Advances in Fuzzy Systems

Volume 2019, Article ID 9365767, 13 pages

https://doi.org/10.1155/2019/9365767

## Predictive Control for Interval Type-2 Fuzzy System with Event-Triggered Scheme

^{1}Chongqing University of Posts and Telecommunications, College of Automation, Chongqing 400035, China^{2}Key Laboratory of Industrial Internet of Things & Networked Control, Ministry of Education, Chongqing University of Posts and Telecommunications, College of Automation, Chongqing 400035, China^{3}Chongqing College of Electronic Engineering, Engineer of Network Center, Chongqing 400037, China^{4}China Coal Technology and Engineering Group Chongqing Research Institute, Chongqing 400037, China

Correspondence should be addressed to Xiaoming Tang; moc.621@eyeymmxt

Received 26 February 2019; Revised 5 June 2019; Accepted 13 June 2019; Published 8 July 2019

Academic Editor: Antonin Dvorák

Copyright © 2019 Siyao Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, a synthesis approach of model predictive control (MPC) is proposed for interval type-2 (IT2) Takagi-Sugeno (T-S) fuzzy system with quantization error, bounded disturbance, and data loss. The novelty lies in the following technical improvements. In order to reduce the redundant data transmission, an event-triggered communication scheme is applied to determine whether the control law should be transmitted into the communication network or not. The IT2 T-S fuzzy model is utilized to address the nonlinearity of plant with parameter uncertainties, which can be captured by the lower and upper membership functions. Furthermore, the phenomena of data loss and quantization error between the controller and the actuator are expressed as Markovian chain and sector-bound uncertainties. The synthesis approach of MPC is provided by solving an MPC optimization problem over an infinite horizon objective function which explicitly considers the input constraints. By applying the quadratic boundedness (QB) technique, the recursive feasibility and quadratic stability of closed-loop system can be guaranteed. A numerical simulation and comparison studies are proposed to illustrate the effectiveness of this approach.

#### 1. Introduction

The networked control systems (NCSs) have played a more and more important role in many areas, which depend on the convenience of communication networks, such as low cost, simple installation and maintenance, and reduced system wiring [1–4]. However, communication networks also bring some challenges to the corresponding control system. The main problems are data loss and quantization error, which may degrade the control performance of NCSs. Caused by the limited capacities of communication networks, the signals always should be quantized before being transmitted into the network. Beyond that, owing to the unreliable features of communication networks, the data may be lost. Thus, the control performance of NCSs will be degraded by the quantization error and data loss inevitably. Many nice works about addressing data loss and quantization error problems under the framework of linear system theory are proposed in the past decades; see, e.g., [5–7].

Recently, the nonlinear characteristics of the NCSs have attracted a lot of attention owing to their practical applications. Takagi-Sugeno (T-S) fuzzy model is a popular method to address the nonlinearity of the NCSs, which bridges the gap between the complex nonlinear systems and linear systems [8]. Almost all the smooth nonlinear systems can be converted into linear systems by using T-S fuzzy model. Reference [9] proposed a new quadratic stabilization condition for the T-S fuzzy system, which based on the linear matrix inequalities (LMIs) technique. Reference [10] designed a method to address stability analysis and stabilization problems for the continuous T-S fuzzy system with time-delay. In [11], a new adaptive terminal sliding mode control method for single-input multi-output T-S fuzzy system with unknown and external disturbance was investigated. It is worth mentioned that there exist no uncertainties in the membership functions for the type-1 T-S fuzzy system; thus, the above researches based on the type-1 T-S fuzzy model may result in conservatism. To our delight, the IT2 T-S fuzzy model is imposed to deal with the parameter uncertainties of nonlinear systems captured by the lower and upper membership function [12–15]. In many aspects, IT2 T-S fuzzy model has better performance than the type-1 T-S model, which can be approved by some applications, such as DC-DC converters [5], autonomous mobile robot [6], airplane flight control [16], and aerospace theory [17]. Moreover, many nice works have begun to design and analyze IT2 T-S fuzzy systems under network environment. In [18], the parameter uncertain in nonlinear networked control systems was described as lower and upper membership functions and relative weighting functions; furthermore, the phenomenon of data loss also has been taken into account. Reference [19] proposed an IT2 T-S fuzzy filter for nonlinear NCSs with data loss and quantization; beyond that, stochastic stability with performance can be guaranteed.

In NCSs, we often exploit time-triggered scheme; however, by this way, almost all data will be transmitted into the network in spite of whether the data is necessary or not. It is obvious that time-triggered scheme will reduce the efficiency of communication resources; see, e.g., [7, 20]. Event-triggered scheme can solve this problem well, which defined a trigger condition to determine whether the data should be transmitted into the network or not. By using this scheme, the burden of the network will be reduced effectively. In [21], an event-triggered communication scheme was exploited to decide whether the data should be transmitted to the controller and achieve better utilization of network resources. In [22], by applying the event-triggered communication scheme and time-triggered periodic communication scheme for T-S fuzzy NCSs, respectively, it proved that the event-triggered scheme would utilize fewer communication resources while preserving the desired control performance. In [23], it considered the networked nonlinear systems with imperfect premise matching, which alleviated the burden of communication networks by using the event-triggered scheme. Reference [24] constructed a fuzzy observer with the imperfect premise matching to estimate the unmeasurable states of networked T-S fuzzy systems and addressed the problem of an event-triggered nonparallel distribution compensation control to achieve higher communication efficiency and less conservation. It is apparently that the above results show the great advantages of event-triggered scheme. However, the state constraints and input constraints also exist in practical NCSs and have not been taken into account.

Model predictive control (MPC) is widely applied in industrial and academic communities since the defining feature of handling the physical constraints in a systematic manner [25–28]. At each sampling time, MPC is to solve a finite horizon optimization control problem based on the current measurements in order to obtain an optimal control sequence; however, only the first control of the optimal sequence is implemented. In the past decades, a great deal of researches [29–32] are based on the different branches of it, which have facilitated the development of this advanced method. Deserved to be mentioned, the synthesis approaches of MPC, as the important part of MPC, have attracted much attention and made some representative achievements; see [33, 34]. Reference [33] proposed a novelty method for linear NCSs based on the classical synthesis approach of MPC, which specified the recursive feasibility and closed-loop stability; both of data loss and quantization error problems are considered in the transmission links. Reference [34] investigated the synthesis approach of MPC for the stochastic system by describing the data loss as Markovian model and the quantization error as bound uncertainties, which guaranteed the mean square stability and recursive feasibility simultaneously. However, neither of them considered releasing the burden of communication networks by using event-triggered scheme.

In this paper, we discuss the synthesis approach of MPC for IT2 T-S fuzzy NCSs via event-triggered scheme. The controller output should be quantized before it is transmitted into the unreliable network, and the phenomenon of data loss and quantization error is taken into consideration simultaneously. In order to reduce the burden of communication networks, an event-triggered scheme is applied to decide whether the data should be released into the network or not. And in the transmission of triggered data, data loss and quantization error are expressed as Markovian chain and bound uncertain, respectively. An online MPC optimization problem that minimizes the upper bound of a quadratic objective function in an infinite time horizon subject to input constraints is proposed via the linear matrix inequality technique. Further, the recursive feasibility and closed-loop stability also can be guaranteed.

*Notation*. Throughout this paper, is an identity matrix with appropriate dimensions. represents which is a real symmetric and positive-definite (semidefinite) matrix. . In block symmetric matrices, the symbol is used to represent a term that is induced by symmetric and diag stands for block-diagonal matrix. denotes the ellipsoid associated with the symmetric positive-definite matrix . Co denotes the convex combination of elements in , with the scalar combining coefficients nonnegative and their sum equal to 1. The notation denotes future state of at time . is the expectation operator.

#### 2. Problem Formulation

##### 2.1. IT2 T-S Fuzzy Model and Controller

Consider a discrete-time IT2 T-S fuzzy model, with th rules. It can be described as follows.

*Plant Rule *. IF is is and and is , THENwhere represents the IT2 T-S fuzzy set of the function , . is the state variable, is the input vector, is the persistent disturbance, and satisfied and are constant matrices of appropriate dimensions. The firing strength of the th rule can be replaced by an interval sets as follows: , where , and , which, respectively, denote the lower and upper grades of membership, and . Then, the IT2 T-S fuzzy model is inferred as follows:where and , , and ; both are nonlinear and satisfying .

The IT2 T-S fuzzy controller with rules is denoted as follows:* Controller Rule *: IF is is and and is , THEN:where is the IT2 T-S fuzzy set of the function , . is the feedback control law, and is the controller output. The firing strength of the th rule can be replaced by interval sets as follows: , where and , which, respectively, denote the lower and upper grades of membership, and . Then, the IT2 T-S fuzzy model is inferred as follows:where , and , and , both are nonlinear and satisfying .

Moreover, hard constraints are always caused by physical constraints and imposed on the manipulated variables, which can be written as follows:

##### 2.2. Event-Triggered Communication Scheme

We focus on the system controller output data transmitted over a communication network. Figure 1 shows the framework of NCSs. In order to save the limited communication resource, an event-triggered scheme is exploited to decide whether the data will be transmitted into network or not, i.e., comparing the current data with the last released data to determine whether should be released. Define the error between the current data and the last released data as , where the is the current time, is the last event-triggered time, and is the next event-triggered time. In trigger time, , where represents the ignored data between and . If the event-triggered condition can be satisfied, the data will be transmitted into the unreliable network and become a new latest event-triggered data; else, the last released data will be maintained and the current data will be ignored. The next event-triggered condition is inferred aswhere is given scalar and ; is a positive-definite matrix with appropriate dimension. We define two sequence and to describe the event-triggered scheme more specifically. The is the sequence of controller output and is the event-triggered sequence. It is apparent that , if all the data are triggered, .