8.1 Introduction

In this chapter, we will study the SMC design problem for discrete-time switched hybrid systems with time-varying delay. First, we transform the original system into a new one with regular form, and then by designing a linear switching function, a reduced-order sliding mode dynamics, described by a switched state-delayed hybrid system, is generated. By utilizing the average dwell time approach and the piecewise Lyapunov function technique, a delay-dependent sufficient condition for the existence of the desired sliding mode is proposed in terms of LMIs, and an explicit parametrization of the desired switching surface is also given. Here, to reduce the conservativeness induced by the time delay in the system, both the slack matrix technique and the delay partitioning method are employed, which makes the proposed existence condition less conservative and more practical. The delay partitioning – also called delay fractioning – has been considered as an effective approach to reduce the conservativeness of the stability condition for time-delay systems. This was initially proposed by Gouaisbaut and Peaucelle in [86], and was then developed in [58, 73, 220, 271]. The basic idea of the delay partitioning method is to evenly partition the time delay into several components (this generally means time-invariant delay), and then take each time-delay component into account individually when constructing a Lyapunov function. In this chapter, the time delay considered is a time-varying one with a known lower bound. In this case, combining with construction of an appropriate Lyapunov–Krasovskii function, the delay partitioning method is used by evenly partitioning the lower bound into several components. It is shown that the conservativeness of the obtained existence condition becomes less as the partitioning gets thinner. Finally, a discontinuous SMC law is designed to drive the state trajectories of the closed-loop system onto a prescribed sliding surface in a finite time and maintained there for all subsequent time.

8.2 System Description and Preliminaries

Consider the following discrete-time switched state-delayed hybrid system:

(8.1a)

(8.1b)

where x(k) ∈ Rⁿ is the system state vector; u(k) ∈ R^m is the control input; f(x, k) ∈ R^p is the nonlinearity; is a family of matrices parameterized by an index set ; and (denoted by α for simplicity) is the switching signal defined as the same in Chapter 5. Also, are the initial conditions, and d(k) denotes the time-varying delays which satisfy h₁ ≤ d(k) ≤ h₂, where h₁ and h₂ are two positive constants representing its lower and upper bounds, respectively.

For each possible value α = i, , we denote the system matrices associated with mode i by A(i) = A(α), A_d(i) = A_d(α), and F(i) = F(α), where A(i), A_d(i), and F(i) are constant matrices. Moreover, we assume that (A(i), B) is controllable for each , and matrix B is of full column rank. For the nonlinearity f(x, k), we suppose that

where η(i) > 0 are scalars.

Since (A(i), B) is controllable, there exists a nonsingular matrix T such that

where B₁ ∈ R^{m × m} is nonsingular. For convenience, choose

where U₁ ∈ R^{n × m} and U₂ ∈ R^{n × (n − m)} are two sub-blocks of a unitary matrix resulting from the singular value decomposition of B, that is,

where Γ ∈ R^{m × m} is a diagonal positive-definite matrix and W ∈ R^{m × m} is a unitary matrix.

By state transformation z(k) = Tx(k), system (8.1a) takes the form

(8.2)

where and . Let with z₁(k) ∈ R^{(n − m)}, z₂(k) ∈ R^m, and

then (8.2) can be expressed in the following regular form:

(8.3)

where , , , , , , , (α) =U^T₂A_d (α)U₂, and B₁ = ΓW^T.

It is obvious that the first equation of system (8.3) represents the sliding motion dynamics of system (8.2), hence the corresponding sliding surface can be chosen as:

(8.4)

where C ∈ R^{m × (n − m)} is the parameter to be designed.

When the system state trajectories reach onto the sliding surface s(k) = 0, that is, z₂(k) = −Cz₁(k), the sliding mode dynamics is attained. Substituting z₂(k) = −Cz₁(k) into the first equation of system (8.3) gives the sliding mode dynamics as

(8.5)

Definition 8.2.1 The sliding mode dynamics (8.5) is said to be exponentially stable under α if the solution z₁(k) satisfies

where η ≥ 1 and 0 < ρ < 1 are two real constants, and

where ξ(k)≜z₁(k + 1) − z₁(k).

8.3 Main Results

8.3.1 Sliding Mode Dynamics Analysis

In this section, we analyze the stability for the sliding mode dynamics in (8.5), and present the following result.

Theorem 8.3.1 Given an integer m and a scalar β > 0, if there exist matrices P(i) > 0, Q_ϱ|κ(i) > 0, R(i) > 0, S(i) > 0, and Z(i) > 0, and matrices L(i) such that for ,

(8.6)

where

then the sliding mode dynamics in (8.5) is exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies that ,

(8.7)

Proof. Choose a Lyapunov function of the following form:

(8.8)

with

where P(α) > 0, Q_1|m(α) > 0, R(α) > 0, S(α) > 0, and Z(α) > 0 are real matrices to be determined, and

Then, as with the solution of (8.5) for a fixed α, we have

(8.9)

Moreover, for any matrix

and any matrices L(α) and Z(α), the following equations are true:

(8.10)

Considering (8.9)–(8.10) and denoting

we have

where Φ(α) is defined in (8.6).

Moreover, from (8.6), it follows that

Then it can be easily seen that

(8.11)

Now, for an arbitrary piecewise constant switching signal α, and for and k > 0, we let k₀ < k₁ < ⋅⋅⋅ < k_l < ⋅⋅⋅, l = 1, 2, …, denote the switching point of α over the interval (0, k). Therefore, for k ∈ [k_l, k_{l + 1}), it holds from (8.11) that

(8.12)

Using (8.7) and (8.8), at switching instant t_k, we have

(8.13)

Therefore, it follows from (8.12)–(8.13) and the relationship ϑ = N_α(0, k) ≤ (k − k₀)/K_α that

(8.14)

Notice from (8.8) that there exist two positive constants a and b (a ≤ b) such that

(8.15)

Combining (8.14) and (8.15) yields

(8.16)

Furthermore, letting , it follows that

(8.17)

By Definition 8.2.1, we know that if 0 < ρ < 1, that is, , the switched system (8.5) is exponentially stable, where function ceil(a) represents rounding real number a to the nearest integer greater than or equal to a. The proof is completed. ▀

Remark 8.1 The matrices Q_ϱ|κ(i) used in the above proof have two advantages: 1) the matrix for each part of the time partition can be chosen respectively according to the constraints of LMI, which decrease the conservativeness of our approach; and 2) it is simple to show the series of matrices from Q_ϱ(i) to Q_κ(i), which makes the result simpler and clearer. ♦

Remark 8.2 It should be pointed out that the switching function defined in (8.4) does not switch with the switching signal α. That is, we design C not C(α) in (8.4). In this way, we can avoid repetitive jumps of the state trajectories of the state components of the closed-loop system between sliding surfaces and hence the possible instability.

Now, we are on the path to solve the parameter matrices in (8.6). Considering the convenience of solving an LMI, more transformation has to be made to turn the inequality in (8.6) into an LMI, and the following theorem is obtained.

Theorem 8.3.2 For a given constant β > 0, suppose that there exist matrices , , , , and , and matrices , , such that for ,

(8.18)

where

Then the sliding mode dynamics in (8.5) is exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies

(8.19)

Moreover, if the conditions above are feasible, the matrix C in (8.4) is given by , that is, the switching function can be designed as

(8.20)

Proof. Defining the following matrices:

and performing a congruence transformation on (8.6) with , we have

(8.21)

Moreover, notice that

which implies

Thus, inequality in (8.21) holds if that in (8.18) holds. This completes the proof. ▀

Remark 8.3 It should be mentioned that the matrices P and in Theorem 8.3.2 do not depend on the switching signal α and are fixed. Since the designed sliding surface in (8.4) is parameter-independent, the parameter in (8.4) is guaranteed to be fixed if P and are fixed. ♦

8.3.2 SMC Law Design

In this section, we design an SMC law to drive the system state trajectories onto the sliding surface s(k) = 0, and have the following result.

Theorem 8.3.3 With the switching function given by (8.20), the state trajectories of the closed-loop system in (8.3) can be driven onto the sliding surface by the following control and finally converges into a residual set of the origin:

(8.22)

where Π is a positive definite matrix.

Proof. We will complete the proof by showing that the control law (8.22) can not only drive the system state trajectories onto the liner sliding surface, but also keeps it there for all subsequent time. From the sliding surface (8.4), we have

where .

Consider the following Lyapunov function:

(8.23)

Then the incremental ΔV(k) is

If ‖B^T₁s(k)‖ > ϵ, with the control law (8.23), holding that ‖B^T₁s(k)‖ ≤ |B₁^Ts(k)|, we have

If ‖B^T₁s(k)‖ ≤ ϵ, with the control law (8.22), we have

Since Π > 0 is to be tuned, an appropriate Π can be selected large enough such that ΔV(k) < 0 as long as s(k) is within a certain bounded region which contains an equilibrium point. Then Δs(k) is reasonably bounded, although it is not asymptotically convergent to zero, which shows that the state trajectories of (8.3) can be driven onto the sliding surface by the control law (8.22) and maintained there for all the subsequent time. This completes the proof. ▀

8.4 Illustrative Example

Consider system (8.1a)–(8.1b) with N = 2 and the following parameters:

and , where round(a) represents the nearest integer to number a. Some other parameters of the system are given as , β = 0.5, m = 3, μ = 1, h₁ = 3, and h₂ = 5. Using the LMI Toolbox in Matlab to solve conditions (8.18)–(8.19) in Theorem 8.3.2, we have and , thus,

Let the initial condition be (k = −5, −4, …, 0). The switching signal is shown in Figure 8.1, and the states of the closed-loop system are illustrated in Figure 8.2. Figure 8.3 depicts the control input, and the switching function is given in Figure 8.4 with Π = 3 and ϵ = 0.2.

8.5 Conclusion

In this chapter, the problem of SMC of a discrete-time switched hybrid system with time-varying delay has been investigated. Within the LMIs framework, a sufficient condition, which is dependent on the maximum and minimum delay bounds, has been established to guarantee the existence of a linear sliding surface. The conservativeness of the obtained results has been reduced by employing the delay partitioning method and the slack matrix technique. An SMC law has been designed to force the closed-loop system to be driven onto a prescribed sliding surface and maintained there for all subsequent time. Finally, a numerical example has been included to demonstrate the usefulness of the developed new design techniques.