2.1 Introduction

In this chapter, we are aiming at the investigation of state estimation and SMC problems for Markovian jump singular systems. Although there has been some existing work on the stability analysis of such systems based on the LMI technique, the results are not all of strict LMI form since there are usually some matrix equality constraints. This may cause a lot of trouble in checking the analysis results numerically. Therefore, a natural question is immediately raised: are there any techniques that can release the matrix equality constraints? In this chapter, we extend the approach proposed in [195] to the stability analysis of Markovian jump singular systems, and a new necessary and sufficient stability condition is established in terms of strict LMI. Also, the analysis and synthesis of singular systems have been extensively investigated in the past decades, but little progress has been made toward solving the SMC problem of singular systems. This problem may become difficult and complicated due to the singular matrix E in the systems. Since the rank of E may not be equal to that of B in a simple singular LTI system of , it is difficult to obtain the so-called ‘regular form’ through conventional model transformation approach. As a result, the linear sliding surface is not suitable for singular systems. Therefore, a key issue in the study of this problem is how to design a suitable sliding surface such that the resulting sliding mode dynamics exists.

In this chapter, a new integral-type sliding surface is designed by taking the singular matrix E into account. Then, by using the integral SMC technique, the sliding mode dynamics described by a Markovian jump singular differential equation can be derived. A necessary and sufficient condition for the stochastic stability of the sliding mode dynamics is presented in terms of strict LMI, by which the sliding surface can be designed. In practice, the system states are not always available owing to the limit of physical conditions or the expense of measuring it. Thus, the estimation problem has become more important in this case. In this chapter, we investigate the state estimation and SMC problems for Markovian jump singular systems with unmeasured states. An observer is first designed to estimate the system states, and then a discontinuous SMC law is synthesized based on feedback of the estimated states, which forces the system state trajectories onto the sliding surface in a finite time.

2.2 System Description and Preliminaries

Consider the continuous-time Markovian jump singular system described by

(2.1a)

(2.1b)

where x(t) ∈ Rⁿ is the state vector; u(t) ∈ R^m is the control input; y(t) ∈ R^p is the measured output. Matrix E ∈ R^{n × n} may be singular, and we assume that rank(E) = r ≤ n. A( · ), B( · ), and C( · ) are known real matrices with appropriate dimensions. These matrices are functions of r_t. Here, let {r_t, t ≥ 0} be a continuous-time Markov process which takes values in a finite state space , and the generator matrix Π = π_ij, with transition probability from mode i at time t to mode j at time t + Δ is given by

(2.2)

where Δ > 0 and lim_{Δ → 0}o(Δ)/Δ = 0; π_ij > 0, i ≠ j, and π_ii = −∑_{j ≠ i}π_ij for each .

For each possible value , A(r_t) = A_i, B(r_t) = B_i, and C(r_t) = C_i. Then, the system (2.1a)–(2.1b) can be described by

(2.3a)

(2.3b)

The following preliminary assumption is made for system (2.3a)–(2.3b).

Assumption 2.1 For each , the pair (A_i, B_i) in (2.3a)–(2.3b) is controllable, the pair (A_i, C_i) is observable, and matrix B_i is full column rank.

Before proceeding, we first consider the unforced system of (2.3a), that is,

(2.4)

Definition 2.2.1

• I. The Markovian jump singular system in (2.4) is said to be regular if is not identically zero for each .
• II. The Markovian jump singular system in (2.4) is said to be impulse free if deg for each .
• III. The Markovian jump singular system in (2.4) is said to be stochastically stable if, for any x₀ ∈ Rⁿ and , there exists a positive scalar T(x₀, r₀) such that
  
• IV. The Markovian jump singular system in (2.4) is said to be stochastically admissible if it is regular, impulse free and stochastically stable.

The following lemma provides a necessary and sufficient condition for the stochastic admissibility of the Markovian jump singular system in (2.4).

Lemma 2.2.2 [244] The Markovian jump singular system in (2.4) is stochastically admissible if and only if there exist nonsingular matrices P_i such that for ,

(2.5a)

(2.5b)

Remark 2.1 Notice that the conditions in Lemma 2.2.2 are not all of strict LMI form due to the matrix equality constraint of (2.5a). This may cause major problems in checking the conditions numerically, since the matrix equality constraint is fragile and is not usually perfectly satisfied. Therefore, the strict LMI conditions are more desirable than non-strict ones from the numerical point of view. ♦

2.3 Stochastic Stability Analysis

In the section, we propose a strict LMI condition (easy to check by using standard software) of the stochastic admissibility for the Markovian jump singular system in (2.4), and present the following result.

Theorem 2.3.1 The Markovian jump singular system in (2.4) is stochastically admissible if and only if there exist matrices X_i > 0, Y_i, U, and W such that for ,

(2.6)

where U ∈ R^{(n − r) × n} and W ∈ R^{n × (n − r)} are matrices satisfying UE = 0 and EW = 0.

Proof. (Sufficiency) Letting P_i≜X_iE + U^TY_iW^T, in (2.6), we can satisfy (2.5a) and (2.5b). Thus, according to Lemma 2.2.2 we know that the continuous Markovian jump singular system in (2.4) is stochastically admissible.

(Necessity) Suppose that the system in (2.4) is stochastically admissible, then there exist nonsingular matrices M and N such that, for each ,

(2.7)

Since the system in (2.4) is regular and impulse free we have that matrices A_4i are nonsingular for . Thus, we can set

Then, it follows that

(2.8)

(2.9)

where . Therefore, it is easy to see that the stochastic stability of system (2.4) implies that the following continuous Markovian jump system is stochastically stable:

It follows that there exist matrices such that, for ,

Now, let and , thus by (2.9) we have , where the partitions of N and are compatible for algebraic operations. Therefore, for a sufficient small α > 0, we have

(2.10)

By Schur complement, (2.10) is equivalent to

(2.11)

where

Furthermore, (2.11) can be rewritten as

(2.12)

Considering (2.7), it follows from (2.12) that

(2.13)

Let in (2.13) (obviously, X_i > 0), we have

Since is of full row rank, it can be written as , where is nonsingular (thus, by (2.8) implies UE = 0). Then, defining and W≜N₂ (it is easily seen from (2.7) that EW = EN₂ = 0), we have

which is equivalent to (2.6). This completes the proof. ▀

2.4 Main Results

In this section, we consider the state estimation and SMC problems for the Markovian jump singular systems with unmeasured states in (2.1a)–(2.1b). First, we design an observer to estimate unmeasured states, and then we design a sliding surface and an SMC law based on the state estimates. The designed observer-based SMC law can drive the system state trajectories onto the predefined sliding surface in a finite time.

2.4.1 Observer and SMC Law Design

The following observer is employed to provide the estimates of the unmeasured states for the system in (2.3a)–(2.3b):

(2.14a)

(2.14b)

where represents the estimate of x(t), and L_i ∈ R^{n × p} are the observer gains to be designed.

Let denote the estimation error. Considering (2.3a)–(2.3b) and (2.14a)–(2.14b), the estimation error dynamics is obtained as

(2.15)

Design the following integral switching function:

(2.16)

where K_i ∈ R^{m × n} are real matrices to be designed such that

(2.17)

is stochastically admissible. The matrices G_i ∈ R^{m × n} are to be chosen so that G_iB_i are nonsingular.

Design the following state estimate-based SMC law:

(2.18)

where ϵ > 0 is a real constant and

The following theorem shows that the sliding motion in the specified sliding surface s(t) = 0 is attained in a finite time.

Theorem 2.4.1 Under the SMC law (2.18), the state trajectories of the observer dynamics (2.14a)–(2.14b) can be driven onto the sliding surface s(t) = 0 in a finite time and remain there in subsequent time.

Proof. Choose the following Lyapunov function:

where Z_i > 0 are matrices to be specified such that B^T_iZ_iB_i > 0. Thus, we choose G_i = B^T_iZ_i in (2.16), and then G_iB_i = B^T_iZ_iB_i > 0 are nonsingular. According to (2.14a)–(2.14b) and (2.16), we have

(2.19)

Substituting (2.18) into (2.19) yields

Thus taking the derivation of V(t) and considering |s(t)| ≥ ‖s(t)‖, we have

(2.20)

where . It can be seen from (2.20) that there exists a time such that V(t) = 0, and consequently s(t) = 0, for t ≥ t*, which means that the system state trajectories can reach onto the predefined sliding surface in a finite time. This completes the proof. ▀

2.4.2 Sliding Mode Dynamics Analysis

When the system operates in the sliding mode, it follows that s(t) = 0 and . Thus, by in (2.19), we can obtain the equivalent control u_eq(t) as

(2.21)

Substituting (2.21) into (2.14a), the sliding mode dynamics can be obtained as

(2.22)

In the following, we will analyze the stochastic admissibility of the estimation error dynamics in (2.15). By Theorem 2.3.1, we give the following result.

Theorem 2.4.2 The estimation error dynamics in (2.15) is stochastically admissible if and only if there exist symmetric positive definite matrices X_i ∈ R^{n × n}, nonsingular matrices Y_i ∈ R^{(n − r) × (n − r)}, and matrices , , U ∈ R^{(n − r) × n}, W ∈ R^{n × (n − r)} such that for ,

(2.23)

where U and W are matrices satisfying UE = 0 and EW = 0. Moreover, the parametric matrices L_i can be computed by

(2.24)

Proof. According to Theorem 2.3.1, we know that the estimation error dynamics in (2.15) is stochastically admissible if and only if there exist matrices X_i > 0,Y_i, U and W such that for ,

(2.25)

Letting and in (2.25) yields (2.23), thus the proof is completed. ▀

Next, we shall analyze the stochastic admissibility of the dynamics in (2.17), and give a solution to parameter K_i. Before proceeding, we give the following lemma.

Lemma 2.4.3 Let X_i be symmetric such that E^T_LX_iE_L > 0 and matrices Y_i are nonsingular, then X_iE + U^TY_iW^T are nonsingular and their inverse are expressed as

(2.26)

where are symmetric matrices and are nonsingular matrices with

Proof. Decompose E as E = E_LE^T_R, where E_L ∈ R^{n × r} and E_R ∈ R^{n × r} are of full column rank. Since UE = 0 and EW = 0, thus we have that UE_L = 0, E^T_RW = 0 and is nonsingular. Then,

(2.27)

According to (2.27), we have

which implies that X_iE + U^TY_iW^T are nonsingular and

(2.28)

where

Define and in (2.28), and we have (2.26)–(2.27). This completes the proof. ▀

Now, according to Theorem 2.3.1, we present the following result without proof.

Theorem 2.4.4 The dynamics in (2.17) is stochastically admissible if and only if there exist matrices X_i > 0, Y_i, U and W such that for ,

(2.29)

where U ∈ R^{(n − r) × n} and W ∈ R^{n × (n − r)} are matrices satisfying UE = 0 and EW = 0.

The following sufficient condition is proposed for the stochastic admissibility of the dynamics in (2.17), by which the parametric matrices K_i can be solved.

Theorem 2.4.5 The dynamics in (2.17) is stochastically admissible if there exist symmetric positive definite matrices , nonsingular matrices , and matrices , , U ∈ R^{(n − r) × n}, and W ∈ R^{n × (n − r)} such that for ,

(2.30)

where and

where U and W are matrices satisfying UE = 0 and EW = 0. Moreover, the parametric matrices K_i are given by

(2.31)

Proof. By Theorem 2.4.4 we know that the dynamics in (2.17) is stochastically admissible if there exist matrices X_i > 0 and nonsingular matrices Y_i such that (2.29) holds for . However, by Lemma 2.4.3, X_iE + U^TY_iW^T are nonsingular and their inverse matrices are . Now, performing a congruence transformation to (2.29) by , we have

(2.32)

Letting and in (2.32), we have

(2.33)

Also, the following fact is true:

Considering π_ii < 0 in (2.2), we have

Therefore, (2.33) holds if the following inequality holds:

(2.34)

where Ψ_11i is defined in (2.30). By Schur complement, LMI (2.30) implies inequality (2.34). This completes the proof. ▀

Remark 2.3 Notice from Definition 2.2.1 that the stochastic admissability implies the stochastic stability of a Markovian jump singular system. Thus, we know that the estimation error dynamics in (2.15) is stochastically stable if LMI (2.23) in Theorem 2.4.2 holds. Also, the dynamics in (2.17) is stochastically stable if LMI (2.30) in Theorem 2.4.5 holds. It is not difficult to show from stochastic stability of dynamics (2.15) and (2.17) that the sliding mode dynamics (2.22) is stochastically stable. ♦

2.5 Illustrative Example

Example 2.5.1 Consider the Markovian jump singular system in (2.1a)–(2.1b) with two operating modes, that is, N = 2, and the following parameters:

Our aim is to design an observer in the form of (2.14a)–(2.14b) to estimate the states of system (2.1a)–(2.1b), and then synthesize an SMC law u(t) as (2.18) (based on the state estimate) such that the closed-loop system is stochastically admissible.

Solving the LMI condition (2.23) in Theorem 2.4.2 by using LMI Toolbox in the Matlab environment and then by (2.24), we have

However, solving the LMI condition (2.30) in Theorem 2.4.5, and then by (2.31) we have

Here, parameters G₁ and G₂ in (2.16) are chosen as

Thus, the switching function in (2.16) can be computed as

Let the adjustable parameter ϵ be ϵ = 0.5, then the observer-based SMC law designed in (2.18) can be obtained as

where .

To prevent the control signals from chattering, we replace sign(s_i(t)) with

For given initial condition of , the simulation results are given in Figures 2.1–2.2. Specifically, in Figure 2.1 shows the states of the closed-loop system, while Figure 2.2 depicts the switching function s(t).

2.6 Conclusion

In this chapter, the state estimation and SMC problems have been investigated for continuous-time Markovian jump singular systems with unmeasured states. First, we have proposed a strict LMI necessary and sufficient condition of the stochastic admissibility for the unforced Markovian jump singular systems. Then, an observer has been designed and an observer-based sliding mode controller has been synthesized to guarantee the reachability of the system state trajectories to the predefined integral sliding surface. Finally, a numerical example has been provided to illustrate the effectiveness of the proposed design scheme.