Abstract
This paper studies the -infinity stochastic control problem for a class of networked control systems (NCSs) with time delays and packet dropouts. The state feedback closed-loop NCS is modeled as a Markovian jump linear system. Through using a Lyapunov function, a sufficient condition is obtained, under which the system is stochastically exponential stability with a desired -infinity disturbance attenuation level. The designed -infinity controller is obtained by solving a set of linear matrix inequalities with some inversion constraints. An numerical example is presented to demonstrate the effectiveness of the proposed method.
1. Introduction
In the past few years, the networked control systems (NCSs) whose control loops are connected via communication networks have received increasing attention due to their advantages, such as reduced cost, low weight, easier installation, and maintenance. Time delay and packet dropout are the two major causes of instability of system and deterioration of system performance. Therefore, the time delay and packet dropout problems have been investigated in the existing literature. In [1], time delays are time-varying in intervals. In [2, 3], the bounds were imposed on the maximum number of successive dropouts. In [4], the sufficient condition that establishes the quantitative relation between the packet-dropout rate and the stability of the NCS with a constant delay is obtained.
Considering the disturbance attenuation problem, there has been much research effort on controller design. In [5–7], the controller dynamics is continuous, but in many NCSs, the system is controlled by a discrete-time controller with sample and hold devices. In [8–10], a discrete-time controller is designed; however, it should be pointed out that the packet dropout or the delay problem is studied separately.
In [11, 12], control of a class of systems with random packet dropout is investigated. It is noticed that the plant is a discrete-time system and the delay is a multiple of the sampling time; therefore, the result of the papers cannot be applied to the NCSs when the plant is a continuous-time system and the delay is smaller than the sampling period. In [13], the plant studied is a continuous-time system; the delay takes values in a finite set at a fixed rate. In fact, the time delays and packet dropouts may be random and modeled as Markov chains in most cases. Unfortunately, they do not take into account the time delay and packet dropout with Markovian characterization in [13].
In this paper, we investigate the stochastic control of a class of NCSs with time delays and packet dropouts. The random time delay and packet dropout are described by a Markov chain. By using a Lyapunov function, we obtain the system with exponential stability with a desired disturbance attenuation level. The designed stochastic controller is obtained by an iterative linear matrix inequality approach. To demonstrate the effectiveness of the method, an illustrative example is presented.
2. Model for Networked Control System
The structure of the NCS is shown in Figure 1. Consider a continuous-time linear system described by where is the state, is the plant input, is the disturbance input, and is the plant output. , , , and are constant matrices of appropriate dimensions.
The following assumptions are made for the considered NCS throughout the paper [13].
The controller is event-driven; both the sensor and the actuator are time-driven. The sampling period of the sensor is . The actuator has a receiving buffer which contains the most recently updated packet from the controller. The actuator reads the buffer periodically at a smaller period than , say for some integer large enough. The sensor and the actuator are time synchronized. Upon reading a new value, the actuator with a zero-order-hold device will update the output value. The network-induced delay satisfies .
Based on the above assumptions, the discrete-time state feedback controller can be expressed as follows: where where is the value of at the sampling time .
Consider where is the value of the at the sampling time .
During each sampling period, several different cases may arise, which leads to the following discrete-time switched system model [13]: where The is called a switching signal. Note that , , implies , while implies packet dropout.
is modeled as Markov chain that takes values in . The transition probability matrices of are . That means that jump from mode to mode , from mode with probabilities : where and .
3. Disturbance Attenuation Analysis
Definition 1. System (5) is said to be stochastically and exponentially stable, if there exist constants and , such that for .
Definition 2. System (5) is said to be stochastically and exponentially stable with an disturbance attenuation level , if system (5) is stochastically and exponentially stable and for the zero initial condition, .
Lemma 3 (see [14]). Define , where is a positive definite matrix; then there exist constant scalars such that
Theorem 4. For given positive scalars , , and , if there exist matrices , , such that where then system (5) is stochastically and exponentially stable with an disturbance attenuation level .
Proof. Let the Lyapunov function
correspond to the subsystem as follows:
where
When , we obtain
where
From (6) and (7), it can be obtained that
can be rewritten as follows:
where
From the Schur complement, we have that (21) is equivalent to
Similarly, we can see that is equivalent to
It can be seen that if (12) holds, is true, which means
When ,
where
From (6) and (8), it can be seen that
can be rewritten as follows:
where
From the Schur complement, we have that (29) is equivalent to
Similarly, it is easy to see that is equivalent to
It can be seen that if (13) holds, is true, which means
It follows from (25) and (33) that
which means
Next, we prove the stochastically and exponentially stable system (5). The perturbation is assumed to be zero.
When , we obtain
From (21), (23), and (24), it can be seen that is equivalent to .
Then, it can be seen from (19) that if (12) holds, we have
and then
which means
When ,
From (27), (29), (31), and (32), it can be seen that is equivalent to .
Then, it can be seen from (27) that if (13) holds, we have
and then
which means
From (36) and (43), we have
From Lemma 3, we get
Then, the result is established.
The conditions in Theorem 4 are a set of LMIs with some inversion constraints. can be solved by an iterative LMI approach which is called the cone complementarity linearization algorithm [15, 16].
4. Numerical Example
Consider the following system [13]. Suppose , . The transition probability matrices of are taken as follow: which means Using Theorem 4 and the cone complementarity linearization algorithm, we obtain Figure 2 is the possible realizations of the mode . Under this mode sequence, the corresponding state trajectories of the closed-loop system are shown in Figure 3. It is shown that the closed-loop system is stochastically and exponentially stable.
5. Conclusions
In this paper, by modeling the random delays and packet dropouts as a Markov chain, a new Markovian jump system model is presented to describe the networked control system with disturbance attenuation. The criteria for the system are stochastically and exponentially stable with an disturbance attenuation level which is derived by an iterative LMI approach.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is supported by Zhejiang Provincial Natural Science Foundation of China (Grant no. LY14G020014), Zhejiang Provincial Key Research Base of Humanistic and Social Sciences in Hangzhou Dianzi University (no. ZD01-201402), the Research Center of Information Technology & Economic and Social Development, and the National Natural Science Foundation of China (no. 71101040). Professor Chunfeng Liu and Kai Li give the authors some useful comments and suggestions on the language and the structure of the paper. The work is supported by Professor Chunfeng Liu and Kai Li.