Finite-Time Asynchronous Stabilization for Nonlinear Hidden Markov Jump Systems with Parameter Varying in Continuous-Time Case

. The ﬁnite-time asynchronous stabilization problem has received great attention because of the wide application of actual engineering. In this paper, we consider the problem of ﬁnite-time asynchronous stabilization for nonlinear hidden Markov jump systems (HMJSs) with linear parameter varying. Compared with the existing research results on Markov jump systems, this paper considers the HMJSs which contain both the hidden state and the observed state in continuous-time case. Moreover, we consider the parameters of the systems are time varying. The aim of the paper is to design a proper observation-mode-based asynchronous controller such that the closed-loop HMJSs with linear parameter varying be stochastically ﬁnite-time bounded with H ∞ performance (SFTB-H ∞ ). Then, we give some suﬃcient conditions to solve the SFTB-H ∞ asynchronous controller by considering the stochastic Lyapunov–Krasovskii functional (SLKF) methods. Finally, a numerical example is used to show the validity of the main results.


Introduction
In recent years, hybrid systems have attracted the great attention and research due to their wide application in the industrial control field. Moreover, with the continuous upgrading and changes in the industrial environment, how to model and control hybrid systems in complex environments has become a major research hotspot in the control field. As a kind of special hybrid systems, Markov jump systems (MJSs) have received researchers' great attention and many constructive results have been made. It should be noted that the synchronous controllers are always considered in many existing results when it comes to the stabilization problem of Markov jump systems. at means the controller of the systems can always track the information of the model of the system in real time. However, such a situation is always impossible to achieve in a Markov jump system with a complex environment. erefore, the asynchronous characteristic between the system modal and the controller has attracted the attention of many researchers and they began to focus on the various characteristics of the HMJSs. e static output constrained control [1], the output feedback control [2], and the observer-based asynchronous fault detection [3] problems of the HMJSs with the discretetime state are studied by the authors, respectively. In [4], the resilient asynchronous H ∞ control problem of the discretetime HMJSs with singularly perturbed was considered. For the nonlinear HMJSs, the quantized control [5] and the finite-time L 2 -gain asynchronous control [6] problems by using T-S fuzzy model approach are studied, respectively. Moreover, many researchers also have carried out some work for the continuous-time HMJSs.
e asynchronous H 2 -controller design and the asynchronous filter design problems are studied for the continuous-time HMJSs in [7,8], respectively. For more particulars of HMJSs, the interested readers can read references [9,10].
It should be pointed out that many results of the MJSs assume that the parameter matrices are constant matrices or uncertain matrices that satisfy some given known conditions. In actual control systems, the system parameters may have the characteristics of convex polygonal linear parameters varying due to the sensor and actuator failures. For the control systems with such a special situation, it is necessary to design the corresponding nonlinear parameter timevarying controller to stabilization of the systems. us, many researchers proposed the linear parameter-varying control methods which laid a solid theoretical foundation for the design of nonlinear parameter time-varying controller. For a kind of parameter-varying system [11], an adaptive algorithm was proposed to achieve the active vibration control by using a new online secondary path estimation method. For bounded parameter variation linear parameter-varying systems, the LMI-based filter design problem is studied in [12]. e robust fault estimation [13] and set-membership fault estimation [14] problem is studied for parameter timevarying systems. For more particulars of parameter-varying systems, the interested readers can read references [15][16][17][18].
In the abovementioned references, the system analysis and/or some control method were concerned only over an infinite-time interval, which portrayed the asymptotic properties of the HMJSs and parameter-varying systems. However, the transient characteristics in a given finite-time interval is significant in many control systems [19][20][21] and should be considered simultaneously. For a class of nonlinear systems, the authors studied the finite-time adaptive fuzzy control problem in [22,23], respectively. In [24], a finite-time control approach was used to solve the problem of accurate trajectory tracking for disturbed surface vehicles. Moreover, many researchers also combine the finite-time control scheme with the networked switched systems [25], nonlinear systems [26,27], and quadrotors control [28] and have carried out lots of excellent works. e authors studied the design and implementation of bounded finite-time control algorithm problem for the speed regulation of permanent magnet synchronous motor in [29]. By using the state-dependent switching method, the adaptive fuzzy finitetime control of switched nonlinear systems is studied in [30]. e problem of fuzzy finite-time control for switched systems via adding a barrier power integrator is considered in [31]. For more particulars of HMJSs, the interested readers can read references [32][33][34].
However, few results have been reported regarding research on finite-time asynchronous control of HMJSs based on the parameter varying, which is the motivation of this work. Different from some existing results on the finite-time control problem [35,36], the problem of SFTB-H ∞ asynchronous control is studied for continuous-time HMJSs via parameter varying in this paper. Compared with the existing results of asynchronous control for discrete-time HMJSs [37][38][39][40], this paper mainly consists of the following threefold contributions: (1) e asynchronous characteristic between the controller modes and the system modes is characterized by the hidden Markov dynamics. Moreover, we firstly consider the asynchronous stabilization problem for the continuous-time HMJSs with parameter varying models. (2) Some sufficient conditions will be given to solve the SFTB-H ∞ observation-mode-based asynchronous controller by considering the SLKF methods and introducing some auxiliary variables. (3) Considering the parameter time-varying, the methods of gridding technique and approximate basis function will be used in order to change the infinite LMIs into finite LMIs which can be solved by MATLAB LMI toolbox to get the finite-time asynchronous controller gain directly. e organization of this paper is made of five parts. In Section 1, the background of the HMJSs, the parameter timevarying systems, finite-time control scheme, and the notation meaning of this paper are introduced and given. Section 2 introduces the system description of the parameter varying hidden Markov jump systems (PV-HMJSs) and designs a suitable asynchronous controller for the studied systems. Moreover, the main definitions and lemmas are also given in this part. Section 3 gives some sufficient conditions to solve the SFTB-H ∞ observation-mode-based asynchronous controller. In Section 4, the simulation experiment of HMJSs via parameter varying with two subsystems is carried out, only to find that the closed-loop HMJSs via parameter varying fulfill the condition of SFTB-H ∞ under the action of the designed asynchronous controller. e conclusion and future research work follow in Section 5.
Notation: the notation meaning throughout this paper is shown in Table 1. Furthermore, we assume that the matrices and notations in this paper are standard with compossible dimensions.
is the transition rate matrix and satisfies π IJ ≥ 0 for ∀I ≠ J, ℵ I � [q I KL ] is the Π-dependent conditional probability matrix and satisfies Remark 1. From relation (3), we can summarize that the same state will be visited of the hidden state h 1 (t) and the observation state In this case, the full information of the hidden state h 1 (t) will be provided for the detector. If α K JL � 1 and α L JL � 0 with K ≠ L, only one jump will occur between the hidden state h 1 (t) and the observation state h 2 (t). Moreover, the observer will not provide any information if M � 1 { }, q I KL � 0, and α 1 I1 � 1 and the observation state is h 2 (t) � 1. For more specific details of the relationship between the hidden state h 1 (t) and the observation state h 2 (t), we can refer to [3,6,35,36].
In actual engineering applications, we often encounter situations where the system mode information accessible by the controller/observer is usually inaccurate. In other words, the actual model of the system cannot be observed by the controller, which makes the information between the controller mode and the system mode asynchronous. erefore, a new controller mode h 2 (t) related to the system mode h 1 (t) needs to be introduced. e PV-HMJSs (1) can be rewritten as the following PV-HMJSs: For any n � 1, 2, . . . , m, the time-varying parameter matrix p(t) and its variation rate _ p(t) are both assumed bounded, i.e., p n (t) ∈ p 1 p 1 and _ p n (t) ∈ p 2 p 2 . Moreover, the time-varying parameter matrix p(t) is also measurable and affine parameter dependent in real time.
at means the following equations hold: e probability measure which defined on Δ Complexity 3 where A I , B I , W 1I , F I , C I , D I , and W 2I with I � 0, 1, . . . , m are known matrices. We suppose the system states of the PV-HMJSs (9) be available, we design the following h 2 (t)-dependent asynchronous controller: where K is the h 2 -dependent controller gain which will be solved in eorem 3.

Main Definitions and Lemmas
Definition 1 (see [32]). Given two positive constants 0 < J 1 < J 2 , a weighting matrix R I K > 0, and a finite-time interval 0 T , the closed-loop PV-HMJSs (8) with Definition 2 (see [32]). Given two positive constants 0 < J 1 < J 2 , a weighting matrix R IK > 0, and a finite-time interval 0 T , the closed-loop PV-HMJSs (12) is sto- Definition 3 (see [6]). Under zero initial condition, the h 2 (t)-dependent controller (9) is said to be a SFTB-H ∞ controller of the PV-HMJSs (7) if there exists a h 2 (t)-dependent controller gain K p K , K ∈ M such that the closedloop PV-HMJSs (10) be SFTB and satisfies the following H ∞ -gain performance index: Lemma 1 (see [22]). We have 2E T XYF ⩽ βE T X T XE+ β − 1 F T Y T YF for any given proper dimensional matrices E and F, positive scalar β > 0, and matrices X and Y.
Assumption 1. For any given positive scalars T and ϖ, the disturbance input

E IV(x(t), H(t), t) { } < ωV(x(t), H(t), t) + φω T (t)ω(t).
(23) en, we have the following relation by integrating both left and right sides of inequality (23) from 0 to t for ∀t ∈ 0 T : us, e above inequality can be rewritten as (26) by considering the Gronwall inequality: us, the E x(t) T R IK x(t) < J 2 for ∀t ∈ 0 T holds by inequality (12). is completes the proof.
In eorem 1, we give some sufficient conditions to ensure the SFTB of the closed-loop PV-HMJSs (12). en, the SFTB-H ∞ condition will be given in the following eorem 2 on the basis of eorem 1.

(29)
Inequality (28) guarantees that inequality (29) is established. en, we use e − ωt to multiply inequality (29) and integrate such inequality from 0 to T under zero initial condition, and it yields We can rewrite inequality (30) as the following inequality by considering E V(x(t), H(t), t) { } > 0: us, we can obtain Recalling Definition 3, we can get the SFTB-H ∞ condition (10) of the closed-loop PV-HMJSs (8). is completes the proof.
rough the analysis of eorem 2, we know that the state feedback controller gain matrix K p(t) h 2 (t) cannot be solved by Matlab LMI tools due to the nonlinear terms in inequality (28). In the following eorem 3, some sufficient conditions will be given to obtain finite-time asynchronous controller gain K p(t) h 2 (t) . □ Theorem 3. Given four positive constants ω > 0, J 1 > 0, T > 0, and ϖ > 0 with t 0 ω T (τ)ω(τ)dτ ≤ ϖ, there exist a finitetime H ∞ -gain asynchronous controller with h 2 -dependent state feedback gain K such that the SFTB-H ∞ condition (10) of the closed-loop PV-HMJSs (12) will be sat- where 6 Complexity Proof. We substitute A p IK and C p IK into inequality (34) and yields where We use diag P − 1 IK , I, I I to pre-and postmultiply inequality (37), and let X IK � P − 1 IK , Y p IK � K p K X IK , and Q IK � X IK X IK , and we use the Schur complement lemma. It can be seen that the following inequality satisfies with positive scalar ξ > 0: where where p(t) is the time-varying parameter. From inequalities (45) and (46), we know that Z denotes an LMI and A p has infinite number of LMIs. en, we select the following basis function F (p(t)) I n f J�1 to solve inequality (45): Furthermore, the infinite LMIs of the A p can be transformed into the finite ones if we divide the space of parameter changes into finite-dimensional grids, which means the following relation satisfied for each set of parameters on the grid for ∀p(t) ∈ Z: □ Remark 3. In eorem 3, the difficulty of calculation will be increased because of the asynchronous characteristic between the controller modes and the system modes. In order to solve such difficulty, the controller modes h 2 (t) is converted h 1 (t). Moreover, some sufficient conditions are obtained to make the closed-loop HMJSs with linear parameter varying be SFTB-H ∞ by introduced auxiliary variables.

Remark 4.
In addition, we have introduced the stochastic Lyapunov-Krasovskii functional methods in eorem 3, which will induce somewhat conservatism of the main results. In future work, we can reduce the impact of conservativeness through replacing the quasi-one-sided Lipschitz condition or one-sided Lipschitz condition with local Lipschitz condition in Assumption 2.

Numerical Example
In this section, we consider a class of PV-HMJSs with two subsystems.
where sin(t) and cos(t) are time-varying parameters and bounded with − 1 1 . We assume the weighted matrix R � I, the initial consideration J 1 � 0.8, the external disturbance ω(t) � 0.1sin 2 (2t) 0.2sin 2 (t) , and the unknown state- Meanwhile, the other relevant solutions are given as J 2 � 5.0894 and φ � 9.8. en, the simulation results are shown in Figures 1-3. e jump modes of h 1 (t) and h 2 (t) are shown in Figure 1. e state trajectory x(t) and y(t) of the closedloop PV-HMJSs (8) are shown in Figures 2 and 3. From Figures 2 and 3, we can seen the SFTB-H ∞ condition is satisfied with E x(t) T R IK x(t) < J 2 in which J 2 � 5.0894 and φ � 9.8.
Remark 5. Similar to some results of reinforcement learning methods [41][42][43][44], this paper also considered the problem of stabilization of stochastic Markov jump systems. Moreover, we also use the hidden Markov model to denote the asynchronous characteristic between system modes and controller modes. Different from some existing results on the finite-time control problem [35,36], the problem of SFTB-H ∞ asynchronous control is studied for continuous-time HMJSs via parameter varying in this paper. Compared with the existing results of asynchronous control for discrete-time HMJSs [37][38][39][40], this paper firstly considers the asynchronous stabilization problem for the continuous-time HMJSs with parameter varying models.
For the future work, we can learn from the reinforcement learning methods to study the problem of finite-time asynchronous online control for the continuous-time PV-HMJSs.

Conclusion
is paper studied the SFTB-H ∞ asynchronous control problem for continuous-time PV-HMJSs. Some sufficient conditions are given to solve the SFTB-H ∞ asynchronous control gain by considering the methods of SLKF and LMIs. e designed SFTB-H ∞ asynchronous controller makes the closed-loop PV-HMJSs satisfy the SFTB-H ∞ condition. Finally, we use a numerical example to show the validity of the main results of this paper. For the future research work, the asynchronous control problem for fuzzy PV-HMJSs apply approximation method will be considered.

Data Availability
e data findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.