Finite-Time H∞ Control for Time-Delay Markovian Jump Systems with Partially Unknown Transition Rate via General Controllers

This paper deals with the problems of finite-time boundedness (FTB) and H∞ FTB for time-delay Markovian jump systems with a partially unknown transition rate. First of all, sufficient conditions are provided, ensuring the FTB and H∞ FTB of systems given by linear matrix inequalities (LMIs). A new type of partially delay-dependent controller (PDDC) is designed so that the resulting closed-loop systems are finite-time bounded and satisfy a given H∞ disturbance attenuation level. The PDDC contains both non-time-delay and time-delay states, though not happening at the same time, which is related to the probability distribution of the Bernoulli variable. Furthermore, the PDDC is extended to two other cases; one does not contain the Bernoulli variable, and the other experiences a disordering phenomenon. Finally, three numerical examples are used to show the effectiveness of the proposed approaches.


Introduction
In actual industrial processes, the transient performance of systems is sometimes particularly important. For example, aircraft control systems require that the states not exceed a given limit [1]; the temperature of a chemical reaction needs to be strictly controlled within a certain range [2]; the angular location of a robot arm should be limited to a particular scope [3]. In recent years, an increasing number of academics have focused on the finite-time stability (FTS) problem. Different from the traditional Lyapunov stability [4][5][6][7], FTS discusses the transient performance of systems in the finite-time interval. In fact, the stable systems in the Lyapunov sense may have very bad transient performances, such as severe oscillation. The definition of FTS (or short-time stability [8]) was first proposed by Kamenkov in [9]. According to FTS, a system state is limited to a certain critical value within a certain time region, if the initial state is norm bounded. The authors of [10] extended FTS to the concept of FTB and took external disturbances into account. The studies of FTS and FTB have been further developed with the evolution of LMI theory [11][12][13][14][15][16][17][18][19]. For example, in [11], sufficient conditions for FTB of closed-loop systems were given in the form of LMIs by designing a dynamic feedback controller. Meanwhile, finite-time H ∞ control/filtering problems [20][21][22][23][24] have received much attention in order to reduce influences on a system caused by external disturbances.
On the other hand, abrupt changes are often encountered in the industrial process due to a component fault, invalidation, an associated change between subsystems, a sudden environmental disturbance [25], and so on. The occurrence of these situations causes the structure and parameters of a system to switch between various subsystems, such as

Problem Statement and Preliminaries
Consider a linear time-delay Itô stochastic Markovian switching system where x(t) ∈ R n is the system state, u(t) ∈ R m is the control input, and z(t) ∈ R q is the control output. S(σ t ), S τ (σ t ), L(σ t ), G(σ t ), U(σ t ), U τ (σ t ), J(σ t ), F(σ t ), H(σ t ), H τ (σ t ), and D(σ t ) are constant matrices, for simplicity. When σ t = i, they are denoted as S i , S τi , L i , The transition rate of the Markovian process {σ t , t ≥ 0} is given by where {σ t , t ≥ 0} takes the values in S = {1, 2, · · · , N}, o( t) is the order of t that satisfies t > 0, and lim is the transition rate of σ(t) from the mode i at the time t to the mode j at the time t + t, such that π ii = − ∑ j =i π ij . All of the transition rates π ij , i, j ∈ S, can be collected into the following transition rate matrix Assume that the transition rate is partially unknown, for example, there is a 2 × 2 transition rate matrix where "?" is an unknown element and π ij is known. For all π ij ∈ S, define S = L i k + L i uk , where If L i k is non-empty, it is described as follows where k i m ∈ S denotes the mth known element in the matrix Π's ith row.

Definition 1 (FTB). For the given scalars c
and (2) Remark 1. FTB can be simplified to FTS with respect to (c 1 , c 2 ,T, R i ) when v(t) = 0. The FTB/FTS can be used to solve some practical problems, such as the chemical reaction process, electronic circuit systems, and medicine. For example, the body's normal systolic blood pressure is 90-140 mmHg. If the body's systolic blood pressure is greater than 140 mmHg, then one suffers from high blood pressure disease. One must take blood pressure medicine.
Definition 2 (H ∞ FTB). For the given scalar γ > 0, system (1) with u(t) = 0 is H ∞ FTB with respect to (c 1 , c 2 ,T, R i , d, γ). If system (1) is FTB and under zero initial condition, for any non-zero disturbance v(t), the control output z(t) satisfies When the control problem is considered, the following definition is needed.
there exists a controller u(t) such that the resulting closed-loop system is H ∞ FTB.
Lemma 1 (Gronwall-Bellman inequality [52,53]). Let g(t) be a nonnegative continuous function. If there are positive constants r, q such that then Remark 2. Lemma 1 can be reformulated with sharp inequalities. The proof is given in Appendix A.
Lemma 2 (Schur's complement lemma [54]). For the real matrix H, the real symmetric matrix S, and the positive-definite matrix U, the below inequalities are equivalent:

Main Results
Firstly, we discuss the FTB problem for system (1) (when u(t) = 0) in this section. where

Proof.
For system (1), we choose a stochastic Lyapunov functional as For each σ t = i ∈ S, let L be the differential generating operator of system (1). According to the Itô formula, it follows that where From (8) and (13), it is easy to obtain Integrating both sides of (14) from 0 to t (t ∈ [0,T]) yields Taking the mathematical expectation on both sides of (15), the following is concluded Applying Lemma 1 or the Gronwall-Bellman-type inequality for the three functions [55] to (16) yields From conditions (17) to (20), it is derived ].
which is (11). The proof is complete.
then Theorem 1 is reduced to Theorem 1 in [29] .
In the following, we propose three novel types of partially delay-dependent controllers. One of the controllers is where K τ (σ t ) and K(σ t ) represent the control gains, and δ(t) is the Bernoulli variable defined as and satisfies Furthermore, Substituting (22) in (1), we have wherê The following theorem gives the sufficient condition of H ∞ FTB for the closed-loop system (23) via controller (22).
Moreover, the gains of controller (22) are Proof. Choosing the Lyapunov functional (12) for system (23), we obtain (24) and the following inequality The following result is obtained and according to Lemma 2, one obtains where , By pre-and post-multiplying (32) and its transpose, respectively, and comparing it with (29), it is seen that Then, one has Under zero initial condition, taking mathematical expectation, and integrating both sides of (33) from 0 to t (t ∈ [0,T]), by applying Lemma 1, it is deduced that It is also clear that (34) implies By (33), we obtain Because of R i > 0, it is easy to see that (9) is the actual condition (26). For (10), it is equivalent to P i < λ i2 I and P i < λ i2 I, that is, and According to Lemma 2, (26) is equivalent to (36), and (37) is acquired by (27) and (30). From Theorem 1, if Q i = γ 2 I, it is concluded that (14) and (35) are equivalent. The rest is similar to the proof of (16)- (21), which is obtained by conditions (9), (10), and (28). This completes the proof.
With the idea behind controller (22), another stabilizing controller without a Bernoulli variable is devised Using controller (38) in system (1), which includes the Bernoulli variable, one obtains which is rewritten as follows whereS The following theorem is developed, which is a sufficient condition of H ∞ FTB for the closed-loop system (39).
The gains of controller (38) are presented by Proof. Choosing the Lyapunov functional (12) for system (39), then LV(x t , σ t = i) satisfies The next steps are the same as those for the proof of Theorem 2. Pre-and post-multiply (40) by diag {X −1 i , X −1 i , I, I, · · · , I} and its transpose, respectively. Then, by Schur's com-plement and pre-and post-multiplying both sides by [x T (t) x T (t − τ) v T (t) z T (t)] and its transpose, respectively, and, by comparing it with (41), one obtains The following step is similar to Theorem 2 and is omitted here. The proof ends.
For system (1), another controller experiencing a disordering phenomenon is described as which implies if δ(t) = 0 or with disordering.
It is easy to see that (42) is the same as Controller (43) is applied to system (1), and let δ t = δ(t) − δ. Then, we have whereS Then, the following theorem is developed.

Numerical Examples
In this part, three examples are given to illustrate the effectiveness of the proposed results. Example 1. Consider system (1) with the following parameters: Mode1:

Mode2
:  Figures 1 and 2, respectively. From Figure 1, it is seen that the minimum value of c 2 is 32.3726 and the corresponding γ = 2.8132 when η = 0.05.
When η = 0.05, the gains of controller (22) are This indicates that under controller (22), According to the conditions mentioned above, Figure 3 shows the state response of system (23), where the small figures represent the curves of a possible Markovian mode evolution and the evolution of the Bernoulli variable δ(t) with δ = 0.6. The evolution of E[x T (t)Rx(t)] is shown in Figure 4, which implies that the closed-loop system (23) is H ∞ FTB.   In order to show the advantages of Theorem 2 and the influence of the probability δ, Figure 5 depicts the relationship between c 2 and δ. It is seen that c 2 takes the minimum value when δ = 0.78. This means controller (22) has less conservatism.  (39) and the evolution of E[x T (t)Rx(t)], respectively. From these figures, it is seen that the closed-loop system (39) is H ∞ FTB by the designed controller (38). This implies that Theorem 3 is valid.    Similar to Example 2, the state response of system (44) is shown in Figure 8, and the evolution of E[x T (t)Rx(t)] is drawn in Figure 9. It is concluded from these plots that the closed-loop system (44) is H ∞ FTB, by the designed controller (42). Therefore, Theorem 4 is valid.

Conclusions
In this paper, the FTB and H ∞ FTB problems of time-delay Markovian jump systems with a partially unknown transition rate have been studied. A sufficient condition of FTB for the given system is obtained by the LMIs technique and the Lyapunov functional method. A new controller that is partially time delay-dependent is designed. This controller has the advantages of strong generality and less conservative property. Based on PDDCs, two new kinds of controllers are derived; one does not contain the Bernoulli variable, and the other describes controllers experiencing a disordering phenomenon. Combined with LMIs, some sufficient conditions of H ∞ FTB for closed-loop systems are given via the designed controllers.
Three numerical examples illustrate that the proposed methods are effective. The results in this paper can be extended to the H ∞ filtering problem for Markovian jump systems with time-varying delays. In the future, the FTB and H ∞ FTB problems of fractional systems will be considered by means of the theories of fractional calculus and negative probabilities [56].