Optimal Cost Control for Variable Sampling Period Network Control System with Actuator Failure
Ling Wang, Nan Xie^{*}
College of Computer Science and Technology, Shandong University of Technology, Zibo, China
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To cite this article:
Ling Wang, Nan Xie. Optimal Cost Control for Variable Sampling Period Network Control System with Actuator Failure. Automation, Control and Intelligent Systems. Vol. 4, No. 4, 2016, pp. 66-72. doi: 10.11648/j.acis.20160404.11
Received: July 18, 2016; Accepted: July 28, 2016; Published: August 17, 2016
Abstract: This paper considers the networked control systems (NCSs) with the varying sampling period and actuator failure. When the NCSs are modeled, the varying sampling period was described by a constant sampling period and a network delay. Base on this and the actuator failed, using the Lyapunov stability theory and linear matrix inequalities to prove the existence of the cost guaranteed performance, and obtain the optimal cost guaranteed performance controller.
Keywords: Network Control System, Varying Sampling Period, Delay, Actuator Failure, Optimal Cost Guaranteed Performance
1. Introduction
Network control system (NCS) is a distributed closed-loop feedback control system which is composed of sensors, controllers and actuators [1]. The controller exchanges the information with sensor and actuator according to the internet [2]. Compare to the traditional control system, the NCS not only can share the information but also can use the remote operate and control. So the NCS become more and more popular. But also because of the use of the internet in the NCS, the NCS has some problems, such as delay, data packet dropout.
In the NCS commonly use continuous controlled objects and discrete controllers. So we can regard it as a sampling system to study [3]. The sensor reads information for a period of time. The time interval is called the sampling period. The selection of the sampling period can impact system performance [4]. So the study of the variable sampling network control system has become meaningful. And if the actor broke down the NCS will can’t work properly. And it will lead to a huge lose [5]. So we can see the importance of the fault-tolerant of NCS [6-8].
In recent years, in order to deal with the sampling period problem most study assuming that the sampling period is a constant [9]. Actually the sampling period is unstable. Yi Jianqiang used the delay to represent the sampling period [10]. Xie Guangming translated the variable sampling period into the uncertainty of system parameters [11]. Yu-Long Wang assumed that the sampling period can chose a random value in a finite set [12]. For the fault tolerant problem, Li Wei used the switch matrix presents the actuator’s state [13]. But it just can present the actuator in the normal working state or the actuator complete failure. It can’t describe the actuator partial failure. Some people also discard the wrong data and still use the last period’s data. But it can’t effectively improve the performance of the system.
Fan Jinrong provided a function to deal with the sampling period. Sampling period described by the delay and a number what is continuous changed in a limited range [9]. And this paper will base on that to do the further study. And in order to deal with the actuator failure problem we refer to Li Yu’s method. In that function not only can describe the normal case and outage case but also can describe the actuator partial degradation [14]. First we will introduce Fan Jinrong’s function to deal with the variable sampling period. It is described as follows
Symbol description: The symbol * indicates the block matrix in a symmetric matrix, is the transpose matrix of .
2. Problem Description and Preparation
Consider the network control system describes by the Figure 1, controlled object is a linear time invariant system, and it is described by the following state equation:
(1)
where and represent the input state, control input and output state respectively.
This article uses the static memory-less state feedback controller, so controller can be moved to the actuator side without affecting the performance of the system. Therefore the time delay from the sensor to the controller , the time delay from controller to the actuator and the time delay for the controller to the calculate can be combined together, we can regard as a constant. In order to facilitate analysis, highlight the characteristics of variable sampling period network control system, the system to do the following assumptions:
(1) Just think about short time delay, where .
(2) Time-varying sampling period , is time-varying and bounded, so the sampling period is time-varying and bounded .
(3) The nominal value of is , so , where represents the uncertain part of variable sampling period.
(4) The sensor in this system is time-driven, sampling instant is and sampling period is . The controller and the actuator are event-driven.
(5) The input state remains unchanged during the time and , and does not synchronously change with sensors in the sampling time owning to the actuator with zero-order holder. So
(2)
According to the sampling period to discretize the controlled object, we can get the discrete state equation of controlled object:
(3)
According to where and are constant matrices of appropriate dimensions, there is
(4)
Introducing augmented variables , (3) can be written as
(5)
where
(6)
(7)
, , ,, , , is a required value.
The system (1) can be described by the following state equation:
(8)
where
, (9)
In order to deal the actuator failure, we refer the function that is producted by Li Yu [14]. And it is described as the follows.
For control input , let denotes the signal from the actuator that has failed. The following failure model is adopted in this paper
(10)
where with
Define the , that is described as follows:
In the above model of actuator failure, if , then it corresponds to the normal case ; When , it covers the outage case. If , it corresponds to the partial failure case, i.e., partial degradation of the actuator.
Denote:
(11)
is said to be admissible if satisfies .
So (8) can be represented by
(12)
where
(13)
For system (12) denote a cost function
(14)
where and are given weighting matrices.
Definition 1: For the uncertain system (12) and cost function (14), if there exist a matrix and a positive number such that the close-loop NCS is stable and cost function satisfies , then is said to be a guaranteed cost control law and is the upper bound of quadratic performance.
Lemma 1 [15]: Set is any square, if exist matrix such that if and only if there exist a matrix is satisfies .
Lemma 2 [15]: If R, S and F are real matrices of appropriate dimensions, and , then for any positive number , the following linear matrix inequality(LMI) satisfies.
Lemma 3 [16]: For a constant matrix , if , so .
Lemma 4 [3]: The system contains an uncertainty sampling period . And it is norm-bounded, , if the real number and satisfy the following condition: , so ,
where , represents the maximum singular value, and is the conjugate transpose matrix of .
3. Main Results
Theorem 1: If any feasible and symmetric positive definite matrix satisfy the following LMI
(15)
then is the guaranteed cost control of system (12) and the upper bound of quadratic performance is
(16)
Proof: Take in the system (12) and the cost function (14). Suppose now there exist symmetric positive definite matrices , such that matrix inequality (15) holds for all admissible uncertainties, then the Lyapunov function candidate is positive definite. The corresponding Lyapunov difference along any trajectory of the close-loop system (12) is given by
(17)
From condition (15), we have . It follows from Lyapunov stability theory that the system (12) is asymptotically stable.
Summing both sides of the above inequality from 0 to ∞, we can get that
(18)
Use the system asymptotically stability and yield . It’s equal to
(19)
The upper bound of the system performance index what conclude from theorem 1 depends on the initial state , if is a zero-mean random variable and satisfies , then system performance index satisfies:
(20)
Define
and (21)
swhere
(22)
and (23)
where
Theorem 2: Consider system (12) with cost function (14), if the following optimization problem
(24)
(I)
(25)
(II)
(26)
has a solution , then is the optimal quadratic guaranteed cost control law for system (12) and the corresponding upper bound of the system performance index is
(27)
Proof: In light of theorem 1, the uncertain system (12) exists an optimal guaranteed cost control law if and only if there exist matrix , symmetric matrix and any feasible satisfythat
(28)
In light of Lemma 1, the inequality exists a matrix satisfies that
(29)
where
(30)
Define a matrix
(31)
The inequality (29) can be written as
(32)
In light of Lemma 2: If the inequality (32) exists, if and only if there exists a constant, such that
(33)
take in the inequality, then
(34)
It follows from the Schur complement that the above inequality is equivalent to
(35)
It follows from the Schur complement that the above inequality is equivalent to
(36)
Define , we can get that
(37)
Take (23) in the inequality
(38)
(39)
(40)
(41)
Using the inequality for any diagonal matrix , it follows that
(42)
It equals to
(43)
It follows from the Schur complement that the above inequality is equivalent to
(44)
We obtain the first condition of the optimization problem.
Following from the Schur complement, the second condition of the optimization is equivalent to , minimizing the will make to be minimized, then the upper bound of the system performance index will be minimized.
4. Conclusion
In this paper, we have derived the existence condition for guaranteed cost control for a class of variable sampling period network control system with actuator failure. The optimal cost controller was obtained through LMI and Lyapunov stability theory.
References