Abstract
The problem of an unmatching observer-based controller design for discrete-time fuzzy systems with time delay is investigated, in which the fuzzy controller shares different membership functions from the fuzzy model. The objective is to design a state observer and unmatching fuzzy controller such that the discrete closed-loop system with time delay is asymptotically stable. A sufficient condition that contains the information of the membership functions of fuzzy model and fuzzy controller for the stabilization via an unmatching observer-based output feedback is presented. The proposed control scheme is well capable of enhancing the design flexibility, and the stability condition is less conservative. Three numerical examples are given to illustrate the effectiveness and advantages of the proposed method.
1. Introduction
Time-delay phenomena intensively exist in natural science and engineering and social life, such as nonlinear and time-delay models. For example, they are in numerous dynamical systems including biology systems, mechanics, economics, chemical systems, and network systems. Generally, time delay can often lead to instability and poor performances. Therefore, it is of great significance to investigate the issue of the stability for time-delay systems. A lot of relevant research work has been reported in [1–3]. However, the results obtained are mainly only based on the linear time-delay systems. It is indeed necessary to generalize them to the nonlinear time-delay systems. As we know, the fuzzy model proposed by Takagi and Sugeno can effectively represent many nonlinear dynamic systems [4]. During the past decade, more and more researchers paid attention to the issue of the stability for the nonlinear fuzzy systems with time delay [5, 6]. The stability problem refers to the design of a controller that can guarantee the fuzzy model stability. State feedback plays an important role in handling stabilization in the control systems. However, sometimes the system states cannot be measured directly, and there are limitations of the measurement equipment in use. It is actually impossible to get all the information of the system state variables. Under this circumstance, the physical form of state feedback is difficult to realize. Therefore, state estimation and observation of nonlinear systems are important but challenging problems in modern control theory. The design of observers and controllers for the T-S fuzzy systems is a demanding topic. For example, the stability analysis and stabilization of the T-S fuzzy models for designing the observers and control laws are addressed in [7, 8]. A single-step linear matrix inequality method is developed for the observer-based controller design for discrete-time fuzzy systems in [9], which overcomes the drawbacks induced by the conventional two-step approach, and yields less conservative results. The problem of fuzzy observer-based controller design is investigated for nonlinear networked control systems subject to imperfect communication links and parameter uncertainties in [10]. Some results are generalized to the observer-based adaptive model in [11]. A novel method based on the nonuniform delay partitioning approach is put forward to analyze the stability of the time-delay T-S fuzzy system, and the observer-based feedback controller design scheme via the parallel distributed compensation (PDC) scheme is also discussed in [12]. A fuzzy functional observer method is proposed to design a controller for the observer-based fuzzy model in [13]. Furthermore, some results related to the observer-based controller design problem have been extended to the MIMO time-delay systems in [14]. Recently, the observer-based controller design problem is investigated in networked control systems [15]. In the abovementioned work, the observer-based feedback controller design problems are all based on the PDC scheme, which means that the observed-based fuzzy model and the observed-based fuzzy controller share the same membership functions. As a matter of fact, if the membership functions of the fuzzy controller in the premise of the fuzzy rules are allowed to be designed arbitrarily, a greater design flexibility can be achieved. Thus, some research studies on the control problems under imperfect premise matching have been carried out in [16–20]. As the states of the systems are sometimes difficult to be obtained or the costs of measurement are too high in many practical problems, it is critical to design observers for the systems. In addition, nonlinear and time delays are inherent and not all the states are available in most practical systems. Therefore, the observer-based stabilization control for the discrete-time T-S fuzzy systems with time delay is an important topic.
In this paper, we focused on the observer-based stabilization control for discrete-time T-S fuzzy time-delay systems and a novel observer-based output feedback controller is investigated for the discrete-time T-S fuzzy time-delay systems. The main contributions of this paper can be summarized as follows:(i)As the membership functions of the fuzzy model and the fuzzy controller are all considered in the analysis, some membership-function-dependent stability conditions are derived, which are less conservative than membership-function-independent stability conditions(ii)As the membership functions of the controller are distinctive from those of the fuzzy model, the membership functions can be chosen arbitrarily, which makes a higher degree of design flexibility of the fuzzy controller achieved, and its structure can be further simplified
The remainder of this paper is organized as follows. In Section 2, the considered problem is described. An unmatching observer-based controller design method is proposed in Section 3. Three numerical examples are utilized to illustrate the conservativeness, effectiveness, and superiority of the proposed method in Section 4. Finally, Section 5 concludes this paper.
Notation 1. In this paper, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation is used to denote a symmetric positive-definite (positive semidefinite, negative, and negative semidefinite, respectively) matrix. denotes the Euclidean norm for vector or the spectral norm of matrix. For convenience, we use instead of .
2. Problem Formulation and Preliminaries
Let be the number of the fuzzy rules describing the time-delay nonlinear plant. The ith rule can be represented as follows.
If is and … and is , thenwhere is the number of the fuzzy rules and , , denotes the fuzzy set. , are the known premise variables which do not depend on the input variables. is the state vector, is the input vector, is the output vector, and are some constant matrices of compatible dimensions. Here, we assume that is a constant.
With the fuzzy inference methods, the final outputs of the fuzzy time-delay model can be formalized as follows:wherein which are the premise variable and is the grade of membership of in .
In engineering practices, the state information of system equation (1) is often not available. Therefore, it is necessary to construct a state observer satisfying the following formal equation to estimate the state of the discrete system.
Observer rule : if is and … and is , thenwhere is the fuzzy observer gain for the subsystem.
The overall fuzzy observer is represented by
With the aforementioned fuzzy observer, an unmatching fuzzy observer control law is defined as follows.
If is and … and is , then
The overall observer-based fuzzy control law is represented by
Combining the fuzzy controller in equation (7) and the fuzzy observer in equation (5) and denoting , we obtain the following system representations:
Therefore, the augmented system can be written aswhere
Remark 1. It can be observed from equation (10) that the controller we design and the observer-based fuzzy model do not share the same membership functions. Thus, for the observer-based T-S fuzzy time-delay systems, the membership functions of the fuzzy controller can be arbitrarily selected in the design phase, which can enhance the design flexibility. Consequently, it is worthy to investigate the stability conditions of this fuzzy time-delay system.
Lemma 1. (see [21]). The linear matrix inequality is equivalent to where , and is a matrix with appropriate dimension.
Lemma 2. For any there exist and such that , where
Proof. For convenience, we use instead of . Define is the eigenvalue of . Assume that are the eigenvectors of . Here, can be regarded as a group of bases of n-dimensional Euclidean space. Assuming that , then for any , we have , and thenEspecially, if we select thenProvided that and , we haveNotice that , so there exist and , such thatDefining we have
3. Main Result
Theorem 1. The equilibrium of the closed-loop discrete fuzzy time-delay system in equation (10) with an unmatching observer-based control law in equation (7) is asymptotically stable, if the membership functions of the fuzzy model and fuzzy controller satisfy for all and , where . There exist , , , , and satisfyingwhere
The state feedback gains and observer gains can be constructed as
Proof. Select a Lyapunov function aswhere and . By Lemma 2, there exist and such thatIn this section, we will prove the asymptotic stability of the discrete time-delay system in equation (10) based on the Krasovskii theorem [22].whereIfthen .
By Schur complement,Ifthen .
It is obvious that equation (25) is equivalent towhere and Letand by substituting equation (27) into equation (26) and defining we can show equation (26) is equivalent to the following LMIs:Premultiplying and postmultiplying equation (29) with and denoting we haveFor arbitrary , we haveThese terms are introduced to equation (28) to alleviate the conservativeness. From equation (28), we obtainWith for all and , letfor all , then
On the other hand, for equation (30), we haveLetfor all , then
If and , then equation (28) and equation (30) hold, so that equation (26) holds, then we have which means system (10) is asymptotic stable.
4. Numerical Examples
In this section, three examples will be given to illustrate the less conservativeness and the effectiveness of the proposed results.
Example 1. From the proposed results, we can see the stability conditions are connected with the values of . In this example, different values of will be testified using the LMI based on the stability conditions in Theorem 1 and then we will illustrate the less conservativeness of the proposed results. Consider the following fuzzy time-delay model [6]:
Rule : if is , thenwhereIt is assumed that the membership functions of the discrete fuzzy model with time delay and fuzzy controller are different and satisfy for all and . The feedback gains of , are determined such that the eigenvalues of , are located at . The stability region given by stability conditions in Theorem 1 with and (denoted by “o”) and and (denoted by “ד) is shown in Figure 1. For comparison purposes, the stability region given by the stability conditions in [5, 6] (denoted by “×” and denoted by “o”, respectively) is shown in Figure 2. From Figure 1, it can be seen that the proposed stability conditions in Theorem 1 become lager with the increase in , , and . Meanwhile, the proposed stability conditions offer a larger stability region than in [5, 6], as can be seen in Figure 2. Based on the simulation, it can also be observed that when and take certain values, the stability conditions in [5, 6] cannot provide feasible solutions.
Example 2. An unmatching observer-based output feedback controller is designed to backing up control of a computer-simulated truck-trailer [23]. Consider the following modified truck-trailer model with time delay:where is the angle difference between the truck and trailer, is the angle of trailer, is the vertical position of the rear end of trailer, and is the control input. The model parameters are given as ,
The fuzzy truck-trailer model can be modeled by a two-rule fuzzy model.
Rule , : if is about 0, thenIf is about , thenwhereThe fuzzy state observer for the T-S fuzzy model in equations (39) and (40) is formulated as follows.
Observer rule : if is , thenThe output of equation (42) is represented as follows:Here, we set and the membership functions are defined asFrom Theorem 1, according to the abovementioned fuzzy state observer model, an unmatching fuzzy controller with two rules is proposed to control the observer model so that the dynamic system equation (43) is asymptotically stable.
Rule : if is , thenThe fuzzy controller is defined as The membership functions are selected asWe assume that such that for all and .
Assuming and applying the MATLAB LMI toolbox to solve the LMIs in Theorem 1, we haveWith the initial condition and , the simulation results under Theorem 1 are shown in Figure 3, which illustrate the effectiveness of Theorem 1.
(a)
(b)
(c)
(d)
Example 3. The same fuzzy state observer model equation (43) is also considered here. Some advantages of the proposed method are shown in this example. Suppose the membership function is chosen as follows:where and , in which is an uncertain variable.
Comparing with (equation (44)) in Example 2, we can observe that the structure of the membership functions (equation (49)) is not only complex but also contains an uncertain variable. By equation (46), under the PDC controller design framework, the membership functions of the fuzzy controller must be selected as the same as those of the fuzzy model, which will lead to the fuzzy controller unrealized. However, based on Theorem 1, some simple and specific membership functions (equation (47)) can be selected instead of the complex and uncertain functions (equation (49)) as the membership functions of the fuzzy controller, which can ensure the closed-loop system (10) is asymptotically stable (refer to Figure 4).
Remark 2. Comparing with the PDC scheme, as the membership functions of the controller are distinctive from those of the fuzzy model in our paper, the membership functions can be chosen arbitrarily. Therefore, the proposed method can enhance the design flexibility of the fuzzy controller as well as retain the robustness property of the T-S fuzzy control systems.
5. Conclusions
In this paper, an unmatching observer-based stabilization control for discrete-time T-S fuzzy systems with time delay is investigated, in which the discrete fuzzy observer model and fuzzy controller do not share the same membership functions. The information of the membership functions of the fuzzy model and controller is considered in the observer design scheme. Furthermore, the design flexibility can be enhanced by arbitrarily selecting simple membership functions for the observer-based controller. The advantages and effectiveness of the proposed method have been illustrated using simulation examples. According to the obtained results, the proposed method only fits for the systems with constant delay. Therefore, how to extend the abovementioned method to time-varying delay systems will be considered in the future.
Data Availability
The process data used to support the findings of this study have not been made available because they form part of an ongoing study.
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
Acknowledgments
This work was supported by projects of the National Natural Science Foundation of China (NSFC) (51805008) and project of the Scientific Research of Jilin Provincial Department of Education (JJKH20190644KJ and JJKH20190639KJ).