Control Design for Discrete-Time Fuzzy Systems with Disturbance Inputs via Delta Operator Approach

This paper is concerned with the problem of passive control design for discrete-time Takagi-Sugeno (T-S) fuzzy systems with time delay and disturbance input via delta operator approach. The discrete-time passive performance index is established in this paper for the control design problem. By constructing a new type of Lyapunov-Krasovskii function (LKF) in delta domain, and utilizing some fuzzy weighing matrices, a new passive performance condition is proposed for the system under consideration. Based on the condition, a state-feedback passive controller is designed to guarantee that the resulting closed-loop system is very-strictly passive. The existence conditions of the controller can be expressed by linear matrix inequalities (LMIs). Finally, a numerical example is provided to demonstrate the feasibility and effectiveness of the proposed method.


Introduction
It is well known that the fuzzy logic control [1][2][3][4] is one of the most effective approaches to handle complex nonlinear systems.Takagi-Sugeno (T-S) fuzzy model has been proposed in [5] and applied to formulate a complex nonlinear systems into a framework.Some affine local models can be interpolated by a set of fuzzy membership functions in this framework.By T-S fuzzy model, a set of complex nonlinear systems can be possibly described as a weighed sum of some simple linear subsystems.Recently, the problem of stability analysis and controller synthesis of nonlinear systems in T-S fuzzy model has been extensively investigated in [5][6][7][8][9][10][11][12][13][14][15][16][17][18].Due to the effect of time delay in systems, the problems of T-S fuzzy systems with time delays have got considerable attention in recent years, and some results have been developed in [9,[19][20][21][22][23][24][25][26][27][28][29][30][31].
Recently, due to the fact that the passive properties can keep the system internally stable and have been frequently used to improve the stability of control systems.The passivity control has been widely applied in various engineering areas, such as electrical circuits systems, complex networks systems, mechanical systems, and nonlinear systems.The problems of passivity analysis and passive control for systems have been widely investigated [32][33][34][35][36][37][38].The passive control problem has been investigated for fuzzy systems [35,39,40].Among them, the authors in [40] are concerned with the very strict passive controller design problem for T-S fuzzy systems with timevarying delay.
It is well known that the best system performances can be obtained with the shorter sampling period.In [41,42], the authors pointed out that the excellent finite word length performance can be achieved under fast sampling via delta operator approach.The authors in [43] introduced the transformations between shift operator and delta operator transfer function models.A technique was developed in [44] to obtain an approximate delta operator system for a given continuous system.In [45], a tabular method was presented for real polynomial root distribution with respect to a circle in the complex plane, which is useful in stability of delta operator Mathematical Problems in Engineering formulated discrete-time systems with a sampling time.By using delta operator formulation, the main design features of a loop transfer recovery controller both at input and output have been overviewed in [46].More recently, many stability analysis and controller synthesis results about delta operator approach have been proposed in [47][48][49].To mention a few, the robust stabilization problem was investigated in [47] for delta operator systems with time-varying delays.The authors in [49] investigated the robust  ∞ control problem for a class of T-S fuzzy systems with time delays by using delta operator approach.However, there have been few results on the passive control for T-S fuzzy systems with time delays and disturbance inputs via delta operator approach, which motivates this study.
In this paper, the problem of passive control is investigated for discrete-time T-S fuzzy systems with time delay and disturbance input via delta operator approach.The discrete-time passive performance index is concerned in this paper for the control design problem.A novel type Lyapunov-Krasovskii functional (LKF) is constructed in delta domain to present a new passive performance condition for discrete-time T-S fuzzy systems.Based on the condition, a state-feedback passive controller is designed to guarantee that the resulting closed-loop system is very strictly passive.The controller existence condition can be obtained in terms of linear matrix inequalities (LMIs), which can be solved by the standard software.A numerical example is given to illustrate the effectiveness of the proposed approach.
At first, the T-S fuzzy model is employed to represent the nonlinear systems.By applying the LMIs techniques and LKF in -domain, the problem of passive control design for the discrete T-S fuzzy system with time delay and disturbance inputs is chewed.Then a new fuzzy state-feedback controller is designed which guarantees that the closed-loop fuzzy delta operator system with time delay is robustly asymptotically stable and satisfies a prescribed passive performance level.And those are some key points of contribution.It is worthwhile to note that a faster sampling method is utilized, and hence a better control effect by applying delta operator approach than shift operator approach is achieved.Finally, a numerical example is shown to indicate the feasibility and effectiveness of the proposed method.
This paper is organized as follows.The problem to be solved is formulated in Section 2. Main results, including passive analysis and passive controller design, are presented in Section 3. Section 4 provides an illustrative example to show the effectiveness and potential of the proposed design techniques.It is concluded this paper in Section 5.
Notation.The notation used throughout the paper is fairly standard. 2 [0, ∞) denotes the space of square-integrable vector functions over [0, ∞).The notation  > 0 (resp.,  ≥ 0), for  ∈ R × , means that the matrix  is real symmetric positive definite (resp., positive semidefinite).The symbol " * " in a matrix  ∈ R × stands for the transposed elements in the symmetric positions.The superscripts "" and "−1" denote the matrix transpose and inverse, respectively.Identity matrices of appropriate dimensions will be denoted by .
Definition 1 (see [47]).The conditions for the asymptotic stability of a delta operator system hold: (a) ((  )) ≥ 0, with equality if and only if (  ) = 0, where ((  )) is a Lyapunov function in -domain.For Lyapunov function both in s-domain and z-domain, the condition () ((  )) ≥ 0 in Definition 1 is given.On the other hand for the condition (b), when  → 0, there exists lim and when  = 1, there exists Obviously, the Lyapunov function in -domain can be reduced to the traditional Lyapunov function in s-domain or z-domain when the sampling period  tends to 0 or is 1.
Definition 2 (see [50]).(i) System ( 5) is said to be passive if there exists constant  such that (ii) System ( 5) is said to be strictly passive if there exist constants  > 0 and  such that (iii) System ( 5) is said to be output strictly passive if there exist constants  > 0 and  such that (iv) System ( 5) is said to be very strictly passive if there exist constants  > 0,  > 0, and  such that Lemma 3 (see [47]).For any of the time functions (  ) and (  ), where  is a sampling period.
Lemma 5 (see [52]).For the given constant matrices ,  and a symmetric constant matrix  of appropriate dimensions, the following inequality holds: where (  ) satisfies   (  )(  ) ≤ , if and only if for  > 0 the following inequality holds: Lemma 6.If there exist scalars  > 0,  > 0, and a differential function ((  )) ≥ 0 such that then system (5) is very strictly passive.
Remark 7. Based on Definition 2, the main objective of this paper is just to prove that T-S fuzzy system ( 5) is very strictly passive via delta operator approach, which can also satisfy other three indexs.The very strictly passive control for system (5) is shown in the next section.

Main Results
This section focuses on designing a sufficient condition for the solvability of the proposed passive control problem and a developed LMI approach for designing the passive controller for fuzzy system (5).Firstly, the passivity analysis criterion is derived for the system (5) in the following theorem.
Remark 10.In this paper, the modeling uncertainties are not taken into account in system (1).It should be pointed out that the robust passive controller design condition can be presented for uncertain discrete-time T-S fuzzy systems with time delay and disturbance input via delta operator approach.In order to present the stability and stabilization condition for discrete-time T-S fuzzy systems with time delay via delta operator approach, it is assumed that the disturbance input (  ) = 0 in fuzzy delta operator system (3), which can be described as Mathematical Problems in Engineering Similar to the proof of Theorems 8 and 9, the following two corollaries can be obtained.

Numerical Example
In this section, one example is presented to demonstrate the effectiveness of the proposed method.
Example 1.Consider the following fuzzy system with timevarying delay and input disturbance: The system matrix parameters in ( 45) and ( 46) are given as follows: Two membership functions are chosen for Plant Rules 1 and 2 as follows: And Figure 1 shows the membership functions.Define the value range of time delay   = , and choosing the lower bound   = 0.01, the sampling period  may be assumed as  = 0.001 and it can be found from Theorem 9 that the maximum upper bound of time delay   and the control gain matrices are listed in Table 1.
Assume that time delay   =  = 40 and the disturbance input (  ) = −1/(2 +   ).It can be calculated  In Figure 2, it can be observed that (  ) decreases finally as the time   increases, which means that there may not exist a scalar  such that (  ) ≥  hold for all   ≥ 0 for  = 0.001.Furthermore, Figure 3 shows that there may not exist a scalar  such that (  ) ≥ , which means that the open-loop system is not passive in the sense of Definition 2, and it is not very strictly passive.Figure 4 illustrates that the open-loop system is not stable for  = 0.001.
Under the control gain matrices in Table 1, Figure 5 plots the responses of (  ) for the closed-loop system for  = 0.001.It can be seen that there may exist a scalar  such that (  ) ≥  holds for all   ≥ 0. It can be seen from Figure 6 that there may exist a scalar  such that (  ) ≥ .Then, it is clear that the closed-loop system is very strictly passive under the control gain matrices in Table 1.In addition, Figure 7 shows that the closed-loop system is stable.Furthermore, Figure 8 depicts the control input responses.These simulation results have demonstrated the effectiveness of the proposed method.

Conclusions
In this paper, the problems of passivity analysis and passive control have been investigated for fuzzy delta operator systems with time delay and disturbance input.By applying some new LKF in -domain and utilizing some fuzzy weighing matrices, the state-feedback controller has been designed to guarantee that the resulting closed-loop system is very strictly passive.The existence conditions for the controller have been expressed as LMIs.Finally, a numerical example has been included to illustrate the effectiveness of the proposed results.In future work, based on T-S fuzzy control method, the problems of control and monitoring in the data-driven framework [53,54] could be further studied.

Figure 1 :
Figure 1: Membership functions of two rules.

Figure 2 :
Figure 2: Response of (  ) of the open-loop system.
State response of the closed-loop system

Figure 7 :Figure 8 :
Figure 7: State response of the open-loop system.