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Stabilization of Semi-Markovian Jump Systems with Uncertain Probability Intensities and Its Extension to Quantized Control
This paper concentrates on the issue of stability analysis and control synthesis for semi-Markovian jump systems (S-MJSs) with uncertain probability intensities. Here, to construct a more applicable transition model for S-MJSs, the probability intensities are taken to be uncertain, and this property is totally reflected in the stabilization condition via a relaxation process established on the basis of time-varying transition rates. Moreover, an extension of the proposed approach is made to tackle the quantized control problem of S-MJSs, where the infinitesimal operator of a stochastic Lyapunov function is clearly discussed with consideration of input quantization errors.
Over the past few decades, considerable attention has been paid to Markovian jump systems (MJSs) since such systems are suitable for representing a class of dynamic systems subject to random abrupt variations. In addition to the growing interest from their representation ability, MJSs have been widely applied in many practical applications, such as manufacturing systems, aircraft control, target tracking, robotics, networked control systems, solar receiver control, and power systems (see [1–9] and references therein). Following this trend, numerous investigations are underway to deal with the issue of stability analysis and control synthesis for MJSs with complete/incomplete knowledge of transition probabilities in the framework of filter and control design problems: [10–13] with a complete description of transition rates and [14–20] without a complete description. Generally, in MJSs, the sojourn-time is given as a random variable characterized by the continuous exponential probability distribution, which tends to make the transition rates time-invariant due to the memoryless property of the probability distribution. The thing to be noticed here is that the use of constant transition rates plays a limited role in representing a wide range of application systems (see [21–23]). Thus, another interesting topic has recently been studied in semi-Markovian jump systems (S-MJSs) to overcome the limitation of this memoryless property.
As reported in [24–26], the mode transition of S-MJSs is driven by a continuous stochastic process governed by the nonexponential sojourn-time distribution, which leads to the appearance of time-varying transition rates. Thus, it has been well recognized that S-MJSs are more general than MJSs in real situations. Further, with this growing recognition, various problems on S-MJSs have been widely studied for successful utilization of a variety of practical applications (see [21, 22, 25, 27–29] and references therein). Of them, the first attempt to overcome the limits of MJSs was made by [21, 22] for the stability analysis of systems with phase-type (PH) semi-Markovian jump parameters, which was extended to the state estimation and sliding mode control by . Besides,  considered the Weibull distribution for the stability analysis of S-MJSs and introduced a sojourn-time partition technique to make the derived stability criterion less conservative. Continuing this,  applied the sojourn-time partition technique to the design of state-feedback control for S-MJSs with time-varying delays. After that, another partition technique of dividing the range of transition rates was proposed by  to derive the stability and stabilization conditions of S-MJSs with norm-bounded uncertainties. Most recently,  designed a reliable mixed passive and filter for semi-Markov jump delayed systems with randomly occurring uncertainties and sensor failures. Also,  considered semi-Markovian switching and random measurement while designing a sliding mode control for networked control systems (NCSs). Based on the above observations, it can be found that their key issue mainly lies in finding more applicable transition models for S-MJSs, capable of a broad range of cases. In this light, one needs to explore the impacts of uncertain probability intensities in the study of S-MJSs and then provide a relaxed stability criterion absorbing the property of the resultant time-varying transition rates. However, until now, there have been almost no studies that intensively establish a kind of relaxation process corresponding to the stabilization problem of S-MJSs with uncertain probability intensities.
This paper addresses the issue of stability analysis and control synthesis for S-MJSs with uncertain probability intensities. One of our main contributions is to discover more reliable and scalable transition models for S-MJSs on the basis of their time-varying and boundary properties. To this end, this paper provides a valuable theoretical approach of constructing practical transition models for S-MJSs () by taking into account uncertain probability intensities and () by reflecting their available bounds in the transition rate description. Further, in a different manner from other works, all constraints on time-varying transition rates are totally incorporated into the stabilization condition via a relaxation process established on the basis of time-varying transition rates. Here, it is worth noticing that our relaxation process is developed in such a way that all possible slack variables can be included therein. In contrast to other works, our relaxation process plays a key role in obtaining a finite and solvable set of linear matrix inequalities (LMIs) from parameterized matrix inequalities (PLMIs) arising from uncertain probability intensities. On the other hand, the quantization module that converts real-valued measurement signals into piecewise constant ones has been commonly used to implement a variety of networked control systems over wired or wireless communications (see [31, 32]). Especially among optical wireless communications, the visible light communication can be applied as a data communication channel to transmit the control input to the S-MJSs under consideration. Thus, as an extension, this paper tackles the quantized control problem of S-MJSs, where the infinitesimal operator of a stochastic Lyapunov function is clearly discussed with consideration on input quantization errors. In addition, this paper proposes a method for reducing the influence of input quantization errors in the control of S-MJSs, which is also one of our main contributions. Finally, simulation examples show the effectiveness of the proposed method.
Notation. The notations and means that is positive semidefinite and positive definite, respectively. In symmetric block matrices, is used as an ellipsis for terms induced by symmetry. For any square matrix , . For , where and denote real submatrices with appropriate dimensions or scalar values. The notation denotes the mathematical expectation and stands for a block-diagonal matrix. The notation denotes the maximum eigenvalue of the argument, and indicates the exponential distribution.
2. System Description
Let us consider the following continuous-time semi-Markovian jump linear systems (S-MJSs): where and denote the state and the control input, respectively. Here, denotes a continuous-time semi-Markov process that takes values in the finite space and further has the mode transition probabilities: where and denotes the transition rate from mode to mode at time . Further, indicates the sojourn-time elapsed when the system stays at mode from the last jump (i.e., is set to 0 when the system jumps). In particular, the transition rate matrix belongs to the following set:Before going ahead, for later convenience, we define the system matrix for the th mode as , , and set . Also, to deal with the stability analysis problem in such a stochastic setting, we consider the following definition.
Definition 1. An S-MJS (2) with is stochastically stable if its solution is such that, for any initial condition and ,
3. Stochastic Stability Analysis
Lemma 2. Suppose that there exists , for all , such that where . Then, S-MJSs (6) with are stochastically stable.
Proof. Let us consider a stochastic Lyapunov function candidate of the following form: where is taken to be positive definite. Then, the infinitesimal generator of the stochastic process acting on is given by where , Here, the probabilities and are described as where denotes the cumulative distribution function of the sojourn-time at mode and stands for the probability intensity such that and . Especially in (11), the probability intensity is taken to have uncertainties. Meanwhile, from (6), it follows that , for , which leads to That is, from (11) and (12), it is clear that Thus, using (14), and become That is, becomeswhere denotes the transition rate of the system jumping from mode . As a result, by defining , for , and , we have In what follows, by the generalized Dynkin’s formula , it is clear that which results in Thus, from (7), it follows that Finally, by Definition 1, the proof can be completed.
In this paper, as a model of probability distribution for the sojourn-time , we utilize the Weibull distribution with shape parameter and scale parameter , since such a distribution has been witnessed as an appropriate choice for representing the stochastic behavior of practical systems. In other words, to represent the probability distribution of , its cumulative function and probability distribution function are given as follows: for all and , which leads to As a special case, let . Then, we can represent MJSs from (22); that is, the transition rate can be reduced to an -independent value as follows: . Accordingly, it can be claimed that (22) expresses a more generalized transition model, compared to the case of MJSs.
Remark 3. As shown in (22), the transition rate is time-varying and depends on the probability intensity . Thus, to derive a finite number of solvable conditions from (7), there is a need to consider the lower and upper bounds of both and , respectively, as follows: and . Then, from , the bounds of are decided as follows: , where
Lemma 4. Suppose that there exists , for all , such that where and . Then, S-MJSs (6) with are stochastically stable.
However, it is worth noticing that solving (25) of Lemma 4 is still equivalent to solving an infinite number of LMIs, which is an extremely difficult problem. Thus, it is necessary to find a finite number of solvable LMI-based conditions from (25). To this end, the following theorem provides a relaxed stochastic stability condition for (6) with .
Theorem 5. Suppose that there exists matrices and symmetric matrices such that where , , Then, S-MJSs (6) with are stochastically stable.
Proof. From the first constraint , we can obtain, under (28), Further, under (27), the second constraint provides Here, note that (30)–(32) can be converted, respectively, into where and . Likewise, according to the form of (33)–(35), we can rewrite in Lemma 4 as follows: As a result, by combining (36) with (33)–(35) through the S-procedure , the stochastic stability condition (25) is given by where , , and are given in the body of Theorem 5. Therefore, as (26) implies (37), the proof can be completed.
4. Control Design
Let us consider the following mode-dependent state-feedback control law: where . Thereby, the resultant closed-loop system under (2) and (38) is given by The following theorem provides a relaxed stochastic stabilization condition for S-MJSs (39) with .
Theorem 6. Suppose that there exist matrices and and symmetric matrices such that where , , , , Then, closed-loop system (39) with is stochastically stable, where .
Proof. In light of (25), the stabilization condition of (39) is given by Also, pre- and postmultiplying (45) by yields where . Here, by employing such that (43) holds (i.e., ), we can convert (46) into Thereupon, is factorized as follows: where .
Next, as in the proof of Theorem 5, the constraint can be represented as where and . As a result, by the S-procedure, combining (48) with (49) results in where , , and are given in the Theorem 6. Therefore, as (40) implies (50), the proof can be completed.
Hereafter, as a practical extension of the proposed approach, we consider the following input-quantized S-MJSs: where stands for a uniform quantization operator with the quantization level ; that is, . Here, note that , where the th element of the quantization error satisfies Thus, (51) can be rewritten as where is known. Continuously, as a mode-dependent state-feedback law, we adopt Then, the resultant closed-loop system is described as The following theorem provides a relaxed stochastic stabilization condition for S-MJSs (55) with input quantization error.
Theorem 7. Let (i.e., the th element of ) be given as follows: where and denotes the th element of . Suppose that there exist matrices and and symmetric matrices such that (40)–(43) hold. Then, the closed-loop system (55) with is stochastically stable, where .
Proof. From (55), it follows that , for , which yields Thus, it is given that Then, based on (11) and (12), we have which leads to Hence, for , letting implies Further, for , letting yields That is, (56) allows (61) to be reduced to (62) for all . In what follows, we need to derive the stabilization condition from (62), which is omitted herein because it is in line with the proof of Theorem 6.
5. Numerical Examples
Example 1. Consider the following system with three different modes: for and , Here, the Weibull distribution for the sojourn-time is set by , , and (see Figure 1). Thereby, a reasonable interval can be obtained such that , from which the lower and upper bounds of can be also found. As a result, by considering the uncertain probability intensities such that , for all and , the following matrices such that are established from Remark 3: Figure 2(a) shows the mode evolution generated from the stochastic setting. Besides, from Theorem 6, the following control gains are obtained: Figure 2(b) shows the behavior of the state response coupled with the mode transition depicted in Figures 2(a) and 2(c) which presents the Monte Carlo simulation result of the settling time for different mode transition configurations, where its mean value and standard deviation are 12.2979 and 3.126, respectively. From Figure 2(b), it can be seen that the states converge from the initial condition to the equilibrium point as time increases. Consequently, Theorem 6 provides an applicable method for deriving a relaxed stabilization condition for S-MJSs with uncertain probability intensities.
Example 2. Consider the following system representing an inverted pendulum: for and : where is the angle of the inverted pendulum, is the angular velocity, is the input current, is the control input voltage, is the acceleration due to gravity, and are the mass and length, respectively, and are the back emf constant and motor torque constant, respectively, and are the inductance and resistance of the DC motor, respectively, and is the gear ratio. Here, we set , , , , , , and Then, the linearized model of (67) is given by Besides, the matrices and such that are taken to be the same as in Example 1. Thereupon, Theorem 7 provides the following control gains:Figure 3 shows the behavior of the state response for the mode transition generated according to , , and . Here, the control input is designed in accordance with (56), and the quantization level is assumed to be . From Figure 3, it can be seen that the states converge from the initial-state condition to the origin as time increases. Consequently, Theorem 7 provides a suitable mode-dependent control for the S-MJSs with input quantization errors as well as uncertain probability intensities.
6. Concluding Remarks
The issue of stability analysis and control synthesis for S-MJSs with uncertain probability intensities has been addressed in this paper. Here, the boundary constraints of probability intensities have been totally reflected in the stabilization condition via a relaxation process established on the basis of time-varying transition rates. Furthermore, as an extension, the quantized control problem of S-MJSs has been addressed herein. Through simulation examples, the effectiveness of the proposed method has been shown.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2015R1A1A1A05001131).
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