3.1 Introduction

It is recognized that the sliding mode of an SMC system is invariant to system perturbations and external disturbances, only if the perturbations/disturbances satisfy the so-called ‘matching condition’. Although many researchers have paid considerable attention to the mismatched uncertainties in SMC design, the obtained results are very conservative. If the undesired uncertainties/disturbances can not be eliminated in the sliding mode, it is possible to attenuate its effect on the system performance. In this chapter, we will consider the disturbance attenuation problem in sliding mode with performance. For this purpose, we design an integral switching function. The plant considered in this chapter is the Markovian jump singular time-delay system, which is a typical kind of hybrid systems of high complexity (including system parameter jumping, time delay in states, and singularity). How to establish a less conservative stability condition is a key issue in SMC design. As is well known, the slack matrix technique [228, 243], usually combined with the Lyapunov–Krasovskii approach, has been proved to be an effective tool to establish less conservative stability conditions for time-delay systems. Unfortunately, little progress has been made in dealing with singular time-delay systems by this technique, probably due to the particularity and complexity caused by the singular matrix E, thus it is difficult to choose a suitable Lyapunov–Krasovskii function.

In this chapter, we will pay particular attention to the singular matrix E in the design of an integral-type switching function, which leads to a full-order Markovian jump singular time-delay system for describing the sliding mode dynamics. We will then apply the slack matrix technique combining with the Lyapunov–Krasovskii approach to derive a delay-dependent sufficient condition, which guarantees that the sliding mode dynamics is stochastically stable with a bounded gain performance. In addition, the analysis result and the solvability condition for the desired switching function are both established. All the obtained results are in terms of strict LMI, which can be solved by efficient interior-point algorithms [25]. Finally, a discontinuous SMC law is designed to drive the system state trajectories onto the predefined sliding surface in a finite time.

3.2 System Description and Preliminaries

Consider a Markovian jump singular time-delay system described by

(3.1a)

(3.1b)

(3.1c)

where {r_t, t ≥ 0} is a continuous-time Markov process on the probability space which has been defined in (2.2) of Chapter 2, and x(t) ∈ Rⁿ is the system state vector; u(t) ∈ R^m is the control input; z(t) ∈ R^p is the controlled output; ω(t) ∈ R^q is the exogenous input (which represents either the exogenous disturbance input or the exogenous reference input) belonging to . Matrix E ∈ R^{n × n} may be singular, and it is assumed that rank(E) = r ≤ n. A( · ), B( · ), C( · ), A_d( · ), C_d( · ), B_w( · ) and D_w( · ) are known real matrices with appropriate dimensions. d represents the constant time-delay and is a compatible vector-valued initial function. In addition, f(x(t), t) ∈ R^m is an unknown nonlinear function (which represents the unmodeled dynamics of a physical plant), and there exists a known constant η > 0 such that

For each , A(r_t) = A_i, B(r_t) = B_i, C(r_t) = C_i, A_d(r_t) = A_di, C_d(r_t) = C_di, B_w(r_t) = B_wi, and D_w(r_t) = D_wi. Then, system (3.1a)–(3.1c) can be described by

(3.2a)

(3.2b)

(3.2c)

Assumption 3.1 For each , the pair (A_i, B_i) in (3.2a) is controllable, and matrix B_i is full column rank.

Before proceeding, we first consider the unforced system of (3.2a)–(3.2c), that is,

(3.3a)

(3.3b)

We introduce the following definition for the Markovian jump singular time-delay system in (3.3a)–(3.3b).

Definition 3.2.1

• I. The Markovian jump singular time-delay system in (3.3a)–(3.3b) is said to be regular and impulse free if the pairs (E, A_i) and (E, A_i + A_di) are regular and impulse free for each .
• II. The Markovian jump singular time-delay system in (3.3a)–(3.3b) is said to be stochastically stable if for any x₀ ∈ Rⁿ and , there exists a positive scalar T(x₀, φ( · )) such that
  
• III. The Markovian jump singular time-delay system in (3.3a)–(3.3b) is said to be stochastically admissible if it is regular, impulse free and stochastically stable.

In addition, we introduce the following definition for the Markovian jump singular time-delay system of

(3.4a)

(3.4b)

(3.4c)

Definition 3.2.2 Given a scalar γ > 0, the Markovian jump singular time-delay system in (3.4a)–(3.4c) is said to be stochastically admissible with a bounded gain performance γ, if the system (3.4a)–(3.4c) with ω(t) ≡ 0 is stochastically admissible, and under zero condition, for nonzero , it holds that

(3.5)

3.3 Bounded Gain Performance Analysis

This section is concerned with the bounded gain performance analysis for the Markovian jump singular time-delay system in (3.4a)–(3.4c) in the sense of Definition 3.2.2, and we give the following theorem.

Theorem 3.3.1 Given a scalar γ > 0, the Markovian jump singular time-delay system in (3.4a)–(3.4c) is stochastically admissible with bounded gain performance γ, if there exist matrices Q > 0, R > 0, , (with 0 < P_11i ∈ R^{r × r} and W_1i ∈ R^{n × r}) such that for each ,

(3.6)

where

Proof First, we consider the nominal case of (3.4a)–(3.4c), that is, ω(t) = 0 in (3.4a)–(3.4c). Without loss of generality, we assume that the matrix E and the state vector x(t) in (3.4a)–(3.4c) have the form of

where x₁(t) ∈ R^r and x₂(t) ∈ R^{n − r}.

In the following, we will consider the stochastic stability of the system in (3.4a)–(3.4c) with ω(t) ≡ 0. To this end, we choose a Lyapunov function as

(3.7)

where x_t≜x(θ), θ ∈ [t − 2d, t], thus {(x_t, r_t), t ≥ d} is a Markov process with initial condition (φ( · ), r₀). Matrices Q and R are positive definite, and

with P₁₁(r_t) > 0 and P₂₁(r_t) = 0 (which can be found from ). Let be the weak infinitesimal generator of the random process {x_t, r_t}. Thus, for each possible value and t ≥ d, we have

(3.8)

On the other hand, Newton–Leibniz formula gives

Thus, for with W_1i ∈ R^{n × r}, it holds that

(3.9)

By (3.8)–(3.9) and noting

we have

(3.10)

where and

with

By Schur complement, LMI (3.6) implies Φ_i < 0. Moreover, noting that the last integral term in (3.10) is semi-positive, thus (3.10) implies that there exists a scalar ϵ > 0 such that for each ,

The rest of the proof on stochastic stability can be found in [230, 231], and so we omit it here.

Moreover, (3.6) implies Ψ_11i < 0. Now partition matrices A_i and Q as

and then substituting them and P_i into Ψ_11i < 0 yields

which implies that A_22i are nonsingular for , thus the pairs (E, A_i) are regular and impulse free for . Therefore, the system in (3.4a)–(3.4c) with ω(t) = 0 is regular and impulse free.

Now, we establish the bounded gain performance of system (3.4a)–(3.4c). Consider the Lyapunov function in (3.7) again and the following index:

Then, under zero initial condition, it can be shown that for any nonzero ,

where and

By Schur complement, LMI (3.6) implies , thus J ≤ 0, and hence (3.5) is true for any nonzero . This completes the proof. ▀

Remark 3.1 It should be noted that Theorem 3.3.1 presents a delay-dependent sufficient condition of the stochastic admissibility with the bounded gain performance defined in Definition 3.2.2 for the Markovian jump singular time-delay system in (3.4a)–(3.4c). Notice that the slack matrix variables W_i are introduced in the derivation of the delay-dependent result in Theorem 3.3.1, which avoids some conservativeness caused by the commonly used model transformation approach when dealing with time-delay systems. ♦

3.4 Main Results

3.4.1 Sliding Mode Dynamics Analysis

We design the following integral-type switching function:

(3.11)

where G_i ∈ R^{m × n} and K_i ∈ R^{m × n} are real matrices. In particular, the matrices G_i are to be chosen such that G_iB_i are nonsingular for . The solution of Ex(t) can be given by

(3.12)

It follows from (3.11) and (3.12) that

(3.13)

When the system state trajectories reach onto the sliding surface, it follows that s(t) = 0 and . Therefore, by , we get the equivalent control as

(3.14)

By substituting (3.21) into (3.2a)–(3.2c), the sliding mode dynamics can be obtained as

(3.15)

For notational simplicity, we define

Thus, the sliding mode dynamics in (3.15) and the controlled output equation in (3.2b) can be formulated as

(3.16a)

(3.16b)

The above analysis gives the first step of the SMC for the Markovian jump singular time-delay system in (3.1a)–(3.1c). Specifically, we design an integral-type switching function as (3.11) so that the dynamics restricted to the sliding surface (i.e. the sliding mode dynamics) has the form of (3.16a)–(3.16b). Thus, the remaining problems to be addressed in this chapter are as follows:

• performance analysis of the sliding mode dynamics. Given all the system matrices in (3.1a)–(3.1c) and the matrices G_i and K_i in the switching function of (3.11), determine under what condition the sliding mode dynamics in (3.16a)–(3.16b) is stochastically admissible with a bounded gain performance defined in the sense of Definition 3.2.2.
• SMC law synthesis. Synthesize an SMC law to drive the system state trajectories onto the predefined sliding surface s(t) = 0 in a finite time and maintain them there for all subsequent time.

By Theorem 3.3.1, we have the following result for dynamics (3.16a)–(3.16b).

Corollary 3.4.1 Given a scalar γ > 0, the sliding mode dynamics in (3.16a)–(3.16b) is stochastically admissible with bounded gain performance γ, if there exist matrices , , (with ), such that for each ,

(3.17)

where

The following theorem is devoted to solving the parameter K_i in the switching function of (3.11).

Theorem 3.4.2 Given a scalar γ > 0, the sliding mode dynamics in (3.16a)–(3.16b) is stochastically admissible with bounded gain performance γ, if there exist matrices , , , (with ), and a scalar σ > 0 such that for each ,

(3.18)

where

Moreover, if the above LMI conditions have a set of feasible solutions then the parametric matrices K_i in (3.11) can be computed by

(3.19)

Proof Letting , , and , and performing a congruence transformation on (3.17) by , we have

(3.20)

where

and is defined in (3.18).

Also notice that

which implies

Similarly, we have

Moreover, noting that , it follows that there exists a sufficient small scalar σ > 0 such that , where

thus,

Therefore, inequality (3.20) holds if

where , , and are defined in (3.18), and

Moreover, letting and by Schur complement, the above inequality is equivalent to (3.18). This completes the proof. ▀

Remark 3.2 Note that we introduced the term in the proof of Theorem 3.4.2. The reason is that is singular (and there is no inversion), while is nonsingular and thus it has inversion.♦

3.4.2 SMC Law Design

In this section, we shall synthesize a discontinuous SMC law, by which the state trajectories of the Markovian jump singular time-delay system in (3.2a)–(3.2c) can be driven onto the predefined sliding surface s(t) = 0 in a finite time and then maintained there for all subsequent time.

Theorem 3.4.3 Consider the Markovian jump singular time-delay system in (3.2a)–(3.2c). Suppose that the switching function is designed as (3.11) with K_i being solvable by (3.19), and matrices G_i in (3.11) are chosen such that G_iB_i are nonsingular. Then, the state trajectories of system (3.2a)–(3.2c) can be driven onto the sliding surface s(t) = 0 in a finite time by the following SMC law:

(3.21)

where ϱ is a positive constant which is adjustable, and μ is a positive constant which satisfies

Proof Suppose matrices G_i are chosen such that G_iB_i are nonsingular. Choose the following Lyapunov function:

According to (3.13), we have

(3.22)

Thus, taking the derivative of W(t), and considering (3.22) and the SMC law designed in (3.21), we have

(3.23)

Substituting (3.21) into (3.23) and noting ‖B^T_iG_i^Ts(t)‖ ≤ |B^T_iG_i^Ts(t)|, we have

(3.24)

where

It can be seen from (3.24) that there exists a time such that W(t) = 0, and consequently s(t) = 0, for t ≥ t*. This means that the system state trajectories can reach onto the predefined sliding surface in a finite time, thereby completing the proof.

3.5 Illustrative Example

Example 3.5.1 Consider Markovian jump singular time-delay system (3.1a)–(3.1c) with two operating modes, that is, N = 2 and the following parameters:

In addition, f(x(t), t) = 0.78exp ( − t)sin (t)x₁(t) (so η can be chosen as η = 0.78), the time delay d = 0.5, and the disturbance input ω(t) = 1/(1 + t²).

Our aim here is to verify the effectiveness of the proposed theoretical results in the previous sections. By solving the LMI condition (3.18) in Theorem 3.4.2 by using LMI-Toolbox in the Matlab environment and then considering (3.19), we have

Here, parameters G₁ and G₂ in (3.11) are chosen as

Thus, the switching function in (3.11) is

Let ϱ = 0.7748, then the SMC law designed in (3.21) can be computed as

where

For a given initial condition of , t ∈ [ − 0.5, 0], the simulation results are given in Figures 3.1–3.2. Specifically, Figure 3.1 shows the states of the closed-loop system, and Figure 3.2 depicts the switching function s(t).

3.6 Conclusion

In this chapter, we have investigated the problems of the bounded gain performance analysis and the SMC of continuous-time Markovian jump singular time-delay systems. The major theoretical findings are as follows. First, the delay-dependent sufficient condition in the form of LMI has been established so as to ensure that the sliding mode dynamics is stochastically admissible with a bounded gain performance. An integral-type switching function has been designed, and then the condition that enables us to solve the parameter in the switching function has been derived. Furthermore, it has been shown that, by synthesizing an SMC law, the system state trajectories can be driven onto the predefined sliding surface in a finite time. Finally, the usefulness of the proposed theory has been verified by the numerical results.