Robust Output Tracking of Uncertain Large-Scale Input-Delay Systems via Decentralized Fuzzy Sliding Mode Control

This paper deals with the robust output tracking problem of uncertain large-scale systems with input delay and the time-delay interconnections. Due to the information transmission between subsystems, time delays are often encountered in large-scale systems and lead to the source of system instability. First, the original nonlinear time-delay systems can be represented by the Takagi-Sugeno fuzzy model, which combines some simple local linear time-delay systems with their linguistic description. Then, a feasible and systematic design scheme is presented to synthesize the decentralized fuzzy-model-based sliding mode controller. The adaptive fuzzy approach is proposed to approximate the upper bound of the uncertainties including the time-delay interconnections and the input delay. Based on the Lyapunov stability theorem, the proposed control scheme can not only guarantee the robust stability of the whole closed-loop system with input delay and time-delay interconnections, but also obtain the good tracking performance. Finally, simulation results are given to confirm the effectiveness of the proposed controller in this paper.


Introduction
Recently, there have been a nu mber of research works on stability analysis and design for a class of large-scale interconnected systems, such as electrical networks, nuclear reactors, and hydraulic systems, etc. Due to the large-scale system with interconnected terms, the decentralized controller is preferred to be adopted as a control methodology such that design procedures can be simplified and the computational burden can be shared by all the subsystem controllers [1][2][3][4][5]. In fact, for the comp lexity of large-scale systems, the uncertainty and time delay are often encountered in these systems. Therefo re, the problem of control design and stabilization for a class of large-scale systems with uncertainties and time delays becomes an important topic.
In practices, due to the in formation t ransmission between subsystems, t ime delays inev itab ly occu r in large-scale systems. Also, the existence of time delays is often a source of instability in various engineering systems. Therefore, the stabilizat ion problem of large-scale systems with delayed states and the time-delay interconnections has been widely studied in the literature [6][7][8][9][10]. It has been shown that the presence of input delays, if not considered in a controller design, may cause instability or serious deterioration in the performance of the resulting nonlinear control systems [11][12][13][14]. In this study, the problem of the decentralized control for a class of input-delayed large-scale systems with the time-delay interconnections is investigated.
On the other hand, fuzzy logic control has been successfully applied to the control design of nonlinear control systems [15][16][17][18]. It is well-known that fuzzy logic control, which is based on fuzzy sets and fuzzy reasoning with a set of linguistic control rules, does not need a rigorous mathematical model and is more insensitive to plant parameter variations and noise disturbance. It has been shown that the method of T-S fu zzy models, in terms of IF-THEN rules with a linear input-output relation, gives an effective and feasible approach to the control problem of complex nonlinear systems [19][20][21][22][23]. Therefore, in this study, the proposed control scheme is based on the T-S fu zzy model to deal with the control design of uncertain large-scale system with delayed input and time-delay interconnections.
Sliding-mode control (SM C) systems have been extensively studied and widely applied to many engineering systems. Due to its good robustness to uncertainties, sliding mode control has proven to be an effective method for robust control of nonlinear systems. In recent literature, so me researchers proposed the design methods of fuzzy logic control based on sliding-mode approaches for a class of ill-defined or poorly modelled systems [24][25][26][27][28][29]. Recently, the stability and control design of large-scale systems has attracted the attention of many control researchers and been studied extensively [30][31][32][33][34]. Wang et al. [33] and Hsiao et al. [34] presented linear state feedback control approaches based on T-S fuzzy model for the large-scale system, respectively. Un like previous works, this paper is to present a different control scheme to tackle the problem of large-scale systems with delayed input and the time-delay interconnections, without the assumption that the upper bounds of the interconnections and modelling errors must be known.
This paper is concerned with the robust stability and output tracking control problem of decentralized fuzzy-model-based sliding mode controller for uncertain large-scale systems with delayed input and the time-delay interconnections. First, in this paper the original nonlinear time-delay large-scale systems can be represented by the Takagi-Sugeno fuzzy model, which co mbines some simp le local linear t ime-delay systems with their linguistic description. Then, an effective and feasible design scheme is developed to synthesize the proposed decentralized fuzzy-model-based sliding mode controller with some adaptive fuzzy laws to approximate the upper bound of the uncertainties including the time-delay interconnections and the delayed input. Finally, simu lation results are given to demonstrate the validity of the proposed controller in this paper.
The rest of this paper are organized as fo llo ws. In Section 2, some properties of the T-S fuzzy system are reviewed, and the large-scale systems with time delays and uncertainties are formu lated in detail. Furthermore, the control design met hod to synthes ize the p ro posed decent ralized fuzzymodel-based sliding mode controller and the analysis of robust stability are included in Section 3. In Section 4, simu lation results are given to verify the effectiveness of the proposed decentralized controller in this paper. At last, a conclusion is given in Section 5.

Problem Formulation
The fuzzy dynamic model, proposed by Takagi and Sugeno, is described by fuzzy IF-THEN rules, which represents local linear input-output relations of nonlinear systems. Let us consider the uncertain input-delayed large-scale systems with time-delay interconnections which can be described by the following fu zzy model: where Accordingly, the main control ob jective of this paper is to utilize the decentralized fuzzy -model-based sliding mode control such that the robust stability of the whole closed-loop system with input delay and time-delay interconnections can be guaranteed.

Decentralized Fuzzy-Model-Based Adaptive Sliding Mode Controller Design
In this section, the control objective is to design a decentralized fu zzy -model-based sliding mode control scheme such that the desired state trajectory of the closed-loop system is ach ieved and the effects of system uncertainty can be attenuated while maintaining the boundedness of all signals inside the control loops.
Using the T-S fuzzy model (4) of the original system, it can be obtained that Fro m (2), (3), and (5), we obtain or equivalently of the form Define the controller as the following form: where ( ) si u t will be determined in the latter.
The control objective is to drive the state

Assumpti on 2:
The desired state  (9), and (10), the error dynamic system of the th i subsystem can be expressed as Define the fuzzy basis function as Then the fuzzy logic system (15) is equivalent to a fuzzy basis function expansion where ij Ω is the convex co mpact set, wh ich contain the feasible parameter set for * ij θ , and and the adaptation error of adaptive gain 0i ζ is defined as According to the proposed decentralized sliding mode control scheme in (10), the co mposite switching hyperplane is defined by letting the composite switching vector , where the sliding surface of each subsystem is selected as the following form: it imp lies that tracking error tends to zero as ∞ → t . Based on Assumption 3, we get the decentralized control law as follows: Now, the following adaptive laws for those unknown parameters in (14) and (19) are chosen as: where 1 i γ , 2 i γ , and 3 i γ are positive constants specified by the designer. The proposed control law will guarantee the asymptotical stability for the error dynamics of (13), and it will be proved in the following theorem. Theorem 1: Fo r the subsystems consisting of (1), the decentralized fu zzy-model-based sliding mode control law is chosen as (10) with (25), and consider the adaptation laws (27) Using the control of ( ) t u si , the slid ing surface may be expressed as Then the derivative of i V 1 can be stated as follows: Thus, we get Then, computing the time derivative of i V 2 , we have By the fact Thus from (27) Then, we know that is convergent, from the above analysis we obtain that the solutions i S , i ω  , ij θ , and ij ξˆ are bound.
Because of the boundedness of all the signals, it is obvious fro m (33) that i S  is bounded. From (40) and based on the above discussion, this imp lies that 2 L S i ∈ . According to Barbalat's Lemma [35], we can get tends to zero at ∞ → t . Thus, we conclude that the asymptotic state tracking can be achieved.

An Example and Simulation Results
In this example, we consider a large-scale system T composed of two fu zzy subsystems i T defined as Subsystem 1: If is about 0 If is about 2 2  11  11  3  3  1  1  1  1  1  1  12  12 If is about 4   If is about 2 The membership functions for ( ) 11 x t , If is about 0 x t

Conclusions
The problem of robust stability and output tracking control for a class of uncertain large-scale input-delay systems with t ime-delay interconnections is investigated in this paper. In addition, a feasible and systematic design method is provided to develop the decentralized fuzzymodel-based adaptive sliding mode controller with so me adaptive laws to approximate the upper bounds of the uncertainties including the time-delay interconnections and the delayed input. Based on the Lyapunov stability theorem, the proposed control scheme not only guarantees the robust stability of the whole closed-loop system, but also achieves the good tracking perfo rmance. An examp le and simu lation results are illustrated to verify the effect iveness of the proposed controller in this paper.