Non-Fragile Control for Variable Sampling Period Network Control System with Actuator Failure

The guaranteed cost control problem is studied in this paper for a class of nonlinear discrete-time systems with both time-varying parametric uncertainty and actuator failures. At present, the researches on related fields of networked control systems are relatively mature, and many achievements have been made. But at the same time, most of the researches are focused on linear networked control systems, and the research on nonlinear networked control systems is relatively less. The goal is to design a non-fragile state feedback control law so that the closed-loop system is asymptotically stable and the closed-loop cost function value is no more than a specified upper bound for all admissible uncertainties. Firstly, the system model is established by using the method of variable sampling period. Secondly, the sufficient conditions for the asymptotic stability of the closed-loop system are given by using Lyapunov stability theory. Thirdly, based on the above researches, a non-fragile state feedback controller is designed by using linear matrix inequality (LMI). In the end, through the study of this paper, the cost function of the system under the designed non-fragile guaranteed cost controller does not exceed the given upper bound. This paper considers the actuator failure, and gives the design method of the non-fragile guaranteed cost controller of the nonlinear network control system, and makes a contribution to the field of network control system.


Introduction
In the modern society, the Internet is widely used in various fields. Many high-tech fields and large enterprises, such as resource sharing, automated factories, robots manufactures, advanced aerospace undertakings and electrified transport vehicle manufacturres, depend heavily on the networks. Since the concept of Network Control Systems (NCS) was put forward in the early 1990s, it has attracted people's attention immediately and posed a new challenge to the traditional control theory and application [1].
Networked control system is a distributed feedback control system. It consists of two parts. On the one hand, sensors, controllers and actuators scattered in different geographical locations are connected by the network, they form a closed-loop control system [2][3], which facilitates the connection and data sharing between each component; On the other hand, the related control theory is introduced into network control to control the network system.
Compared with the traditional control system, the NCS has a tremendous impact on the industrial automation and intelligent technology level. Firstly, it has high control efficiency, and can display the real-time operation of the controlled objects, the statistical processes and results. Secondly, precision wise, NCS is more accurate and flexible. And the optimal control can be achieved. Finally, the participation of the network greatly improves the automation degree of the control system and realizes the integration of management and control [4][5]. However, due to the introduction of the network, many challenges are posing, such as time delay and packet loss, bring great difficulties to the analysis and design of the network control system [6][7]. Therefore, the research on network control system has drawn wide attention from scholars both at home and abroad. In recent years, many research results have been obtained on NCS, such as system modeling and stability analysis, robustness and H∞ control analysis, optimal guaranteed uncertainty, Li Yuan modeled it as a discrete system. The period uncertainty was transformed into the matrix uncertainty by Taylor formula, and the system was transformed into a dynamic interval system [16]. Zhao Yan used the method of active variable sampling period to model the nonlinear continuous network control system with multi-packet transmission as a discrete switching system. Sufficient conditions for the closed-loop system to be asymptotically stable were obtained [17]. Aiming at the time-varying sampling period and delay in networked control systems, Fan Jinrong transformed the uncertainty of sampling period and delay into the uncertainty of system structural parameters through matrix Jordan transformation and decomposition, and established the discrete-time convex polyhedron uncertain system model [18].
In networked control systems, since industrial instruments and measurement control components are affected by their own physical and environmental factors, there are parameter disturbances during implementation. Even small disturbances may cause instability of the system. Therefore, the effect of parameter changes on the system performance must be considered in the design of the controller [19]. In recent years, the non-fragile problem of controller parameter uncertainty has attracted wide attention. Ma Weiguo studied the non-fragile guaranteed cost control problem for nonlinear systems with quantized and Markov chain losses. The networked control system was described as a Markov jumping system. Sufficient conditions for the existence of non-fragile guaranteed cost controllers for nonlinear systems with additive and multiplicative disturbances were given in the form of linear matrix inequalities [20]. Aiming at the control input constraints and controller gain disturbances, Gao Xingquan proposed a non-fragile guaranteed cost state feedback control method for a class of norm-bounded parameter uncertain linear systems, and derived a new sufficient condition for solving the constrained non-fragile guaranteed cost control law [21]. Yu Shuiqing studied the non-fragile guaranteed cost control problem for a class of nonlinear networked control systems with stochastic delay and controller gain disturbances [22]. Su Yakun studied the guaranteed cost control problem for a class of uncertain stochastic systems with interval delays. The purpose was to design a non-fragile guaranteed cost control rate and find the upper bound of the cost function [23].
This paper, designs a non-fragile controller for a class of nonlinear networked control systems with variable sampling period. In order to solve the problem of actuator failure, this paper refers to Li Yu's method. The function not only can describe the normal case and outage case but also describe the actuator partial degradation [24]. The sufficient conditions for the asymptotic stability of the closed-loop system are given by using Lyapunov stability theory. Based on the above researches, a non-fragile state feedback controller is designed by using linear matrix inequality (LMI). In the end, through the study of this paper, the cost function of the system under the designed non-fragile guaranteed cost controller does not exceed the given upper bound.
Symbol description: The symbol * indicates the block matrix in a symmetric matrix, A is the transpose matrix of A.

Problem Description
Consider the following non-linear controlled system: Where x t ∈ R 、u t ∈ R and y t ∈ R represent the input state, control input and output state respectively. A, B and C are known matrices of appropriate dimensions. f t, x is an in determinate nonlinear term that satisfies the Lipschitz condition: Where G is a known constant matrix.
In this system, it is assumed that the state variable x t is measurable, and its measurements are first discrete and then transmitted in a single package. 0 < d ! ≤ d expressed the bounded random delay from the sensor to the controller.
In this article, we use a variable sampling period method. If the time delay at sampling time k is d ! , then it is assumed that the variable sampling period T ! at sampling time k is equal to d ! . Therefore, to sample system (1) with variable sampling period T ! , the discretization model is: where: In order to deal the actuator failure, we refer the function that is proposed by Li Yu [24]. And it is described as the follows.
In the state feedback controller (9), K is the nominal controller gain and ∆K is the disturbance of the gain. ∆K = DEF, F F ≤ I. Where D and E are known matrices of appropriate dimensions, F is uncertain parameter matrix.
Lemma 1 [25]: If Q、H、E and R are real matrices of appropriate dimensions, Q and R are symmetrical, R > 0, Then Q + HFE + E F H < 0 holds for all the matrices F that satisfy F F < P. If and only if there exists a constant ε > 0 such that Q + ε > HH + ε R> E RE < 0.
Summing both sides of the above inequality from 0 to ∞, we can get that: Use the system asymptotically stability and J = ∑ =x (k)Qx(k) + u (k)α Rαu(k)?

Design of Non-Fragile Guaranteed Controller
On the basis of Theorem 1, the design method of non-fragile guaranteed cost controller is given below.
Theorem 2: For the closed-loop system (7) and the cost function (8), if there exists a matrix X, Y > 0 and a scalar Γ > 0, ε 0 , ε > , ε d , ε m , ε n , ε o > 0, the following matrix inequality holds: Proof: Matrix inequality(11) can be transformed into: Using Schur complement theorem, inequality(24) is equivalent to: It equals to: It follows from the Lemma1 complement that the above inequality is equivalent to: Using Schur complement theorem, the upper form can be transformed into: Reusing Schur complement theorem, inequality (29) is equivalent to: The above mentioned:∆K = DFE, the above inequality can be written as: The above inequality can be written as: Using Lemma1, (32)is equivalent to: At this point, we use the Schur theorem to get the following inequality: The left and right sides of (34) are respectively multiplied by the diagonal diagBP R0 , ΓI, I, … , ID and let X = P R0 , Y = KX , the following inequality is obtained. p q q q q q q q q r −X * * * * * *