j 6¼ 2 > : y2 ¼ C22 x2 þ E22 u2 0 n4 0 n4 ¼ diagðn1 ; n3 Þ; F 1a2 ¼ ; F 2a2 ¼ ; F 1a3 where. 0 n1 n5 0 1= d4 4 d4 0 0 A^12 ¼ 3 ; A^22 ¼ , ¼ 2 d1 d5 1 0 n2 0:5 0 B12 ¼ ; B22 ¼ , 0 1=3 H 1a1 ¼ 0:25 0 ; H 2a1 ¼ 0:1 0:4 ; H 3a1 1 0 0:1 0 0 1 0 D12 ¼ ; D22 ¼  =3 ; a12 ¼ a22 ¼ 0:25 0 0:15 0 0:5 0 0 ¼ ; H 1a2 ¼ 0 0:4 0 0:25 0 0 5 0 E21 ¼ 1; E22 ¼ 1:5; C21 1 ¼ 1 ; C23 ¼ ; 0:15 0 0 0 1 1 0 0 0 5 H 2a2 ¼ ; H 1a3 ¼ ; L1a1 ¼ ; T 0 0:5 0 0:1 1 0 2 1 1 2 0 0 0:5 C21 ¼ ; C23 ¼ ; C21 ¼ 1= ; 1 0 0 0 0 2 3 L2a1 ¼ L3a1 ¼ L1a2 ¼ L2a2 ¼ L1a3 ¼ T 0 1 1 C22 ¼ : where ni ; i ¼ 1;    ; 5 are random numbers on interval 1 ½1; 1. Now, by using the MATLAB LMI Toolbox, The- Subsystem S3 : orem 3, and assuming c ¼ 2:432, the following solution is Rule1: obtained. 8 > b 1 x3 þ B1 u3 þ D1 d3 ðtÞ þ a1 þ P3 < x_3 ¼ A 1 3 3 3 3 j ¼ 1 C3j xj IF x31 is M13 and x32 is M23 THEN > j 6¼ 3 d12 ¼ 2; d21 ¼ 1; d31 ¼ 2; d13 ¼ 2; d23 ¼ 2; d32 ¼ 4; c1 : y3 ¼ C13 x3 þ E13 u3 ¼ c2 ¼ c3 ¼ 2; K 11 ¼ ½ 68:4747 36:2931  where 123 I. Zamani et al.: Nonlinear Pseudo State-Feedback Controller Design… K 21 ¼ ½ 1:00610:5009 K 12 ¼ ½ 27:6127  6:8022 ; K 13 uncertainties. First, a set of asymptotic stability conditions ¼ ½ 69:5951  44:8869 ; 1a1 ¼ 0:60 was derived for an AFLSS. It was shown that stabilization can be determined by solving a set of matrix inequalities. 2a1 ¼ 0:40; 3a1 ¼ 0:80; 1a2 ¼ 1:0; 2a2 ¼ 0:90; 1a3 ¼ 1:2; g11 ¼ g21 ¼ g31 ¼ 118:0179 Second, this approach was used to stabilize an AFLSS in the presence of parametric uncertainties. For this purpose, a g12 ¼ g22 ¼ 139:4786; g13 ¼ 188:2272; q11 ¼ 128:9055; set of stabilization conditions and H 1 controllers were q31 ¼ 141:5489; P1 ¼ ½ 5:5420 0:39370:39372:8138; P2 ¼ presented. Through these conditions, it was shown that there is no need to determine controller gains by solving 16:626 1:1811 16:626 1:1811 matrix inequalities. It was also shown that these conditions ; P3 ¼ . 1:1811 8:4414 1:1811 8:4414 could be considered as an alternative to BMI or LMI (by predefining decision variables). An example was illustrated Remark 7 According to Algorithms A and B and also as by the proposed control method. stated in Page 9 (part iv), for the system without uncer- tainties, ci is optional for controller design and K 1i are extracted from Theorem 1 although the other K ki are Funding Open Access Funding provided by Universitat Autonoma optional. It is one of the most important merits of this de Barcelona. paper. When we consider uncertainties, ci is extracted from Open Access This article is licensed under a Creative Commons Theorems 2 and 3. In here, it is not optional and in the best Attribution 4.0 International License, which permits use, sharing, case, the minimization problem (16) can be solved with a adaptation, distribution and reproduction in any medium or format, as predefined ci . long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate Remark 8: By proposing this example, it has been shown if changes were made. The images or other third party material in this that the algorithm is entirely practical. Here in this exam- article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not ple, as is evident in Fig. 4, Fig. 5, and Fig. 6, the trajec- included in the article’s Creative Commons licence and your intended tories of the three subsystems are leading to zero and they use is not permitted by statutory regulation or exceeds the permitted are fixed during the time horizon. So, this means that the use, you will need to obtain permission directly from the copyright overall closed-loop system is stable during the time. And holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/. the proposed method is applicable. Remark 9: In this paper, we remark that the other feedback gains K 22 and K 23 are optional and this means by Appendix (Proof of Theorems 1 and 2) having three gains out of six gains this algorithm is able to stabilize the system. By referring to the cost side of the Proof of Theorem 1 Consider the function V ð x; tÞ, for an engineering it means decreasing in costs. This shows the AFLSS in (6), expressed as effectiveness of the proposed approach. P P V ð x; tÞ ¼ Ji¼1 V i ðxi ; tÞ ¼ Ji¼1 xTi Pi xi , where V_i ðxi ; tÞ ¼ x_Ti Pi xi þ xTi Pi x_i . For l 2 I i1 , it can be shown that 5 Conclusion the derivative of V i ðxi ; tÞ along the trajectory of ð6Þ can be written as. This paper was dealt with asymptotic stability, robust sta- X J X ri X ci V_i ðxi ðtÞ; tÞ ¼ lli mki bilization, and H 1 control of AFLSS, where each sub- j¼1 l2Ii1 k¼1 system includes offset terms, disturbances and j 6¼ i  1  T lkT  x Y Pi xi þ DlT lT T lT i di ðt ÞPi xi þ ai Pi xi þ xj Cij Pi xi 30 X31 J1 i i XJ X ri X ci  X32 1  T  20 þ lli mki xi Pi Yilk xi þ xTi Pi di ðtÞDli þ xTi Pi ali þ xTi Pi Cijl xj J  1 j ¼ 1 l2Ii1 k¼1 10 j 6¼ i 0 ðA:1Þ -10 Note that here J [ 1. By using the following inequality -20 for two vectors x and y, 2xT y  xT P1 x þ 1 yT Py in -30 which P [ 0 is a real matrix, we obtain 0 10 20 30 40 50 60 70 80 90 100 T 1 T time Dli di ðtÞPi xi þ xTi Pi di ðtÞDli  sli xi T Pi xi þ sli bi 2 Dli Pi Dli , Then, it follows that Fig. 6 Trajectories of subsystem 1 123 International Journal of Fuzzy Systems X j X ri X ci ri X X ci X J  V_i ðxi ; tÞ  lli mki 1 j¼1 l2Ii1 k¼1 V_i ðxi ; tÞ   lli mki xi T Qlk l i xi  F ij  j6¼i l¼1 k¼1 J1 1  T  lkT x Y Pi þ Pi Yilk þ sl1  lT T l l 2 lT l  T lT T l i Pi xi þ ai Pi xi þ xi Pi ai þ si bi Di Pi Di þ xj Cij Pi xi þ xi Pi Cij xj j¼1 J1 i i 0 j 6¼ i 1 ðA:2Þ X BX  C T Because sli bi 2 Dli Pi Dli  0 and equation (A.2) should be ci B ri X J 1 C ¼ mki B B l l i x i T lk Q x i i  F l C ij C given in terms of matrix inequalities, the following positive k¼1 @ l¼1 j ¼ 1 J1 A scalars are added to (A.2). This implies that each rule of I i1 P l  j 6¼ i l qi is encircled by a hyper-ellipsoid. l2I i1 li J1 1  xTi  T T X ri X J 1 Hli xi þ xTi Hli xli þ xli Hli xi  xli Hli xli Þ where qli is a positive ¼ m1i lli xi T Ql1 i xi  F ij l   J  1 T T scalar and 1  xTi Hli xi þ xTi Hli xli þ xli Hli xi  xli Hli xli is l¼1 j ¼ 1 0j 6¼ i 1 the definition for a hyper-ellipsoid that includes the l th rule (l 2 I i1 ) of the i th subsystem. Therefore, we obtain BX  C X ci B ri X J 1 C X J X ri X ci  mki B B ll i xi T Qlk l i xi  F ij C C V_i ðxi ; tÞ  lli mki k¼2 @ l¼1 j ¼ 1 J  1 A j¼1 l2I i1 k¼1 j 6¼ i j 6¼ i  ðA:5Þ 1   T  l 1 lT xTi Y lk lk i Pi þ Pi Y i þ s i Pi x i þ a i P i x i Because Hlh 1 T lh l J1 ij ¼ J1 xi Qi;cont xi  F ij , (A.5) can be rewrit- T T ten as þ xTi Pi ali þ sli bi 2 Dli Pi Dli Þ þ xTj C lij Pi xi þ xTi Pi C lij xj Xri X J   ql  l T  V_i ðxi ; tÞ   m1i lli Hl1 þ i Oi  xTi Hli xi þ xTi Hli xli þ xli Hli xli ij J1 l¼1 j ¼ 1 ðA:3Þ j 6¼ i 0 0 12 1 X J X ri X ci  T lk l k xi Qi xi B B Pr i PJ C C ¼ li mi þ F lij B @ l¼1 j ¼ 1 lli Hlk J1 B ij A CC j ¼ 1 i1l2I k¼1 ci B C X B i 6¼ j  BP P P  C ðA:6Þ j 6¼ i B ci ri J  l lh C C X ri Xci   T  k¼2 B B h¼1 l¼1 j ¼ 1 li Hij C C T þ lli mki xTi Pi xi þ ali Pi þ qli xli Hli xi @ j 6¼ i A l¼1 k¼1   T  þxi Pi ali þ qli Hli xli þ sli bi 2 Dli Pi Dli þ qli Oli T Therefore, we obtain V_i ðxi ; tÞ   X J ri X X ci  xTi Qlk i xi ¼ lli mki þ F lij j ¼ 1 l2I i1 k¼1 J1 Pr i PJ l l1 m1i l¼1 j ¼ 1 li Hij and as a result j 6¼ i Xri Xci  T Pi  xi j 6¼ i þ lli mki xTi 1 T T T l2I i1 k¼1 ali Pi þ qli xli Hli sli bi 2 Dli Pi Dli þ qli Oli 1 X J J X X ri X J V_ð x; tÞ ¼ V i ðx i ; t Þ   m1i lli Hl1 ij ðA:4Þ i¼1 i¼1 l¼1 j¼1 Here F lij ¼ T xTj Clij Pi xi þ xTi Pi Clij xj and Qlk are as defined 0 j 6¼ i 1 i Pi in Theorem 1. Now, let T ali Pi T þ qli xli Hli T sli bi 2 Dli Pi Dli þ qli Oli 0 which BX C PJ Pri Pci   X J B ri X ri X J C gives V_i ðxi ; tÞ  j ¼ 1 l¼1 k¼1 li mi  J1 xi Qi xi þ F ij . l k 1 T lk l Then, ¼ m1i B B lli xi T Ql1 i xi  lli F lij C C: j 6¼ i i¼1 @ l¼1 l¼1 j¼1 A we conclude that (7c) implies (A.4). We remark that since j 6¼ i Dli di ðtÞ ¼ 0; ali ¼ 0 for l 2 I i0 , there is no need to check ðA:7Þ (A.4) for l 2 I i0 . Thereby, for both l 2 I i0 and l 2 I i1 ; we have 123 I. Zamani et al.: Nonlinear Pseudo State-Feedback Controller Design… l1 It is clear that Using (7b), we obtain lli xi T Qi xi   lli gi I i kxi k2 . Pi l   Z 1 Because rl¼1 li ¼ 1 and gi ¼ min gli , we conclude that dV ð x; tÞ l Jt  yT ðtÞyðtÞ  c2 dðtÞT dðtÞ þ dt P i l T l1 Pi l dt  rl¼1 li xi Qi xi   gi kxi k2 rl¼1 li ¼ gi kxi k2 . Also Z 01 X J  dV i ð x; tÞ X ri X ri  T    ¼ y i T y i  c2 d i ð t Þ T d i ð t Þ þ dt lli F lij ¼ lli xTj Clij Pi xi þ xi T Pi C lij xj  r i kPi xi kkClij xj k ni þ nj 0 i¼1 dt l¼1 l¼1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðA:10Þ   T    ri Clij xj C lij xj T kmax ðPi Pi Þkxi kkxj k ni þ nj  r i For l 2 I i1 , using yi expressed in (13), ð A.10) becomes    X J (" Xci X ri    #T " Xci X ri    # ) kxi kkxj kkPi k2 max kC lij k ni þ nj r1 0 i¼1 8 k¼1 l2Ii1 lli mki Cil  Eil Kik xi k¼1 l2Ii1 lli mki Cil  Eil Kik xi  c2 di ðtÞT di ðtÞ dt 9 l > > > > > > > > J > X < X J ri X X ci  T lkT > = ðA:8Þ þ r1 0 i¼1 > > > l2Ii1 k¼1 lli mki xi Y^i Pi xi þ DlT J1 ^lT i di ðt ÞPi xi þ ai Pi xi xT Pi Y^ilk xi þ xTi Pi di ðtÞDli þ xTi Pi a^li þ xTj CijlT Pi xi þ i J1 þ xTi Pi Cijl xj > > > dt > > j¼1 > > : ; j 6¼ i where ni and nj are the number of states in the i th and j th ðA:11Þ subsystems, respectively. Now, based on (A.7) and the above inequalities, it is clear that Similar to (A.2), by using the inequality 0 1 2xT y  xT P1 x þ 1 yT Py, for the two vectors x and y, PJ B P C V_ð x; tÞ  i¼1 m1i @gi kxi k2 þ Jj ¼ 1 r i kxi kkxj kkPi k2 dij A we obtain T T xTi Pi d i ðtÞDli þ Dli di ðtÞPi xi  c2 xTi Pi Dli Dli Pi xi þ   j 6¼ i c2 d i ðtÞT di ðtÞ. Similar to (A.2)-(A.3), we conclude that  R 1 PJ wheredij ¼ max kClij k ni þ nj ,kPi k2 ¼ kmax ðPi Þ ¼ 1=kmin ðTi Þ, Jt  0 i¼1 fU i þ V i þ W i gdt, in which l P  P " #T " # and T i 1 ¼ Pi . ThereforeV_ðx; tÞ  Ji¼1 m1i gi kxi k2 þ Jj ¼ 1 X ci X ri  l  X ci X ri  l  Ui ¼ lli mki C i  Eli K ki xi lli mki C i  Eli K ki xi r i k xi kkxj kkmax ðPi Þdij Þ. The right-hand side this inequality k¼1 l2I i1 k¼1 l2I i1 is quadratic in terms off kx1 k kx2 k    kxJ k g, and 0   1 j X X ri X ci xTi Y^ilkT Pi þ Pi Y^ilk þ c2 Pi Dli DlT l l l1 l lT l i Pi þ Pi  qi Hi þ 2ai Pi Hai Rai Hai Pi xi can be rewritten as mi ¼ j¼1 l2Iil k¼1 lli mki @ J1 þ xTj CijlT Pi xi þ xTi Pi Cijl xj A j6¼i ½ kx1 k kx2 k    kxJ k  M ½ kx1 k kx2 k    kxJ k T . Now, if ð A:7Þ is negative and (7b)-(7c) hold, then we X ri X ci   T T    T  Wi ¼ lli mki xTi Pi xi þ ali Pi þ qli xli Hli xi þ xTi Pi ali þ qli Hli xli þ qli Olci þ lai Llai Llai obtainV_ð x; tÞ\0. This procedure was considered l2I i1 k¼1 P i Pri PJ    l lh  ðA:12Þ when ch¼1 l¼1 j ¼ 1 i ij  6¼ 0. In the case that  l H j 6¼ i Now, from (A.1)-(A.4), we recall that P ci P r i P J   Pr i PJ  l lh  j X X ri X ci h¼1 l¼1 j¼1  li ij  ¼ 0 H and l¼1 j¼1 V_i ðxi ; tÞ  lli mki j¼1 l2Ii1 k¼1 j 6¼ i j 6¼ i     j6¼i l lk k 1 l 1 li H ij  0 in (4), we obtainmi ¼ ci ,, but sinceli J1 xi T 1 T  ^lkT   x Y Pi þ Pi Y^ilk þ sil1 þ 1 Pi  qli Hli xi J1 i i Qlk l i xi  F ij Þ ¼ 0, the above procedure is much simpler and  Xri X ci  þxTj CijlT Pi xi þ xTi Pi Cijl xj þ lli mki xTi Pi xi results _ tÞ  0. inVðx; In the case of l2Ii1 k¼1 Pr i PJ      l   l lk þ a^lT l lT l T ^i þ qli Hli xli þ sli b2i DlT i Pi þ qi xi Hi xi þ xi Pi a l l l i Pi Di þ qi Oi l¼1 j ¼ 1 li H ij \0; in (4), the result is easy to ðA:13Þ j 6¼ i obtain, although the derivation is omitted here. By using lk l l Pi where Ybi ¼ Abi  Bbi K ki . Therefore V_i ðxi ; tÞ  rl2I LaSalle’s principle, since the limit set includes only the P ci   T i1 l k lT T l l l l trivial trajectoryx  0, the origin is asymptotically stable. k¼1 li mi ai Pi xi þ xi Pi ai þ H ai F ai ðtÞLai Pi xi þ Thus the proof is complete. xTi Pi H lai F lai ðtÞLlai Þ þ Qi where Qi are the remaining terms Proof of Theorem 2 Consider the following cost function of (A.13). We know the for the given matrices Q; H; R; E of for AFLSS expressed in (12) and (13). appropriate dimensions, with Q ¼ QT ; R ¼ RT and R [ 0; Z 1  then Q þ HFE þ ET F T H T \0 for all F satisfying F T F\R, Jt ¼ jyðtÞj2  c2 jdðtÞj2 dt if and only if there exists some  [ 0 such that Z 01  dV ð x; tÞ Q þ HH T þ 1 ET RE\0. It can be concluded that ¼ yT ðtÞyðtÞ  c2 dðtÞT dðtÞ þ dt 0 dt V_i ðxi ; tÞ\0 if and only if the following inequality holds:  V ð xð1Þ; 1Þ ðA:9Þ 123 International Journal of Fuzzy Systems  T  X ri X ci 1 T T lli mki ali Pi xi þ xTi Pi ali þ lai xTi Pi H lai Rlai H lai Pi xi þ lai Llai Llai þ Qi  0 Because 0  lli  1; 0  mki  1, it is clear that l2I i1 k¼1 R 1 PJ n P ci P r i Jt  0 i¼1 ci r i k¼1 l2I i1 ðA:14Þ  l k 2 T  l   l  l k T li mi xi Ci  Ei K i C i  Eli K ki xi þ V i gdt. We know that results in that for given matrices Q; H; R; E of appropriate dimen- J X X ri X ci V_i ðxi ; tÞ  j¼1 l2Ii1 k¼1 lli mki sions, with Q ¼ QT ; R ¼ RT and R [ 0; then Q þ HFE þ 0  j6¼i  xTi Y^ilkT Pi þ Pi Y^ilk þ sl1  þ 1 Pi  qli Hli þ 2l1 l l lT  1 ET F T H T \0 for all F satisfying F T F\R, if and only if i ai Pi Hai Rai Hai Pi xi @ þ xTj CijlT Pi xi þ xTi Pi Cijl xj A J1 there exists some [0 such that   T 1 T þ J X X ri X ci  lli mki xTi Pi xi þ alT l lT l  T  l l l l  l 2 lT l l l l lT l i Pi þ qi xi Hi xi þ xi Pi ai þ qi Hi xi þ si bi Di Pi Di þ qi Oi þ 2ai Lai Lai Q þ HH þ  E RE\0. Using this and analogous to j¼1 l2Ii1 k¼1 j6¼i (A.5)-(A.8), if the following inequality holds ðA:15Þ T T T l Ali Pi þ Pi Ali  K ki Bli Pi  Pi Bli K ki þ Pi  qli Hi The second term of (A.15), can also be written as 1 T T þ lai Pi H lai Rlai H lai Pi þ lai Llai Llai  T T    xTi Pi xi þ ali Pi þ qli xli Hli xi þ xTi Pi ali þ qli Hli xli 1 T T T þlai Pi H lai Rlai H lai Pi þ lbi K ki Llbi Llbi K ki T T þ sli bi 2 Dli Pi Dli þ qli Oli þ lai Llai Rlai Llai þ 1 T T lbi Pi H lbi Rlbi H lbi Pi þ c2 Pi Dli Dli Pi   T  l  ¼ xTi  1 T T Pi T T T xi þ ci r i C li  Eli K ki C i  Eli K ki   gli I i ðA:21Þ ali Pi þ qli xli Hli sli bi 2 Dli Pi Dli þ qli Oli þ lai Llai Llai 1 ðA:16Þ we obtain U i þ V i  0. Now considering (A.21) as T  1 Qlk lk i;cont ¼ Qi þ D1 ð5; 1Þ Diag11 ð Þ D1 ð5; 1Þ þ Using the Schur complement, if the following inequality T  T  l  1 holds, then W i becomes negative c2 Pi Dli Dli Pi þ ci r i C li  Eli K ki C i  Eli K ki  sli Pi Pi   gli I i , and by continuing the same procedure, and using T T T  0ðA:17Þ the following inequality, we obtain J t  0. ali Pi þ qli xli Hli qli Oli þ lai Llai Llai 2 3 gli Ii  Qlk i Now by considering (A.13)-(A.17), we conclude that 6 lT 2 7 6  Dl i Pi l k  c I 7  0 ; l 2 Ii1 ; k ¼ 1 4 ci ri C  E K 0 ci ri Ii 5 !li i i i  0 ; l 2 Ii1 where !li ¼ D1 ð5; 3Þ Diag11 ðÞ rð4; 1Þ Diag21 ðÞ  T ðA:22Þ T T qli Oli þ qli ali Hli xli þ qli ali Hli xli implies (A.17). By set- For l 2 I i0 , the procedure is simpler and similar to ting sli ¼ 0 inhere, we can obtain a less restrictive (A.19)-(A.22). Consequently, it yields condition. Consequently, if the following inequality holds T T T T then we obtainW i  0. gli I i þ Ali Pi þ Pi Ali  K ki Bli Pi  Pi Bli K ki þ lai Llai Llai 1 T T T !li þ lai Pi H lai Rlai H lai Pi þ lbi K ki Llbi Llbi K ki 0; l 2 Ii1 ðA:18Þ 1 T rð4; 1Þ Diag21 ðÞ þ lbi Pi H lbi Rlbi H lbi Pi  T  l  Pr i   þ ci r i C li  Eli K ki C i  Eli K ki  0 Now, By defining vlk i ¼ l2I i1 lli mki Cli  Eli K ki xi , we obtain ðC:37Þ X ci X ri   X ci Let Qlk lk T 1 i;cont ¼ Qi þ D0 ð4; 1Þ ðDiag0 ð ÞÞ D0 ð4; 1Þþ lli mki C li  Eli K ki xi ¼ vlk i ðA:19Þ  T  l  k¼1 l2I i1 k¼1 ci r i Cli  Eli K ki C i  Eli K ki . To satisfy this latter inequality, the following matrix inequality should hold. Chebyshev inequality indicates 8vi 2 Rn n we have 2 3  Pm  T  Pm  Pm l lk T gi Ii  Qi  i¼1 vi i¼1 vi  m i¼1 vi vi : Using Chebyshev 4 ci ri Ck  Ek K k i i i ci ri Ii 50 l 2 Ii0 ; k ¼ 1 inequality, we have D0 ð4; 2Þ Diag0 ðÞ !T ! " #T " #! X ci X ci X ci X ri   X ri   ðC:38Þ vlk i vlk i  ci lli mki Cli  Eli K ki xi lli mki C li  Eli K ki xi k¼1 k¼1 k¼1 l2I i1 l2I i1 With regards to remarks of Theorem 2, we obtain J t  0; X ci X ri  2  T  l  and the proof is complete.  ci r i lli mki xi T C li  Eli K ki C i  Eli K ki xi k¼1 l2I i1 ðA:20Þ 123 I. Zamani et al.: Nonlinear Pseudo State-Feedback Controller Design… References 20. Zamani, M.H.: Zarif, On the continuous-time Takagi-Sugeno fuzzy systems stability analysis. Appl. Soft Comput. 11(2), 1. Siljak, D.D.: Large-Scale Dynamic Systems: Stability and 2102–2116 (2011) Structure. Elsevier North-Holland, New York (1978) 21. Zamani, N., Sadati, M.H.: Zarif, On the stability issues for fuzzy large-scale systems. Fuzzy Sets Syst. 174(1), 31–49 (2011) 2. Sadati, N., Ramezani, M.H.: Optimization of large-scale systems using gradient-type interaction prediction approach. Electr. Eng. 22. Zamani, M.H.: Zarif, Nonlinear controller for fuzzy model of 91(4–5), 301–312 (2009) double inverted pendulums, World Academy of Science, Engi- 3. Sadati, N., Ramezani, M.H.: Novel interaction prediction neering and Technology, International. Journal of Electrical and Computer Engineering 1(10), 1588–1594 (2007) approach to hierarchical control of large-scale systems. IET Control Theory and Application 2(4), 228–243 (2010) 23. Zamani, N. Sadati, Fuzzy large-scale systems stabilization with 4. Xiaohua, L., Xiaoping, X., Bo, L.Y.: Adaptive neural network nonlinear state feedback controller, in: Proceedings of IEEE decentralized stabilization for nonlinear large scale intercon- International Conference on Systems, Man and Cybernetics, nected systems with expanding construction. J. Franklin Insti. SMC, 2009, pp. 5156–5161. 354(1), 233–256 (2017) 24. Wang, W.J., Lin, W.W.: Decentralized PDC for large-scale T-S 5. Yang, Y., Yue, D., Xue, Y.: Decentralized adaptive neural output fuzzy systems. IEEE Trans. Fuzzy Syst. 13(6), 779–786 (2005) feedback control of a class of large-scale time-delay systems with 25. Wang, W.J., Luoh, L.: Stability and stabilization of fuzzy large- input saturation. J. Franklin Inst. 352(5), 2129–2151 (2015) scale systems. IEEE Trans. Fuzzy Syst. 12(3), 309–315 (2004) 6. Zhong, Z., Fu, S., Hayat, T., Alsaadi, F., Sun, G.: Decentralized 26. Phong, V.V., Wang, W.J.: Polynomial controller synthesis for piecewise H? fuzzy filtering design for discrete-time large-scale uncertain large-scale polynomial TS fuzzy systems. IEEE trans- actions on cybernetics (2019). https://doi.org/10.1109/TCYB. nonlinear systems with time-varying delay. J. Franklin Inst. 352(9), 3782–3807 (2015) 2019.2895233 7. Zhong, Z., Zhu, Y.: Observer-based output-feedback control of 27. M. Hosseinzadeh, N. Sadati, I. Zamani, H? disturbance attenu- large-scale networked fuzzy systems with two-channel event- ation of fuzzy large-scale systems, in: Proceedings of IEEE triggering. J. Franklin Inst. 354(13), 5398–5420 (2017) International conference on Fuzzy Systems 2011, 2364–2368. 8. Leong, W.Y., Trinh, H.: An LMI-based functional estimation 28. Younsi, E., Benzaouia, L.A., Hajjaji, A.E.: Decentralized control scheme of large-scale time-delay systems with strong intercon- design for switching fuzzy large-scale T-S systems by switched nections. J. Franklin Inst. 353(11), 2482–2510 (2016) lyapunov function with H? performance. Int. J. Fuzzy Syst. 9. Sun, Y., Fu, M., Wang, B., Zhang, H.: Distributed dynamic state 21(4), 1104–1116 (2019) estimation with parameter identification for large-scale systems. 29. Qiang, Z., Zhai, D., Dong, J.: Observer-based adaptive fuzzy J. Franklin Inst. 354(14), 6200–6216 (2017) decentralized control of uncertain large-scale nonlinear systems with full state constraints. Int. J. Fuzzy Syst. 21(4), 1085–1103 10. Wenqiang, J., Fu, S., Chen, H., Qiu, J.: Asynchronous decen- tralized fuzzy observer-based output feedback control of non- (2019) linear large-scale systems. Int. J. Fuzzy Syst. 21(1), 19–32 (2019) 30. Huang, Y.S., Wu, M., He, Y., Yu, L.L., Zhu, Q.X.: Decentralized 11. Zhao, J., Lin, C., Huang, J.: Decentralized H? sampled-data adaptive fuzzy control of large-scale non-affine nonlinear systems by state and output feedback. Nonlinear Dyn. 69(4), 1665–1677 control for continuous-time large-scale networked nonlinear systems. Int. J. Fuzzy Syst. 19(2), 504–515 (2017) (2012) 12. Win, K.N., Chen, J., Chen, Y., Fournier-Viger, P.: PCPD: A 31. Moradvandi, I., Shahrokhi, M., Malek, S.A.: Adaptive fuzzy parallel crime pattern discovery system for large-scale spa- decentralized control for a class of MIMO large-scale nonlinear tiotemporal data based on fuzzy clustering. Int. J. Fuzzy Syst. state delay systems with unmodeled dynamics subject to (2019). https://doi.org/10.1007/s40815-019-00673-3 unknown input saturation and infinite number of actuator failures. 13. Emamzadeh, M.M., Sadati, N., Gruver, W.A.: Fuzzy-based Inf. Sci. 475, 121–141 (2019) interaction prediction approach for hierarchical control of large- 32. Tong, S., Huo, B., Li, Y.: Observer-based adaptive decentralized scale systems. Fuzzy Sets Syst. 329, 127–152 (2017). https://doi. fuzzy fault-tolerant control of nonlinear large-scale systems with org/10.1016/j.fss.2017.05.018 actuator failures. IEEE Trans. Fuzzy Syst. 22(1), 1–15 (2014) 14. Wang, H., Yang, G.H.: Decentralized dynamic output feedback 33. Liu, X., Zhang, H.: Delay-dependent robust stability of uncertain fuzzy large-scale systems with time-varying delays. Automatica control for affine fuzzy large-scale systems with measurement errors. Fuzzy Sets Syst. 1(314), 116–134 (2017) 44(1), 193–198 (2008) 15. K. Zhu (2006) Stability analysis and stabilization of fuzzy state 34. Chang, W., Wang, W.J.: Fuzzy control synthesis for fuzzy large- space models, PhD thesis, Dept. of Mathematics, Duisburg-Essen scale systems with weighted interconnections. Control theory and University application, IET 7(9), 1206–1218 (2013) 16. Lam, H.K., Leung, F.H., Tam, P.K.S.: Nonlinear state feedback 35. Zhang, H., Li, C., Liao, X.: Stability analysis and H? controller controller for nonlinear systems: stability analysis and design design of fuzzy large-scale systems based on piecewise Lyapunov based on fuzzy plant model. IEEE Trans. Fuzzy Syst. 9(4), functions. IEEE Trans. Syst. Man Cybern. B Cybern. 36(3), 657–661 (2001) 685–698 (2005) 17. Sonbol, I.H., Sami Fadali, M.: TSK fuzzy systems type II and 36. Zhang, H., Feng, G.: Stability analysis and H? controller design type III stability analysis: continuous case. IEEE Trans. Syst. of discrete-time fuzzy large-scale systems based on piecewise Lyapunov functions. IEEE Trans. Syst. Man Cybern. B Cybern. Man Cybern. B Cybern. 36(1), 2–12 (2006) 18. Zamani, M. H. Zarif, S. R. Musawi, A new approach to relaxed 38(5), 1390–1401 (2008) stability conditions of fuzzy control systems, in: Proceedings of 37. Lin, W.W., Wang, W.J., Yang, S.H.: Anovel stabilization crite- Conference on of Control, Automation and Systems ICCAS’07, rion for large-scale T-S fuzzy systems. IEEE Trans. Syst. Man Cybern. B Cybern. 37(4), 1074–1079 (2007) 2007, pp. 126–131. 19. Zamani, M. Shafie, Fuzzy affine impulsive controller, in: Pro- 38. Wang, T., Tong, S.: Decentralised output-feedback control design ceedings of IEEE International Conference on Fuzzy Systems for switched fuzzy large-scale systems. Int. J. Syst. Sci. 48(1), FUZZ-IEEE, 2009, pp. 361–366. 171–181 (2017) 123 International Journal of Fuzzy Systems Iman Zamani was born in Iran. Control, Adaptive Control Systems, Neural Networks, Deep Learn- He received the PH.D. from ing, and Machine Learning. Amirkabir University of Tech- nology (Tehran Polytechnic), Mohammad Sarbaz was born Tehran, Iran. Now, he is assis- in Tehran, Iran, on October 26, tant professor of the Department 1993. He received the B.Sc. in of Control Systems, Shahed control engineering from Isla- University, Tehran, Iran. His mic Azad University, south research interests include branch, Tehran, Iran, in 2016. Hybrid Systems, Biological He received the M.Sc. in control Systems, Fuzzy Systems, and engineering from Shahed Singular Systems. University, Tehran, Iran. His research interests include Model Predictive Control, Large-Scale Systems, Time-Varying Delay, Fuzzy Control, and Neural Networks. Mohsen Shafieirad received the PH.D. from Amirkabir University of Technology Asier Ibeas was born in Bilbao, (Tehran Polytechnic), Tehran, Spain, on July 7, 1977. He Iran. Currently, he is assistant received his MSc degree in professor of control engineering Applied Physics and his PhD at University of Kashan, Iran. degree in Automatic Control His research interests include from the University of the Bas- System Identification, Multi- que Country, Spain, in 2000 and agent Systems and bBiological 2006, respectively. He is cur- systems. rently Associate Professor of Control Systems at Autonomous University of Barcelona, Spain. His research interests include time-delayed systems, robust adaptive control, applications of Mohammad Manthouri was artificial intelligence to control born in Khoy, Iran, on October systems design and nonconventional applications of control such as to 12, 1983. He received the B.Sc. epidemic systems, supply chain management and financial systems, in control engineering from fields where he has published more than 130 contributions in inter- Shiraz University, Shiraz, Iran, national journals and conferences. in 2005. He received the M.Sc. and Ph.D. degrees in control engineering from K. N. Toosi University of Technology, Teh- ran, Iran, in 2008 and 2015, respectively. Now he is assistant professor of the Department of Control Systems, Shahed University, Tehran, Iran. His research interests include Fuzzy 123