Delay-Dependent H ∞ Control for Descriptor Markovian Jump Systems with Time-Varying Delay

This paper is concerned with the delay-dependent H ∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay. By constructing various Lyapunov-Krasovskii functionals for different subsystems, together with delay decomposition method, a new delay-dependent bounded real lemma (BRL) is derived, under which descriptor Markovian jump time-delay systems are regular, impulse-free, and stochastically stable and satisfy a prescribed H ∞ performance level. Since the reciprocally convex combination approach is adopted to estimate the upper bound of the integral terms, the BRL obtained in this paper is less conservative than some existing ones. Based on the proposed BRL, a sufficient condition for the existence of state feedback controller is provided. Finally, three numerical examples are provided to demonstrate the validity of the proposed methods.


Introduction
Over the past decades, stability problems for time-delay systems have received considerable attention.Many methods have been presented to determine the stability.In the published results, two types of criteria have been widely investigated (see [1]), that is, delay-independent one and delay-dependent one.As we know, the delay-dependent criteria are less conservative than delay-independent ones, especially when the time delay is fairly small.Therefore, in order to reduce the conservatism of stability conditions, a great number of delay-dependent results have been reported, including, descriptor modeled transformation method (e.g., [2]), parameterized modeled transformation one (e.g., [3]), free-weighting matrix one (e.g., [4]), discretized Lyapunov functional one (e.g., [5]), delay decomposition one (e.g., [6]) and so on (e.g., [7,8]).Among the aforementioned methods, since discretized Lyapunov functional method is complicated and it is difficult to apply this method to design controller, the delay decomposition method is less conservative than others except the discretized Lyapunov functional method.
Markovian jump systems are modeled by many stochastic systems, which arise abrupt random changes in their structure, such as component failures or repairs and unexpected environmental changes.A great deal of attention has been devoted to Markovian jump systems (see [9,10]).Recently, many efforts have been made to study the stability analysis and controller synthesis problems for Markovian jump systems with time-varying and time-invariant delay.Reference [11] divides the time-invariant delay into  segments and uses the delay division method to obtain a less conservative criterion than [12].In [13], stability and stabilization are discussed for Markovian jump systems with time-varying delay by selecting distinct Lyapunov matrices for different subsystems and introducing a triple-integral term.Reference [14] studies the  ∞ analysis for Markovian jump stochastic systems with time-varying delay.By dividing the delay interval into two parts, two improved delay-dependent conditions have been presented to guarantee the stochastic stability.
Descriptor systems are also referred to as singular systems, semistate systems, which are a natural representation of dynamical systems and describe a much wider range of systems than standard state-space systems (see, e.g., [10,15]).When descriptor systems are subject to time delay and stochastic disturbance in their structure, it is very natural to cast this model into descriptor Markovian jump time-delay systems.Very recently, a large number of researches have been published to study descriptor Markovian jump timedelay systems.By constructing more complicated Lyapunov-Krasovskii functionals, [16] studies the passivity and stochastic stability of descriptor Markovian jump systems with timevarying delay.Based on the delay decomposition method, [17] investigates the  ∞ control problem for descriptor Markovian jump time-delay systems.To the best of our knowledge, the time-varying delay in most published results has not been tackled adequately, which leaves us much room to improve.All of these motivate the current study.
In this paper, we investigate  ∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay.Motivated by delay decomposition method, through designing various Lyapunov-Krasovskii functionals for different subsystems, a delay-dependent BRL is established, which guarantees that the considered system is stochastically admissible and satisfies a prescribed  ∞ performance level.The reciprocally convex combination method is used in the proof of the proposed results, which is one of the contributions to this paper.The other is the employed Lyapunov-Krasovskii functionals.Thanks to all of these, the obtained results are less conservative than some existing results.
The rest of this paper is organized as follows.Section 2 gives problem description and some essential lemmas.Section 3 focuses on  ∞ control problem for continuoustime descriptor Markovian jump time-delay systems.A new delay-dependent BRL is proposed, and the design method for the state feedback controller, which guarantees that the resultant closed-loop system is stochastically admissible with a prescribed  ∞ performance level, is provided.Section 4 provides numerical examples to illustrate the effectiveness of our methods, and Section 5 concludes the paper.
Notation.The notation used throughout this paper is quite standard.R  stands for the -dimensional Euclidean space, and R × represents the set of all  ×  real matrices.The superscripts  and −1 stand for matrix transposition and matrix inverse, respectively.L 2 [0, ∞) represents the space of square integrable vector functions over the interval [0, ∞).(Ω, F, P) is the probability space, Ω is the sample space, F is the -algebra of subsets of the sample space, and P is the probability measure on F. E{⋅} is the expectation operator with respect to some probability measure P. N + represents the set of positive integers.The notation  > 0 ( ≥ 0) implies that  is a real symmetric and positive definite (semipositive definite) matrix.For simplicity,   is often adopted to denote ().The asterisk * in matrix block implies that it can be induced by symmetric position.A block diagonal matrix with diagonal blocks  1 ,  2 , . . .,   will be denoted by diag{ 1 ,  2 , . . .,   }.Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.
To facilitate exposition, the unforced descriptor Markovian jump time-delay systems are given as follows: In the sequel, we recall some indispensable definitions.
(1) The continuous-time descriptor Markovian jump time-delay systems (4) are said to be regular and impulse-free for any time-varying delay satisfying (3) if the pairs (,   ) are regular and impulse-free for each   ∈ ℓ.
(2) The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically stable if there exists a scalar ((⋅),  0 ) > 0 such that lim where (,   ,  0 ) denotes the solution to systems (4) at time  under the given initial conditions.(3) The continuous-time descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible if they are regular, impulse-free, and stochastically stable.
Definition 2 (see [17]).For a given scalar  > 0, the descriptor Markovian jump time-delay systems (4) are said to be stochastically admissible with  ∞ performance level ; if the system with () ≡ 0 is stochastically admissible and under zero initial condition, it satisfies for any nonzero () ∈ L[0, ∞).
In this paper, the purpose is to design a mode-dependent state feedback controller for the continuous-time descriptor Markovian jump time-delay systems (1) with a prescribed scalar  > 0 such that the resultant closed-loop systems are stochastically admissible with  ∞ performance level .Specifically, the desired state feedback controller can be expressed as To this end, some essential lemmas are introduced to expedite the following discussion.Lemma 3 (see [18]).For any constant matrix  ∈ R × ,  > 0, scalar  > 0, and vector function  : [−, 0] → R  such that the following integration is well defined, then Lemma 4 (see [8]).Let  1 , Based on Lemma 4, we present the following lemma, which can be regarded as an alternative form of the reciprocally convex combination.Lemma 5.For any constant matrix  ∈ R × ,  ≥ 0, and a variable  satisfying  ≤  ≤ , if there exists a matrix Ŵ where  1 () and  2 () are vector functions with compatible dimensions. Proof.Taking it can be easily verified that the above choice satisfies constraint (10) By Lemma 4, ( 13) and ( 14) yield that This completes the proof.
Lemma 6 (see [10]).Given any real matrices  1 ,  2 , and  3 with   1 =  1 and  3 > 0, then one has if and only if or equivalently Lemma 7 (see [10]).Given matrices Φ = Φ  < 0,  and  with appropriate dimensions for all () satisfying   ()() ≤  if and only if there exists scalar  > 0 such that Lemma 8 (see [19]).Let  be symmetric such that      > 0, and let Φ be nonsingular.Then,  +   Φ  is nonsingular and its inverse is expressed as where   and   are full rank with  =      and   and  are of full column rank and composed of bases of null space of  T and , respectively. is symmetric and

Main Results
In this section, the delay-dependent  ∞ control problem will be studied for the descriptor Markovian jump time-delay system (1) by virtue of the delay decomposition method.To this end, by dividing the delay interval (0, ℎ] into two disjoint subintervals (0, ℎ] and (ℎ, ℎ], (0 <  < 1), the following delay-dependent bounded real lemma (BRL) for the system (4) is firstly proposed, which will play an indispensable role in the derivation of our main results.
where  1 = 1/ℎ,  2 = 1/(ℎ − ℎ), and Proof.Assume that there exist matrices such that LMIs ( 22)-( 27) hold.With these conditions in hand, we firstly show the regularity and absence of impulse of the system (4).To this end, it is from [10] that there exist a couple of nonsingular matrices  and  such that for each Decomposing the associated matrices with a compatible dimension, we have where  1 ∈ R (−)×(−) and  2 ∈ R (−)×(−) are arbitrarily nonsingular matrices.
From the (1, 1) block in the left-hand matrices of ( 22) or ( 23), and observing that  1 ,  2 ,  1 , and  2 are positive definite, it can be easily seen that where (  ,  2 ) is the function of   and  2 and makes no difference to the subsequent proof.Pre-and postmultiplying (31) by   and , respectively, together with (29) and (30), we get According to [10], we can know that the pairs (,   ) are regular and impulse-free for each  ∈ ℓ.Therefore, by Definition 1, the system ( 4) is regular and impulse-free.
Case 2 (ℎ < ℎ() ≤ ℎ).The upper bound can be evaluated for the last five integral terms in L 2 (  , , ), which can be shown as Likewise, by applying Jensen inequality of Lemma 3 and reciprocally convex combination method of Lemma 5, the upper bound for the two definite integrals in L 3 (  , , ) can be shown as Combining ( 35)-( 37) and ( 43)-( 44), it yields that where and From the Schur complement lemma of Lemma 5, it can be easily verified that Ψ 2 < 0 holds if (23) holds for ℎ < ℎ() ≤ ℎ.
Based on the above analysis, we can know that there exists a scalar  > 0 such that for each  ∈ ℓ L (  , , ) ≤ −‖ ()‖ 2 . (47) Then, by Dynkin's formula, we can have for each  ∈ ℓ and  ≥ −ℎ which implies that Following a similar line in [12], we can find that the system (4) is stochastically stable under the condition that () = 0.By Definition 1, the system (4) is stochastically admissible.
Remark 10.By virtue of the novel Lyapunov-Krasovskii functional, Theorem 9 provides a delay-dependent and delayderivative-dependent  ∞ criterion to guarantee the system (4) to be stochastically admissible with  ∞ performance level.The novelty of Theorem 9 lies in the new Lyapunov-Krasovskii functional and the reciprocally convex combination method.By dividing the delay interval (0, ℎ] into two subintervals (0, ℎ] and (ℎ, ℎ] the following modedependent definite integrals and mode-dependent double integrals are introduced.Obviously, the above integral terms take the mode   into full consideration and contain the modeindependent one [12] as a special case.Since ℎ() is restricted to the subinterval (0, ℎ] or (ℎ, ℎ] and various Lyapunov-Krasovskii functionals are selected for different subsystems, our method can lead to reduction of conservatism.For the unforced system (4), if disturbance signal () ≡ 0, then we have the following time-delay systems: Based on Theorem 9, the following stochastic admissibility result can be given.
To show how to decide the parameter  optimally, we construct the following iterative algorithm for Theorem 9. Following a similar way, we can obtain the iterative algorithms for other theorems and corollaries.Algorithm 15.Select optimal parameter  for maximum delay ℎ.
From the above algorithm, the output  0 is the optimal  to ensure the delay upper bound ℎ to achieve its maximum value ℎ 0 .

Numerical Examples
In this section, we provide three examples to clarify the applicability and less conservatism of the results proposed in this paper.
The transition rate matrix is given as It is noted that if  = , descriptor Markovian jump time-delay systems (55) reduce to the standard state-space Markovian jump time-delay systems.For the time-invariant delay, the stochastic stability problem can be solved by Corollary 12. Table 1 shows the compared results on the maximum allowable upper bound for different  11 with the recent results [12,14].For ℎ  = 0.5 and ℎ  = 1.5, the maximum upper bound on allowable time delay is shown with given  11 in Tables 2 and 3, respectively.
As shown in the above tables, the methods in this paper have advantages over some previous ones in that the obtained maximum upper bound for allowable time delay is much larger.Therefore, the developed results have less conservatism than references [12,14].(79) The transition rate matrix is given as Compared with the existing results [17,20], we find that Theorem 14 in this paper reaches the much larger bounds than the aforementioned results.Therefore, we can conclude that Theorem 14 in this paper is less conservative than the results in references [17,20].The compared results can be seen in Table 4.
The transition matrix is given as When  = , we assume that  1 =  2 = 0,  1 =  2 = 0, and ℎ  = 0; the compared results for different  > 0 are given in Table 5. Obviously, the BRL obtained in this paper is less conservative than that in [12].
When we take  = [ 1 0 0 0 ], then   =   = [1 0] satisfies  =      .The coefficient matrices of the input vector can be given as Therefore, it is easy to see that our results are effective.

Conclusions
This paper has studied the  ∞ control problem for continuous-time descriptor Markovian jump systems with time-varying delay.By constructing novel Lyapunov-Krasovskii functionals, delay-dependent stability criteria and BRL have been developed under which the underlying systems are regular, impulse-free, and stochastically stable with a prescribed  ∞ performance level.Compared with other time-varying delay results based on Jensen inequality, the obtained results in this paper have advantages over some previous results, which is partially attributed to the reciprocally convex combination method.The illustrated examples show that the maximum allowable upper bound of time delay is much larger than some latest results.

Example 2 .
Consider the continuous-time descriptor Markovian jump time-delay systems (59) with two modes and the following parameters:

Table 1 :
The maximum allowable upper bound for different  11 and time-invariant delay.

Table 2 :
The maximum allowable upper bound for different  11 and ℎ  = 0.5.

Table 3 :
The maximum allowable upper bound for different  11 and ℎ  = 1.5.

Table 5 :
The maximum allowable upper bound for different  with ℎ  = 0.Example 1.Consider the nominal Markovian jump systems (55) with two modes and the following parameters: