Abstract
For large-scale systems which are modeled as interconnection of networked control systems with uncertain missing measurements probabilities, a decentralized state feedback controller design is considered in this paper. The occurrence of missing measurements is assumed to be a Bernoulli random binary switching sequence with an unknown conditional probability distribution in an interval. A state feedback controller is designed in terms of linear matrix inequalities to make closed-loop system exponentially mean square stable and a prescribed performance is guaranteed. Sufficient conditions are derived for the existence of such controller. A numerical example is also provided to demonstrate the validity of the proposed design approach.
1. Introduction
With the advances in network technology, more and more control systems have appeared whose feedback control loop is based on a network. This kind of control systems are called networked control systems (NCSs) [1–4]. Owing to the data communication errors in network and the temporarily disabled sensor, missing measurements and transmission time delay usually occur, which can degrade the system performance and even make the system unstable. There have been significant research efforts on the design of controllers and filters for system with missing measurements. There are two main approaches to handle missing measurements. One approach is to replace the missing measurements with an estimated value [5], and the other approach is to view missing measurements as “zero” [6], such as Markov chains [7] and Bernoulli binary switching sequence [8–13]. Fault detection is considered for NCS with missing measurements probabilities being known in [8]. Furthermore, still fault detection is considered for NCS with delays and missing measurements in [9]. In [10], the robust control problem is investigated for stochastic uncertain discrete time-delay systems with missing measurements. In [11], an observer-based controller is designed for NCS with missing measurements, where the missing measurements are assumed to obey the Bernoulli random binary distribution. The controlled systems in references [8–11] are linear discrete systems and the missing measurements probabilities are known constants. A robust fault detection method is proposed for NCS with uncertain missing measurements probabilities in [12].
In most existing results, the controlled NCS is usually treated as isolated one and the missing measurement probability is known [13–18]. However, on one hand, in practice the missing measurements probability usually keeps varying and cannot be measured exactly. On the other hand, in many practical applications, controlled systems are large-scale systems which are composed of discrete-time NCSs. Each discrete-time NCS is influenced not only by missing measurements, but also by interconnection terms generated by the other NCSs. At the same time, due to the dispersion of some large-scale systems such as power systems, it is impossible to feed back all states of whole large-scale systems to design the controller. So the decentralized controller that only feed back local information is more practical. In [19], for large-scale systems composed by discrete-time NCSs with missing measurements, where the missing measurements are modeled as Bernoulli distribution with a known conditional probability, the control problem is considered using linear matrix inequality (LMI) method. In summary, to study the decentralized control for large-scale systems composed by discrete-time NCSs with uncertain missing measurements probability is of important significance. But as far as the authors know, such research is seldom to be found.
In this paper, the decentralized control problem is studied for linear discrete-time large-scale systems composed of discrete-time NCSs with missing measurements, where the occurrence of missing measurements is assumed to be a Bernoulli random binary switching sequence with an unknown conditional probability distribution that is assumed to be in an interval. Decentralized stabilization controller design is proposed for such systems. Sufficient conditions are established by means of LMI, which can be solved conveniently by MATLAB LMI toolbox.
2. Problem Formulation
Consider the linear large-scale systems composed of discrete-time NCSs with missing measurements. The th NCSs are assumed to be of the formwhere , , , , and denote the state vector, the control input, the controlled output, the measuring output, and the disturbance of th subsystem, respectively; ; , and are known real matrices with appropriate dimensions; is the interconnection between theth subsystem and th subsystem.
The measurements with packet loss are described bywhere is the actual measured states, is a Bernoulli distributed white sequence taking the values of 0 and 1 with certain probabilityand the unknown positive scalar : means the occurrence probability of the missing measurements. Without loss of generality, we assume where and are the upper limit and lower limit of the probability, respectively, and satisfyChoose and ; we can obtain another expression about as follows:
Remark 1. The missing measurements probability usually keeps varying and cannot be measured exactly. However, it can be estimated by a value region shown as (4), which is much more practical. In (5), means that no measurement is lost and means that measurements are lost completely.
For system (1), the control input can be chosen aswhere , are gain matrices to be designed. Submit (7) into (1); we can get the following closed-loop system:
Definition 2 (see [11]). Closed-loop system (8) with is said to be exponentially mean-square stable if there exist constants and such thatwhere .
The objective of this paper is to design the state feedback controller (7) for system (1), such that closed-loop system (8) satisfies following requirements:
(1) When , closed-loop system (8) is exponentially mean-square stable.
(2) Under the zero-initial condition, the controlled output satisfieswhere , , and is a prescribed scalar.
We first give following useful two lemmas.
Lemma 3 (see [20]). Let be a Lyapunov functional. If there exist real scalars , , , and such that then sequence satisfies
Lemma 4 (see [21]). For any parameter and matrices , , and with appropriate dimensions, if , then
3. Main Results
At first, for the case of system (1) without disturbance, that is, , we have the following two theorems.
Theorem 5. Closed-loop system (8) with is exponentially mean-square stable if there exist positive definite matrices and the controller gain matrices satisfyingwhere is an arbitrary given constant,
Proof. Consider the following Lyapunov functional:when , we haveBy virtue of Lemma 4 and and , we have where . By Schur complement, (14) implies and we obtainwhere . Definite ; we getwhere .
By Definition 2 and Lemma 3, closed-loop system (8) is exponentially mean-square stable. This completed the proof.
It should be noted that matrix inequality (14) is not a linear matrix inequality and difficult to be solved. For this, we have following Theorem 6.
Theorem 6. Closed-loop system (8) with is exponentially mean-square stable if there exist positive definite matrix and gain matrix satisfying the following linear matrix inequality:where is an arbitrary given constant,
Proof. Through left-and-right multiplication of (14) bywe can getwhich is equivalent to LMI (21). By solving (21), we can obtain matrices and . Furthermore, from (21), we can get matrices and . This completed the proof.
For the case of system (1) with disturbance, that is, , we have the following two theorems.
Theorem 7. Closed-loop system (8) is exponentially mean-square stable and achieves the prescribed performance ifthere exist positive definite matrix and gain matrix satisfying the following LMI:where is a given parameter and is an arbitrary given constant, , , and , , , , , and are the same as in (14).
Proof. When , inequality (25) is equivalent to (14). From Theorem 5, closed-loop system (8) is exponentially mean-square stable.
When , choose the Lyapunov functional asthen, we havewhereBased on the Schur complement, inequality (25) implies , and then we getNow summing (29) from to with respect to yieldsSince system (8) is exponentially mean-square stable. Under the zero-initial condition, it is straightforward to see thatThis completed the proof.
Theorem 8. Closed-loop system (8) is exponentially mean-square stable and achieves the prescribed performance if there exist positive definite matrix and gain matrix satisfying the following LMI:where is a given parameter and , , , , , , , , and are the same as in (21).
Proof. Through left-and-right multiplication (25) bywe haveThen matrix inequality (32) is equivalent to (25). From Theorem 7, we can conclude that closed-loop system (8) is exponentially mean-square stable and achieves the prescribed performance. This completed the proof.
4. Simulation Example
Consider a linear discrete-time large-scale system which is composed of two NCSs as follows:
Assume that . We can obtain the Lyapunov function solution matrices and controller parameters as follows:
Choose the disturbance input . The initial state values are and . The simulation results are shown in Figure 1 and the closed-loop systems are stable.
When , the simulation results are shown in Figure 2 and the closed-loop systems are unstable. From Figures 1 and 2, we can conclude that the closed-loop systems cannot be guaranteed to be stable when the missing measurements probabilities are large enough. For the limit of space, the detailed design procedure is omitted here.
When are uncertain and , we can get the following parameters in Theorem 8 by using the YALMIP toolbox in MATLAB:According to and , we have the Lyapunov function solution matrices and controller parameters as follows:The simulation results are shown in Figure 3 and the closed-loop systems are stable. It can be verified that .
In summary, the closed-loop stability cannot be guaranteed using the method where probability is known to deal with the missing measurements. However, when the probability varies within a given interval, the closed-loop stability can be guaranteed through the controller designed by the method proposed in this paper.
5. Conclusions
In this paper, the decentralized controller has been designed for a class of large-scale systems with uncertain missing measurements probabilities. The random missing measurements are modeled as a stochastic variable satisfying Bernoulli distribution with uncertain probabilities. Sufficient conditions for the existence of a stable controller are presented via LMI, and the designed controller enables the closed-loop system to be exponentially mean-square stable and achieve the prescribed performance.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work is supported by the National Natural Science Foundation of China (nos. 61104103, 61302155, and 61304089).