DHMM-Based Asynchronous Finite-Time Sliding Mode Control for Markovian Jumping Lur’e Systems

—This paper is concerned with the asynchronous problem for a class of Markovian jumping Lur’s system (MJLSs) via sliding mode control (SMC) in continuous-time domain. Speciﬁcally, the discrete hidden Markovian model (DHMM) is employed to describe the nonsynchronization between the controller modes and the MJLSs modes. In particular, considering the nonlinearity of MJLSs, a novel Lur’e-integral-type sliding surface is constructed. In order to ensure the ﬁnite-time stability of sliding mode dynamics and the accessibility of the speciﬁed sliding surface, the asynchronous Lur’e-type SMC law of the detector mode is presented. Finally, an example of DC motor is provided to demonstrate the effectiveness of the proposed technique.


I. INTRODUCTION
N ONLINEAR systems have received extensive attention and a series of gratifying results are available in the past decades, see, e.g., [2], [3], [4], [5] and many references therein. As an important kind of nonlinear systems, Lur'e system is composed of a linear system and an unknown static nonlinear system with certain constraints through negative feedback [6]. For more topics of Lur'e systems one can refer to [7], [8]. Additionally, many nonlinear dynamic systems in engineering and economic fields, such as manufacturing process, network communication system, power circuit system and economic system, are susceptible to random variations and abrupt changes caused by external environment, internal structure or human intervention in the operation process. Thus, Markovian jumping systems (MJSs) are developed to model these more general nonlinear systems and notable results have been produced [9], [10], [11]. Recently, the stability analyses and control problems of MJSs have received great interests among researchers [12], [13], [14]. However, there are few researches on the Markovian jumping Lur'e systems (MJLSs), which makes this paper have certain research significance.
It should be noted that in the ideal MJLSs, each state of the Markov chain corresponds to an observable physical quantity. But the practical situation is complex, so all the R. Nie, W. Du, Z. Li  real states observation is notoriously difficult to conduct. In this case, the hidden Markov model (HMM) is introduced to describe such phenomenon. As a statistical analysis model, HMM has became a hot topic over the last few years, and its application including but not limited to speech processing [15], target tracking [16], digital communication [17], biomedical engineering [18] and finance [19]. It's worth mentioning that the aforementioned HMM includes two random processes, i.e., the random process caused by the Markov chain, and the other process brought from the observed variables. According to the observed values, the HMM can be divided into two types. When the observed value is discrete, it is called the discrete HMM (DHMM). Otherwise, when the observed value is continuous and can be described as a function, such HMM is called continuous HMM (CHMM). However, to the best of our knowledge, most of the existing research are mainly focused on interpreting DHMM in discrete time domain, see [20], [21], [22], and rarely studied in continuous time domain. Therefore, this paper addresses a longstanding open issue on DHMM in continuous time domain.
Based on DHMM, the asynchronous control scheme [23], [24] offers an effective alternative for dynamic system fully utilized the potential information in the presence of time delays and data loss. In [25], when the switching between the candidate controller and the system mode is asynchronous, the stability of a class of linear systems with mean residence time is studied by using asynchronous switching control. In [26], an asynchronous elastic controller is designed to ensure the stability of the nonlinear switched system subject to time delay and uncertainty for the asynchronization problem. In [27], considering the asynchronous problem between the controller and the system modes, the stability of uncertain time-delay switched nonlinear systems is studied by using asynchronous switching control. However, for nonlinear systems with external disturbance, the control objective is not only to guarantee system stability but also to accelerate the convergence and improve the robustness. Sliding mode control (SMC) has been widely applied in practical engineering in terms of its fast response speed and strong robustness to uncertainties and disturbances. In [28], the ∞ control problem in commodity pricing has been solved by SMC method. Reference [29] develops an adaptive SMC scheme integrating fuzzy control with SMC control for interval type-2 fuzzy systems in the presence of uncertainty parameters, and the issue of SMC for fuzzy singularly perturbed systems with application to electric circuit is investigated in [30]. Furthermore, the extended application to missile system, communication network system and robot control system, has enabled finite-time boundedness to become considerably embraced [31], [32], [33]. (2) An asynchronous Lur'e-type SMC law is designed to drive state trajectories onto the specified sliding surface during the prescribed finite-time interval.
(3) Nonlinear terms are added to the integral-type sliding surface in order to reduce the conservatism. (4) Compared with [34], we avoid the coupling problem caused by mode in the derivation process by selecting the appropriate Lyapunov function in the proof of reachability.
The rest of this paper is organized as follows. The MJLSs with stochastic perturbation is decsribed in Section II followed by the main results in Section III. The effectiveness of the proposed method is illustrated in Section IV. Finally, Section V draws a remarking conclusion. . The transition probability from mode at time to mode at time + can be described by:
Some assumptions and the corresponding definitions as well as lemmas are given to facilitate the stability analysis of MJLSs (2).

A. Asynchronous Sliding Surface and SMC Law Design
Due to packet loss, transmission delay or random disturbance involved in practice along with the inaccurate model information of the system, there is a need of the asynchronization between the controller modes and the system modes. In this section, we introduce the DHMM ( ,ˆ) to describe the asynchronous phenomenon described above. More precisely, a stochastic variableˆcan be utilized to estimate with a known Π-dependent conditional probability matrix Φ △ = [ ], in which the probability is defined as: where 0 ≤ ≤ 1 and ∈ ℳ, ∑ =1 = 1 for all ∈ . Remark 1: It should be mentioned thatˆis introduced as a detector to represent the controller mode, and take values in a finite set ℳ = {1, 2, · · · , }. Notice thatˆdepends on according to the given conditional probability. By reserving the index for the modes of the Markov chain and for the detectorˆ, we design an Lur'e-integral-type sliding surface function as: in which ∈ ℜ × is a real matrix such that is nonsingular.
Then, the asynchronous Lur'e-type SMC law can be achieved by: where is the controller gain and is the nonlinear feedback gain.

B. FTS Analysis of Sliding Mode Dynamics
By applying the partitioning strategy, the FTS problem of MJLSs (2) can be addressed in two phases via asynchronous SMC, i.e., the reaching phase within [0 1 ] and the sliding motion phase within [ 1 ].
According to Newton-Leibniz Formula, we have Under the condition of (16), we can obtain the following inequality by substituting (18) into (17) Under Assumption 1, for any given Ω, there exists Δ such that Recalling to (19) and (20), we know that Next, we define an auxiliary variable as: Further defining the auxiliary variable as follows: From (10), one can infer Φ > 0 and the following inequality holds: From (22) and (24), it is easily checked that Next, we perform a series of equivalent transformations on (25). By multiplying the left and right sides of the (25) with − , and then integrate the result from 0 to with ∈ [0 1 ] on both sides, we obtain More specifically, it follows: Based on (13) and (14), it yields: Hence, it is easily to find that T ( ) ( ) < * for all ∈ [0 1 ] under condition (11). Thus, we complete the proof of FTB for MJLSs (2) within [0 1 ].
Next, we give the FTS proof for MJLSs (2) during the finitetime interval [ 1 ], i.e., the sliding motion phase. According to ( ) = 0, the equivalent control law can be written as: By substituting the equivalent control law (30) into the MJLSs (2), the closed-loop MJLSs become in which¯= + , = − ( ) −1 . Theorem 2: For a given finite-time interval [ 1 ], MJLSs (2) is FTB with the equivalent control law (33), i.e., the closedloop MJLSs (34) is FTB respect to ( * , 2 , , ), for any ∈ and ∈ ℳ, if there exists a scalar > 0, such that for all positive-definite symmetric matrix and positive-definite symmetric matrix 3 satisfies Proof : Consider the following LyapunovKrasovskii functional candidate for MJLSs (31): Along the state trajectories of MJLSs (31), we can get: According to the Assumption 1, we have in which Define an auxiliary function as: Then, under condition (32), the following inequality can be guaranteed from (38): Repeating the equivalent transformation of the (25) to the (39): multiplying both sides of (39) with − , and then integrate the result from 1 to with ∈ [ 1 ] on both sides. We can get: From Theorem 2, we immediately have On the other hand, it follows: Based on (41) and (42), it yields: Accordingly, when conditions (33) and (34) (2) is FTB with respect to ( 1 , * , 2 , , ), if there exists scalars > 0, 1 > 0, 2 > 0 such that the following inequalities hold for all ∈ and : < 0 (46) . Thus, the asynchronous controller gains can be obtained by = −1 and = −1 ( − ). Proof: Generally, the matrix inequalities in Theorem 2 and Theorem 3 are difficult to solve directly. Based on certain equivalent transformation and linear matrix inequalities (LMIs) techniques, we give a sufficient condition.

C. Analysis of Reachability
The control objective in this paper is to design a asynchronous Lur'e-type SMC law (8) that ensures system trajectories reach the predetermined sliding surface within a finite time interval.
Theorem 4: Consider MJLSs (2), the reachability of the specified sliding surface (7) can be guaranteed in a finite time 1 with 0 < 1 < by the designed asynchronous Lur'e-type SMC law (8). The robust term ( )) is chosen as in which , and the adjust scalar is determined by: Proof : Consider the following LyapunovKrasovskii functional as: Then, it follows from (7) that whereˆ= As shown in (56), there exists a positive scalar 1 : such that 3 ( ( )) = 0 when ⩾ 1 , and we have ( ) = 0. By (55), it follows that Substituting (58) into (57), it yields: Under condition of (54), the following inequality can be derived by (59) Thus, we can also find a time 1 within [0 ] to guarantee the accessibility of the sliding mode surface. This completes the proof.

IV. ILLUSTRATIVE EXAMPLE
In this section, a DC motor circuit [37] is used to evaluated the effectiveness of the proposed methodology, which is shown in Fig. 2. The dynamics is described by: The physical meaning of each symbol is listed in Table I.
Then the dynamic nonlinear model can be described in the form of The parameters are given in Table II.

V. CONCLUSION
This paper investigates the asynchronous SMC problem of a class of MJLSs. Firstly, the asynchronous problem of system modes and controller modes is modeled by HMM. Then, asynchronous sliding mode controller with Lur'e nonlinear information is designed to guarantee the reachability of sliding surface. Finially, according to the partitioning strategy, sufficient conditions are given to ensure the finite-time stabilization of the MJLSs. Future work will be directed at integrating the CHMM theory with SMC and fuzzy control.
Funding The work is supported by the National Natural Science Foundation of China (Basic Science Center Program: 61988101), National Natural Science Foundation of China (61725301), International (Regional) Cooperation and Exchange Project(61720106008).
Data availability statement The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflict of interest The authors declare that they have no conflicts of interest.