Abstract
This paper is concerned with model predictive control (MPC) problem for continuous-time Markov Jump Systems (MJSs) with incomplete transition rates and singular character. Sufficient conditions for the existence of a model predictive controller, which could optimize a quadratic cost function and guarantee that the system is piecewise regular, impulse-free, and mean square stable, are given in two cases at each sampling time. Since the MPC strategy is aggregated into continuous-time singular MJSs, a discrete-time controller is employed to deal with a continuous-time plant and the cost function not only refers to the singularity but also considers the sampling period. Moreover, the feasibility of the MPC scheme and the mean square admissibility of the closed-loop system are deeply discussed by using the invariant ellipsoid. Finally, a numerical example is given to illustrate the main results.
1. Introduction
As a class of stochastic hybrid systems, Markov Jump Systems (MJSs) are suitable to describe many practical systems whose structures and parameters suffer from random abrupt changes [1–5]. Furthermore, MJSs, which contain a non-full rank system matrix, are usually referred to as singular MJSs, and they have obtained much attention from mathematical and control community. It should be mentioned that a singular MJS can better describe some physical systems than standard MJS model and has extensive applications, such as microelectronic circuits and economics [6–9].
In recent years, an ever increasing number of scholars have paid much attention in the study of singular MJSs [6, 10–12]. In [13], Boukas investigated the state feedback control for singular MJSs with norm-bounded uncertainties. By considering the full and partial knowledge of transition rates, sufficient conditions which guarantee that the closed-loop system is piecewise regular, impulse-free, and mean square stable are given in [14]. Due to the regularity and pulse phenomenon, the stability and stabilizability of singular MJSs are intensively studied [10]. In [11], the authors aimed to study the mean square stability of singular MJSs and extended the main criterion to frequency domain. The robust mean square stability problem for uncertain singular MJSs with actuator saturation is further considered [15], in which sufficient conditions for such systems to be regular, causal, and mean square stable are derived in LMI form.
On another research frontline, model predictive control (MPC), which is also named as receding horizon control, has received tremendous attention in practical applications [16–19]. At each time instant, MPC requires the online solution of an optimization problem and then gets the optimized control inputs. We calculate a control sequence, but only the first element is implemented. At the next sampling time, we resolve the optimization problem again. Therefore, the control gain is updated at each sampling time and can naturally compensate the model uncertainties and exogenous disturbance to some extent. MPC has been successfully employed in industry since it can cope with hard constraints, control moves, system states, and outputs. As MJS is considered, MPC strategy has also received considerable research efforts on its analysis and synthesis from a theoretical point of view [20–23].
Almost all of the above-mentioned studies are based on the assumption that the transition probabilities are completely known. However, in practice, obtaining the complete knowledge of transition probabilities is difficult and costly. Until very recently, some efforts start to focus on MJSs with incomplete transition probabilities. Zhang and Boukas firstly proposed an analysis and design method for MJSs with partly unknown transition probability, which does not require the information of unknown elements [24]. Subsequently, necessary and sufficient conditions for the stability analysis of MJSs with incomplete transition descriptions are further given in [25]. Based on the results of [24, 25], Zheng and Yang [26] investigated the robust stabilization problem with a sliding mode control approach. Shen et al. proposed a new method for the state feedback control [27] and established sufficient conditions for the existence of the static output feedback controller [28].
It is worth noting that MPC for MJSs are usually based on discrete-time version. However, in the real world, we take the controlled objects as continuous-time models. To the best of our knowledge, few results about continuous-time singular MJSs are addressed due to the difficulty in stability analysis and controller design, especially further considering incomplete transition rates. In this paper, a new MPC method is developed for continuous-time singular MJSs with incomplete transitions rates. Our main contributions include the following three aspects. A sampled-period MPC law is introduced to deal with the continuous-time plant. A modified Lyapunov function approach is employed to deal with analysis and design problems for the underlying system concerning incomplete information. The approach not only refers to the knowledge of system state but also considers the sampling period and the singularity. The feasibility of the MPC scheme for the continuous-time singular MJS and the mean square admissibility of the closed-loop system are discussed by using invariant ellipsoid.
2. Problem Statement
Consider a continuous-time singular MJS with uncertainties described as follows:where is the system state, is the control input, is a real constant matrix with , and are known mode-dependent constant matrices with appropriate dimensions, and and are matrices with unknown uncertainties satisfyingwhere , , and are known constant matrices and is a time-varying matrix function satisfying . is a continuous-time Markov process which takes values in a finite set with the following transition rate:where and and is the switching rate from mode at time to mode at time .
For the sake of convenience, , , , and could be denoted by , , , and , respectively, where .
Moreover, in this paper, the transition rates are considered to be partly available; namely, some elements in matrix are unknown. Taking singular MJS (1) with four operation modes as an example, the transition rate matrix could be expressed aswhere the elements with “” represent the unknown transition rate.
In this paper, we denote , , with
In addition, if , it can be written as where represent the th known element in the th row of matrix . Also, we denote .
In the case that is unknown, providing a lower bound with is necessary.
MPC for (1) is studied over an infinite horizon. At each sampling time for , we could obtain control gains by making use of the LMI techniques and we will only implement the first calculated control input , where represents the fixed sampling interval. At the next sampling time, we will calculate the state and then compute the feedback control gain once again. Let denote the state measured at sampling time ; the predicted state at time is denoted by ; is the control action at time . can be noted as for simplicity.
About the singular MJS (1), we give the following definitions.
Definition 1 (see [10]). The continuous-time singular MJSs (1) are said to be(1)regular if is not identically zero for each ;(2)impulse-free if for each ;(3)mean square stable if there exists a scalar , for any , , such that(4)mean square admissible if it is regular, impulse-free, and mean square stable.
Definition 2 (see [29] (invariant ellipsoid)). Given a continuous-time dynamical system , a subset of the state space is said to be an invariant ellipsoid, if it has the property that whenever , then for all times .
Before giving the main results of the singular MJS (1), some useful lemmas are presented below.
Lemma 3 (see [30]). Given matrices and with appropriate dimensions and a symmetric matrix , , for all , if and only if there exists a scalar such that
Lemma 4 (see [31]). Given symmetric and positive definite matrices and , if , then .
Lemma 5 (see [14]). If a mode-dependent symmetric matrix and a real constant matrix with are given a priori, then we will have for any , where presents the identity matrices with compatible dimensions.
In this paper, at each sampling time , we aim to design a state feedback controllerby solving the following optimization problem:where and are symmetric weighting matrices, is the performance index of system (1) at time , and is the state feedback control gain.
3. Model Predictive Controller Design
In this section, we will design MPC controller for the singular MJSs with incomplete transition rates by presenting sufficient conditions, which could be efficiently solved by LMI toolbox.
Theorem 6. Consider the singular MJS (1) with incomplete transition rates and let be the state of uncertain singular system (1) at sampling time . One will obtain and by solving the following optimization problem:satisfyingwhere The mode-dependent state feedback gain is given by
Proof. Define a quadratic functionHere, we use to denote the weak infinitesimal generator of random process for each mode ; it is defined asAt each sampling time , suppose that satisfy the following inequality:Because the robust cost function will eventually become , we will getby integrating both the left and the right sides of inequality (19) with .
Considering Lemma 5, for any , we will haveletthen, we have Using the fact that for any and Lemma 4, we will obtainObviously, via the well-known Schur complement, it is easy to get that conditions (12) and (24) imply (23).
Applying the control law (9) and the definition of weak infinitesimal generator, inequality (19) becomesPre- and postmultiplying (25) by and its transpose, replacing and , by and , respectively, one will getconsidering the unknown uncertainty (2), (26) becomesAccording to Lemma 3, (27) holds if and only if there exists satisfying According to Lemma 5, (28) holds if (29) is satisfied for any satisfyingReplacing and by and , respectively, we will have Subsequently, we take transition rate into account and separate it into two cases, and , respectively.
Case 1 Considering that , the left side of (30) is rewritten aswhere the elements () are unknown, , and . Noting that, in this case, is known, , and , , thenwhere and , because of and ; we will obtain thatSince , we can have the following inequality:Using (24), we can easily get (34), ifholds. Applying Schur complement to the above inequality, one can obtain (13).
Case 2 Likewise, considering , we rewrite the left side of (30) asNoting that, in this case, is unknown, , thenSimilarly, since we have and , we can easily obtainTherefore, is equivalent to , ,As is lower bounded by , we know that . Thus, (39) holds if the following inequality holds:Similarly, (40) is implied by (24) andApplying Schur complement to (41), we can obtain (14).
Certainly, MPC for singular MJSs without incomplete transition descriptions and the normal MJS with incomplete transition descriptions can be viewed as two special cases of singular MJS. Then, we have the following corollaries.
Corollary 7. Consider the singular MJS (1) without incomplete transition rate and let be the state of singular system (1) at sampling time . One will obtain and by solving the following optimization problem:satisfying where Then, the mode-dependent state feedback gain can be obtained by (16).
Corollary 8. When is nonsingular and can be simplified as an identity matrix, system (1) becomes let be the state of uncertain MJS (45) at sampling time . One will obtain and by solving the following optimization problem:satisfyingwhere The mode-dependent state feedback gain is obtained by (16).
4. Mean Square Admissibility Analysis
Previous section gives sufficient conditions, which could guarantee the existence of model predictive controller. In this section, we mainly study the mean square stability of the closed-loop singular MJS (1).
Lemma 9 (feasibility). The solution of the optimization problem in Theorem 6 is feasible over any time interval with , if it has the property that the solution is feasible over the time interval .
Proof. At sampling time , from inequality (12), we limit the measured state in the ellipsoid region . According to the proof of Theorem 6, we can getThen, for any sampling time with , the future state remains in the ellipsoid set. Therefore, by referring to Definition 2, is an invariant ellipsoid.
The solution of the optimization problem in Theorem 6 is feasible over the time interval . By observing (12), (13), and (14), it could be concluded that the feasibility of the MPC strategy only depends on (12). Thus, to prove Lemma 3, we only need to prove that the LMI (12) is feasible for all future states , . Considering the invariance of the ellipsoid, we obtainwhich means LMI (12) is feasible with state and then the feasibility of optimization problem (11) could be achieved over the time interval . Similarly, we can complete the proof on the time intervals , ,.
Theorem 10. The feedback predictive control gain computed from Theorem 6 guarantees that the closed-loop singular MJS (1) is mean square admissible.
Proof. Firstly, we show that the singular MJS (1) is regular and impulse-free. There exist two orthogonal matrices and such that Based on singular value decomposition, system (1) can be written as the following differential and algebraic form:where , and it can be decomposed as .
Assuming the optimal solution of optimization problem (11) at sampling time can be denoted as and , from the proof of Theorem 6, we can obtain that there exists satisfying (25) andLetBy (53), we have (because is nonsingular) and . Pre- and postmultiplying the left and right sides of (25) by and , we obtainFrom (55), we can easily get that is invertible. Hence, for any ,from which we can see that is not identically zero; by Definition 1, the closed-loop system is regular and impulse-free.
What is more, we consider the mean square stability of the singular MJS (1). Here, the piecewise Lyapunov function is described as From (25), we havewhich yieldsTherefore, is strictly decreasing over each time interval .
Note that represents a feasible solution at time while is an optimal solution at time . Since is a piecewise function, we further considerfor all . Inequalities (59) and (60) guarantee that is strictly decreasing when ; system (1) is mean square stable.
Therefore, according to the property of being regular and impulse-free, the closed-loop singular MJS is mean square admissible.
5. Numerical Example
To illustrate the efficiency of the proposed MPC scheme for continuous-time MJSs, a numerical example is presented in the following.
Consider a system with the form of (1), where . The system matrices are , , , , , , , , , , , and . The weighting matrices are and
The TRM is given as follows:where the diagonal element is unknown and its lower bound is a priori given value. The initial value of the state is and the initial mode is .
The simulation step is taken as 150 time units and each unit length is taken as 0.01. We get the jump mode (Figure 1) and apply the designed MPC algorithm and normal state feedback control to system (1), separately. Then, we get the simulation results in Figures 2–5.
Obviously, the unstable system (1) becomes more easily stable by using MPC strategy.
6. Conclusions
In this paper, MPC strategy is presented for continuous-time singular MJSs with incomplete transition rates. The controller design problem is formulated as LMI optimization algorithm. The predictive control strategy is proved to be feasible at every sampling time and it also can guarantee the mean square admissibility of the closed-loop system.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This project was jointly supported by NSFC (61203126), NSFC (61374047), BK2012111, and 111 Project (B12018).