Abstract
This study focuses on the static output feedback control of nonlinear Markov jump singularly perturbed systems within the framework of Takagi–Sugeno fuzzy approximation. From a practical point of view, the phenomenon of asynchronous switching between the plant and the controller is considered and characterized by a finite piecewisehomogenous Markov process. Particularly, for facilitating the controller synthesis, the closedloop system is transformed into a fuzzy Markov jump singularly perturbed descriptor system by adopting descriptor representation. In order to fully accommodate the system features, an appropriate stochastic Lyapunov function is constructed. Afterwards, by combining Finsler’s lemma, the mean square exponential admissibility of the system is analyzed. The conditions ensuring the existence of the predesigned controller are given and further solved by designing a brief search algorithm. Finally, a typical circuit system is used to demonstrate the application potential of the developed control technology and the effectiveness of the control strategy.
1. Introduction
Analysis and synthesis of singularly perturbed systems (SPSs) have attracted significant research attention since many practical dynamical systems including the wellknown power systems and nuclear reactor systems can be referred to these kind of systems [1]. Among the existing singular perturbation analysis approaches, the one based on the singularperturbationLyapunov stability theory and matrix inequalities has received considerable research interests due to its positive features such as the avoidance of the system decomposition into fast and slow subsystems and the convenience for solution and estimation of the stability bound [2]. It is worth pointing out that most available results on SPSs are only applicable to linear models. Moreover, restrictive assumptions are usually employed in the existing results on nonlinear SPSs in order to satisfy the specified analysis conditions, which reduce seriously their applicability. As an alternative, there has been a growing research attention on nonlinear SPSs by Takagi–Sugeno fuzzy approximation approach, which establish an intimate connection between the fuzzylogic technique and the perturbation analysis approach for linear SPSs [3–6]. On another research front, Markov jump systems (MJSs) are often utilized in modeling the dynamical systems with randomly changed structures governed by a Markov chain [7–10]. For the MJSs subject to multiple disturbances and uncertain transition rates, G. Zong proposed a composite antidisturbance resilient controller to guarantee the performance of the systems. The antidisturbance control for switching Markovian systems was studied and a bumpless transfer controller was proposed. For linear or fuzzy SPSs with random variable structures, it is natural to classify the mass Markov jump singularly perturbed systems (MJSPSs), see, for instance, [11–13]and the references therein.
In control community, the synthesis of static output feedback controller (SOFC) for various complex dynamical systems is a fundamental and challenging problem [14]. Compared with other output control methodologies, the main appealing feature of SOFC lies in its simple structure and its ease of implementation in practice. Despite the surge on SOFC design, when it comes to SPSs, the available results are scattered due to the additional complexity induced by singular perturbation structure. The recent progress is achieved in [15], where two SOFC design methods were developed for fuzzy SPSs. The first method imposes a condition that all the output matrices share the same specified matrix, which may be, unfortunately, hard to be fulfilled in some practical applications. To solve this problem, another method based on matrix transformation is further provided. However, this method is only suitable for the selfstable fuzzy SPS, which is very restrictive. Motivated by the existing results in literature, the first motivation of the current study is to propose an appropriate SOFC design method for fuzzy MJSPSs exempt from the above restrictions. On the contrary, most of the existing results on MJSPSs are based on the synchronous mode switching between the plant and controller. In practice, however, it may be difficult or costly to fully access the mode information of the plant on time due to the factors such as packet dropouts and transmission delays [16]. Therefore, it is more realistic that incorporating asynchronous factors into the SOFC design has positive significance for improving efficiency, etc., which has caused the second motivation of the research.
In this paper, we focus on the SOFC synthesis problem for fuzzy MJSPSs subject to asynchronous switching. By resorting to an appropriate stochastic Lyapunov function and Finsler’s lemma, the criterion to determine the gain parameters of the designed asynchronous fuzzy SOFC is put forward. A suitable search algorithm is designed to solve the stabilization bound problem. Finally, a circuit system is utilized to verify the application potential of the developed new control strategies.
The key challenges to be solved for the current research topic mainly embody in the following two folds:(1)To circumvent the restrictive condition employed on output matrices and facilitate the controller design, the descriptor representation is adopted to transform the closedloop system into a fuzzy Markov jump singularly perturbed descriptor system (MJSPDS). For such a complicated system, how to fully exploit the mixed system features and achieve SOFC synthesis is a challenging task.(2)A finite piecewisehomogenous Markov process is utilized to characterize the phenomenon of asynchronous operation between the plant and the controller. The involvement of asynchronous switching brings more challenges to the system analysis and synthesis.
The main arrangements of the study are as follows: Section 2 presents the problem researched in the article, Section 3 analyzes the system to lay the foundation for solving asynchronous ambiguity, Section 4 determines the design of the asynchronous fuzzy SOFC, and in Section 5, a simulation study is carried out to prove the validity of the obtained results. Section 6 summarizes the main content of this article.
2. Problem Statement
Fix a probability space and consider a class of fuzzy modelbased nonlinear MJSPSs, which is given by fuzzy rule : IF is and is , THENwhere , denotes the singular perturbation parameter, are the premise variables depending on the measured variables, represent the fuzzy sets, indicates the number of IFTHEN rules, and denote the state vectors, denotes the measured output, denotes the control input, , , , , , and are appropriately dimensioned matrices, and denotes a continuoustime Markov chain taking values within a finite set and is controlled by a transition probability matrix (TPM) subject towhere and , , for , and , for each .
To simplify the notation, the matrix will be rewritten as . Denote as the membership grade and as the normalized membership functions satisfying
The overall dynamics of plant (1) can be further rewritten aswhere and
For fuzzy MJSPS (4), an asynchronous fuzzy SOFC is designed for stabilizing the plant, which is formulated in parallel distributed compensation (PDC).
Controller rule : IF is and is , THENwhere are the gain parameters to be determined.
A more compact representation of the fuzzy controller is
The controller mode is subject to a finite piecewisehomogeneous Markov chain that takes values within a set and is governed by a TPM satisfyingwith and , , for , and , for each . To simplify the notation, will be rewritten as .
Remark 1. It is noted that the jumping modes of adopted fuzzy SOFC are synchronous to the system modes. In general, the hidden Markov model [17] or piecewisehomogenous Markov process [18] is employed to characterize the phenomenon of asynchronous jumping. The former means the designed controller/filter only depends on the plant modes and the latter implies the variation of the controller/filter subjects to a lowerlevel TPM. In this paper, a finite piecewise homogeneous Markov process is used to describe the mismatch phenomenon. From (8), we can see that the TPM is time varying but invariant during the whole plant mode .
To facilitate the solution of the desired gain matrices, the closedloop system is described by us in the following description form, that is,where
Remark 2. If we directly substitute controller (7) into system (9), the system matrices for the closedloop system become . In this case, it is hard to get a strict matrix inequality solution for the gain matrices if the output matrices do not share a common matrix under the same operation mode or uncertainties affect them. To overcome this issue, the descriptor representation [25] is utilized to study the asynchronous SOFC design problem for fuzzy MJSPSs, which may take the profit of induced redundancy. As a consequence, the corresponding closedloop plant is transformed into a fuzzy MJSPDS. For SPDSs, however, only stability results have been reported [19, 20], and the stabilization problem is not well addressed to the best of the authors’ knowledge.
The following lemma and definition are helpful for the derivation of the main result.
Lemma 1. For the given scalar and symmetric appropriated dimensioned matrices , , and , inequality holds for all if the inequalities , , and hold simultaneously.
Definition 1 (see [21]). (1)MJSPDS (9) has regularity and impulse and impulse elimination if, for every and , the pairs are regular and impulsefree, where (2)MJSPDS (9) has mean square exponential stability for any initial state if two positive scalars and exist such that , where and are the decay rate and decay parameter, respectively(3)MJSPDS (9) has mean square exponential admissibility (MSEA) if it has regularity, impulse elimination, and mean square exponential stability
3. Admissibility Analysis
In this section, MSEA of system (9) will be analyzed, which establishes the basis for solving the asynchronous fuzzy SOFC (7).
Theorem 1. For a prescribed , if there exist symmetric matrix variables , , , and and matrix variables , , , , and , , such that the following conditions hold, for each and ,wherewiththen, the MSEA for the fuzzy MJSPDS (9) is ensured for any .
Proof. It is easy to obtain that the characteristic polynomials are not identically zero and the equation is satisfied for each , . Thus, based on Definition 1, the regularity and impulse elimination of the fuzzy MJSPDS (9) is ensured.
Recall the definition of , and in the light of (11)–(13), we can deduce from Lemma 1 thatThen, the following stochastic Lyapunov function is proposed:Set as the weak infinitesimal operator of the stochastic process . Then, along the fuzzy MJSPDS (9), taking the infinitesimal operator on and emanating from the point yieldswhereOn the contrary, the fuzzy MJSPDS (9) can be written asThen, by applying Finsler’s Lemma, the following inequality is obtained:Then, taking into account expressions of (14), (15), and (20)–(23), we readily obtainwhere .
By resorting to Dynkin’s formula, for all , we obtainOn the contrary,Thus,where .
Correspondingly, it is easy to see thatwhich means by combining (27) the MSEA of the fuzzy MJSPDS (9) is ensured. The proof is completed.
Remark 3. To accommodate the mixed system features of plant (9), an appropriate stochastic Lyapunov function is proposed in (19). The distinguished Lyapunov matrix variables fully adapt the mixed singular perturbation and descriptor structure of the system and can yield stabilization conditions that are convenient to compute the stabilization bound. Moreover, it is an exponentially decaying function, and the decay rate can be adjusted freely to cater design specifications.
Remark 4. It is noted that the proposed Lyapunov matrices depend on two kinds of modes and their structures are complicated, which bring a big difficulty to the controller synthesis. By resorting to Finsler’s lemma, equality (16) is presented to separate the system matrices from the Lyapunov variable matrices, which is helpful to the computation of the controller gain matrices and can introduce extra design freedom.
4. Asynchronous Fuzzy SOFC Synthesis
Through the previous section, establish a foundation for solving asynchronous fuzzy SOFC (7). In this section, the gain values of the designed asynchronous fuzzy SOFC (7) will be determined.
Theorem 2. For a prescribed , if there exist symmetric matrix variables , , , and and matrix variables , , , , , , , and , , such that (11)–(13), the following conditions hold, for each and ,wherethen, the MSEA for the fuzzy MJSPDS (9) is ensured for any . Furthermore, the gain values for the asynchronous fuzzy SOFC (7) are computed by
Proof. To obtain the designed fuzzy SOFC, we choose the structures of matrices and aswhere is nonsingular and is a given arbitrary matrix. Thus, based on the structure of , it can be derived thatLet , and substitute expression (38) into Theorem 1, and (29)–(34) can be easily obtained by Lemma 1. The proof has been completed.
Remark 5. By specifying a suitable form of matrix variables and in (37), Theorem 2 presents a SOFC design scheme for system (9).The involved arbitrary matrices need to be preset. Different can lead to different matrix inequalities with different conservatisms. An initial and simple choice is to define .
Remark 6. For fuzzy MJSPDS (9), a fundamental problem is to determine the stabilization bound . To efficiently evaluate it, the following brief algorithm is proposed.
Remark 7. The achieved contributions of the paper are summarized as follows. (1) It is for the first time that the asynchronous switching between the plant and controller is investigated for MJSPSs, which may reflect more reality in some cases. (2) The existing SOFC results on fuzzy SPSs impose restrictive conditions on output or system matrices. In contrast, the results are not constrained by these conditions by the virtue of descriptor representation. (3) A rigorous proof for the MSEA of the transformed fuzzy MJSPDS and a solution for the designed asynchronous fuzzy SOFC are presented.
5. Simulation Study
During this section, tunnel diode circuit (demonstrated in Figure 1) is adopted to demonstrate the validity of the obtained results.
The whole dynamic equation of the circuit system iswhere is the “parasitic” inductor, is the capacitor, and denote, respectively, the voltage of the capacitor and current from the inductor, and is the resistor, and its value varies abruptly between and , and the abrupt variation is characterized by a Markov process . TPM is assumed to satisfy
Define , , and , and suppose that . Then, the nonlinear plant (44) can be described by the following uncertain fuzzy MJSPS [4]:where the basic functions are demonstrated in Figure 2, and
Moreover, it is assumed that the measurement value of the resistor is not accurate and real time, and it varies stochastically according to different actual values, and TPM is assumed to satisfy
Note that the membership functions in this example are not differentiable, setting . Under the scalars and , by solving Algorithm 1 for Theorem 2 with , the achieved stabilization bound is , with the corresponding gain matrices:

By implementing the asynchronous fuzzy SOFC (7) with gain parameters achieved in (44), the simulation responses with initial state value and initial mode are depicted in Figures 3–5. Among the figures, Figure 3 demonstrates the state trajectories, Figure 4 illustrates the input response, and Figure 5 depicts the evolution signal of the two modes.
6. Conclusion
The topic of static output control is addressed for fuzzy modelbased nonlinear MJSPSs. One feature lies in which the asynchronous operation between the plant and the controller is considered and expressed by a finite piecewisehomogenous Markov process. In order to facilitate the controller synthesis, the closedloop plant is transformed into a fuzzy MJSPDS, after which the mean square exponential stabilization criterion determining the designed asynchronous fuzzy SOFC is provided, by which the stabilization bound can be evaluated. Finally, a practical circuit system is offered to show the validity of the developed control strategies.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.