Abstract
This work investigates the asynchronous control for fuzzy Markov switching systems (MSSs) with randomly occurring fading channel. By resorting to a T-S fuzzy model, the nonlinear MSSs can be handled. Meanwhile, in the unreliable network, the Rice fading model is proposed to capture the randomly occurring channel fading, which covers packet dropouts and network-induced delays as special cases. In light of the hidden Markov model, the asynchronous phenomenon of controller is taken into consideration, and asynchronous fuzzy controller is obtained. In the end, a numerical example and a single-link robotic arm model are applied to demonstrate the validity of the derived results.
1. Introduction
In physical applications, such as power station monitoring systems, fire-fighting operating systems, and electric networks, the signals are transmitted through a shared wireless/wired communication channel. The stability of the systems is weakened by uncontrollable accidents on some occasions and external complex environment, and it is of importance to make the systems have certain robustness. However, because of the limited resources, the communication quality and computing load are affected, and the unexpected phenomenon occurs, such as networked-induced delay, quantization, and fading channel (FC). Normally, when signals are transmitted by wireless communication link, the phenomena of reflection/diffraction can be encountered, which results in multichannel-based fading. Recently, many scholars have devoted their attention to hybrid systems with FC [1–3]. It is remarkable that, compared with packet dropouts, FC is more general, which contains the finite coefficients. Lately, FC is widely applied in time-dependent probabilistic process, and many valuable results are reported in robust control [4], sliding mode control [5], and filtering [6, 7].
As a special type of switching systems, Markov switching systems (MSSs) consist of finite subsystems, and different subsystems are activated by a random Markov process [8–10]. Note that MSSs have been successfully applied in practical systems with probabilistic parameter changes, for instance, component failures. Added by its powerful application, many fruitful results have been gained in the issues of admissibility and stabilization, robust control, and filtering [11–14]. As one can be seen from [11–14] that, the control designed scheme has major concerns on synchronous situation. Owing to the networked mechanism and time delay, it is unreasonable to consider the synchronous scenario. By resorting to the hidden Markov model (HMM) [15–17], the asynchronous controller/filter can be solved in an application perspective.
On the contrary, it can be observed that aforementioned literatures are constrained to linear systems, which limited the practical application [18–23]. In general, MSSs tend to nonlinearities for external nonlinear environments and other factors, and it is urgent to take nonlinear MSSs into consideration [24–27]. Following this trend, many efforts have been devoted to nonlinear systems. Among them, Takagi–Sugeno (T-S) fuzzy model has been proved to be an efficient tool in tacking with nonlinear systems, in which by resorting to fuzzy inference strategy, the nonlinear systems are separated into finite local linear parts [28–30]. Recently, many meaningful results of T-S fuzzy MSSs (FMSSs) have been expressed in [20–22]. However, the FC issue has not been extended to FMSSs, not to mention asynchronous scenario.
Inspired by the above observation, we focus on the asynchronous control for FMSSs with randomly occurring FC. The major contributions are summarized as follows: (1) the FC model with disturbance is absorbed, which covers packet dropouts and network-induced delays as special cases; (2) added by parallel distributed compensation strategy and HMM scheme, the asynchronous control law is developed, which relaxes the limitation in existing results; and (3) by means of Lyapunov functional and stochastic analysis, sufficient criteria are gained and SLRAM is applied to illustrate the effectiveness of the proposed asynchronous control law.
The rest of this study is organized as below. The system descriptions are given in Section 2, and main results including stochastic stability of closed-loop system and controller gains are provided in Section 3. Computational experiments are expressed in Section 4. In the end, conclusions are shown in Section 5.
Notations: the notations utilized in this work are same as that in [10]. Furthermore, and means the block-diagonal matrix.
2. Problem Formulations
Consider the T-S FMSS (1) as follows:
Plant rule : IF is is , THENwhere is the state vector, is the control input, is the output vector, and is the external disturbance. Stochastic variable (SV) is a discrete Markov chain (DMC) and . For any , , , , , and indicate the predetermined matrices. are the fuzzy sets, and means the premise variable. Here, and implies the IF-THEN rules number.
In (1), for any , the transition probability matrix (TPM) of original state is inferred by :where , , and , . For , one has , where .
By means of the T-S fuzzy model, FMSS (1) can be deduced as follows:where symbolizes the grade of membership of in , , and ; here, , and . For simplification, denotingapparently, the FMSS (3) can be rewritten as
It is remarkable that the signals are sending out via unreliable network; some unpredictable factors occur, such as channel fading (CF) and packet dropout. To model the randomly occurring CF, th-order CF model is proposed:where is the signal arrived at controller and are mutually independent SVs with subject to variances . represents the external disturbance and is the known matrix.
Remark 1. In the unreliable network, aiming to depict the unexpected networked phenomenon including scattering and other factors, fading channel is proposed, which is characterized by the Rice fading model. The stochastic variables can describe the variety of fading channel, which are applied and make the signal transmission more general.
In this work, by considering the unreliable of network medium, one establishes the following asynchronous control law (ACL):
Controller rule : IF is , THENwhere indicates the unsolved controller gains. The SV is another DMC. DMC represents HMM and obeys the conditional probability matrix CPM :where and . Accordingly, the ACL (7) is inferred aswhere .
Remark 2. Note that asynchronous phenomenon comes from the network-induced delay, quantization, and other factors, which results in unreasonable of traditional synchronous control law [17, 31]. To tackle with such shortage, HMM is employed in designing controller, which is elaborated by .
Substituting (6) and (9) into (5), the closed-loop FMSS can be acquired:where .
Definition 1. (see [12]). Under the initial conditions , , the FMSS (10) is called stochastically stable (SS), such that
Definition 2. (see [12]). The FMSS (10) is SS with performance such that
3. Main Results
Theorem 1. If there exists a scalar , matrices , , and , the FMSS (10) is SS with performance , for any , , such thatwhere
Proof. It follows from (14) and (15) thatwhereConstruct the Lyapunov functional for FMSS (10):whereCalculating the difference of and taking expectation, one hasAlong system (10), it yields thatAdded by the variances in fading channels , (22) can be reformulated aswhereOn the contrary, for term , one hasFor simplification, denoting . Firstly, we will prove the SS of system (10) with . Combining with (23) and (25), one obtainsApplying Schur complement to (26), it is clear thatwhereIn light of (27), it achieves thatwhere , . By (17), it is clear that , which indicatesFor , it yieldsBy Definition 1, it is obviously that FMSS (10) is SS with .
Next, the FMSS (10) is SS with the performance index will be verified. For under zero-initial condition, denoting one hasDeploy Schur complement to (32), which results inwhere .
In light of (17) and (33), it can be concluded thatwhich indicates . By Definition 2, the FMSS (10) is SS with performance .
Theorem 2. If there exists a scalar , matrices , , , and matrices , , , the FMSS (10) is SS with performance , for any , , such thatwhereThe controller parameters can be obtained by
Proof. DefineApplying a congruence transformation to (14), it yields thatwhereFrom , it follows thatBy resorting to (35), (39), and (43), (35) can be guaranteed if (14) holds. By the same way, (37) and (39) can be guaranteed if (15) and (17) hold.
4. Computational Experiments
4.1. Numerical Example
Consider the FMSS (1) with 3 modes, its parameters are chosen as follows:
The TPM and CPM are inferred as
One assumes that the th-order of CF model (6) with and , , , and . , . By solving the inequalities of Theorem 2, a set of controller gains are acquired:
By applying the aforementioned controller gains, Figures 1 and 2 demonstrate the state mode and controller mode, respectively. The state trajectories with CF are shown in Figure 3, and the control input is depicted in Figure 4. Defining , Figure 5 plots the evolution of .
4.2. Practical Example
Robot arms are essential parts in fire-fighting operating systems. In this case, we consider a single-link robotic arm model (SLRAM) stemmed from [12], which is inferred bywhere the meaning of , , , , and are explained in [12] and , , , , , , and .
According to [12], for sampling period , the SLRAM is reestablished as the T-S FMSSs.
Plant rule 1: IF is about 0 rad, THEN
Plant rule 2: IF is about rad, THENwhere
The TPM and CPM are expressed as follows:
Assuming that and , , , and . Meanwhile, and are chosen as Example 4.1. Solving the inequalities of Theorem 2, the controller gains are achieved:
For initial condition , by resorting to the obtained filter gains, Figures 6 and 7 demonstrate the state mode and controller mode, respectively. The state trajectories with CF are shown in Figure 8, and control input is depicted in Figure 9. In the end, Figure 10 plots the evolution of . It can be observed from Figures 8–10 that the proposed controller is effective.
5. Conclusions
In this work, the asynchronous control for FMSSs with randomly occurring FC is designed. By resorting to the T-S fuzzy model, the nonlinear MSSs can be handled. Furthermore, the Rice fading model is proposed to capture the randomly occurring channel fading, which covers packet dropouts and network-induced delays as special cases. In light of HMM, the asynchronous phenomenon of controller is taken into consideration, and asynchronous fuzzy controller is obtained. In the end, two examples are applied to demonstrate the validity of the derived results. It can be observed that, compared with the existing approach, the obtained methodology is more general, and less conservatism results are expressed. In the networked scenario, sliding mode control for FMSSs with FC is significant in future works [32].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Science and Technology Project of State Grid Jiangxi Electric Power Co. Ltd. (521820200036).