Estimation and Synthesis of Reachable Set for Singular Markovian Jump Systems

The problems of reachable set estimation and state-feedback controller design are investigated for singularMarkovian jump systems with bounded input disturbances. Based on the Lyapunov approach, several new sufficient conditions on state reachable set and output reachable set are derived to ensure the existence of ellipsoids that bound the system states and output, respectively.Moreover, a state-feedback controller is also designed based on the estimated reachable set. The derived sufficient conditions are expressed in terms of linear matrix inequalities. The effectiveness of the proposed results is illustrated by numerical examples.


Introduction
The research on singular systems has attracted significant attention in the past years due to the fact that singular systems can better describe a larger class of physical systems such as robotic systems, electric circuits, and mechanical systems.When singular systems experience abrupt changes in their structures, it is natural to model them as singular Markovian jump systems [1,2].The analysis and synthesis of such class of systems have gained considerable attention because of their importance in applications (see, e.g., the literature [3][4][5][6][7][8][9][10] and the references therein).
Reachable set is one of the important techniques for parameter estimation or state estimation problems [11].Reachable set for a dynamic system is the set containing all the system states starting from the origin under bounded input disturbances.However, the exact shape of reachable sets of a dynamic system is very complex and hard to obtain; for this reason a number of researchers began to turn their attention to the reachable set estimation problem.The common strategies for reachable set estimation are ellipsoidal method [12] and polyhedron method [13].The main idea of these methods is to detect simple convex shapes like ellipsoid or polyhedron, which contains all the system states.Compared with polyhedron method, the primary advantage of ellipsoidal method is that the ellipsoid structure is simple and directly related to quadratic Lyapunov functions.As a result, linear matrix inequalities (LMIs) techniques can be used to determine bounding ellipsoids.In the framework of bounding ellipsoid, the reachable set estimation problem for linear time delay systems has received significant research attention in recent years.In [14] sufficient conditions for the existence of bounding ellipsoids containing the reachable set of continuous-time linear systems with time-varying delays were derived by using the Lyapunov-Razumikhin function.
In [15], by using Lyapunov-Krasovskii functional method, the author derived some less conservative conditions than those in [14].In [16] the reachable set of delayed systems with polytopic uncertainties was investigated by using the maximal Lyapunov-Krasovskii functional approach, and some new conditions bounding the set of reachable states are derived.Interesting results on reachable set of delayed systems with polytopic uncertainties can also be found in [17][18][19][20].In addition, some other strategies without using Lyapunov-Krasovskii functional have been provided to estimate the reachable set of continuous-time linear time-varying systems [21] and nonlinear time delay systems [22,23].The authors in [24] extended the ideas of reachable set estimation of continuous-time systems to discrete-time systems, wherein a fundamental result (Lemma 2.1 [24]) for the reachable set estimation of discrete-time systems was proposed.The authors in [25] improved the fundamental result obtained 2 Complexity in [24] and provided a basic tool (Lemma 4 [25]) for the reachable set estimation of discrete-time systems.On the basis of the general ideas proposed in [25], the reachable set estimation problem was also extended to some classes of complicated systems, such as singular systems [26], Markovian jump systems [27], switched linear systems [28], and T-S fuzzy systems [29,30].For the reachable set estimation of discrete-time systems, the other important contributions can be found in [31,32].On the other hand, the problem of controller design for specifications involved with the reachable set of a control system is also a very important issue [33].The controller design problems concerning reachable set were studied in [34] and [35] by using ellipsoidal method and polyhedron method, respectively.Two issues were raised in [34]: the first one is to design a controller such that the reachable set of the closed-loop system is contained in an ellipsoid, and the admissible ellipsoid should be as small as possible; the second one is to design a controller such that the reachable set of the closed-loop system is contained in a given ellipsoid.By constructing suitable Lyapunov-Krasovskii functional, LMI-based sufficient conditions for the existence of controller guaranteeing the ellipsoid bounds as small as possible have been derived for continuous-time delay systems [34] and discrete-time periodic systems [36].It is obvious that the LMI-based controller design is quite simple and numerically tractable.However, it should be pointed out that the reachable set estimation and synthesis problems of singular Markovian jump systems are much more difficult and challenging than that for nonsingular Markovian jump systems since the ellipsoid containing the reachable set is not directly related to quadratic Lyapunov functions.To the best of the authors' knowledge, no related results have been established for reachable set estimation and synthesis of singular Markovian jump systems, which has motivated this paper.
In this paper, we consider the problems of reachable set estimation and synthesis of singular Markovian jump systems.By using the Lyapunov approach, the estimation conditions on state reachable set and output reachable set are derived, respectively.Moreover, the desired state-feedback controller is designed based on the estimated reachable set.
Notation.Throughout this paper, R  denotes the dimensional Euclidean space;   represents the transpose of ; Sym() stands for  +   ;  > 0 (<0) means  is a symmetric positive (negative) definite matrix; E{⋅} refers to the expectation;  × denotes the matrix composed of elements of first  rows and  columns of matrix ; ‖ ⋅ ‖ refers to the Euclidean vector norm; the symbol " * " in LMIs denotes the symmetric term of the matrix;  is the unit matrix with appropriate dimensions.
In this paper we are interested in determining ellipsoids that contain, respectively, the state reachable set and output reachable set.In the reachable set analysis, it is required that systems should be asymptotically stable.When this requirement is not met, we will further design a statefeedback controller such that the reachable set of the closedloop system is contained in the smallest ellipsoid.
The state reachable set of the free system in (1) is defined by An ellipsoid E() bounding the reachable set can be always represented as follows: Particularly, when  =  for ∀ > 0, the ellipsoid E() will become a ball which is denoted by B().
Since rank() =  < , there exist two nonsingular matrices  and  such that Let . Then the free system can be rewritten as the following differential-algebraic form: The following definition and lemma are also useful in deriving the main results.
Definition 1 (see [2]).(I) The free system is said to be regular if det( −   ) is not identically zero for each  ∈ S.
(II) The free system is said to be impulse-free if deg(det( −   )) = rank() for each  ∈ S.

State Reachable Set Estimation.
In this subsection, we will focus our attention on determining a ball which contains the state reachable set of the free system.
Theorem 4. If there exist nonsingular matrices   ∈ R × and a scalar  > 0 such that the following LMIs hold for each  ∈ S, then the state reachable set of free system starting from the origin is mean-square bounded within the following set: where Proof.We first prove the regularity and nonimpulsiveness of the free system.Let Then, by (10), we obtain that  12 = 0. From (11), it is easy to show that Pre-and postmultiplying ( 14) by   and , respectively, we get where ⋆ will be irrelevant to the results of the following discussion; thus the real expressions of these two variables are omitted.It follows from (15) that which implies that  22 is nonsingular for each  ∈ S. Therefore, by Definition 1, we have that the free system is regular and nonimpulsive.
It should be noted that inequality (10) represents a nonstrict LMI.This may lead to numerical problems since equality constraints are usually not satisfied perfectly.Below, we will develop a numerically tractable and nonconservative LMI condition.Theorem 5.If there exist symmetric positive definite matrices   ∈ R × , nonsingular matrices   , and a scalar  > 0 such that the following LMI holds for each  ∈ S, then the state reachable set of free system is mean-square bounded within the following set: where ∈ R (−)× is any matrix with full row rank and satisfies  = 0;  ∈ R ×(−) is any matrix with full column rank and satisfies  = 0.
Remark 6.In order to make the ellipsoid E( P ) as small as possible, we require trace( P ) → max.For this purpose, we can add the additional requirement P >  and then maximize a positive scalar , which is equivalent to the following minimization problem: where  = 1/.

Output Reachable
then the output reachable set of free system is mean-square bounded within the following set: where Ψ  , , , and P are defined in Theorem 5.
Remark 8.The output reachable set is also expected to be as small as possible.To achieve this goal, we first solve LMI (30) and get P satisfying trace( P ) → max, which can be implemented by using (29).Then we add the additional requirement        >  and maximize a positive scalar , which is equivalent to the following minimization problem: where  = 1/.

State-Feedback Controller Design.
In this section, we turn our attention to the state-feedback control problem.Our goal here is to find a state-feedback controller, which not only stabilizes the closed-loop system, but also makes the ellipsoid bound on the reachable set of closed-loop system as small as possible.Now, consider the state-feedback controller () =   (), where   is a gain matrix to be determined later.By using this controller, the closed-loop system can be obtained as where Ã =   +     .
Theorem 9. Consider singular Markov jump system (1).If there exist nonsingular matrices   ∈ R × , matrices   , and scalars  > 0,   > 0 such that the following LMIs hold for each  ∈ S, then the reachable set of system ( 1) is mean-square bounded within the set , and the desired controller gain matrix is given by   =    −1  , where Proof.Denote   =  −1  and   =     for each  ∈ S.Then, pre-and postmultiplying (39) by diag(   , , ) and its transpose, respectively, we obtain where Using Lemma 2, we have From ( 41) and (42), it is easy to obtain that where . By Schur complement, the previous matrix inequality becomes where where Φ  = Sym( Ã    ) + ∑  =1       +     .Pre-and postmultiplying (37) by    and   , respectively, we obtain From the above discussion, we show that if (37)-(39) hold, then (45) and (46) hold.Thus, it follows from Theorem 4 that the closed-loop system can be stabilized by the designed statefeedback controller.

Numerical Examples
In this section, two numerical simulation examples are given to show the effectiveness of the main results derived above.
The switching between two modes is described by the following transition rate matrix: In this example, we choose  = ,  = .By solving optimization problem (29) with the aid of fminsearch, the minimal  and the corresponding  are 0.1312 and 1.7419, respectively.Using the above parameter values, we can obtain P1 = (1/r 2 1 ) = 2.1414 and P2 = (1/r 2 2 ) = 0.9992 by solving (26).Owing to Theorem 5, the state reachable set R  is meansquare bounded within the set B( P1 ) ∩ B( P2 ) = B( P1 ).
By applying Theorem 7, we have the following results: Therefore, the output reachable set of free system is meansquare bounded within the set B( 1 ) ∩ E( 2 ).
For simulation we assume that  0 = [0.20.1303]  and the disturbance is chosen as () = sin().A case for stochastic variation with transition rate matrix Π is shown in Figure 1.The state reachable set R  and the ball B( P2 ) are depicted in Figure 2. Figure 2 shows that the trajectory of the system is mean-square bounded within the region B( P2 ).
The output reachable set R  , the ball B( 1 ), and the ellipsoid E( 2 ) are depicted in Figure 3. Figure 3 shows that the output reachable set R  is mean-square bounded within the region B( (59) The switching between two modes is described by the following transition rate matrix: (62) The corresponding parameter values r2 1 and r2 2 are, respectively, 27.7747 and 314.7993, which imply  1 = (1/r 2 1 ) = 0.0360 and  2 = (1/r 2 2 ) = 0.0032.Applying this controller makes the state reachable set of closed-loop system (36) mean-square bounded within the region B( 1 ).
For the purpose of the simulation, we assume the initial condition  0 = [0.30.9763]  and the disturbance is chosen as () = sin(0.2).Figure 4 shows one possible switching between two modes.Figure 5 depicts the state reachable set of closed-loop system (36).

Conclusions
This paper has dealt with the problems of reachable set estimation and state-feedback controller design for singular Markovian jump systems.New sufficient conditions for the state reachable set estimation and output reachable set estimation have been, respectively, derived in terms of linear matrix inequalities.Based on the estimated reachable set, the state-feedback controller has also been designed.Numerical examples and simulation results have been provided to demonstrate the effectiveness of the proposed methods.

Figure 2 :Figure 3 :
Figure 2: The state reachable set R  and the bounding ball B( P1 ).