Stochastic H ∞ Finite-Time Control of Discrete-Time Systems with Packet Loss

This paper investigates the stochastic finite-time stabilization and H∞ control problem for one family of linear discrete-time systems over networks with packet loss, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, the dynamic model description studied is given, which, if the packet dropout is assumed to be a discrete-time homogenous Markov process, the class of discrete-time linear systems with packet loss can be regarded as Markovian jump systems. Based on Lyapunov function approach, sufficient conditions are established for the resulting closed-loop discrete-time system with Markovian jumps to be stochastic H∞ finite-time boundedness and then state feedback controllers are designed to guarantee stochastic H∞ finitetime stabilization of the class of stochastic systems. The stochastic H∞ finite-time boundedness criteria can be tackled in the form of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also give sufficient conditions on the robust stochastic stabilization of the class of linear systems with packet loss. Finally, simulation examples are presented to illustrate the validity of the developed scheme.


Introduction
Networked control systems NCSs are feedback control systems with control closed loops via digital communication channel. Compared with traditional point-to-point controller architectures, the advantages of NCSs include low cost, high reliability, less wiring, and easy maintenance 1 . In recent years, NCSs have found successful applications in broad range of modern scientific areas such as internet-based control, distributed communication, and industrial automation 2 . However, the insertion of the communication channels creates discrepancies between the data records to be transmitted and their associated remotely transmitted images, which hence makes the traditional control theory confronts new challenges. Among these challenges, random communication delay, data packet dropout, 2 Mathematical Problems in Engineering and signal quantization are known to be three main interesting problems for the stability and performance degradation of the controlled networked system. In view of this, many researchers have made to study how to design control systems by packet loss, delay, and quantization, see 3-6 and the references cited therein. Among a number of issues arising from such a framework, packet losses of NCSs are an important issue to be addressed and have received great attention, see 7-15 . Meanwhile, Markovian jumps systems are regarded to be as a special family of hybrid systems and stochastic systems, which are very appropriate to model plants whose structure is subject to random abrupt changes, see [16][17][18][19][20][21][22] and references therein.
It is well known that classical Lyapunov theory focuses mainly on the state convergence property of the systems in infinite time interval, which, just as was mentioned above, does not usually specify bounds on the trajectories in finite interval. However, the main attention in many practical applications is the behavior of the dynamic systems over a specified time interval, for instance, large values of the state are not acceptable in the presence of saturations 23 . To discuss this transient performance of control dynamics, finitetime stability or short-time stability was presented in 24 . Then, some appealing results were found in 25-32 . However, to date and to the best of our knowledge, the problems of stochastic finite-time stability and stabilization of network control systems with packet loss have not fully investigated and still remain challenging, although results related to systems over networks with packet loss are reported in the existing literature, see 6-15, 33-36 . Motivated by the above discussion, in this paper, we address the stochastic H ∞ finite-time boundedness SH ∞ FTB problems for linear discrete-time systems over networks with packet dropout, parametric uncertainties, and time-varying norm-bounded disturbance. Firstly, we present dynamic model description studied, which, if the data packet loss is assumed to be a time homogenous Markov process, the class of linear discrete-time systems with packet loss can be referred as Markovian jump systems. Thus, the class of linear systems investigated could be studied by the theoretical framework of Markov jumps systems. Then, the concepts of stochastic finite-time stability, stochastic finite-time boundedness, and SH ∞ FTB and problem formulation are given. The main contribution of this paper is to design a state feedback controller which guarantees the resulting closed-loop discrete-time system with Markovian jumps SH ∞ FTB. As an auxiliary result, we also give sufficient conditions on the robust stochastic stabilization of the class of linear systems with packet loss. The SH ∞ FTB criteria of the class of Markovian jump systems can be addressed in the form of linear matrix inequalities LMIs with a fixed parameter.
The rest of this paper is organized as follows. Section 2 is devoted to the dynamic model description and problem formulation. The results on the SH ∞ FTB are presented in Section 3. Section 4 presents numerical examples to demonstrate the validity of the proposed methodology. Finally, in Section 5, the conclusions are given. Notation 1. The notation used throughout the paper is fairly standard, R n , R n×m , and Z k≥0 denote the sets of n component real vectors, n × m real matrices, and the set of nonnegative integers, respectively. The superscript T stands for matrix transposition or vector, and E{·} denotes the expectation operator with respective to some probability measure P. In addition, the symbol * denotes the transposed elements in the symmetric positions of a matrix, and diag{· · · } stands for a block-diagonal matrix. λ min P and λ max P denote the smallest and the largest eigenvalue of matrix P , respectively. Notations sup and inf denote the supremum and infimum, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations. Engineering   3

Problem Formulation and Preliminaries
Let us consider a linear discrete-time system LDS as follows: where x k ∈ R n is the state, z k ∈ R l 1 is the measure output, and u k ∈ R m is the control input. The noise signal w k ∈ R l 2 satisfies The matrices ΔA k and ΔB k are uncertain matrices and satisfy where Δ k is an unknown, time-varying matrix function, and satisfies Due to the existence of the packet dropout of the communication during the transmission, the packet dropout process of the network can be regarded as a timehomogenous Markov process {γ k , k ≥ 0}. Let γ k 1 mean that the packet has been successfully delivered to the decoder, while γ k 0 corresponds to the dropout of the packet. The Markov chain has a transition probability matrix defined by 1} are the state of the Markov chain. Without loss of generality, let γ 0 1 and the failure rate p and the recovery rate q of the channel satisfy p, q ∈ 0, 1 . It is worth noting that a smaller value of p and a larger value of q indicate a more reliable channel.
Remark 2.1. When the above transition probability matrix is p 1−p p 1−p with 0 ≤ p ≤ 1, the above two-state Markov process is reduced to a Bernoulli process 37 .
Consider the control law for the LDS 2.1 in the form where L is to be designed the control gain matrix. {γ k , k ≥ 0} is a Markov packet dropout process satisfying 2.5 . Then, the resulting closed-loop LDS follows that where A A ΔA k and B B ΔB k . Now, we define two models according to the value of γ k . If γ k 1, we define the Model 1 at time k 1 as follows:

2.8
If γ k 0, we define the Model 2 at time k 1 as follows: where the selection of L ζ in 2.7 is according to the model of x k for all ζ ∈ {1, 0}, that is to say, if x k is at Model 1, which is γ k − 1 1, L ζ L 1 , otherwise, if x k is at Model 2, which is γ k − 1 0, L ζ L 0 . Then, 2.7 can be regarded as a closed-loop LDS with Markovian jumps described by where a , a ∈ {1, 2} denotes the mode indicator function. 1 corresponds to a mode with feedback, and 2 corresponds to a mode without feedback. It is noted that it yields a 1 when at time k 1 be a ∈ {1, 2} and b 0 for b / a. The mode transition probabilities of Markovian jump LDS 2.10 is given by where π uv ≥ 0 for all u, v ∈ {1, 2} and 2 v 1 π uv 1. η v k 1 implies 1 1, 2 0, which the communication transmission succeeds, and η v k 2 implies 1 0, 2 1, which the communication dropout occurs. Thus, compared to 2.5 , it follows that π 11 1 − p, π 12 p, π 21 q, π 22 Definition 2.2 stochastic finite-time stability SFTS . The closed-loop Markovian jump LDS with w k 0 2.10 is said to be SFTS with respect to δ x , , R, N , where 0 < δ x < , R is a symmetric positive-definite matrix and N ∈ Z k≥0 , if for any nonzero w k which satisfies 2.2 , where γ is a prescribed positive scalar.

Lemma 2.5 see 38 .
The linear matrix inequality X 11 * In this paper, the feedback gain matrices L 1 and L 0 with Markov packet dropout of failure rate p and recovery rate q will be designed to guarantee the states of the closed-loop Markovian jump LDS 2.10 SH ∞ FTB.

Main Results
In this section, for the given failure rate p and recovery rate q with p, q ∈ 0, 1 , we will design a state feedback controller that assures SH ∞ FTB of the Markovian jump LDS 2.10 . Mathematical Problems in Engineering γ > 0, two symmetric positive-definite matrices P 1 , P 2 , and a set of feedback control matrices {L ζ , ζ ∈ {1, 0}}, such that the following inequalities hold: where P a R −1/2 P a R −1/2 for all a ∈ {1, 2}.
Proof. Assume the mode at time k be a ∈ {1, 2}. Taking into account that if a 1, then we have γ k − 1 1 and L ζ L 1 , otherwise if a 2, then γ k − 1 0 and L ζ L 0 . Consider the following Lyapunov-Krasovskii functional candidate for the Markov jump LDS 2.10 : Then, we have

3.6
Taking into account that if a 1, then L ζ L 1 , otherwise a 2, then L ζ L 0 . Noting that π 11 1 − p, π 12 p, π 21 q, π 22 1 − q. Thus, when a 1, it follows that By Lemma 2.5, it follows from 3.1 and 3.7 that When a 2, taking into account condition 3.2 , the similar to 3.9 , we can derive the following inequality: Thus, for all a ∈ {1, 2}, we have Θ x k , w k , a < 0.

3.11
That is to say, for all a ∈ {1, 2}, it follows that By 3.12 , it is obvious that From 2.2 and 3.13 and noting that μ ≥ 1, we have 3.14 8

Mathematical Problems in Engineering
Let P a R −1/2 P a R −1/2 and noting that E{x T 0 Rx 0 } ≤ δ 2 x , we have

3.15
On the other hand, for all a ∈ {1, 2}, we have

3.17
Noting condition 3.3 , it is obvious that E{x T k Rx k } < 2 for all k ∈ {1, 2, . . . , N}. This completes the proof of this theorem.

Mathematical Problems in Engineering 9
Proof. Noting that Thus, when a 1, we have where Φ 1 , Υ 1 are the same as the above. Thus, according to Lemma 2.5, we can obtain from 3.18 and 3.23 When a 2, taking into account condition 3.19 and 3.21 , the similar to the above deduction, we can derive that the following inequality holds: Thus, for all a ∈ {1, 2}, we can obtain According to 3.26 , it is obvious that From 3.27 , we have Under the zero-value initial condition and noting that V k ≥ 0 for all k ∈ Z k≥0 , we have From 3.29 and noting that μ ≥ 1, we have

3.30
Thus, this completes the proof of the theorem.
and applying Lemmas 2.5 and 2.6, one can obtain from Theorem 3.2 the following results on the stochastic H ∞ finite-time stabilization.

Remark 3.4.
It is easy to check that condition 3.33 is guaranteed by imposing the conditions for all a ∈ {1, 2}: It follows that conditions 3.31 , 3.32 , and 3.34 are not strict LMIs; however, once we fix the parameter μ, the conditions can be turned into LMI-based feasibility problem: Remark 3.5. From the above discussion, we can obtain that the feasibility of conditions stated in Theorem 3.3 can be turned into the following LMIs based feasibility problem with a fixed parameter μ. Furthermore, we can also find the parameter μ by an unconstrained nonlinear optimization approach, which a locally convergent solution can be obtained by using the program fminsearch in the optimization toolbox of Matlab.
Remark 3.6. If we can find feasible solution with the parameter μ 1, by the above discussion, we can obtain that the designed controller can ensure both stochastic finite-time boundedness and robust stochastic stabilization of the family of network control systems.

Numerical Examples
In this section, we present two examples to illustrate the proposed methods. Furthermore, let R I 3 , δ x 1, and N 5, by Theorem 3.3, the optimal bound with minimum value of γ 2 2 relies on the parameter μ. We can find feasible solution when 1.12 ≤ μ ≤ 34.13. Figures 1 and 2 show the optimal value with different value of μ. Then, by using the program fminsearch in the optimization toolbox of Matlab starting at μ 1.8, the locally convergent solution can be derived as  Then, let R I 3 and δ x 1, by Theorem 3.3, we can find feasible solution when μ 1. Furthermore, when μ 1, it yields the optimal value γ 4.7057 and 5.0791 and the following optimized state feedback controller gains: Thus, the above LDS with Morkovian jumps is stochastically stable and the calculated minimum H ∞ performance γ satisfies T wz < 4.7057.

Conclusions
This paper addresses the SH ∞ FTB control problems for one family of linear discrete-time systems over networks with packet dropout. Under assuming packet loss being a time homogenous Markov process, the class of linear discrete-time systems can be regarded as Markovian jump systems. Sufficient conditions are given for the resulting closed-loop linear discrete-time Markovian jump system to be SH ∞ FTB, and state feedback controllers are designed to guarantee SH ∞ FTB of the class of linear systems with Markov jumps. The SH ∞ FTB criteria can be tackled in the form of linear matrix inequalities with a fixed parameter. As an auxiliary result, we also give sufficient conditions on the robust stochastic stabilization of the class of linear discrete-time systems with data packet dropout. Finally, simulation results are also given to show the validity of the proposed approaches.