Int. J. Fuzzy Syst. https://doi.org/10.1007/s40815-021-01217-4 Hierarchical Optimization-Based Model Predictive Control for a Class of Discrete Fuzzy Large-Scale Systems Considering Time-Varying Delays and Disturbances Mohammad Sarbaz1 • Iman Zamani1 • Mohammad Manthouri1 • Asier Ibeas2 Received: 15 April 2021 / Revised: 27 August 2021 / Accepted: 30 September 2021  The Author(s) 2022 Abstract In this manuscript, model predictive control for (LMIs) method is performed for our computations. So the class of discrete fuzzy large-scale systems subjected to closed-loop large-scale system is asymptotically stable. bounded time-varying delay and disturbances is studied. Ultimately, by two examples, the effectiveness of the The considered method is Razumikhin for time-varying proposed method is illustrated, and a comparison with delay large-scale systems, in which it includes a Lyapunov other papers is made by remarks. function associated with the original non-augmented state space of system dynamics in comparison with the Kra- Keywords Time-varying delay  Large-scale systems  sovskii method. As a rule, the Razumikhin method has a Model predictive control  Hierarchical optimization  perfect potential to avoid the inherent complexity of the Input-to-state stability  Lyapunov-Razumikhin  Fuzzy Krasovskii method especially in the presence of large Takagi–Sugeno systems delays and disturbances. The considered large-scale system in this manuscript is decomposed into several subsystems, each of which is represented by a fuzzy Takagi–Sugeno (T- 1 Introduction S) model and the interconnection between any two sub- systems is considered. Because the main section of the Large-scale systems have become interesting in control model predictive control is optimization, the hierarchical engineering. Many systems become large in scope and scheme is performed for the optimization problem. Fur- complex in scale since decades ago and coping with these thermore, persistent disturbances are considered that robust systems requires time and efforts since many problems are positive invariance and input-to-state stability under such appeared like existence of disturbances, nonlinear param- circumstances are studied. The linear matrix inequalities eters and complex dynamic. Many algorithms have pro- posed for large-scale systems since years ago [1, 2]. The decentralized controller is an interesting approach that & Mohammad Manthouri scholars aimed to solve these problems [3]. In this mmanthouri@shahed.ac.ir approach, the considered large-scale system is decomposed & Asier Ibeas into several subsystems and a controller is applied to each asier.ibeas@uab.cat subsystem separately, and the interconnections between Mohammad Sarbaz subsystems are considered. So the overall system is stabi- mohammad.sarbaz@shahed.ac.ir lized by adopting this approach. In [4], a decentralized Iman Zamani fault-tolerant tracking control is applied to a large-scale zamaniiman@shahed.ac.ir system with sensor and actuator faults that its problem is 1 decomposed into several problems. The decentralized Electrical and Electronic Engineering Department, Shahed University, Tehran, Iran adaptive fuzzy controller is proposed for a class of uncer- 2 tain nonlinear large-scale systems considering random Departament de Telecomunicació i Enginyeria de Sistemes, Escolad’Enginyeria, Universitat Autònoma de Barcelona, sensor delays and random sensor nonlinearities in [5]. By Barcelona, Spain using a group of sensor blocks two new distributed data- 123 International Journal of Fuzzy Systems driven optimal fault detection approaches are applied to a linear delay difference equations in [21]. The stability of large-scale system in [6]. delay coupled systems with hybrid switching diffusions As it was mentioned above, one of the important issues in based on Razumikhin method is studied in [22] and it com- large-scale system is its complex dynamic. To design a bines Lyapunov method with graph theory for typical sys- suitable controller and solve problems, it is significant to tems. To stabilize these systems that are involved time- have the exact model of the system. But it is almost impos- varying delay, many control systems have been applied like sible when the system has many nonlinear parameters and adaptive, fuzzy, robust, or so. One of the most interesting severe dynamic. In this case, scientists use some methods controller has been model predictive control. like fuzzy logic or neural networks to identify the model of Model predictive control (MPC) has attended popularity the system [7, 8]. For instance, in [9], the neural network is as a reliable control approach. By properly using the system used to estimate the inverse dynamic of the da Vinci surgical model to predict the output response, MPC methods allow to robot to enables estimation of the external environment choose the optimal control action that minimizes a desired forces. In this paper, since the considered system is large- cost function. In many applications, MPC has been recog- scale, the fuzzy model of the systems is used instead of the nized as a viable alternative to many other classic schemes dynamic model. Nowadays, fuzzy systems with IF–THEN based. For such applications, the advantages introduced by rules have become popular and useful approach for systems MPC are the higher dynamic performance, the possibility of with large, complex, and nonlinear relations and several performing a multivariable controller design, the inclusions researches have been done [10–13. Introduces a type of fuzzy of constraints on input and output variables, and the possi- inference system well-known Takagi–Sugeno, fuzzy model, bility of including nonlinearities in both the model and the and model complex nonlinear system to arbitrary degrees of constraints. Although the need for an accurate model may accuracy. On the other hand, another vital issue that is con- represent a drawback in some applications, it is remarked sidered in this paper is time-varying delay that is unavoidable how reliable model of systems are usually available for problem. The time delay phenomena can be seen in many control design [23, 24]. The gist of the model predictive real and industrial systems [14], and in many cases it causes control is an optimal control sequence, which is computed by instability and might lead outputs to unexpected goal. Since minimizing a finite horizon cost function at each sampling the time delay causes instability, attentions have been time. Studying on nonlinear robust model predictive control attracted for many years. Totally, Lyapunov theory for sys- for Takagi–Sugeno fuzzy systems can be seen in [25]. As tems with time-varying delay can be proposed into two mentioned in [26, 27], invariant set theory can provide suf- approaches, the Lyapunov–Krasovskii functional (LKF) and ficient conditions to make sure the recursive feasibility and Lyapunov-Razumikhin function (LRF) [15]. For discrete the closed-loop system stability. In [28] both online and off- time systems, the Krasovskii method makes use of an aug- line robust fuzzy model predictive control with structured mentation of the state vector with all delayed states, which uncertainties and persistent disturbances is studied. In [23] yields the application of classical Lyapunov methods to an robust fuzzy model predictive control with nonlinear local augmented system without delay. In the Krasovskii method, models is introduced, in which, for both nonlinear and linear an augmentation of the state vector with all delayed states are states of system, separated controllers are designed. In [29], used, which yields the applications of classical Lyapunov an LMI-based MPC is proposed for a large-scale system with methods to an augmented system without delay. So the disturbances and uncertainties. Obviously, it can be seen that computational burden in this method will be heavy, espe- hierarchical-based optimization MPC for large-scale sys- cially for large-scale systems [16]. On the contrast, the tems with time-varying delay, persistent disturbances, and Razumikhin method includes a Lyapunov function for the with respect to Lyapunov-Razumikhin function has not been original non-augmented system despite of the fact that it studied yet and several problems remain unsolved. In this might be a bit conservative. Thereby, Razumikhin method study, the considered controller is model predictive control. has a better potential to avoid the complexity of the Kra- Since the considered system is a large-scale, the hierarchical sovskii method. Many studies have been done by adopting scheme is applied to solve the optimization problem based on Razumikhin method [17–19. In [16], the author proposed an decomposition and coordination concept [30]. On the other LMI-based model predictive control for fuzzy T-S system hand, no research has been done regarding the MPC for fuzzy with time-varying delay and used Razumikhin method for large-scale systems with time-varying delay adopting the the delay problem and the LMI-based method is proposed for Lyapunov-Razumikhin function. Therefore, the MPC is not interconnected system. In [20], a model predictive con- applied to a large-scale system that its dynamic is modeled trol strategy for nonlinear time-delay systems with unknown by T-S fuzzy, and the Lyapunov-Razumikhin function is time varying delay is considered instead of the traditional considered for the time-varying delay problem. Besides, the constant delay. Nonlinear discrete time-delay systems are hierarchical scheme is assumed for the optimization problem represented by T–S fuzzy systems comprised of piecewise of the system. 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control So the contribution of this paper can be summarized as: Now, for time-delay systems some definitions related to the robust positive invariance and input-to-state stability are • Proposing the fuzzy model predictive control for proposed [31]: discrete nonlinear large-scale system. • Considering the time-varying delay and persistent Definition 1 (Robust Positively Invariant (RPI) set) For disturbances simultaneously. the Lyapunov-Razumikhin function conditions, consider • Applying hierarchical optimization to the considered system ð1Þ; a set Xis is an RPI set for the closed-loop large-scale system. system corresponding to the control law, and for the d ðkÞ 2 • Studying Lyapunov-Razumikhin function. Z½h;1 ; if 8xi ; xid 2 Xis ; and 8di ðkÞ 2 D; the control effort • Analyzing the input-to-state stability. assures that xþ i 2 Xis : • Using the fuzzy Takagi–Sugeno model of the large- scale system. Definition 2 (Input-to-state-Stability (ISS)) A discrete- • Inspiring the H 1 performance to encounter the effec- time nonlinear system xðk þ 1Þ ¼ f ðxi ðkÞ; xid ðkÞ; d i ðkÞÞ tiveness of disturbances. where d symbolizes the disturbance vector, is input-to-state stable (ISS) if there exists an HL-function b and a H- The remainder of this paper is: in Section 2, some pre- function c such that for each input liminaries information about fuzzy model predictive con-     d,kxðkÞk  b kxi ½h;0 k; k þ c k di ½0;k1 k ; where trol of Takagi–Sugeno large-scale systems with time- xi ½h;0 2 Xhþ1 is the initial (delayed) state vector, varying delay and hierarchical optimization scheme are introduced. In Section 3, robust positive invariance (RPI), di ½0;k1 2 Wk is the disturbance sequence, k 2 Z þ . the computation of terminal constraint set for nonlinear Definition 3 (ISS Lyapunov-Razumikhin Function [16]) A model-based fuzzy systems is provided. In Section 4, two continuous positive definite function V ð xðkÞÞ is called an numerical examples are illustrated. Finally, the conclusion ISS-Lyapunov-Razumikhin function for system xðk þ 1Þ ¼ is given in Section 5. f ðxi ðkÞ; xid ðkÞ; di ðkÞÞ; if there exist H1 function o1 ; o2 ; and Notations: R; Rþ ; Z; and Z þ depict sets of real numbers, H-function q such that: positive real numbers, integers, and positive integers, o1 ðk xðkÞ kÞ  V ð xðkÞÞ  o2 ðk xðkÞ kÞ ð3Þ respectively. Z ½m;n is symbol of set of integers in the   interval ½m; n for convenience. kxk depicts the norm of V ð xðk þ 1ÞÞ  max V ð xðkÞÞ; qðk dðkÞ kÞ ð4Þ vector xi 2 Rc : A real-valued scalar function o : Rþ ! Rþ where V ð xðkÞÞ ¼ maxfV ðxi ðkÞÞ; V ðxid ðkÞÞg is an H-function ðo 2 HÞ, if it is continuous, rigidly increasing and oð0Þ ¼ 0: So we can say, o 2 H1 if o 2 H Lemma 1 [32] If system xðk þ 1Þ ¼ f ðxi ðkÞ; xid ðkÞ; d i ðkÞÞ and lim oðsÞ ¼ 1: A function b : Rþ  Rþ ! Rþ is an admits an ISS-Lyapunov- Razumikhin function, then it is s!1 HL-function ðb 2 HLÞ; if for each fixed s [ 0; bð:; sÞ is a ISS. H-function, and for each fixed r [ 0, bðr; :Þ is rigidly decreasing and bðr; sÞ ! 0 as s ! 1. 2.2 Time Delay Takagi–Sugeno System Description The fuzzy Takagi–Sugeno large-scale system composed of 2 Preliminaries N subsystems with time-varying delay is: 8 > IFzi1 is Fli1 and .. . and zig isFlig > < 2.1 System Description Sli : l l l l PN > THEN xi ðk þ 1Þ ¼ Ai xi ðkÞ þ Bi ui ðkÞ þ Aid xid ðkÞ þ wi di ðkÞ þ j ¼ 1 f ij xj ðkÞ > : i 6¼ j The following discrete-time nonlinear large-scale system is ð5Þ assumed: in which Ali ; Bli ; and wli ði ¼ 1; 2;.. .;N; l ¼ 1;2;. ..;ri Þ are the xðk þ 1Þ 2 f ðxi ðkÞ; xid ðkÞ; ui ðkÞ; d i ðkÞÞ k 2 Z þ ð1Þ system matrices and disturbances of rule-l in subsystem Si : where xi ðkÞ; xid ðkÞ 2 X; ui ðkÞ 2 U and d i ðkÞ 2 D; with Here, xi ðkÞ 2 Rc ; ui ðkÞ 2 Rn ; and di ðkÞ 2 Rm are state vector, X  Rn ; U  Rm ; and D  Rc , are system current and input vector, and disturbance vec- delayed states, inputs, and disturbances, respectively. xid ðkÞ l tor.r i ;f ij ; zi ðkÞ ¼ ½zi1 ;zi2 ;. ..; zig , and F iq ðq ¼ 1; 2;...;gÞ; is defined as follows: respectively, illustrate the number of the fuzzy rules in xid ðkÞ ¼ xid ðk þ dðkÞÞ; dðkÞ 2 Z½h;1 : ð2Þ subsystem Si ; the interconnection between subsystem Si and SJ ; some measurable premise variables, and the lin- where dðkÞ denotes the number of delays, h denotes the guistic fuzzy sets of the rulel. The fuzzy large-scale system upper bound of delay, and the minimal delay is set as 1. ð5Þ with time-varying delay can be shown as: 123 International Journal of Fuzzy Systems X N X N X N  xþ i ¼ Ail xi þ Bil ui þ Aidl xid þ wil d i þ f ij xj ; i P ðk Þ ¼ P i ðk Þ ¼ xTi ðk þ tjkÞQxi ðk þ tjkÞ j¼1 i¼1 i¼1 i 6¼ j :þ uTi ðk þ tjkÞRui ðk þ tjkÞ  si d Ti ðk þ tjkÞd i ðk þ tjkÞÞ ¼ 1; 2; . . .; N ð6Þ ð13Þ Pr i l   l Pr i l   l Ail ¼ Pl¼1 li ziq  Ai ; Aidl ¼ l¼1 li ziq Aid ; where R and Q are real fixed matrices, and si is a positive Bil ¼ Prl¼1 i lli ziq Bli Pi l ð7Þ scalar. It is evident the cost function includes the distur- wil ¼ l¼1 lli ziq wli ; f ijl ¼ rl¼1 ri li ðziq Þf ij bance, and is influenced by the H 1 control [34]. Conse-   quently, the cost function cannot be optimized directly as and lli ziq represents the normalized membership the disturbance is included. Instead, a min–max technique function. is chosen that minimizing the worst-case cost function [35]. Control law is shown as:  Furthermore, at the end of the prediction, it is mostly l IF zil is Fli1 and . . . and zig is Flig required that the states enter a terminal constraint set to Ci : ð8Þ THEN ui ðkÞ ¼ kli xi ðkÞ attain asymptotic stability, since it is almost impossible   task to achieve the asymptotic stability to the origin, in the With the same weight notation lli ziq and respect to ð8Þ, presence of persistent disturbance [36]. xi ðk þ TjkÞ 2 Xis the final output of the controller will be: shows the terminal constraint set. The online optimization X ri problem is: ui ð k Þ ¼ lli ðziq Þkli xi ðkÞ ð9Þ l¼1 min max J i ðkÞ; ui ðkþtjkÞdi ðkþtjkÞ The closed-loop system with time-varying delay will be: s:t:ui ðk þ tjkÞ 2 U i Xri Xri     l  xi ðk þ 1Þ ¼ lli ziq lm i ziq Ai þ Bli km i xi ðk Þ di ðk þ tjkÞ 2 Di l¼1 m¼1 X ri X N xi ðk þ TjkÞ 2 Xis   þ Alid xid ðkÞ þ wli d i ðkÞÞ þ lli ziq f ij xj ðkÞ where t 2 Z ½1;T1 and Xis shows the terminal constraint set. l¼1 j¼1   Heredi 2 Di :¼ di jd Ti di  ci 2 , ui 2 U i :¼ i 6¼ j fui jjuim j  uim:max g and ci 2 is a positive scalar, uim is the m- ð10Þ th element of the inputs, m 2 Z ½1;w . 2.3 Model Predictive Control As it was mentioned, the model predictive control is a kind of controller which computes the input vector by The prediction model and cost function are proposed here. minimizing a specified cost function. In this manuscript, The prediction form is: the considered system is large-scale and the form of the controller is decentralized. Thus, to avoid the uncertainty in xi ðk þ n þ 1jkÞ ¼ Ail xi ðk þ tjkÞ þ Aidl xid ðk þ tjkÞ cost function and reach the best solution, the hierarchical þ Bil ui ðk þ tjkÞ þ wil di ðk þ tjkÞ scheme is applied to the optimization problem [37]. X N þ f ijl xj ðk þ tjkÞ ð11Þ j¼1 i 6¼ j 3 Main Results and the cost function is: 3.1 RPI Set for Fuzzy Takagi–Sugeno Large-Scale System with Time-Varying Delay X N J ðk Þ ¼ j i ðk Þ i¼1 ! First, the RPI property and terminal constraint set are XN X T 1 defined. Second, the recursive feasibility is analyzed. ¼ P i ðK Þ þ Pi ðk þ tjkÞ þ V it ðxi ðk þ TjkÞÞ Finally, the ISS will be provided. The RPI set is illustrated i¼1 n¼0 by Xis ; and Ki ð xÞ as the corresponding control law. The ð12Þ RPI set is defined as, 8xi ; xid 2 Xis ; the control effort where Pi ðk þ njkÞ and V it ðxi ðk þ TjkÞÞ are stage cost at the assures that xþ i 2 Xis ; for all allowable disturbance. For the predicted time instant and terminal cost, and T is the pre- fuzzy Takagi–Sugeno large-scale system ð3Þ with time- diction length. It is highly notable that V it ðÞ must be a varying delay, define Xis as: positive function [33], and the stage cost is chosen as: 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control ( ( ! !) ) X ri   T X ri   T Zi kTil Xis :¼ fxi ; xid g max lli ziq xi Pil xi ; lli ziq xid Pil xid  1i  0; Z iss  u2is;max ; s 2 Z ½1;m ð17Þ l¼1 l¼1 kil 1 ð14Þ    Pr i l then the set Xis :¼ fxi ; xid g max l¼1 li Pr i   T Pr i l   T  where Pil ¼ l¼1 lli ðziq ÞPi and 1i is a positive scalar and ziq xi Pil xi Þ; l¼1 li ziq xid Pil xid g  1i :g is a RPI set the corresponding control law is Pi l for the fuzzy system ð5Þ corresponding to the feedback Ki ðxi ; xid Þ ¼ rl¼1 li ðziq Þkli xi ðkÞ. Pi l control law Ki ðxi ; xid Þ ¼ rl¼1 li ðziq Þkli xi ðkÞ: Where hi ¼   Lemma 2 [28]: The set Xis is an RPI set if there exists a Ail þ Bil kil ; Z iww is w  th diagonal element of matrix positive scalar ki ; ð0\ki \1Þ; such that, Z i ; N represents the number of subsystems, .i and .id are XN    ki fixed positive values which satisfy .i þ .id ¼ 1, X i ¼ 1 þ T þ þ 1  ki xi Pil xi  max xTi Pil xi ; xTid Pil xid  2 d Ti di 1i0Pi ; X il ¼ 1i Pil ; X j ¼ 1i Pj ; -i ¼ c1i2 ; a  2,and i¼1 1i 1i ci i i; j; l; N; a 2 Rþ : ð15Þ Pr i Proof See Appendix A. with Pþ il ¼ l þ þ l¼1 li ðxi ÞPi ; for all xi 2 Ail xi þ Aidl xid þ PN Bil ui þ wil di þ j ¼ 1 f ijl xj ; ui 2 U i ; and di 2 Di : 3.2 The Coordination Sub-Network i 6¼ j In this section the hierarchical optimization will be carried Remark 1 Here, by proposing the Theorem 1, two LMIs on. For this, the Hamiltonian function is defined as: are solved and if they are feasible, two results are achieved. (1) It will be proved that the considered large-scale fuzzy X N H ðÞ ¼ H i ð Þ system with time-varying delay is stable in the sense of i¼1 Lyapunov. (2) Controller’s gains are computed in the ( X N X K 1 restricted bound, optimally. ¼ P i ðK Þ þ xTi ðkÞQxi ðkÞ þ uTi ðkÞRui ðkÞ i¼1 k¼0 Theorem 1 Consider the fuzzy system ð5Þ; if there exist 0 1 positive definite matrices X i X j ; and X il , positive scalar ð18Þ B C ki ð0\ki \1Þ such that the following matrix inequalities B X N C T T B C are met:  si di ðkÞdi ðkÞ þ di ðkÞBzi ðkÞ  f ij xj ðkÞC B C @ j¼1 A i 6¼ j þ pTi ðk þ 1Þðxi ðk þ 1Þ þ gi ðxi ðkÞ; ui ðkÞ; zi ðkÞÞÞÞg ð16Þ 123 International Journal of Fuzzy Systems where gi ðxi ðkÞ; ui ðkÞ; zi ðkÞÞ ¼ Ail xi þ Aidl xid þ Bil ui þ now, provided the desire convergence obtained, stop. wil di þ C i zi ðkÞ and Pi ðK Þ is continuously differentiable. Otherwise, repeat the steps again. Here, di ; pi ; and C i are the Hamiltonian multipliers, the co-state variables, and fixed value matrix. By the interac- 3.4 The Terminal Constraint Set tion prediction strategy, the coordination for hierarchical optimization is to find an approach for updating the coor- The terminal constraint set Xis should satisfy two condi- dination values di and zi ðkÞ: tions. First, it should be RPI set. Second, a positive definite oH ðÞ oH ðÞ function (terminal cost function)V i ð xÞ should exist such ¼ 0; ¼ 0; i ¼ 1; . . .; N; k ¼ 0; . . .K  1 ð19Þ that 8xi 2 Xis ; ozi ðkÞ odi ðkÞ c3 ðk xi kÞ  Vðxi Þ  c4 ðk xi kÞ ð23Þ thereby, X N   X N og ðxi ; ui ; zi Þ V xþ  V ð x Þ\  H i ð:Þ ð24Þ di ð k Þ ¼  i pi ðk þ 1Þ ¼ C Ti pi ðk þ 1Þ; i i i ozi i¼1 i¼1 ¼ 1; . . .; N; k ¼ 0; . . .K  1 ð20Þ where 13 and 14 are H 1 functions, V ð xÞ is given as: X N X N N X X ri zi ðk Þ ¼ f ij xj ðkÞ; i ¼ 1; . . .; N; k ¼ 0; . . .K  1 ð21Þ V ð xÞ ¼ V i ð xÞ ¼ lli ðziq ÞxTi Pi xi ð25Þ j¼1 i¼1 i¼1 l¼1 i 6¼ j Remark 2 In this section, by proposing the Theorem 2, the 3.3 The Local Optimization Sub-Networks concept of the robust positively invariant and constraint set are achieved through solving a LMI, and it is ensured that In local sub-networks, exchanging the information between trajectories of the large-scale system with time-varying the coordination and local sub-networks will make up the delay are stable robustly. hierarchical scheme as shown in Fig. 1. In this fig- For convenience, a table of notations is provided: ure di ¼ ri .in the final step of hierarchical optimization, the Table 1 convergence must be checked by evaluating the overall interaction error: Theorem 2 Consider the fuzzy Takagi–Sugeno system 8 9T 8 9 ð5Þ; if ð16Þ; ð17Þ and the following matrix inequality are > > > > > > > > >> > > feasible, K1 > N X X < X N = > > < X N > = eð k Þ ¼ zi ðk Þ  f ij xj ðkÞ zi ðk Þ  f ij xj ðkÞ > > > > > > > > i¼1 k¼1 > > j¼1 > > > j¼1 > : ; > : > ; i 6¼ j i 6¼ j ð22Þ 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control ð26Þ then Xis is a terminal constraint set considering the ter- X N minal cost function. Where N represents the number of V ð x ðk ÞÞ ¼ V i ð x ðk ÞÞ   T i¼1 subsystems,hi ¼ Ail þ Bil kil ,X i ðkÞ ¼ 1i pTi ðkÞ, Hi ¼ 1i R, XN X ri X i ¼ 1i Pi ; X il ¼ 1i Pil ; X j ¼ 1i Pj , a  2; and ¼ lli ðziq ÞxTi ðk þ dÞPi xi ðk þ dÞ  1i ; d ðkÞ i; j; l; N; a 2 Rþ : i¼1 l¼1 2 Z½h;1 : Proof See Appendix B. ð27Þ 3.5 Control Algorithm the following optimization problem is for minimizing 1i : min 1i subject to V i ð xðkÞÞ  1i Based on the recent results and to finalize the control design, now, the online control algorithm is studied. furthermore, a proper condition for V i ð xðkÞÞ  1i is: Therefore, the terminal constraint set V ð xðkÞÞ; see ð24Þ; should satisfy the following condition, 123 International Journal of Fuzzy Systems p1 ,u 1 , x 1 p N ,u N , x N V ,z 1 Coordination unit V ,z N Upper level V ,z 2 p 2 ,u 2 , x 2 Lower level Subsystem Subsystem Subsystem ... 1 2 N Fig. 1 Architectures of interactions xTi ðk þ dÞPi xi ðk þ dÞ  ni ; i ¼ 1; . . .; N ð28Þ must be modeled while the engineer is modeling the dynamic of the system. Interconnections in a large-scale which is 1i  xTi ðk þ d ÞPi xi ðk þ dÞ  0 and equal to 1i  system are same as system matrices like Ali and Bli . These xTi ðk þ dÞ 1i1Pi i xi ðk þ dÞ  0 by defining symmetrical matrix are computed when the system’s dynamic is linearized. X i ¼ 1i Pi , is guaranteed by the following LMIs, Remark 3 As mentioned in introductory section, an 1i xTi ðk þ dÞ important issue in MPC is optimization problem. The  0; dðkÞ 2 Z½h;1 : ð29Þ x i ðk þ d Þ X 1 i 1i considered system in this paper is large-scale and values of the LMIs are computed and controlled the closed-loop system based on the minimized value of 1i : In the previous Note 1: The symmetric matrix X i is defined as a matrix theorems, the bilinear matrix inequalities have been pro- with an achievable inversion. So, the inversion of X i posed. Since the significant parts of this paper are gains and must be considered in ð29Þ. minimized 1i ; the values of X i ; X il ; ki ; -i ; and H i are Note 2: Since the first Example is a mathematical defined until the feasible solutions for gains and minimized Example, the interconnection between two subsystems value of 1i : are considered as mathematical matrices. But in a real example, the interconnection between two subsystems Algorithm Step 1 (Coordination optimization): Set initial values for , and different values of ,Η , and . Step 2: Solve the following optimization problem min , 16 , 17 , 26 , 29 , and for each subsystem, find the values of , ̅ , ̅ , , ,Η . Step 3: Check for the converging by the overall interaction error. ⎧ ⎫ ⎧ ⎫ ⎨ ⎬ ⎨ ⎬ ⎩ ⎭ ⎩ ⎭ ̅ Step 4: If desired values of coordination achieved, stop. Otherwise, Set ∑ and go step 2. 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control Table 1 Abbreviations and Notation Definition Notation Definition notation Ali ; Bli ; and wli System matrices and disturbances RandQ Positive weights Xi Xi ¼ ni Pi Hi 1i R Xj Xj ¼ ni Pj N Number of subsystems Pi Positive matrix a Positive scalar fij Interconnection between subsystem c2i Positive scalar 1i 1i Positive scalar variable -i c2i si Positive scalar .i Positive value ki Positive scalar di Hamiltonian multiplier T X i ðk þ 1Þ 1i pTi ðk þ 1Þ pi ð k þ 1Þ Co-state variable eðkÞ Overall interaction error Ci Fixed value matrix One of the significant part of the model predictive and d ðk Þ defined in (2). control is recursive feasibility. In the following theorem, it in which will be proven that the recursive feasibility can be achieved , here, is an optimal value of and the constrained optimization problem has the solution Pi ðkÞ at time k; then it can be obtained: at any time. Theorem 3 For the system ð5Þ, it will always be solvable ð31Þ if the solution of the optimization problem can be achiev- in which able at time 0, and the recursive feasibility can be obtained. , . where wmax ðÞ and wmin ðÞ are, respectively, the maximal and Proof If ð30Þ is feasible at time k, then resorting to the minimal eigenvalues. xi ; xid 2 Xis ; and according to the Theorem 1, which has 0 0 1 B B C B B X N C B T T T T B C V k ðxi ðk þ 1ÞÞ  V ðxi ðkÞÞ\  Bxi ðkÞQxi ðkÞ þ ui ðkÞRui ðkÞ  si d i ðkÞdi ðkÞ þ di ðkÞBzi ðkÞ  f ij xj ðkÞC B B C ð32Þ @ @ j¼1 A i 6¼ j þpTi ðk þ 1Þðxi ðk þ 1Þ þ gi ðxi ðkÞ; ui ðkÞ; zi ðkÞÞÞÞ been implied that xi ðk þ 1Þ 2 Xis ; then it can be concluded In addition, ð24Þ implies that, that ð30Þ can be solvable at time k þ 1. Besides, the solu- tion which achieved at the time k, is feasible at time k þ 1, and also the optimization problem is feasible at all time. where , and if Theorem 4 should the constrained optimization problem V k ðxi ðk þ 1ÞÞ  V ðk; xi Þ\xTi ðkÞQxi ðkÞ þ si dTi ðkÞdi ðkÞ is feasible at the initial time 0; system ð5Þ is ISS due to the ð33Þ disturbance d. due to the ð30Þ at time ðk þ 1Þ, it will be: Proof Assume the Lyapunov-Razumikhin function , consider that it gets its maxi- V kþ1 ðxi ðk þ 1ÞÞ  V k ðxi ðk þ 1ÞÞ ð34Þ mum values at the delayed states xi ðk þ DÞ, D 2 fdðkÞ; 0g, finally, it achieves that 123 International Journal of Fuzzy Systems V kþ1 ðxi ðk þ 1ÞÞ  V ðk; xi Þ\xTi ðkÞQxi ðkÞ þ si d Ti ðkÞdi ðkÞ Subsystem S2 ð35Þ 0:325 0 1 A21 ¼ ; A21d ¼ 0:5A21 ; B21 ¼ ; w21 corresponding definition ð3Þ, Lemma 1, ð31Þ, and ð35Þ, the 0:4 0 1 V ðxi ðkÞÞ is an ISS Lyapunov function. And the closed-loop 0:1 0:1 0:1 ¼ ; f 21 ¼ system is ISS due to disturbances. So, the proof is 0 0 0 completed. 0 0 0:018 0 f 23 ¼ ; k2 ¼ 0:488; X 2 ¼ 0:1 0:1 0 0:018 4 Numerical Example 0:6 0:2 1 A22 ¼ ; A22d ¼ 0:5A22 ; B22 ¼ ; w22 0:1 0 1 In this section, to prove the effectiveness of the proposed 0 ¼ : algorithm, two examples are illustrated, the mentioned 0:2 algorithm is applied, and results with explanations are Subsystem S3 depicted. Consider a large-scale system S composed of three fuzzy subsystems Si ; i ¼ 1; 2; 3ðN ¼ 3Þ; as follows, in 0:2 0:4 1 A31 ¼ ; A31d ¼ 0:5A31 ; B31 ¼ ; w31 which each state of each subsystem has two dimensions. 0:2 0 1 Let -i ¼ 0:5; Q ¼ diagf5; 5g; Hi ¼ 5: In this example, the 0:3 0:03 0 ¼ ; f 31 ¼ ; delay time is considered as 1: The membership function, 0 0 0:02 l1i ¼ cos2 ðxi2 ðkÞÞ, l2i ¼ 1  l1i : The T-S fuzzy large-scale 0:1 0 0:027 0 model with time-varying delay is: f 32 ¼ ; k3 ¼ 0:487; X 3 ¼ 0:1 0 0 0:027 8 l l > < IF zi1 is Fi1 and . . . and zig is Fig P THEN xi ðk þ 1Þ ¼ Ai xi ðkÞ þ Bli ui ðkÞ þ Alid xid ðkÞ þ wli di ðkÞ þ Nj ¼ 1 f ij xj ðkÞ l 0:3 0 2 > : A32 ¼ ; A32d ¼ 0:5A32 ; B32 ¼ ; w32 i 6¼ j 0 0:4 1 0 ¼ Subsystem S1 0:4 0:55 0:05 1 where x1 ðkÞ ¼ ½ x11 ðkÞ x12 ðkÞ T ; x 2 ðk Þ ¼ A11 ¼ ; A11d ¼ 0:5A11 ; B11 ¼ ; 0 0:42 0 T T ½ x21 ðkÞ x22 ðkÞ ; x3 ðkÞ ¼ ½ x31 ðkÞ x32 ðkÞ  , and we 0:1 0:08 0:05 w11 ¼ ; f 12 ¼ ; have: 0 0:05 0:05 0:09 0:06 0:015 0 Si ¼ x i ð k þ 1Þ f 13 ¼ ; k1 ¼ 0:5; X 1 ¼ 0:06 0:09 0 0:015 X ri X ri    h l i ¼ lli ziq lmi z iq A il þ B l m il i xi ðk Þ k 0:4 0 0 l¼1 m¼1   A12 ¼ ; A12d ¼ 0:5A12 ; B12 ¼ ; w12 þ lli ziq Alidl xid ðkÞ þ wlil d i ðkÞ 0 0:08 1 0 X ri X N   ¼ 0:1 : þ lli ziq f ij xj ðkÞ: l¼1 j ¼ 1 i 6¼ j 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control Remark 4 As it is evident in simulation results, Fig. 2, the trajectories of the subsystem 1, at first, the states of the two rules went to instability due to the existence of the time- varying delay, which is 1 s. But as they go on, the proposed controller has learned to compute efficient gains, which converged the states to 0. Same as Fig. 2, in Figs. 3, and 4, it is clear that the states of the subsystems were attempting to have overshoot and undershoot. But controllers have successfully stabilized states as time keeps going. The Fig. 2 Trajectories of subsystem 1 settling time in the first subsystem is before 10, but in subsystems 2 and 3 is before 5, which means the subsystem 1 is more complex than two others. Remark 5 According to Figs. 5, 6, and 7, it will be noticeable that the trajectories of the input vector ui 2 Rn are converged to 0. Therefore, there is no need to input vector and costs will be reduced. And also by referring to Fig. 8, after 2 or 3 iterations, the interaction error leads to 0; means that, local subsystems are fully optimized. Finally, Figs. 9, 10, and 11, show 3 cost functions corre- sponding to 3 subsystems and it is obvious that despite of Fig. 3 Trajectories of subsystem 2 the fact that cost functions involve negative terms, they are definitely positive during the time. It would be necessary to say that the mentioned numerical example is illustrated to show the effectiveness and usefulness of the proposed method, and it will be definitely a practical method in real conditions. Remark 6 In comparison, in [23] and [28], the model predictive control is applied to a usual system with per- sistent disturbances and uncertainties. But a noticeable issue is that the considered system is a usual one and the proposed algorithm is not efficient for large-scale systems. Fig. 4 Trajectories of subsystem 3 On the other hand, the mentioned paper has not considered the time-varying delay that is an inevitable part in real systems. In [21], an algorithm is proposed for typical systems with time-varying delay. The model predictive control is assumed for the mentioned research but as mentioned above, it is not considered for large-scale sys- tems. In this research, model predictive control is applied to a fuzzy large-scale system with time-varying delay and persistence disturbances, which the proposed algorithm covers the weaknesses of the mentioned researches. Example 2: A double inverted pendulum is considered Fig. 5 Trajectories of controller 1 due to the [38] to show the effectiveness of the method. All configurations and parameters are chosen same as [38], but a 0.5 s time-varying delay is considered for the system. All other configurations and considered assumptions are same as the previous example. Subsystem S1 123 International Journal of Fuzzy Systems Fig. 6 Trajectories of controller 2 Fig. 10 Trajectory of cost function 2 Fig. 7 Trajectories of controller 3 Fig. 11 Trajectory of cost function 3 1 0:005 A11 ¼ A13 ¼ ; A13 0:0262 1 1 0:005 ¼ ; A11d ¼ 0:5A11 ; A12d 0:0441 1 1 ¼ 0:5A12 ; A13d ¼ 0:5A13 ; B11 ¼ B12 ¼ B13 ¼ ; w11 0 0:1 0:08 0:05 ¼ w12 ¼ w13 ¼ ; g12 ¼ ; k1 0 0:05 0:05 0:015 0 ¼ 0:5; X 1 ¼ ; H 1 ¼ ½10: Fig. 8 Interaction error 0 0:015 Subsystem S2 1 0:005 A21 ¼ A23 ¼ ; A23 0:0272 1 1 0:005 ¼ ; A21d ¼ 0:5A21 ; A22d 0:0451 1 1 ¼ 0:5A22 ; A23d ¼ 0:5A23 ; B21 ¼ B22 ¼ B23 ¼ ; w11 1 0:1 0:08 0:05 ¼ w22 ¼ w23 ¼ ; g21 ¼ ; k2 0 0:05 0:05 0:018 0 Fig. 9 Trajectory of cost function 1 ¼ 0:448; X 2 ¼ ; H 2 ¼ ½10: 0 0:018 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control Xri X ri    h l i stabilized with lower cost and this is the efficient of the x i ð k þ 1Þ ¼ lli ziq lm l m i ziq Ail þ Bil ki xi ðkÞ proposed approach.   l¼1 m¼1 þ lli ziq Alidl xid ðkÞ þ wlil di ðkÞ Xri X N   þ lli ziq f ij xj ðkÞ: 5 Conclusion l¼1 j ¼ 1 i 6¼ j Here, fuzzy model predictive control and hierarchical optimization for a class of discrete large-scale systems with yi ðkÞ ¼ H i xi ðkÞ time-varying delay and disturbance is investigated. The Subsequently, we can obtain the feedback gains: considered method in this paper is Razumikhin for time- k11 = [-4.54 -6.06] k12 = [-6.009 -8.79] varying delay systems, in which it includes a Lyapunov k13 = [-15.15 -19.585]. function associated with the original non-augmented state k21 = [-5.14 -.01] k22 = [-1.049 -4.14] space of system dynamics. The Razumikhin method has the k23 = [-28.255 -12.252]. perfect potential to avoid the inherent complexity of the Krasovskii method especially in the presence of large Remark 7 Figure 12 shows output responses of the delays and disturbances. Model predictive control is closed-loop discrete-time nonlinear large-scale system with applied to the fuzzy Takagi–Sugeno large-scale system, time-varying delay. It is evident that the proposed con- and the system is composed of several subsystems to troller in this paper based on the fuzzy dynamic model not consider the decentralized scheme for controller. Many only stabilizes the original nonlinear large-scale system but similar methods have been done before in this area and also effectively attenuates the disturbances as expected. Razumikhin approach was applied to. But the novelty of And as it is clear, for the first few seconds there is no this paper is that this method is applied to a large-scale response due to the delay that is in the nature of the system. system and simultaneously the optimization problem is Remark 8 To prove the effectiveness of the proposed considered. On the other hand, by robust performance and method, a comparison is made due to [39]. In [39], the PI LMI algorithm, the stability of the results is guaranteed. In controller is considered for a large-scale system. At first, a this paper, the uncertainties and disturbances that are control scheme using PI controllers, which are currently always seen in systems are considered. As the assumed used in industry, is used. although this scheme is very system in this paper is large-scale and there are some simple to implement, its performance has often limitation subsystems, the hierarchical approach is applied to the and tuning the P and I gains are a tedious process. Besides, optimization problem which divides the system into two the mentioned controlled is applied to a system without upper and lower orders. At the upper order, coordination, time-varying delay. Todays, many systems in industry and and at the lower order, some units are established. The academic face many delays in their natures and the pro- exchange information between two orders, finally, leads to posed method in [39] is not able to stabilize the system. the most efficient in the optimization problem. This algo- rithm is assumed for systems with known time-varying Remark 9 By referring to gains, a notable point in this delay systems and in future works it can be applied to example is results of gains. To stabilize the system with nonlinear dynamics with unknown time-varying delay. At time-varying delay, the computed gains have little values least two examples are submitted to investigate the effec- and this means the double inverted pendulum can be tiveness of the proposed method. Fig. 12 Output responses of the closed-loop system 123 International Journal of Fuzzy Systems 6 Appendix A Proof Applying Schur complement, the inequality ð16Þ can be written as: ð36Þ where pffiffiffi P ui ¼ NHTi X i Hi þ N a Nj ¼ 1 f Tij X j f ij  .i ðki þ 1ÞX il . i 6¼ j By considering X i ¼ 1i Pi ; X il ¼ 1i Pil ; X j ¼ 1i Pj ; -i ¼ c1i2 , i now, the inequality (36) is: ð37Þ pffiffiffi PN ui ¼ 11i NHTi Pi Hi þ 11i N a j ¼ 1 f Tij Pj f ij  i 6¼ j 1 . 1i i ð k i þ 1 ÞP il ; now, if the inequality (37) multiplied h i from both sides by dTi xTi xTid xTj    xTN and its transpose, respectively, 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control ð38Þ 123 International Journal of Fuzzy Systems according to [40], the inequality (38) is equivalent to: ð39Þ 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control The inequality (39) can be written: ð40Þ 123 International Journal of Fuzzy Systems it is assumed that .i þ .id ¼ 1; so it will be easily the inequality (40) is equivalent to: confirmed that .i xTi Pil xi þ 0 1T  T  T T .id xid Pil xid  max xi Pil xi ; xid Pil xid : Substituting into B C (42), thus, ð15Þ is obtained. Furthermore, the input con- 1B B  X N C C B Ail þ Bil kil xi þ Aidl xid þ wil di þ f ij xj C straint can be acknowledged by ð17Þ; and the proof is 1i B C @ j¼1 A shown here: i 6¼ j By multiplying diagfI; xi g and its transpose from both 0 1 sides of ð17Þ : B C xTi Z i xi T xTi kil  0 B  X N C ð43Þ þB C kil xi 1 Pil B Ail þ Bil kil xi þ Aidl xid þ wil d i þ f ij xj C B C @ j¼1 A applying Schur complement to (43), then, i 6¼ j  T   xTi Z i xi  kil xi kil xi  0 ð44Þ ki 1  ki  T   2 d Ti di  .i xi Pil xi þ .id xTid Pil xid ci 1i as ui ¼ kil xi ; the following is got: ð41Þ ðui ÞT ðui Þ  xTi Z i xi ¼ H i ¼ positivevalue ð45Þ the equation xþ i ¼ Ail xi þ Aidl xid þ Bil ui þ wil d i þ thus uTi ui  H i . The proof is, thereby, completed. PN j ¼ 1 f ij xj is results: 7 Appendix B i 6¼ j N  X Proof Applying schur compliment to the inequality ð36Þ: 1 T ki T 1  ki  T  xþ þ þ i Pil xi  d di  .i xi Pil xi þ .id xTid Pil xid i¼1 1i ci 2 i 1i ð42Þ ð46Þ 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control pffiffiffi P where vi ¼ Nhi T X i hi þ N a Nj ¼ 1 f Tij X j f ij  .i X il þ i 6¼ j 1i Q þ kTil H i kil . The inequality (46) is equivalent to: ð47Þ 123 International Journal of Fuzzy Systems pffiffiffi PN where Ni ¼ Nhi T Pi hi þ N a T j ¼ 1 f ij Pj f ij  .i Pil þ i 6¼ j  T Q þ kTil Rkil .by multiplying di xTi xTid xTj    xTN I Ti  and its transpose from both sides of the matrix in the inequality (47), respectively, the following inequality is obtained: ð48Þ 123 M. Sarbaz et al.: Hierarchical Optimization-Based Model Predictive Control according to [40], the inequality (48) is equivalent to: ð49Þ 123 International Journal of Fuzzy Systems the inequality (49) is equivalent to: 82 3T 2 3 > > > >6 7 6 7 XN ><6   X N 7 6 X N 7 6 7 þ 6  7 6 Ail þ Bil kil xi þ Aidl xid þ wil di þ f ij xj 7 Pil 6 Ail þ Bil kil xi þ Aidl xid þ wil di þ f ij xj 7 > >4 i¼1 > 6 7 6 7 > j¼1 5 4 j¼1 5 > : i 6¼ j i 6¼ j  T T  T T T T T  T T   .i xi Pil xi þ .id xid Pil xid þ xi Qxi þ ui kil Rkil ui  si d i di þ I i di ðkÞzi ðkÞþpi ðk þ 1Þzi ðkÞ I i 1X N  1X N T  I Ti pTi ðk þ 1Þ þ dTj ðkÞ f ij xj  xTj f Tij pTi ðk þ 1Þ þ dTj ðkÞ I i  ða  1Þ 2 2 j¼1 j¼1 i 6¼ j i 6¼ j 0 1 B C X N X N pffiffiffi T BBX N C C xTj f Tij Pi f ij xj þ ða  1Þ xTj f Tij Pi f ij xj þ xTi a Ail þ Bil kil Pi B f ij xj C B C j¼1 j¼1 @j ¼ 1 A i 6¼ j i 6¼ j i 6¼ j 0 1 0 1 0 1 B C B C B C pffiffiffi T BBX N C pffiffiffiB X C B N T TC C   pffiffiffiB B X T TC  N C   xTi a Ail þ Bil kil Pi B f ij xj C þ aB xj f ij CPi Ail þ Bil kil xi  aB xj f ij CPi Ail þ Bil kil xi g\0 B C B C B C @j ¼ 1 A @j ¼ 1 A @j ¼ 1 A i 6¼ j i 6¼ j i 6¼ j ð50Þ the inequality (50) can be written as: 82 3T 2 3 > > >6 > 7 6 7 XN ><6   X N 7 6 X N 7 6 7 þ 6  7 6 Ail þ Bil kil xi þ Aidl xid þ wil di þ f ij xj 7 Pil 6 Ail þ Bil kil xi þ Aidl xid þ wil di þ f ij xj 7 > i¼1 > 6 7 6 7 >4 > j¼1 5 4 j¼1 5 > : i 6¼ j i 6¼ j  T  ð51Þ T T T T T  .i xi Pil xi þ .id xid Pil xid þ xi Qxi þ ui Rui  si di di þ di ðkÞzi ðkÞ X N  dTj ðkÞf ij xj þ pTi ðk þ 1Þðxi ðk þ 1Þ þ gi ðxi ðkÞ; ui ðkÞ; zi ðkÞÞÞg\0 j¼1 i 6¼ j 123 M. 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Zamani, I., Sadati, N., Zarif, M.H.: On the stability issues for tion. Int. J. Comput. Intell. Appl. 12(03), 1350018 (2013) fuzzy large-scale systems. Fuzzy Sets Syst. 174(1), 31–49 (2011) 31. Prieur, C., Mazenc, F.: ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math. Control Signals Sys- tems 24(1–2), 111–134 (2012) 123 International Journal of Fuzzy Systems 32. Li, H., Liu, A., Zhang, L.: Input-to-state stability of time-varying Iman Zamani was born in Iran. nonlinear discrete-time systems via indefinite difference Lya- He received the PH.D. from punov functions. ISA Trans. 77, 71–76 (2018) Amirkabir University of Tech- 33. Zamani, I., Zarif, M.H.: An approach for stability analysis of TS nology(Tehran Polytechnic), fuzzy systems via piecewise quadratic stability. International Tehran, Iran. Now he is assis- Journal of Innovative Computing, Information and Control 6(9), tant professor of the Department 4041–4054 (2010) of ControlSystems, Shahed 34. Magni L, Scattolini R (2007) Robustness and robust design of University, Tehran, Iran. His MPC for nonlinear discrete-time systems. In: Assessment and research interests include future directions of nonlinear model predictive control. Springer, Hybrid Systems,Biological Sys- pp 239–254 tems, Fuzzy Systems, and Sin- 35. Limon D, Alamo T, Raimondo D, De La Peña DM, Bravo J, gular Systems. Ferramosca A, Camacho E (2009) Input-to-state stability: a uni- fying framework for robust model predictive control. In: Non- linear model predictive control. Springer, pp 1–26 36. Ghaemi, R., Sun, J., Kolmanovsky, I.V.: Robust control of con- strained linear systems with bounded disturbances. IEEE Trans. Mohammad Manthouri was Autom. Control 57(10), 2683–2688 (2012) born in Khoy, Iran, on October 37. Shin S, Hart P, Jahns T, Zavala VM (2019) A hierarchical opti- 12, 1983. He received the B.Sc. mization architecture for large-scale power networks. IEEE incontrol engineering from Shi- Transactions on Control of Network Systems raz University, Shiraz, Iran, in 38. Zhang H, Feng G (2008) Stability Analysis and $ H_ {\infty} $ 2005. He received the M.Sc. and Controller Design of Discrete-Time Fuzzy Large-Scale Systems Ph.D.degrees in control engi- Based on Piecewise Lyapunov Functions. IEEE Transactions on neering from K. N. Toosi Systems, Man, and Cybernetics, Part B (Cybernetics) 38 University of Technology, Teh- (5):1390–1401 ran, Iran, in 2008and 2015, 39. Pagilla, P.R., Dwivedula, R.V., Siraskar, N.B.: A decentralized respectively. Now he is assistant model reference adaptive controller for large-scale systems. professor of the Department of IEEE/ASME Trans. Mechatron. 12(2), 154–163 (2007) Control Systems,Shahed 40. Yougang, Z., Bugong, X.: Decentralized robust stabilization of University, Tehran, Iran. His discrete-time fuzzy large-scale systems with parametric uncer- research interests include Fuzzy tainties: a LMI method. J. Syst. Eng. Electron. 17(4), 836–845 Control, Adaptive ControlSystems, Neural Networks, Deep Learning, (2006) and Machine Learning. Asier Ibeas was born in Bilbao, Spain, on July 7, 1977. He Mohammad Sarbaz was born received his MSc degree in in Tehran, Iran, on October 26, AppliedPhysics and his PhD 1993. He received the B.Sc. degree in Automatic Control incontrol engineering from from the University of the Bas- Islamic Azad University, south que Country,Spain, in 2000 and branch, Tehran, Iran, in 2016. 2006, respectively. He is cur- Hereceived the M.Sc. in control rently Associate Professor of engineering from Shahed Control Systems atAutonomous University, Tehran, Iran. His University of Barcelona, Spain. researchinterests include Model His research interests include Predictive Control, Large-Scale time-delayed systems,robust Systems, time-varying delay, adaptive control, applications of FuzzyControl, and Neural artificial intelligence to control Networks. systems design andnonconventional applications of control such as to epidemic systems, supply chain managementand financial systems, fields where he has published more than 130 contributions in inter- nationaljournals and conferences. 123