#### Abstract

A fuzzy predictive fault-tolerant control (FPFTC) scheme is proposed for a wide class of discrete-time nonlinear systems with uncertainties, interval time-varying delays, and partial actuator failures as well as unknown disturbances, in which the main opinions focus on the relevant theory of FPFTC based on Takagi-Sugeno (T-S) fuzzy model description of these systems. The T-S fuzzy model represents the discrete-time nonlinear system in the form of the discrete uncertain time-varying delay state space, which is firstly constructed by a set of local linear models and the nonlinear membership functions. The novel improved state space model can be further obtained by extending the output tracking error to the constructed model. Then the fuzzy predictive fault-tolerant control law based on this extended model is designed, which can increase more control degrees of freedom. Utilizing Lyapunov-Krasovskill theory, less conservative delay-range-dependent stable conditions in terms of linear matrix inequality (LMI) constraints are given to ensure the asymptotically robust stability of closed-loop system. Meanwhile, the optimized cost function and H-infinity performance index are introduced to the stable conditions to guarantee the robust performance and antidisturbance capability. The simulation results on the temperature control of a strong nonlinear continuous stirred tank reactor (CSTR) show that the proposed control scheme is feasible and effective.

#### 1. Introduction

With the continuous social and economic progress, living standards of the people have been improved greatly. The demands for higher product quality also increase continuously. In order to satisfy these demands, more strict operation conditions are needed for the manufacturing industry, which is a huge challenge. Under such a context, more efforts should be paid to study the advanced control technique.

Industrial production is the backbone of many of the world’s economies [1] and plays an important role in daily life of people. However, most industrial processes have nonlinear characteristics. The traditional control methods have been unable to achieve the goal of higher product quality in industrial production [2]. As a result, a great number of advanced control technologies [2–7] have been proposed over the past decades. Among them, fuzzy model-based approaches that were pointed out clearly in [8] are very effective and feasible to deal with the system nonlinearities. However, the focus is put on the control methods based on Takagi-Sugeno (T-S) fuzzy model [9, 10]. In T-S fuzzy model, a set of local linear models are weighted by nonlinear membership functions in terms of IF-THEN rules to approximate a large class of nonlinear processes well [11]. In virtue of this, many mature linear theories are able to be applied fully to the stability analysis and control synthesis of nonlinear processes [12–16], which will make great progress in the advanced control theory.

Due to the increasing demands for industrial products, the scale of industrial production is growing rapidly, which makes the industrial equipment operated under more complex environments. With the long-running industrial production, the failure may occur. If a failure cannot be coped with instantaneously in such environments through the suitable corrective action, it will make the control performance deteriorate and even expose the equipment and personnel to serious damage. Thus, fault-tolerant systems [17] and advanced process control algorithms [18] were studied to deal with failure using two soft computing approaches, i.e., fuzzy systems and neural networks. Especially, the study of fault-tolerant control (FTC) based on T-S fuzzy model for a class of nonlinear processes with failure [19–24] has attracted a lot of interests over the last few decades. However, the effect of time delay is not fully considered in these references.

Time delay usually occurs in many industrial processes, such as petrochemical furnace, continuous stirred tank reactor, biochemical fermentation process, injection molding batch process, fractionating tower, rolling process, cement grinding system, industrial flotation process and so on. It is well known that time delay is one of the major sources of the instability and performance degradation of system. From the perspective of the term of time delay, it can be divided into two categorizers: constant delay and time-varying delay. The interval time-varying delay is more common, i.e., the delay varies in the upper and lower bounds of the interval and the lower bound is not limited to be zero, which has been identified from many practical industrial processes. But only a few results of FTC for T-S fuzzy system with failure and interval time-varying delay simultaneously can be seen [25–29] during the recent years. However, these methods do not consider the uncertainties or external disturbances that are very important in practical industrial application. Meanwhile, most of them are active FTC. Their main ideas are to apply the proposed observer-based fault estimation (FE) method to obtain the online accurate fault information by solving linear matrix inequality (LMI) conditions and then the fault information is integrated to the design of FTC. Though these methods have made some progress, FE methods need to meet the observer matching condition, which is very difficult for many practical processes. In addition, the obtained LMI-based observer gains are unable to restrain the derivatives of faults effectively within the fault error dynamics. Compared to the active FTC, the passive FTC does not need the accurate fault information, which is simple to implement. The passive FTC can also guarantee the system stability and expected performance within the admissible range of faults. Therefore, the passive FTC is studied for the industrial processes with the partial actuator faults in our work.

There are some researches [30–34] for the industrial process with the partial actuator faults. Based on a two-dimensional Fornasini-Marchsini model, an iterative learning FTC method for batch processes was proposed by Wang et al. [30] in early stage. Wang et al. further extended his result and proposed robust delay dependent iterative learning FTC for batch processes with state delay and actuator failures [31] and robust iterative learning FTC for multiphase batch processes with uncertainties [32], respectively. In addition, Tao et al. [33] proposed a nonminimal state space model predictive control method for linear systems with partial actuator failure; Zhang et al. [34] proposed FTC scheme for flexible spacecraft, which can guarantee the closed-loop system to be asymptotically stable with a H-infinity performance and partial loss of actuator effectiveness; Zhang et al. [35, 36] proposed a linear quadratic FTC method for industrial processes with partial actuator faults and unknown disturbance to ensure the stability and optimal performance; Shi et al. [37] proposed a robust constrained model predictive fault-tolerant control method for uncertain time-delay system with unknown disturbances and partial actuator failures. However, the above methods do not consider the nonlinearity or the time delay.

In fact, the robust performance for the practical industrial applications is vital because it is always expected to control the industrial processes with uncertainty and time delay using the designed control system. As a result, the optimal performance for control system is also an open topic. Model predictive control (MPC) [38–40] is a selection to improve the control performance of system because it can achieve receding optimization in real-time. It is recognized as the only advanced control method which has made a significant impact on industrial control engineering [41]. For the industrial processes with time-varying delay and actuator faults, the study of FTC integrated with MPC is effective and feasible.

Motivated by the aforementioned references, a fuzzy predictive fault-tolerant control (FPFTC) scheme is proposed, which is suitable for discrete-time nonlinear systems with partial actuator failures, interval time-varying delays, uncertainties and unknown disturbances. To the best of our knowledge, there is no result for our work in this paper. The main contributions of this paper are summarized as follows.

With the local-sector nonlinearity approach, a T-S fuzzy model is established by weighting a set of local linear models using the nonlinear membership functions, which can approximate a large class of nonlinear industrial processes. Then the output tracking error is extended to T-S fuzzy model. Therefore, the performance of the convergence and tracking can be improved for the design of control law based on this extended fuzzy model due to independent adjustment for the process state variables and output tracking error.

Without using some redundant free-weighting matrices and cross terms with differential inequality, less conservative delay-range-dependent stable conditions in terms of linear matrix inequality (LMI) constraints are further given to ensure the closed-loop system to be robustly asymptotically stable.

In order to guarantee the robust performance and overcome any unknown bounded disturbances, the optimized cost function and performance index are introduced into the derivation of stability for proposed FPFTC, respectively.

The simulation results on the temperature control of a strong nonlinear continuous stirred tank reactor (CSTR) demonstrate that the proposed control scheme performs excellent tracking and antidisturbance capabilities.

The paper is organized as follows: A process description is shown in Section 2. Section 3 details the fuzzy predictive fault-tolerant control method. Section 4 presents a case study in a CSTR. Conclusions are summarized in Section 5.

The notations used throughout the paper are quite standard. denotes the* i*th fuzzy rule. is the -dimensional, Euclidean space and are the set of real matrices. denotes the transpose of . is used as an ellipsis for the terms that are implied by symmetry.

#### 2. Process Description

A discrete-time nonlinear system with uncertainties, interval time-varying delay, partial actuator failures, and unknown disturbances is described by a great many of T-S fuzzy rules as follows.

*Rule *. If is and is is thenwhere are the state, input, output and unknown external disturbances at the discrete-time* k*, respectively*. * are the premise variables. is the* h*th fuzzy set for the* i*th fuzzy rule. denotes the time-varying delay depending on discrete-time* k* which satisfiesin which and are the upper and down bounds of delay, respectively. is uncertainty set. and are the constant matrices corresponding to appropriate dimensions for the* i*th fuzzy rule, and are uncertain perturbations at discrete-time* k *that followswithwhere and are known constant matrices with appropriate dimensions. are uncertainties depending on discrete-time* k*. stands for the actuator fault that cannot be avoided in actual engineering applications. Therefore, the predefined cannot be obtained, which can be expressed as or . is the complete actuator fault and refers to the stuck fault of actuator. For the aforementioned two faults, the system is unable to be controlled. The appropriate measures must be done to deal with the two faults. is the partial actuator fault that is here studied in this work. is unknown but assumed to vary within a known range, i.e., where and are known scalars.

Then the nonlinear system can be transformed into the following form of discrete-time-varying delay state space by weighting a great many of local linear submodels (1):where and . , in which is the membership function for the* i*th fuzzy rule.

The main objective in this work is to design a fault-tolerant controller for the T-S fuzzy system (5) with the partial actuator failures so that the measurement of output is able to track the desired set point . In view of this, (6) is defined as follows:

From (6), we can see that there has an unknown matrix such as with

#### 3. Fuzzy Predictive Fault-Tolerant Control

##### 3.1. Novel Extended T-S Fuzzy System

Pre- and postmultiplying (5) by back shift operator, the T-S fuzzy model can be transformed into the following incremental state space form:where *,* + + + + , . Defining the set point as , then the output tracking error is

The following form is presented by synthesizing (6) and (7):

By extending the tracking error to T-S fuzzy model (9), the novel extended T-S fuzzy model can be expressed as follows:where

*Remark 1. *From (12), the output error is introduced to the state variables, which can increase more control degrees of freedom. Meanwhile, the control performance with respect to convergence and tracking will be improved by the following design of control law because this extended fuzzy system can independently adjust the process state variables and output tracking error. Hence, using the parallel distributed compensation (PDC) method, the control law of system is designed as follows:where is the constant gain that can be solved by the subsequent theorems. Substituting (15) to (12), the extended T-S fuzzy closed-loop system can be obtained as follows:where

*Definition 2 (robust MPC problem). *Considering fuzzy closed-loop system with uncertainty set , and the state measurement at discrete-time* k* as , the robust MPC problem is feasible if the ‘min-max’ optimization problemis solvable.

*Definition 3. *The fuzzy closed-loop system has performance, if there exists a scalar for any and the following conditions hold: (1)the resulting fuzzy closed-loop system with is asymptotically stable;(2)the system output satisfies under the zero initial condition.

##### 3.2. Main Theorems

The main objective of this section is to solve the predictive fault-tolerant law that ensures the robust stability of fuzzy closed-loop system. The main lemmas and theorems are presented as follows.

Lemma 4 ((Schur complements lemma) [42]). *Let and be matrices of appropriate dimensions in which are real matrices, then for if and only if such that*

Lemma 5 (see [43]). *For any vector , two positive integers , and matrix , the following inequality holds:*

Lemma 6 (see [44]). *Let , and be real matrices of appropriate dimensions with satisfying , then for all ,if and only if there exists such that*

*Remark 7. *Based on the Definition 3, the following Theorems 8 and 11 are given for the fuzzy closed-loop system with disturbance and without disturbance, respectively.

Theorem 8. *Given some scalars , , the delay-range-dependent sufficient conditions for the proposed controller that guarantee the fuzzy closed-loop system with to be asymptotically stable and controllable are that there exists symmetric positive matrices , matrices , and positive scalars , , such that the following LMI holds: and the robust state-feedback fault-tolerant controller gains are given by .**where , and denotes the transposed elements in the symmetric position.*

*Proof. *To ensure robust stability of the fuzzy closed-loop system with , letting that satisfies the following robust stability constraint:Summing up both sides of (28) from* i*=0 to and needing that or , it haswhere is the upper bound of . For notational simplicity, define , *, **,*, and the following Lyapunov-Krasovskii function candidate is established:where, and are positive definite matrices. Define , , , (30) can be written as follows:ThenwhereUsing Lemma 5, the following expression is presented:From (28), it can obtainwhereSynthesizing (33)-(36), it haswhereUsing Lemma 4, the condition holds and is equivalent to the following: where

Thanks to , , (41) can be transformed into the following LMIs using Lemmas 6 and 4:where and .

The delay-range-dependent sufficient conditions (23) and (24) can be obtained by using to pre- and postmultiply (42) and letting .

Moreover, to obtain the invariant set of the fuzzy closed-loop system , taking the maximum value of , one haswhere Letting , it can obtain the sufficient condition (25) using the Lemma 4.

In order to ensure the matrix to be invertible, letting , we can obtain , i.e., . Then, by the Lemma 4, the sufficient condition (26) can be obtained.

This completes the proof of Theorem 8.

*Remark 9. *Different from the traditional methods [45–47], a differential approach is applied to build up the Lyapunov-Krasovskii function candidate without some redundant free-weighting matrices. This constructed Lyapunov-Krasovskii function can take use of the information of the lower and upper bounds of the interval time-varying delay. Meanwhile, the bounding and model transformation techniques for cross terms using differential inequality are averted during the derivation of the obtained stable conditions (23) and (24).

*Remark 10. *Theorem 8 provides the sufficient conditions, together with the expression of the constant gains , for the asymptotic stability of the fuzzy closed-loop system with that depends on the upper and lower bounds of the time-varying delay. First, Lyapunov-Krasovskii function is used, then the constant gains are acquired letting based on the expression of (28); second, it is converted into LMIs shown in (23) and (24). In addition, the invariant set (25) and condition (26) are acquired through some relax technologies. Therefore, by solving these LMIs in the Theorem 8, the corresponding control gains can be obtained.

Theorem 11. *Given some scalars , , , the delay-range-dependent sufficient conditions for the proposed controller that ensure the fuzzy closed-loop system with to be asymptotically controllable and to be a performance less than is that there exists the symmetric positive matrices , matrices , and positive scalars , , such that the following LMI holds:and the robust state-feedback controller gains are given by .**where*

*Proof. *Based on the derivation of Theorem 8, (38) can be rewritten due to as follows:whereIn order to make the fuzzy closed-loop system to be the performance, the following expression is presented: whereUsing Lemma 4, the condition holds and is equivalent to the following: