An improved approach to reliable H_∞ guaranteed cost control for discrete time LPV systems with control input constraints

Abstract

This paper investigates the problem of reliable $H_{\infty }$ guaranteed cost control with multiple criterion constraints for discrete time linear parameter varying (LPV) systems against sensor faults and control input constraints. New sufficient conditions for the existence of a reliable state-feedback controller is proposed to guarantee the closed-loop system against sensor faults satisfying D-stability, the $H_{\infty }$ performance index, the quadratic performance index and control input within constraints for all admissible parameter uncertainties and external disturbances. Due to the introduction of the affine parameter dependent Lyapunov matrix and many extra degrees, these results are less conservative than the existing results, especially for handling the wider range of uncertain parameters and improving both the quadratic performance and the $H_{\infty }$ performance. A simulation example is presented to show the effectiveness and superiority of our proposed approach.

Introduction

The problem of guaranteed cost control (GCC) for uncertain systems has drawn considerable attention in [1], [2], [3], [4], [5], [6], [7]. A guaranteed cost controller can be used to guarantee the closed-loop system not only robust stability but also an adequate level of performance as well. In the practical control system, actuator faults, sensor faults or some component faults may happen, which lead to dissatisfactory performance, even loss of stability, thus the research on reliable guaranteed cost control (RGCC) is necessary. The algebraic equation approaches have been proposed to study RGCC in [8], [9], and the LMI approach has been introduced in [10]. Ref. [11] has focused on the problem of robust reliable guaranteed cost control for a class of uncertain Takagi–Sugeno fuzzy neutral systems with actuator faults. The decentralized RGCC controller has been introduced for discrete time systems with actuator faults in [12]. However, the results on RGCC for discrete time systems with sensor faults are very limited.

On the other hand, the performance requirements of practical systems are usually multi-objective in tolerant faults cases, such as D-stability, the quadratic performance index, the $H_{\infty }$ performance index, etc. (in [13], [14], [15], [16]). The saturation of actuators, which is always met in practical systems, may lead to poor dynamic performance, even loss of stability. Then the magnitude of actuators should be restricted into a given limit. When we take these performance and constraints into account, the traditional RGCC problem becomes the problem of RGCC with multiple criterion constraints. Though [13], [14], [15], [16], [17], [18] have studied the problem of RGCC with multiple criterion constraints, these results are conservative because of the following reasons: firstly, they choose parameter independent Lyapunov functions; secondly, they neglect the influence of parameter rates of variation; thirdly, they have no extra freedom degree to handle the wider range of uncertain parameters and improve both the quadratic performance and the $H_{\infty }$ performance.

Based on the above consideration, we propose an improved approach to design a reliable $H_{\infty }$ guaranteed cost controller to guarantee the closed-loop system against sensor faults satisfying D-stability, the quadratic performance index, the $H_{\infty }$ performance index and control input within constraints. Uncertain parameters and parameter rates of variation are supposed to belong to polytopic domains. New sufficient conditions for the existence of the reliable state feedback controller are derived to guarantee the closed-loop system satisfying the corresponding performance constraints in terms of LMIs. Compared with the existing results, our results are less conservative because of the following reasons: firstly, we introduce parameter rates of variation and make use of bounds of parameter rates of variation, as extra freedom degrees, to handle the wider range of uncertain parameters and improve the quadratic performance and the $H_{\infty }$ performance; secondly, the affine parameter dependent Lyapunov function in this paper is proposed to reduce the conservatism; thirdly, the Lyapunov matrix and the system matrices are separated via additional variables, which can lead to two extra freedom degrees handling the wider range of uncertain parameters and improving both the quadratic performance and the $H_{\infty }$ performance. A numerical example is presented to demonstrate the effectiveness and superiority of the proposed approach.

This paper is organized as follows. The problem statement of the paper is formulated in Section 2. Main results of discrete time LPV systems are presented in Section 3. In Section 4, a numerical example is used to certify the effectiveness of our proposed approach. Section 5 closes the paper.

Notation: Through this paper, for real symmetric matrices X and Y, the notation $X>Y$ means the matrix $X-Y$ is positive definite. ${\mathbb{R}}^n$ denotes the n-dimensional Euclidean space and ${\mathbb{R}}^{m\times n}$ is the set of all m×n real matrices. The superscript “T” represents the transpose. If no explicitly stated, matrices are assumed to have compatible dimensions. I denotes the unit matrix. ${\lambda }_{\mathit{max}}$ denotes the maximum eigenvalue of a matrix. I denotes the unit matrix. The symbol $⁎$ represents blocks that are readily inferred by symmetry in some matrix expression. $L_2[0,\infty$) for discrete time systems is the space of square-summable vector functions over $[0,\infty )$; $\parallel .{\parallel }_2$ stands for the usual $L_2[0,\infty )$ norm. $L_2[0,\infty )$ for discrete time systems is the space of square-summable vector functions over $[0,\infty )$; $\parallel .{\parallel }_2$ stands for the usual $L_2[0,\infty )$ norm. ${\delta }_{\mathit{max}}$ is the largest value of $\delta ≔\max (|{\delta }_1|,\dots ,|{\delta }_N|)$. $\mathrm{\Delta }{\delta }_{\mathit{max}}$ is the largest value of $\mathrm{\Delta }\delta ≔\max (|\mathrm{\Delta }{\delta }_1|,\dots ,|\mathrm{\Delta }{\delta }_N|)$. ${\gamma }_{\mathit{min}}$ is the smallest value of $\gamma$ satisfying given constraints. ${\sigma }_{\mathit{min}}$ is the smallest value of $\sigma$ satisfying given constraints. Without special expression, $x(k),u(k),w(k),z(k),\delta (k),\mathrm{\Delta }\delta (k),P(\delta (k))$ are substituted for $x,u,w,z,\delta ,\mathrm{\Delta }\delta ,P(\delta )$ for the sake of simplicity.