H 2 / H ∞ Control Design of Detectable Periodic Markov Jump Systems

An infinite horizonH 2 /H ∞ control problem is addressed for discrete-time periodicMarkov jump systems with (x, u, V)-dependent noise. Above all, by use of the spectral criterion of detectability, an extended Lyapunov stability theorem is developed for the concerned dynamics. Further, based on a game theoretic approach, a state-feedbackH 2 /H ∞ control design is proposed. It is shown that under the condition of detectability H 2 /H ∞ feedback gain can be constructed through the solution of a group of coupled periodic difference equations.


Introduction
∞ control has been one of the most active areas of modern control theory since the 1970s.Owing to the introduction of state-space approach [1], many researchers have been inspired to extend the deterministic  ∞ control theory to various stochastic systems; see [2][3][4][5].In the development of stochastic  ∞ theory, [2] can be regarded as a pioneering work, which firstly established a stochastic version of bounded real lemma for linear Itô-type differential systems.Besides, initialed from [6], considerable progress has been made in the study of stochastic  2 / ∞ control.By combing  ∞ index with an quadratic cost performance, the resulting multiobjective control strategy is more attractive than the sole  ∞ control in engineering applications.
The main objective of this paper is to settle an infinite horizon  2 / ∞ control problem for periodic Markov jump systems with multiplicative noises.By now, Markov jump systems have been extensively investigated [7][8][9].For example, stochastic and robust stability have been elaborately discussed in [10,11] for networked dynamics with Markovian jump.As concerns  ∞ theory, an  ∞ estimation problem was tackled in [12] for a class of discrete homogeneous Markov jump systems.On the other hand, an infinite horizon  ∞ control problem was handled in [13] for nonlinear Itô systems with homogeneous Markov process.However, few results have been reported for  2 / ∞ control of periodic Markov jump systems.To some extent, this study will generalize the work of [14] to the case of periodically timevarying coefficients and transition probabilities, as in [15][16][17].
The remainder of this paper is organized as follows.Section 2 gives basic preliminaries and problem formulations.In Section 3, the intrinsic relationship between asymptotic mean square stability and detectability is addressed.As a result, a Barbashin-Krasovskii-type theorem is established for periodic Markov jump systems with state-dependent noises.Section 4 contains an internally stabilizing control design, which can not only fulfill the prescribed disturbance attenuation level, but also minimize the output energy.To verify the effectiveness of the proposed approach, a numerical example is supplied in Section 5. Finally, Section 6 concludes this paper with a concluding remark.
Notations.  (C  ) is -dimensional real (complex) space with the usual Euclidean norm ‖ ⋅ ‖;  × is the space of all  ×  real matrices with the operator norm ‖ ⋅ ‖ 2 ;   is
Particularly, the induced matrix of T   is denoted by A   := A +, .
Next, we will give two useful lemmas, which have been shown in [19].( We are prepared to establish the following Barbashin-Krasovskii stability criterion for (4).

Theorem 5. If (A, C; P) is detectable, then (A; P) is AMSS if and only if the PLE
has a unique -periodic solution   ∈  +  .
Remark 6.In [18], a similar result has been proven under the condition of stochastic detectability.According to [19], (uniform) detectability is a weaker prerequisite than stochastic detectability.Therefore, Theorem 5 has improved the result of Theorem 4.1 [18] within the concerned framework.

𝐻 2 /𝐻 ∞ Control
In this section, a game theoretic approach will be employed to deal with the infinite horizon  2 / ∞ control problem of (1).Under the assumption of detectability, a necessary and sufficient condition can be provided for the existence of  2 / ∞ controller.

Numerical Example
Consider the following two-dimensional Markov jump system with the periodic coefficients listed as follows: By Lemma 4, it can be verified that (A, C; P) and (A + BF 1 , C; P) are both detectable.Applying ( * , V * ) to the periodic Markov jump system, we get the closed-loop state trajectory and corresponding  2 performance.Figure 1(a) has displayed 50 sampled state trajectories originating from ( 1 (0),  2 (0)) = (10, 20), while Figure 1(b) demonstrates the cumulative energy of the system output.

Conclusion
In this paper, an infinite horizon  2 / ∞ control problem has been settled for discrete-time periodic Markov jump systems with multiplicative noise.Under the condition of (uniform) detectability, a game theoretic  2 / ∞ control is produced by solving a group of CPDEs.Note that there remain some open topics on this issue.For example, it is interesting as well as challenging to investigate the  2 / ∞ control problem with input or output saturation constraint [23], which no doubt deserves a further study.