Control for Large-Scale Systems with Uncertain Missing Measurements Probabilities

For large-scale systems which are modeled as interconnection of N networked control systems with uncertain missing measurements probabilities, a decentralized state feedback H ∞ 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 H ∞ controller is designed in terms of linear matrix inequalities to make closed-loop system exponentially mean square stable and a prescribedH ∞ 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.


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][2][3][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][9][10][11][12][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][9][10][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][14][15][16][17][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 largescale 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.

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 form where   () ∈ where x () ∈   is the actual measured states,   () ∈  is a Bernoulli distributed white sequence taking the values of 0 and 1 with certain probability and the unknown positive scalar   : 0 <   < 1 means the occurrence probability of the missing measurements.Without loss of generality, we assume where  max and  min are the upper limit and lower limit of the probability, respectively, and satisfy Choose  0 = ( min +  max )/2 and  1 = ( max −  min )/2; 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),  max = 1 means that no measurement is lost and  min = 0 means that measurements are lost completely.
For system (1), the control input can be chosen as where   ,  = 1, . . ., , 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 () = 0 is said to be exponentially mean-square stable if there exist constants  > 0 and 0 <  < 1 such that where 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 () = 0, closed-loop system ( 8) is exponentially mean-square stable.
(2) Under the zero-initial condition, the controlled output where ]  , and  > 0 is a prescribed scalar.We first give following useful two lemmas.

Main Results
At first, for the case of system (1) without disturbance, that is, () = 0, we have the following two theorems.
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 () = 0 is exponentially mean-square stable if there exist positive definite matrix  and gain matrix  satisfying the following linear matrix inequality: where  > 0 is an arbitrary given constant, Proof.Through left-and-right multiplication of ( 14) by we can get which 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.
When () ̸ = 0, choose the Lyapunov functional as where Based on the Schur complement, inequality (25) implies  2 < 0, and then we get Now summing (29) from 0 to ∞ with respect to  yields Since system ( 8) is exponentially mean-square stable.Under the zero-initial condition, it is straightforward to see that This completed the proof.
Proof.Through left-and-right multiplication (25) by we have Then 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.

Simulation Example
Consider a linear discrete-time large-scale system which is composed of two NCSs as follows:    When  1 =  2 = 0.4, 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  1 ,  2 are uncertain and  1 =  2 ∈ [0.4 1], we can get the following parameters in Theorem 8 by using the YALMIP toolbox in MATLAB: The simulation results are shown in Figure 3 and the closed-loop systems are stable.It can be verified that ∑ ∞ =0 {‖()‖ 2 } <  2 ∑ ∞ =0 {‖()‖ 2 }.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.

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.

( 37 )
According to  =  −1 and  = , we have the Lyapunov function solution matrices and controller parameters as follows: