Robust H∞ Control of Markov Jump Linear Systems with Uncertain Switching Probabilities

This paper the problem of robust H∞ control for Markov jump systems with uncertain transition rates is investigated. A robust H∞ performance criterion is established for a given Markov jump system. The robust H∞ control performance analysis in terms of coupled linear matrix inequalities is proposed, then convex optimization problem is solved with constraints defined in terms of the solvability of the linear matrix inequalities. Based on the solution of the optimization problem, the condition of robust stochastic stability for closed-loop systems is found, which minimizes disturbance attenuation level. Depending on the developed performance criterion, the H∞ state-feedback controller is designed too, which warranties the robust H∞ control of the closed-loop system. All the conditions are linear matrix inequalities, and therefore they can be solved by any linear matrix inequalities solver. Finally, a numerical example is given to show the effectiveness of the method of robust H∞ control for Markov jump systems.


Introduction
Many dynamic systems undergo sudden random changes, which may be caused by random component failures and repairs, sudden environmental changes, changes in the interconnectedness of subsystems, etc. Usually most conventional dynamic systems are powerless to overcome with these abrupt random changes. Markov jump systems (MJSs) are special class of stochastic hybrid systems (dynamical systems that exhibits both continuous and discrete dynamic behavior). MJSs have applied in many fields, uncrewed aerial vehicle [1], solar power stations [2], communication protocols [3], control of power systems [4], economic systems, [5].
MJSs have been investigated extensively and many beneficial results have been obtained, such as the stabilizability, and continuous-time MJL quadratic control [6], controller design for MJL [7][8][9] and robust linear filtering for MJL [10][11][12]. The nonlinearity in systems may lead to unstable behavior of the systems, robust stabilization and H ∞ control for nonlinear systems with Markovian jump [13][14][15].
The transition rates are essential to set the MJSs. So, the main investigation on MJSs is to assume that the transition rates are well known. In application, the estimated values of transition rates are only available, and estimation errors, i.e., in the transition rates, the uncertainties may be given instability or deterioration of a system. There have been some works regarding control of this type of system [16], the robust stabilization and control problems are considered for MJS with uncertain switching probabilities by using restrictive Young inequality. In [17] results by using general Young inequality less conservative than those of [16] are proposed. Because of the use of Young inequality, the proposed controller design methods in [16,17] need to solve a set of nonlinear matrix inequalities (NLMIs). It is still not possible to fully resolve these NLMIs yet. The H ∞ control problem for nonlinear MJSs with uncertain transition rates has not been completely scrupulous [18]. It remains important and hard.
This study is interested with the robust H ∞ control for MJLS with uncertain transition rates. First, the robust H ∞ performance criterion is found. Therewith, the method for designing the H ∞ controller based on the proposed performance criterion is presented. We assume an improved bounding for the uncertain terms instead of using the traditional Young inequality. As an advantage, the design method of obtained controller only needs to solve a set of linear matrix inequalities (LMIs) instead of NLMIs, we can easily solve by any LMI solution. At last, a numerical example is given to confirm the efficacy of the proposed methods. Notations where ( ) R n x t ∈ is standing for the state variable We assume the set is a continuous-time Markov chain on the probability space, with transition rate matrix  . The linear state-feedback control law is: where the controller gain matrices to be design. The closed- Now, we introduce the following definitions [20]. (1) is exponentially mean square stable (EMS-stable) if there exist α and β are positive real scalar such that

Robust Stochastic H ∞ Performance Analysis
In this section, analysis performance the problem H ∞ of system is considered in terms of LMI, and then dealt with in terms of the solvability of a set of LMI with equality constraints The following theorem [20] gives a robust stochastic H ∞ criterion for MJLS of (1). Theorem 1. Let the MJS (1) with uncertain transition rates. The controller gains , such that for i S ∀ ∈ the following LMIs are feasible:

Robust Stochastic H ∞ Controller Design
In this section, design problem the H ∞ controller of system (1) is studied. The following theorem has been proposed for designing the robust stochastic H ∞ controller for system (1) Furthermore, a controller gain is given by Proof. From Theorem 1, we have that system (4) with uncertain transition rates is robustly stochastically stable if inequalities (6)-(8) holds with disturbance attenuation level γ . By applay the Schur complement and noting (6)-(8) are equivalent to the relations (13)-(15).
, we find the inequality in (10). and transformation the congruence to the inequality in (15) by { } diag , , i Q I we can get the inequality in (11). In addition, we have , wanted controller is given by 1 .
Remark. From Theorem 2, the H ∞ control problem for MJLS with uncertain transition rates can be solved in terms of the LMIs in (9)- (11). The inequalities in (9)- (11) are not only linear with the variables , , , Such that, the minimum cost is given by The uncertain transition rates given by: The robust H ∞ controller is designed such that the closed-loop system is robustly stochastically stable with γ over all the and uncertain transition rates. We obtain minimum disturbance attenuation level is 0 0.753 γ = by Theorem 2 with the corresponding controller gain matrices  In Fig. 1, the effect of weak noise with an abrupt change in intensity leads to the emergence of unstable modes and an increase in the "stopping distance".
The state response of the resulting closed-loop system with uncertain switching probability is given in Fig. 2, which are switching two models  Fig. 1, obviously, the disturbance observers are effect to handle the random switching disturbances. Based on the above analysis, it can be asserted that the desired controller has good robust performance. On Fig. 3 we seen that the closed-loop system without stabilizing control is unstable.