H ∞ Control of Singular Markovian Jump Systems with Bounded Transition Probabilities

This paper discusses H ∞ control problems of continuous-time and discrete-time singular Markovian jump systems (SMJSs) with bounded transition probabilities. Improved sufficient conditions for continuous-time SMJSs to be regular, impulse free, and stochastically stable with γ-disturbance attenuation are established via less conservative inequality to estimate the transition jump rates, so are the discrete-time SMJSs. With the obtained conditions, the design of a state feedback controller which ensures the resulting closed-loop system to be stochastically admissible andwithH ∞ performance is given in terms of linearmatrix inequalities (LMIs). Finally, illustrative examples are presented to show the effectiveness and the benefits of the proposed approaches.


Introduction
Singular systems, also referred to as generalized systems, descriptor systems, implicit systems, differential-algebraic systems, or semistate systems [1], have been extensively studied in the past years due to the fact that singular systems better describe physical systems than regular ones.Many practical problems, such as those in Leontief 's dynamic system [2] and electrical and mechanical systems [3,4], are modeled as singular systems.Considerable attention has been paid to investigate such continuous-time singular systems [5][6][7][8] and discrete-time singular systems [9][10][11], respectively.On the other hand, Markovian jump systems (MJSs) can be regarded as a special class of hybrid systems with finite operation modes whose structures are subject to random abrupt changes.The studies of MJSs are important in practical applications such as manufacturing systems, aircraft control, target tracking, robotics, solar receiver control, and power systems.Therefore, a great deal of attention has been devoted to the study of this class of systems in recent years such as [12][13][14][15][16][17][18][19].
When singular systems experience abrupt changes in their structure and parameters, it is natural to model them as SMJSs.Reference [20] considered the stochastic stability and robust stochastic stability conditions for continuous-time SMJSs.The delay-dependent  ∞ control problem of continuous-time SMJSs was studied in [21] via Jensen's integral inequality approach after introducing free weighting matrices, while [22] gave sufficient conditions for uncertain discrete-time SMJSs with mode-dependent time delays through transforming them into a standard linear system.Particularly, [23] proposed more general condition of  ∞ control for discrete-time SMJSs, which does not require any decomposition of the original system, and the construction of  ∞ controller gain does not need system decomposition.However, all the above-mentioned criteria on MJSs require the values of transition probabilities to be exactly know.Recently, Boukas discussed  ∞ control of discrete-time normal state-space MJSs with bounded transition probabilities in [24] and stabilization of continuoustime SMJSs with full or partial knowledge of transition rates in [25].
Following the work of [24,25], in this paper, the  ∞ control problem is considered for both continuous-time and discrete-time SMJSs with bounded transition probabilities.Sufficient criteria guaranteeing SMJSs being stochastically admissible with an  ∞ performance are presented in terms of LMIs, which are obtained by using an improved inequality to obtain less conservative estimation on transition probabilities.Based on these, a state feedback controller such that Mathematical Problems in Engineering the resulting closed-loop system is stochastically admissible and possesses a prescribed  ∞ performance is given.Numerical examples are provided to demonstrate the validity of developed methods.
Notation.R  denotes the  dimensional Euclidean space.R × is the set of all  ×  real matrices.| ⋅ | denotes the Euclidean norm.{⋅} is the expectation operator with respect to some probability measure.In symmetric block matrices, we use " * " as an ellipsis for the terms induced by symmetry, diag{⋅ ⋅ ⋅ } for a block-diagonal matrix, and () ⋆ ≜  +   .
Assumption 1.The transition probabilities for system (1) are assumed to be unknown but vary between two known bounds satisfying the following: where  ̸ = .
From Assumption 1, when we only know that   = min  ̸ = ∈S   and   = min  ̸ = ∈S   , we have   ≤   ≤   , which is the same as in [25].Thus, we may say that Assumption 1 is more natural.Definition 2 (see [26]).(1) The nominal system (1) is said to be regular if det(−  ) is not identically zero for every  ∈ S.
(3) The nominal system in (1) is said to be stochastically stable if, when () = 0 and () = 0, there exists a constant ( 0 ,  0 ) such that (4) The nominal system in ( 1) is said to be stochastically admissible if it is regular, impulse free, and stochastically stable.
Definition 3. Given  > 0, the nominal system in ( 1) is said to be stochastically admissible with -disturbance attenuation if it is stochastically admissible and satisfying the following such that holds for zero-initial condition and any nonzero In this paper, the  ∞ controller such that the resulting closed-loop system is stochastically admissible with ( 6) is where   is to be determined.
Lemma 4. Given a scalar  > 0, the unforced system (1) is stochastically admissible with  ∞ performance if there exist matrices   such that

Discrete-Time Markovian Jump Linear Systems. Consider the following discrete-time SMJS described by
where () ∈ R  is the state vector, () ∈ R  is the control input, () ∈ R  is the disturbance input, and () ∈ R  is the control output.Matrix  may be singular assumed rank() =  ≤ .Parameter () is a discrete-time, discretestate Markovian chain taking values in a finite set S with transition probabilities: where   ≥ 0, for ,  ∈ S, and Assumption 5.The transition probabilities for ( 10) are assumed to be unknown but vary between two known bounds satisfying the following: From [24], we know that ∑  =1   and ∑  =1   may not equal 1.It is not necessary to exactly know the transition probabilities but only the bounds of   , which are   and   .Definition 6 (see [26]).(1) The nominal system in ( 9) is said to be regular if det( −   ) is not identically zero for every  ∈ S.
(3) The nominal system in ( 9) is said to be stochastically stable if, when () = 0 and () = 0, there exists a constant ( 0 ,  0 ) such that (4) The nominal system in ( 9) is said to be stochastically admissible if it is regular, causal, and stochastically stable.Definition 7. Given  > 0, the nominal system in ( 9) is said to be stochastically admissible with -disturbance attenuation such that holds for zero-initial condition and any nonzero () ∈ L 2 [0, ∞).
The corresponding  ∞ controller for system (9) is where   is to be determined.
Lemma 8. Given a scalar  > 0, the unforced system (9) is stochastically admissible with  ∞ performance if there exist matrices   =    such that where P = ∑  =1     .
Lemma 9 (see [27]).Given any real matrices , , and  with appropriate dimensions and such that  > 0 and is symmetric, then, one has

Continuous-Time Markovian Jump Linear Systems
Theorem 10.Given a scalar  > 0, the unforced system (1) with constraint (4) is stochastically admissible with  ∞ performance if there exist matrices   and   > 0 such that where hold for all  ∈ .
Remark 11.Since to the term     gives some problems, [25] presented a method to deal with problem; that is,     ≤ (1/4) Theorem 12.Given a scalar  > 0, there exists a state feedback controller in the form of (7) such that the resulting closedloop system is stochastically admissible with (6), if there exist matrices   ,   > 0, and   such that where hold for all  ∈ S with constraint (4).The controller gain   can be constructed as Proof.Substituting   with   +    , then by similar method in Theorem 10 with (28), we can obtain Theorem 12. Thus, it is omitted here.
Remark 13.It is seen that such conditions have equation constraint (28), which are not strict LMIs.In addition, the desired controller ( 7) is mode dependent, which requires its system mode to be available online.This makes the scope of application limited.In order to deal with this practical condition, mode-independent control is usually used.Based on the methods in [28,29], such problems can be solved easily.

Discrete-Time Markovian Jump Linear Systems
Theorem 14.Given a scalar  > 0, the unforced system (9) with constraint ( 12) is stochastically admissible with  ∞ performance if there exist matrices   ,   , and   > 0 such that where hold for all  ∈ .
Remark 15.It should be remarked that when  =  in ( 9), [24] gave important criteria under the constraint (12) where hold for all  ∈ S with constraint (12).The controller gain   can be constructed as

Numerical Examples
In this section, numerical examples are given to demonstrate the effectiveness of proposed theory., By the method in [25], we find that it is infeasible for this system no matter  takes any values.However, for  = 0.  (49) Example 3. In order to do some comparison, consider the following discrete-time normal state-space MJS which can be obtained by (9) with  =  and is described as By the method in [24], we have the minimum  * = 2.45, while via Theorem 16, we obtain  * = 0.43.By this example, it is seen that our method is less conservative.

Conclusions
In this paper, the problems of  ∞ control for both continuous-time and discrete-time SMJSs with bounded transition probabilities have been investigated.Less conservative sufficient criteria for SMJSs to be stochastically admissible with  ∞ performance are given by the LMI approach.
Based on the obtained conditions, a kind of state feedback controller, such that not only stochastic admissibility but also a prescribed  ∞ performance level is guaranteed, can be computed.Finally, numerical examples are provided to illustrate the advantage and effectiveness of the presented results in this paper.

Example 1 .
Consider the following continuous-time SMJS as (