Abstract

The performance of a class of T-S fuzzy systems with Markovian-jump parameters is analyzed. A state feedback controller is designed for T-S fuzzy systems with Markovian-jump parameters. First, a new modal-dependent Lyapunov function composed of closed-loop functions is constructed, which can fully use system status information. Based on this function, the stability conditions with less conservatism are given by linear matrix inequalities (LMIS). At the same time, a design algorithm for a state feedback controller is proposed for the Markovian-jump-parameters T-S fuzzy systems, which ensures the system’s stability under the condition of performance. Simulation results demonstrate that the mentioned method is accurate and practical.

1. Introduction

Takagi–Sugeno (T-S) fuzzy systems can provide an effective description of complex nonlinear system problems [1]. The essence of T-S fuzzy systems is nonlinear systems, but each corresponding rule is linear, which is convenient for people to analyze the systems’ stability and study the control problem. In recent decades, T-S fuzzy systems have become a research hotspot and an important method for nonlinear system control. The stability of a static output controller for a discrete-time T-S fuzzy system was studied by introducing relaxation variables and considering multiple Lyapunov matrices in [2]. The equivalence of the weight product index set for fuzzy rules was defined, and a new stability criterion with less conservatism was proposed to guarantee the system’s stability using the equivalence relation in [3]. Two efficient design methods for fuzzy static output feedback controllers were proposed by combining Markovian Lyapunov functions and matrix inequality convexification techniques, and the robust stabilization of a nonlinear hyperbolic partial differential equation (PDE) system represented by a T-S fuzzy system is successfully solved in [4]. At the identical time, in order to make the system stable and ensure its good performance, many scholars have studied comprehensive control methods such as control and control. In particular, control can effectively suppress interference, so it is widely used in electric ground vehicle systems [5, 6] and electric train asynchronous motor systems [7]. A new augmented Lyapunov function was proposed to ensure the tracking performance of the T-S fuzzy networked control systems [8]. The nonlinear systems were transformed into continuous-time T-S fuzzy systems with uncertain parameters using the Wertinger integral inequality and the extended inverse convex matrix inequality, and the asymptotic stability condition with less conservativeness was proposed in [9]. A new bounded real lemma was constructed for a class of fuzzy time-delay systems, and a new Lyapunov–Krasovskii functional (LKF) was proposed in [10]. The robust output feedback control problem for a class of interval type 2 T-S fuzzy systems with multiple delays and disturbances was studied, and a delay-dependent dynamic output feedback controller was designed using the generalized redundancy method to make the closed-loop system asymptotically stable and obtain the performance in [11]. On the other side of research, the structure or parameters of increasingly complex automatic control systems inevitably jump randomly after a long period of operation, so stochastic hybrid systems with Markovian-jump parameters have received extensive attention. Markovian-jump systems are very effective in describing environmental changes, system failures, and repairs and therefore are widely used in circuit systems, aircraft systems, manufacturing systems [1216], and networked control systems under cyber-attacks [17]. In recent years, many scholars are interested in the Markovian-jump fuzzy systems, and these studies have yielded a large number of meaningful results [1820].

Moreover, as digital technology expands rapidly, research on sampled-data control has received high attention. The sampled-data control methods only need to use the instantaneous sampling information of the system, which greatly reduces the information transmission, so it can guarantee a better system performance and reduce the system operating costs. Sampled-data control methods include input delay method [21], discrete-time system method [22], and closed-loop function method [23]. It is worth mentioning that the looped function mentioned in the closed-loop function method is not necessarily a positive function that fully utilizes the information on the intervals to . At the jumps, this Lyapunov function does not increase, and the whole sampling interval is sufficiently considered to yield less conservative conditions. These functionals are suitable for impulsive systems analysis as they allow one to express the discrete-time stability conditions in an affine manner, thus enabling the consideration of uncertain and time-varying systems. The research results of sampled-data control are numerous and significant [2428]. However, the application of the looped function method to Markovian-jump T-S fuzzy systems with sampled data has been rarely investigated. In [29], a new Lyapunov function whose time scheme consists of an exponential closed-loop function was proposed, and linear interpolation and binning techniques were proposed to obtain criteria for mean-square exponential stability, but the conclusions were less conservative. A new input delay-dependent vector method proposed for fuzzy Markovian systems with sampled data was first mentioned in [30].

Based on the above discussion, our work is motivated by the issue of exploring the control of Markovian-jump T-S fuzzy systems based on sampled data. Some of the topics to be pondered are, for example, how to construct the Lyapunov functions to efficiently utilize the system information and how to modify the integral inequality to make the results less conservative. The nonfragile controllers [3137], which are highly resistant to uncertainty in their parameters, have been widely used in many practical systems, but how we can apply them to Markovian-jump T-S fuzzy nonperiodic sampling control systems.

The contributions of this work consist of the following aspects: (1) A new mode-dependent Lyapunov function consisting of bilateral closed-loop functions is used in the T-S fuzzy systems with Markovian-jump parameters, this function fully covers the information on both intervals and , whereas the Lyapunov function constructed in the literature [38] only contained and does not contained , however, our constructed function includes both, thus the results derived in this study are less conservative. And we use a modified integral inequality with a free matrix to help deal with an integral part of the derivative of the closed-loop function, enabling the systems to be less conservative. (2) A state feedback controller design algorithm is proposed to ensure the system’s stability under the performance condition and a truck-trailer model is adopted to verify the effectiveness of the proposed method.

The rest of the article is organized as follows: some necessary lemmas and systems depiction are presented in Section 2; the stochastic stability condition for Markovian-jump T-S fuzzy nonperiodic sampling control systems is given under the condition and a sampled-data controller with state feedback is designed in Section 3; and the effectiveness of our proposed method is verified with a truck-trailer model in Section 4; finally, we conclude this work with a summary in Section 5.

2. Preliminaries

2.1. Two Useful Lemmas

Lemma 1 (see [23]). Let be a differentiable function: . If there is a vector , matrix , and , then the following inequality holds:where .

Remark 1. This inequality is based on the free weight matrix inequality and is improved based on the inequality in [39]. The inequality contains instead of , and this will facilitate the deduction of the main results and degrade the conservatism of the main results.

Lemma 2 ((Schur complement lemma) [40]). If there are symmetric matrixes , B, and , then the following three conditions are equivalent:

2.2. Depiction of T-S Fuzzy Systems with Markovian-Jump Parameters

Concerning the continuous-time T-S fuzzy systems with Markovian-jump parameters as follows, whose IF-THEN rules are described:(1)Rule i: IF is , is ,…, is , THENwhere , is the number of IF-THEN rules, , , , are premise variables, are fuzzy sets, is the vector describing status of system, is the vector indicates the control input, is the perturbation input taking value in range , is the vector of control output, is a Markovian process in continuous time taking value in a finite set , and with switching transition probability matrix is described as follows:where , . When , indicate the rate of mode transition from mode at time to mode at time , and .According to fuzzy logic inference method, system (3) can be reconstructed as follows:where , , and , denote the membership degree of in , we can easily find out that for all , , , , and . For all the possible , we can simplify fuzzy system (5) as follows:whereConsider the zero-order hold function, which gets a set of hold times , and for any , , indicate the lower boundary of the sampling periods, and indicate the upper bound of the sampling periods, then, the state feedback controller for system (3) is designed as follows.(2)Rule :IF is , is ,…, is , THENBased on the fuzzy logic inference method, the state feedback controller is reconstructed as follows:where

Substituting the into system (6), the closed-loop system is as follows:

3. Main Results

We propose a sufficient condition to solve the control problem formulated in the previous section.

First, two definitions are necessary.

Definition 1 (see [41]). When , the system (10) is stochastic stable, if the continuous function defined on the interval , and the initial mode , then the following inequality holds:where denotes the mathematical expectation and indicates the solution of system (10) at the instant under the initial condition and .

Definition 2. For a control law that any , , system (10) is stochastic stable under the condition. That is to say, for some scalar , system (10) is stochastic stable if the inequalityholds under zero initial condition.
Next, we focus on the following system:If the controller is a constant, we will give the stochastic stability conditions as follows:

Theorem 1. Given a scalar , T-S fuzzy system (13) is stochastic stable if there are matrices , , , , , , and , and arbitrary matrices , , , , , , and with appropriate dimensions such that the following inequation is true for each , , :where , , , , , , , , , , and .

Proof. Let us take the Lyapunov function for system (13) as follows:where is a quadratic Lyapunov function defined as , and with , , where , , and .
Then, for each , we say that is a weak infinitesimal generator of , it follows that , , ,
By Lemma 1, the following two inequalities hold:where , , , and.
Moreover, based on the system (13), for arbitrary matrices , , and with appropriate dimensions, the following equation is true:Observing the two inequalities (16) and (17), let’s add to the right side of the equation (18), thus, for any , there is the following conclusion:where and .
We can conclude from Lemma 2 and (14) thatThis meansBy Dynkin formula, for any , we get the following equation:Hence,After the analysis above, we finally reach the conclusion.Then, we can conclude from Definition 1 that system (13) is stochastically stable.

Remark 2. Inspired by literature [38], we propose a new Markovian modal-dependent Lyapunov function, this function fully utilizes the system sampled-date information from both intervals and , which satisfies the closed-loop function condition (). However, the Lyapunov function constructed in the literature [38] only contained and does not contain whereas our constructed function includes both, thus the results derived in this study are less conservative.
Next, based on Theorem 1, we give the stochastic stability condition of system (10) under the condition.

Theorem 2. For a given scalar , T-S fuzzy system (10) is stochastic stable under the condition, if there are matrices , , , , , , and , and arbitrary matrices , , , , , , such that inequation (25) is true for each , , and :where , , , , , , , , , , , , , and, , and the definitions of , , , are similar to that in Theorem 1.
Next, we give the proof of Theorem 2.

Proof. Observe the statement in Theorem 2, assume that there is a scalar satisfyingwhere , , , , , , , and , .
Even more, the terms and in inequation (14) can be described as follows:Hence, we can conclude from inequation (26) that inequation (14) is true, that is to say, , then we know by Theorem 1 that system (10) is stochastically stable when .
Next, we will prove that the system (10) is stochastic stable under the condition.
LetUsing the same proof as in Theorem 1, for any , the following inequation holds:where , , , , , , , , , and.
We takewhere ; then, we know thatObserving inequation (29), we conclude thatTherefore, for any , there must be a non-negative integer such that ; then let us observe formula (28) and inequation (32), and we obtain thatBy the Dynkin formula, for any , the following inequation holds:As stated in Definition 2, we know that system (10) is stochastic stable under the condition.
Next, let’s give the sufficient condition for the existence of state feedback controllers for sampled-data follows according to Theorem 2.

Theorem 3. 3.For a given scalar , the T-S fuzzy system (10) is stochastic stable under the condition, if there are matrices , , , , , , , , and , and arbitrary matrices , , , and such that inequations (30) are true for each , , , andwhere , , , , , , , , , ,, and are the same definitions as in Theorem 2; in this case, we give the expected controller gains as follows:

Proof. .Let , , , , , , , , , , , , , , , andBy the statements in Theorem 2, we know that , let’s pre-multiply by , and postmultiply by , we can conclude that inequations (35) are accurate. This concludes the proof process.

4. Numerical Simulation Example

We provide an example in this work to verify that the proposed approach is effective.

Example: The Truck-Trailer model is mentioned in the literature [42, 43]. Here, we take it as an example to verify the results. The Truck-Trailer model is shown as follows:where is the angle difference between the trailer and the truck, is the vertical position of the rear of the trailer, is the angle of the trailer, and are disturbance input, and the model parameters can be set as , , , , , , . Assume that the switching process between the three modes follows a Markov process, and the transfer probability matrix is :

Let , the membership function is defined as follows: where , and let .

The T-S fuzzy systems of the model (38) can be described as follows:

The parameters of which we refer to the literature [42, 43] are defined as follows: