10.1 Introduction

In this chapter, we will study the problems of stability and stabilization, and the control of discrete-time switched stochastic hybrid systems with time-varying delays. Similar to Chapter 9, the main results proposed in this chapter are obtained by employing the average dwell time approach and the piecewise Lyapunov function technique. However, the development of such methods in the continuous and discrete cases has certain technical differences. A sufficient condition, which guarantees the considered system mean-square exponentially stable, is first proposed in terms of LMIs, and by this the stabilization problem is then solved. The weighted performance condition is then also established, and control is designed. It is shown that the control problem can be converted into a convex optimization problem with a set of LMI constraints which can be solved by applying interior-point algorithms.

10.2 System Description and Preliminaries

Consider the discrete-time switched stochastic hybrid system with time delays, which is described by the following dynamical equations:

(10.1a)

(10.1b)

(10.1c)

where x(k) ∈ Rⁿ is the system state vector; u(k) ∈ R^m represents the control input; ω(k) ∈ R^p is the noise signal that belongs to ℓ₂[0, +∞); z(k) ∈ R^q is the controlled output; and ϖ(k) is a zero-mean real scalar process on a probability space relative to an increasing family of σ-algebras generated by (ϖ(k))_{k ∈ N}. The stochastic process {ϖ(k)} is independent, which is assumed to satisfy E{ϖ(k)} = 0 and E{ϖ²(k)} = δ, k = 0, 1, …, where δ > 0 is a known scalar. In addition, φ(k) denotes the initial conditions and (denoted by α for simplicity) is a switching signal, which was defined in the same way in Chapter 5. Here, we assume that the switch signal α(k) has an average dwell time. The time-varying delay d(k) satisfies 1 ≤ d₁ ≤ d(k) ≤ d₂, where d₁ and d₂ are two constant positive scalars representing its lower and upper bounds, respectively.

Assumption 10.1 For the nonlinear function f(x): Rⁿ → Rⁿ, there exist matrices Π₁ and Π₂ such that

(10.2)

We design a stabilization controller and an state feedback controller with the following general structure:

(10.3)

where K(α) ∈ R^{m × n} are parameter matrices to be designed. Substituting the controller u(k) into system (10.1a)–(10.1c), we obtain the closed-loop stabilization system as

(10.4)

and the closed-loop control system as

(10.5a)

(10.5b)

Remark 10.1 For each possible value α = i, , we will denote the system matrices associated with mode i by A(i) = A(α), A_d(i) = A_d(α), A_τ(i) = A_τ(α), C(i) = C(α), C_d(i) = C_d(α), B_u(i) = B_u(α), B_ω(i) = B_ω(α), D_ω(i) = D_ω(α), L(i) = L(α), and K(i) = K(α), where A(i), A_d(i), A_τ(i), C(i), C_d(i), B_u(i), B_ω(i), D_ω(i), L(i), and K(i) are constant matrices. ♦

Definition 10.2.1 The discrete-time switched stochastic hybrid system in (10.1a) with u(k) = 0 and ω(k) = 0 is said to be mean-square exponentially stable under α if the solution x(k) satisfies

for constants η ≥ 1 and 0 < ρ < 1, and

where ς(θ)≜x(θ + 1) − x(θ).

Definition 10.2.2 For 0 < β < 1 and γ > 0, the system in (10.1a)–(10.1c) with u(k) = 0 is said to be mean-square exponentially stable with a weighted performance level γ under α, if it is mean-square exponentially stable with ω(k) = 0, and under zero initial condition, it holds for all nonzero ω(k) ∈ ℓ₂[0, ∞) that

(10.6)

10.3 Stability Analysis and Stabilization

In this section, we apply the average dwell time approach combined with the piecewise Lyapunov function technique to investigate the mean-square exponential stability and stabilization problems for the system (10.1a).

Theorem 10.3.1 Given a constant 0 < β < 1, suppose that there exist matrices P(i) > 0, Q(i) > 0, and R(i) > 0 such that for ,

(10.7)

where

Then the discrete-time switched stochastic time-delay system in (10.1a) with u(k) = 0 and ω(k) = 0 is mean-square exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies

(10.8)

Moreover, an estimate of the state decay is given by

(10.9)

where d≜ − max {τ, d₂} and

(10.10)

Proof. Choose a Lyapunov function of the following form:

(10.11)

where P(α) > 0, Q(α) > 0, and R(α) > 0 are real matrices to be determined.

For k ∈ [k_l, k_{l + 1}), define E{ΔV_j(x, α)}≜E{V_j(x(k + 1), α) − V_j(x(k), α)}, j = 1, 2, 3, 4, and thus we have with

(10.12)

(10.13)

(10.14)

(10.15)

Moreover, Assumption 10.1 gives

(10.16)

where H₁ and H₂ are defined in (10.7) of Theorem 10.3.1.

Considering (10.12)–(10.15), and (10.16), we have

(10.17)

where

and Φ(α) is defined as

where Φ₁₁(α)≜ − βP(α) + β(d₂ − d₁ + 1)Q(α) − H₁.

Moreover, by Schur complement to (10.7) it follows that Φ(α) < 0, then one can easily obtain

(10.18)

Now, for an arbitrary piecewise constant switching signal α, and for any k > 0, we let k₀ < k₁ < ⋅⋅⋅ < k_l < ⋅⋅⋅, l = 1, …, denote the switching points of α over the interval (0, k). As mentioned earlier, the i_lth subsystem is activated when k ∈ [k_l, k_{l + 1}). Therefore, for k ∈ [k_l, k_{l + 1}), it holds from (10.18) that

(10.19)

Using (10.8) and (10.11), we have

(10.20)

Therefore, it follows from (10.19)–(10.20) and the relationship ϑ = N_α(k₀, k) ≤ (k − k₀)/T_a that

(10.21)

Notice from (10.11) that there exist two positive constants a and b (a ≤ b) such that

(10.22)

Combining (10.21) and (10.22) yields

(10.23)

Furthermore, letting , it follows that

(10.24)

By Definition 10.2.1, we know that if 0 < ρ < 1, that is, , the discrete-time switched stochastic time-delay system in (10.1a) with u(k) = 0 and ω(k) = 0 is mean-square exponentially stable, where function ceil(h) represents the rounding real number h to the nearest integer greater than or equal to h. The proof’ is completed. ▀

Remark 10.2 In Theorem 10.3.1, we propose a sufficient condition for the mean-square exponential stability condition for the considered discrete-time switched stochastic time-delay system in (10.1a) with u(k) = 0 and ω(k) = 0. Here, the parameter β plays a key role in controlling the lower bound of the average dwell time, which can be seen from ; specifically, if β is a smaller value, the lower bound of the average dwell time becomes smaller with a fixed μ, which may result in the instability of the system. ♦

Remark 10.3 Note that when μ = 1 in we have T_a > T*_a = 0, which means that the switching signal α_k can be arbitrary. In this case, (10.8) turns out to be P(i) = P(j) = P, Q(i) = Q(j) = P, R(i) = R(j) = P, , and the proposed approach becomes a quadratic one thus conservative. In this case, the system in (10.1a) with u(k) = 0 and ω(k) = 0 turns out to be a discrete-time stochastic system with time delays. However, when β = 1 in , we have T_a = ∞, that is, there is no switching. ♦

Theorem 10.3.2 Given a constant 0 < β < 1, suppose that there exist matrices X(i) > 0, Z(i) > 0, R(i) > 0, and Y(i) such that for ,

(10.25)

where

Then the closed-loop stabilization system in (10.4) is mean-square exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies

(10.26)

In this case, a robustly stabilizing state feedback controller can be chosen by

(10.27)

Proof. By performing a congruence transformation on (10.7) with matrix diag{X(i), X(i), . .I, I, X(i), X(i)} (where X(i) = P^{− 1}(i)), it follows that

(10.28)

where

However, the following matrix inequality holds:

thus,

Therefore, matrix inequality (10.28) holds if the following LMI holds:

(10.29)

where is defined in (10.25). From Theorem 10.3.1 and the above derivation we know that the discrete-time switched stochastic time-delay system in (10.1a) with ω(k) = 0 is mean-square exponentially stabilizable, that is, the closed-loop stabilization system in (10.4) is mean-square exponentially stable if the matrix inequality, that is, (10.29) with A(i) replacing by A(i) + B_u(i)K(i)) holds.

Furthermore, we define Y(i) = K(i)X(i), we have (10.25), and we know that K(i) = Y(i)X^{− 1}(i). The proof is completed. ▀

10.4 Control

In this section, we will investigate the weighted performance for system (10.1a)–(10.1c) with u(k) = 0. A sufficient condition of the weighted performance will be established, and based on which the controller will be synthesized.

Theorem 10.4.1 For given constants β > 0 and γ > 0, suppose that there exist matrices P(i) > 0, Q(i) > 0, and R(i) > 0 such that for ,

(10.30)

where

Then the system in (10.1a)–(10.1c) with u(k) = 0 is mean-square exponentially stable with a weighted performance level γ for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies (10.11).

Proof. The proof of mean-square exponential stability can be referred to the proof of Theorem 10.3.1. Now, we will establish the weighted performance defined in (10.6). To this end, introduce the following index:

where the Lyapunov function V(x_k, α_k) is given in (10.11). By employing the same techniques used as those in the proof of Theorem 10.3.1, for k ∈ [k_l, k_{l + 1}), we have

where and

By Schur complement, LMI (10.30) is equal to Π(α_k) < 0, thus . Let Γ(k)≜z^T(k)z(k) − γ²ω^T(k)ω(k), then we have

(10.31)

Therefore, for k ∈ [k_l, k_{l + 1}), it holds from (10.31) that

(10.32)

Considering (10.8) and (10.11), it follows that

(10.33)

Thus by (10.32)–(10.33) we have

Therefore, it follows from the above inequalities and the relationship ϑ = N_α(k₀, k) ≤ (k − k₀)/T_a that

(10.34)

Under zero initial condition, that is, x(θ) = φ(θ) = 0, ( − max {τ, d₂} < θ ≤ 0), (10.34) implies

Multiplying both sides of the above inequality by yields

(10.35)

Notice that N_α(0, s) ≤ s/T_a and , so we have . Thus (10.35) implies

which yields

By Definition 10.2.2, we know that system (10.1a)–(10.1c) with u(k) = 0 is mean-square exponentially stable with a weighted performance level γ under α_k. This completes the proof. ▀

Remark 10.4 Note that Theorem 10.4.1 gives a weighted performance for the discrete-time switched stochastic time-delay system in (10.1a)–(10.1c) with u(k) = 0. The term ‘weighted’ refers to the weighting function β^s in the left-hand side of (10.6). This is also the characteristic of the mean-square exponential stability result to the switched stochastic hybrid system by using the average dwell time approach combining with the piecewise Lyapunov function technique. When setting β = 1, from the analysis in Remark 10.3, there is no switching. Thus, the result in Theorem 10.4.1 becomes a mean-square asymptotic stability condition with an performance for the deterministic system. ♦

Now, we are in a position to present a solution to the control problem for the discrete-time switched stochastic time-delay system in (10.1a)–(10.1c).

Theorem 10.4.2 For given constants β > 0 and γ > 0, suppose that there exist matrices X(i) > 0, Z(i) > 0, R(i) > 0, and Y(i) such that for ,

(10.36)

where

Then the closed-loop system in (10.5a)–(10.5b) is mean-square exponentially stable with a weighted performance level γ for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies

(10.37)

In this case, an state feedback controller can be chosen by

(10.38)

Proof. The result can be carried out by employing the same techniques used as those of Theorems 10.3.2 and 10.4.1. ▀

10.5 Illustrative Example

Example 10.5.1 (Stabilization problem) Consider the switched stochastic hybrid system in (10.1a) with N = 2 and

and , β = 0.7. A straightforward calculation gives d₂ = 3, τ = 1, and δ = 0.314. Give β = 0.7 and set μ = 1.6, thus . Checking the conditions in (10.7) by using LMI Toolbox, a set of feasible solutions is found. Therefore, the switched stochastic hybrid system (10.1a) with the above parametric matrices is mean-square exponentially stable for T_a > T*_a = 2. Moreover, taking T_a = 3 > T*_a = 2 and according to (10.9) and (10.10), we obtained a = 2.4410, b = 5.6846, , and , thus, an estimate of the mean-square of the state decay is given by

Now, we further simulate the stabilization problem. As analyzed above the open-loop system is mean-square exponentially stable when T_a ≥ T*_a = 2. Here, to show the effectiveness, we will design a stabilization controller in (10.3) such that the closed-loop system in (10.4) is mean-square exponentially stable for T_a = 1 (in this case, the allowable minimum of μ is μ_min  = 1.0314). Solving LMI conditions in Theorem 10.3.2, and considering (10.27), we have

Therefore, the controller in the form of (10.3) with above control gains can stabilize the open-loop system when T_a = 1.

Example 10.5.2 ( control problem) Consider the switched stochastic hybrid system in (10.1a)–(10.1c) with N = 2 and the following parameters:

Let β = 0.7 and

then it is easy to verify that there exist

such that (10.2) is satisfied. It can be shown that the switched stochastic hybrid system with the above two subsystems is not stable for a switching signal given in Figure 10.1 (which is generated randomly; here, ‘1’ and ‘2’ represent the first and second subsystems, respectively). Here, our aim is to design a state feedback controller such that the resulting closed-loop system is mean-square exponentially stable with a weighted performance level γ > 0 for T_a > T*_a. Here, for example, we set (in this case, the allowable minimum of μ is μ_min  = 1.6). Letting d₂ = 3, τ = 1, and δ = 0.314. Solving (10.36)–(10.37) in Theorem 10.4.2, we have γ = 2.1204 and

Suppose the disturbance input ω(k) is

Figure 10.2 shows the Brownian path, and Figure 10.3 gives the states of the closed-loop system.

10.6 Conclusion

In this chapter, the problems of stability, stabilization and the control have been considered for discrete-time switched stochastic hybrid systems with time-varying delays. By applying the average dwell time method and the piecewise Lyapunov function technique, sufficient conditions have been proposed for the mean-square exponential stability with a weighted performance for the considered hybrid system. Then the stabilization and the control problems have also been solved. Finally, two numerical examples have been provided to illustrate the effectiveness of the proposed design methods.