Abstract
This work is concerned with the control for Markov switching singularly perturbed systems with the stochastic communication protocol. To coordinate the data transmission and save the bandwidth usage, the stochastic communication protocol with a compensator is applied to schedule the information exchange. The goal of this work is to design a joint-Markov-process-based controller such that the resulting system is stochastically stable with prescribed performance. Based on the Lyapunov functional technique, a sufficient condition is derived to ensure the existence of the achieved controller. Finally, the effectiveness and correctness of the developed results are verified by the simulation example.
1. Introduction
As a significant component of hybrid systems, Markov switching systems (MSSs) have gained extensive interest due to their capability in modeling subsystems [1–4]. Note that MSSs consist of a finite number of subsystems, and some abrupt variations can be depicted by a Markov process, which is recognized as a key feature of MSSs. Nowadays, owing to their potential practical application, much effort has been devoted, and wonderful fruitful achievements have been gained for both continuous-time MSSs and discrete-time cases [5–8]. Nevertheless, as pointed out in [5, 6], the most existing results are concerned with Markov switching linear systems. Due to the widespread of nonlinear characteristics, it is natural to investigate the Markov switching nonlinear systems. Compared with the standard MSSs, the Markov switching nonlinear systems are more general as they contain high nonlinearity. Lately, the T-S fuzzy model has been tendered to deal with the system’s nonlinearities [9, 10]. Benefit from the T-S fuzzy model, many Markov switching nonlinear systems can be approximated as T-S fuzzy MSSs (FMSSs). Following this excellent result, quantities of valuable results have been forwarded on T-S FMSSs [11–13]. For instance, in [11], a dropout compensation approach has been studied for T-S FMSSs. With respect to the network-induced phenomena, the cyber attack has been considered in FMSSs [13].
In many dynamic systems, the system behaviors are involved in multiple-time-scale property. The parasitic parameters, for instance, small-time constants and inductances, may result in the numerical ill-conditioned issues of physical systems. In this regard, the singular perturbation strategy has been employed to tackle the above obstacles. Thanks to singularly perturbed systems (SPSs), the multiple-time-scale-based systems can be transformed into a framework model. Note that the examples of SPSs can be widely found in power systems, airplane systems, etc. Recently, many scholars have drawn their attention to both continuous-time SPSs and discrete-time cases [14–16]. When investigating the SPSs, an extra phenomenon can be encountered, for example, the sudden changes of parameters. To tackle this occurrence, Markov switching SPSs have been studied in [17, 18]. However, the aforementioned results are concerned with linear systems, little attention has been devoted to T-S fuzzy Markov switching SPSs (FMSSPS) except for [19, 20], and this issue remains open and a challenge, which deserves further research.
In the networked control systems (NCSs), massive signals are communicated through a shared wireless network. As an unavoidable phenomenon, the NCSs always experience data collisions, fading channels, and input saturation [21]. To prevent the above shortage and mitigate the side effects, many communication protocols have been addressed to govern which sensors can obtain access to send signals such as the popular communication schedule called round-robin protocol [22, 23], try-once-discard protocol [24], and stochastic communication protocol (SCP) [25, 26]. Among them, the SCP is known as an effective method to schedule the signal exchange via a shared channel, in which only one sensor is activated to transmit data. Nevertheless, to our knowledge, no one carries out the exploration of FMSSPSs with the SCP mechanism, which motivates us to this work.
Inspired by the aforementioned discussions, our attention focuses on the control issue for FMSSPSs with the communication protocol. The main contributions are outlined as follows: in light of discrete-time FMSSPSs, to coordinate the data transmission and save the bandwidth usage, the SCP is applied to schedule the information exchange. Benefit from the novel Markov process, a mode-dependent Lyapunov functional is formulated such that the resulting system is stochastically stable, and the controller is designed.
2. Problem Formulations
Consider the th discrete-time Markov switching system modeled by the T-S fuzzy model.
: IF is , and is , and , and is , THENwhere and are the fast state and the slow state, respectively. and are the output vector and control input, respectively. means the disturbance signal. The sequence renders a discrete-time Markov chain (DTMC) subject to a finite set . Here, describes a homogeneous DTMC with the transition probability matrix of FMSSPS (1) inferred aswhere , , , and TPM .
For technique analysis, , , , , , , , , , , , , and are denoted by , , , , , , , , , , , and , respectively.
Recall the fuzzy weighting function , where refers to the grade of the membership degree of in . In general, assume and .
Let ; by virtue of T-S fuzzy technique, FMSSPS (1) is derived aswhere , , , , and .
In the NCSs, some redundant signals are communicated in the conventional data transmission manner, which may result in unfavorite phenomena, for instance, data collisions. The control signal and the actuators share the same communication network (CN). To prevent such unfavorite factors, the SCP scheduling is used to regulate the node order in transmitting data. Note that only one sensor is borrowed to release the signal each time, and the sensors are chosen in a stochastic way. In general, letting signifies the chosen actuator which gains the permission to access the CN at the time interval . Notably, can be recognized as a stochastic process regulated by another DTMC obeying a set , and TPM is determined bywhere , , and .
Let and , where denotes the th control input vector and signifies the th actuator. Firstly, assume that a set of zero-order hold is employed in the signal transmission. Accordingly, the th actuator is updated by the following principle:
Aiming at describing the data transmission strategy of actuators mathematically, a Kronecker sign function is inferred as
As indicated from the updating principle (5), the th actuator is updated when . Consequently, for , the updated actuator can be devised aswhere and .
The control law in this work is constructed as follows:where are matrices to be designed.
Substituting (8) into (3), the closed-loop FMSPS (9) is formulated aswhere
Before proceeding further, some lemmas and definitions are provided.
Definition 1 (see [27]). The FMSSPS (9) with is named stochastic stable (SS) if for any , one has
Definition 2 (see [27]). The FMSSPS (9) is named SS with a prescribed performance level if the FMSSPS (18) is SS and under zero initial condition such that
Lemma 1 (see [18]). For given a scalar , , , and are matrices with suitable dimensions. For any , such that
Lemma 2 (see [18]). For any symmetric matrices and matrix which meetsone has for any , where
3. Main Results
In this section, sufficient conditions are elicited to ensure the SS and a prescribed performance level of the FMSSPS (9).
Theorem 1. The closed FMSSPS (9) is called SS with a prescribed performance level if there exist symmetric matrices , , , , , and and matrices , , , and such that
Meanwhile, the -dependent controller gains are achieved aswhere
Proof. Combining with the linear matrix inequalities (LMIs) (17)–(19) and Lemma 2, for any , it yields thatwhereRecalling Lemma 2 and LMIs (17)–(19), for any , it is clear that . On the contrary, with respect to the fact that inequality , , , and , which yieldswherePremultiplying and postmultiplying (25) with and its transpose, where , yieldwhereIn the following, a Lyapunov functional for FMSSPS (9) is established:By calculating the difference of , one hasRecalling FMSSPS (9), can be derived aswhereIn (31), the first term can be further devised asBesides, the third term in (31) can be rewritten asCombining (29)–(34) yieldswhereWhen , it follows from inequality (35) thatwhere , , , , , and . Clearly, recalling (27), one gets . Consequently, one concludes thatRecalling Definition 1, when , FMSSPS (16) is SS.
Next, in the case of , we will provide the analysis of performance for FMSSPS (16). Define the performance index:Substituting (35) into (39), can be formulated aswhereAdditionally, by applying the Schur complement to (27) and (40), one getsLetting , it is directly attained from (42) thatAccordingly, by means of Definition 2, one concludes that FMSSPS (16) is SS with performance index . This completes the proof.
4. Simulation Examples
Example 1. Consider FMSSPS (16) with the following parameters:The TPM of the corresponding FMSSPS (1) is selected as . For another MP in SCP (7), the TPM is chosen as .
Let , , , and . In view of the LMIs of Theorem 1, the controller gains can be derived as follows:To carry on the simulation study, the external disturbance and the initial condition are selected as and , respectively. The possible mode switching of Markov process and the evolution of states , , and over 100 realizations are depicted in Figure 1, respectively. The control input over 100 realizations is plotted in Figure 2. Furthermore, the evolution of the output over 100 realizations is shown in Figure 3.
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5. Conclusions
In this work, the control problem has been discussed for FMSSPSs with the SCP. In order to coordinate the data transmission and save the bandwidth usage, the SCP with a compensator is applied to schedule the information exchange. Furthermore, some sufficient criteria have been forwarded such that the resulting system is SS. Finally, one example is exhibited to show the effectiveness and correctness of the developed results. In addition, some advanced techniques including the sliding mode-based filter will be researched in our following work [28, 29].
Data Availability
No data were utilized to support this work.
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 China Southern Power Grid Company Ltd. (nos. 080037KK52190037 and GZHKJXM20190108).