Finite-Time Passivity and Passification Design for Markovian Jumping Systems with Mode-Dependent Time-Varying Delays

This paper investigates the finite-time passivity and passification design problem for a class of Markovian jumping systems with mode-dependent time-varying delays. By employing the Lyapunov-Krasovskii functional method, delay-dependent sufficient criteria are derived to ensure themean-square stochastically finite-time passivity. Based on the established results, mode-dependent passification controller is further designed in terms of linear matrix inequalities, such that the prescribed passive performance index of the resulting closed-loop system can be satisfied. Finally, two illustrative examples are given to show the effectiveness of the obtained theoretical results.


Introduction
During the past decade, Markovian jumping systems have received much attention since they can provide a unified framework in modeling practical systems in the real world including networked control systems, chaos systems, power systems, and biological systems (e.g., [1][2][3] and the references therein).It should be pointed out that time delays have inevitably emerged in Markovian jumping systems, which would affect the dynamical characteristics and system performances.In particular, substantial studies have found that time delays in Markovian jumping systems often vary according to different system modes.As a consequence, a large number of effective approaches for analysis and synthesis of Markovian jumping systems with mode-dependent timevarying delays have been reported in the literature [4][5][6][7].
On another research front line, as a particular form of dissipativity, passivity theory has played a significant role in dealing with many types of dynamical systems by relating the system inputs and outputs, especially for complex systems [8,9].Currently, many efforts have been devoted to the researches on control systems in the passivity framework due to its features, which have achieved satisfying results among different performance indices such as  ∞ index and  2 −  ∞ index.As a result, controllers can be designed to guarantee the passivity of the resulting closed-loop system, which gives rise to the so-called passification problems [10][11][12].It is noted that the relation between stability and passivity can also be revealed from the energy point of view, as well as the controller design problems.While considering Markovian jumping systems, mode-dependent controllers are often designed to reduce the conservatism due to practical conditions.Encouragingly, some burgeoning results in this aspect have been available so far [13][14][15][16].Nevertheless, it should be pointed out that the above results were all established in the context of an infinite-time interval.
Recently, finite-time control issues of dynamical systems have been extensively studied due to their practical backgrounds.Different from the asymptotic stability in the merely infinite-time interval, the finite-time stability theory can ensure the asymptotic behavior over a finite-time interval (often a short interval), which means that the system states should be below the prescribed upper bound during the given time-intervals.To mention a few, [17] addressed the finite-time  ∞ control problem of uncertain switched linear neutral systems with time-varying delays.Reference [18] concerned the finite-time stochastic synchronization of genetic regulatory networks.Reference [19] investigated the finite-time sampled-data control for switching T-S fuzzy systems.Reference [20] solved the finite-time filtering for switched linear systems with a mode-dependent average dwell time.These results show that, by utilizing the finite-time techniques, the transient performance of the dynamical systems can be well analyzed and is more practical for actual conditions.However, to the best of the authors' knowledge, the finite-time passivity and passification design problems for Markovian jumping systems with mode-dependent timevarying delays still have not been fully explored and remain challenging, let alone those under the finite-time theory framework.
Motivated by the aforementioned observations, in this paper, we aim at solving the finite-time control problem of Markovian jumping systems with mode-dependent timevarying delays and disturbances by a passivity-based scheme.Compared with the previous results, the main contributions of this paper can be summarized in two points: (1) this paper makes one of the first steps to deal with the passivity and passification for Markovian jumping systems with mode-dependent time-varying delays; (2) the corresponding mode-dependent controller is designed in the finite-time framework, which is more applicable and practical in the applications.By applying the finite-time theory and constructing appropriate mode-dependent Lyapunov-Krasovskii functionals, the resulting closed-loop system can be stochastically finite-time bounded with the prescribed passivity performance level.Furthermore, the mode-dependent controller is designed by using the matrix transformation approach.The rest of this paper is organized as follows.In Section 2, the Markovian jumping system model is introduced and some essential lemmas and definitions are provided for the formulated problem.Our main results on the finite-time passivity and passification design are given in Section 3. Section 4 presents two numerical examples to demonstrate the effectiveness of our proposed control method.The paper is concluded in Section 5.
Notation.The notations throughout this paper are quite standard.R  denotes  dimensional Euclidean space, and R × represent the set of all  ×  real matrices.The notation  > 0 means  is real symmetric and positive definite, and the superscript "" denotes matrix transposition. min () and  max () denote the minimum and the maximum eigenvalues of the corresponding matrix , respectively.sym{} denotes  +   .L 2 [0, ∞) denotes the space of square-integrable vector functions over [0, ∞).(Ω, F, P) is a 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{⋅} denotes the mathematics expectation of the stochastic process or vector.Moreover, in symmetric block matrices, * is used as an ellipsis for the terms that are introduced by symmetry and diag{⋅ ⋅ ⋅ } denotes a block-diagonal matrix.In the sequel, if not explicitly states, all matrices are assumed to have compatible dimensions.
In this paper, the following mode-dependent statefeedback controller is designed: where (()) is the controller gain matrix with appropriate dimensions to be determined later.
Then, the closed-loop system can be obtained by combining ( 1) and (3) that For notational simplicity, each possible () is denoted by the index ,  ∈ I, such that matrix (()) is denoted by   and so on.Consequently, (4) can be rewritten as follows: Before proceeding, some essential definitions are provided as follows.
Definition 1 (MSSFTS).For some given constants  1 > 0,  > 0, and  > 0 and symmetric matrix  > 0, system (5) with () ≡ 0 is said to be mean-square stochastically finite-time stable (MSSFTS) with respect to ( 1 ,  2 , , ), if there exist constants  2 >  1 , such that Definition 2 (MSSFTB).For some given constants  1 > 0,  > 0, and  > 0 and symmetric matrix  > 0, system ( 5) is said to be mean-square stochastically finite-time bounded (MSSFTB) with respect to ( 1 ,  2 , , , ), if there exist constants  2 >  1 , such that Definition 3 (MSSFTP).For some given constants  1 > 0,  > 0, and  > 0 and symmetric matrix  > 0, system ( 5) is said to be mean-square stochastically finite-time passive (MSSFTP) with respect to ( 1 ,  2 , , , ), if there exist constants  2 >  1 , such that system (5) is MSSFTB and there exists a positive constant , such that Remark 4. The MSSFTP index is introduced to deal with the external disturbances.It is worth mentioning that the concept of being Lyapunov mean-square stochastic stable and being MSSFTS is different, since a MSSFTS system may not be Lyapunov mean-square stochastically stable and vice versa.When there are no external disturbances, the MSSFTB system can be reduced to MSSFTS system.By introducing the MSSFTP performance index , the formulated passification problem can be solved in finite time.
The purpose of this paper is to design a mode-dependent passification controller for system (1) such that the closedloop system (5) is MSSFTP.To this end, the following lemma is introduced for the subsequent analysis.

Main Results
In this section, delay-dependent sufficient conditions are first established for the proposed finite-time control problem based on passivity theory.Then, the corresponding controller gain design procedure is provided in terms of LMIs.Theorem 6.For given positive constants  1 ,  and a symmetric matrix  > 0, system ( 5) is MSSFTB with respect to ( 1 ,  2 , , , ), if there exist a positive constant  2 and modedependent symmetric matrices   > 0, symmetric matrices  > 0 and  > 0, such that Π  < 0 holds for each mode  ∈ I and , and  4 ≜  min (P  ).
Proof.Choose the Lyapunov-Krasovskii function candidate as follows: where Then, define the infinitesimal operator L of (, ) as follows: For each mode  ∈ I, the derivative of (11) along the solution of system (5) can be obtained as The following function is defined: Consequently, it can be deduced that where Pre-and postmultiplying (19) by Moreover, by integrating (20) between 0 and , this yields which means that Denoting , and R =  −1/2  −1/2 , it can be obtained that  Then, by using Dynkin's formula [22], it can be derived that which means that system ( 5) is MSSFTB with respect to ( 1 ,  2 , , , ) in the sense of Definition 2 and this completes the proof.
Remark 7. It should be noted that the positive matrix  is used to address the state elements with different weights.Moreover, it can be observed that when () ≡ 0, the concept of MSSFTB system will lead to MSSFTS system.The differences between the Lyapunov stability and MSSFTS system can be further shown from the above proof procedure.
Based on the obtained results in Theorem 6, the following theorem is presented to deal with the MSSFTP problem.Theorem 8.For given positive constants  1 ,  and a symmetric matrix  > 0, system ( 5) is MSSFTP with respect to ( 1 ,  2 , , , ), if there exist a positive constant  2 and modedependent symmetric matrices   > 0, symmetric matrices  > 0 and  > 0, such that Π  < 0 holds for each mode  ∈ I and  1 ( 1 +  2 +  3 ) + (1 −  − ) <  2  4  − , where with ,  1 ,  2 ,  3 , and  4 defined in Theorem 6.By letting  =  − , the closed-loop system (4) can be MSSFTP with respect to ( 1 ,  2 , , , ) according to Definition 3, which completes the proof.Remark 9.It should be pointed out that the relation between MSSFTB system and MSSFTP index is revealed by the proof of Theorem 8. Compared with the normal stability problem, the solutions derived from the theoretical results in the form of strict LMIs are more complex with MSSFTP index.Moreover, the mode-dependent time-varying delays are introduced such that the mathematical model can be more applicable.The obtained results show that the passivity theory can be effectively utilized to deal with the disturbance.Moreover, the minimum value of parameter  can be obtained by the convex optimization procedure in MATLAB.
In the following, the mode-dependent controller gain design procedure will be presented with standard matrix manipulation based on the results of Theorem 8. Theorem 10.For given positive constants  1 ,  and a symmetric matrix  > 0, system ( 5) is MSSFTP with respect to ( 1 ,  2 , , , ), if there exist positive constants  2 ,  and symmetric matrices   ,  > 0, and  > 0 and matrices K  , such that Π < 0 holds for each mode  ∈ I and with ,  1 ,  2 ,  3 , and  4 defined in Theorem 6.Moreover, the desired mode-dependent passification controller gains can be obtained by   = K  P−1  .

Numerical Examples
In this section, two numerical examples are presented to verify our designed mode-dependent passification controller.
Example 1.Consider the Markovian jumping system in the form of (1) with two modes.For mode 1, the dynamics of the system are described as and the time-varying delay satisfies  1 () = 0.2 + 0.2 sin .
For mode 2, the dynamics of the system are described as and the time-varying delay satisfies  2 () = 0.3 + 0.1 cos .
With the above parameters, the feasible solutions can be found by solving the LMIs in Theorem 10 and the desired mode-dependent controller gains can be obtained as (39) In order to demonstrate the validity of the obtained results, the designed controller is applied to the above system.Given initial states as [0.2, −0.2]  , Figures 1 and 2 show the simulation results of the corresponding state trajectories of the open-loop and closed-loop system dynamics with transition probability modes, respectively.
Figures 3 and 4 illustrate the corresponding state finitetime evolution curves, respectively.Furthermore, Figure 5 depicts the effect on () from the disturbance () within [0, 3].It can be calculated that 2 ∫  Example 2. Another practical example of VTOL (vertical take-off and landing) helicopter is provided.The system dynamics can be modeled by [23,24] ẋ The scalars ( 1 ,  2 , , , ) are given as (50, 5000, 4, 5, ).The scalars  and  are chosen as  = 0.1 and  = 60, respectively.The MSSFTP performance index  can be calculated by 36.3918.The initial states are set as [4, 5, 2, 2]  .The disturbance input () is assumed to be 0.5 sin .Similarly, the feasible solutions can be found by solving the LMIs in Theorem 10 and the desired mode-dependent controller gains can be obtained as Moreover, it can be obtained that 2 ∫ (45) Correspondingly, it can be seen from Figures 6-9 that the designed finite-time passification controller can achieve the prescribed passivity performance of the resulting closed-loop VTOL helicopter system.

Conclusion
In this paper, the finite-time passivity and passification problem of Markovian jumping systems with mode-dependent  time-varying delays is considered.By applying the Lyapunov-Krasovskii functional method, delay-dependent passivity conditions are established to guarantee the MSSFTP index with respect to ( 1 ,  2 , , , ).For the finite-time passification problem, mode-dependent controller is designed to achieve the prescribed passivity performance of the resulting closed-loop Markovian jumping system.In the end, two numerical examples are presented to illustrate the effectiveness of our developed findings.Our interesting further research direction includes extending our current obtained results to the cases with deficient mode transition.

Figure 1 :
Figure 1: State response of the open-loop system.