9.1 Introduction

Stochastic systems play an important role in many branches of science and engineering applications, thus they have received much attention during recent decades. Many results reported about stochastic systems can be found in the literature, for example, stochastic stability analysis, stabilization, optimal and robust control, filtering, and model reduction. Recently, a great deal of work has been reported on stochastic systems with Markovian switching. These results motivate us to study some interesting topics on stochastic systems whose parameters operate by a switching signal, that is, the switched stochastic hybrid systems. This work is interesting and challenging since this kind of hybrid system integrates the switched hybrid systems into that of the stochastic systems, and thus is theoretically significant.

In this chapter, we investigate the control (including state feedback control and DOF control) problems for continuous-time switched stochastic hybrid systems. The average dwell time approach combined with the piecewise Lyapunov function technique are applied to derive the main results. There are two main advantages of using this approach to the switched system. First, this approach uses a mode-dependent Lyapunov function, which avoids some conservativeness caused by using a common Lyapunov function for all the subsystems. The other main advantage is that the obtained result is not just an asymptotic stability condition, but an exponential one. Therefore, by this approach, a sufficient conditions is first proposed, which guarantees the mean-square exponential stability of the unforced switched stochastic hybrid system. When system states are available, a state feedback controller is designed such that the closed-loop system is mean-square exponentially stable with an performance. However, when system states are not all available, a DOF controller is designed, and the mean-square exponential stability with an performance is also guaranteed. Sufficient solvability conditions for the desired controllers are proposed in terms of LMIs.

9.2 System Description and Preliminaries

Consider a class of switched stochastic hybrid systems of the form:

(9.1a)

(9.1b)

where x(t) ∈ Rⁿ is the state vector; u(t) ∈ R^m is the control input; ω(t) ∈ R^p is the disturbance input which belongs to ; z(t) ∈ R^q is the controlled output; ϖ(t) is a one-dimensional Brownian motion satisfying E{dϖ(t)} = 0; and E{dϖ²(t)} = dt. Also, {(A(α(t)), B(α(t)), C(α(t)), D(α(t)), E(α(t))): . is a family of matrices parameterized by an index set and (denoted by α for simplicity) is a switching signal defined the same as in Chapter 5. In addition, we assume that the switch signal α(t) has an average dwell time.

Here, we design a stabilization controller and an state feedback controller with the following general structure:

(9.2)

where K(α) ∈ R^{m × n} are parametric matrices to be designed. Substituting the controller u(t) into the system (9.1a)–(9.1b), we obtain the closed-loop stabilization system as

(9.3)

and the closed-loop control system as

(9.4a)

(9.4b)

The above state feedback controller requires that the system states are fully accessible. However, in practical applications, it is usually either not accessible or hard to access. In such a case, one option is to assume the availability of a measured output signal vector given by

(9.5)

where y(t) ∈ R^r is the measured output, and G(α), H(α), and F(α) are real constant matrices.

For each possible value , we will denote the system matrices associated with mode i by A(i) = A(α), B(i) = B(α), C(i) = C(α), D(i) = D(α), E(i) = E(α), F(i) = F(α), G(i) = G(α), and H(i) = H(α), where A(i), B(i), C(i), D(i), E(i), F(i), G(i), and H(i) are constant matrices.

We are also interested in designing a DOF controller in the form of

(9.6a)

(9.6b)

where is the controller state vector; A_c(α), B_c(α), and C_c(α) are matrices to be determined.

Augmenting the model of (9.1a)–(9.1b) to include the states of the DOF controller (9.6a)–(9.6b), we obtain the closed-loop system as

(9.7a)

(9.7a)

where and

First we present the following definitions.

Definition 9.2.1 The switched stochastic hybrid system in (9.1a)–(9.1b) with u(t) = 0 and ω(t) = 0 is said to be mean-square exponentially stable under α(t) if its solution x(t) satisfies

for constants η ≥ 1 and λ > 0.

Definition 9.2.2 For β > 0 and γ > 0, the switched stochastic hybrid system in (9.1a)–(9.1b) with u(t) = 0 is said to be mean-square exponentially stable with a weighted performance level γ under α(t), if it is mean-square exponentially stable with ω(t) = 0, and under zero initial condition, that is, x(0) = 0, if it holds for all nonzero that

(9.8)

Therefore, the problems to be addressed in this chapter can be formulated as:

1. Stability analysis and stabilization. Propose a condition guaranteeing the mean-square exponential stability of the unforced switched stochastic hybrid system. Then, design of a stabilization controller such that the resulting closed-loop system is mean-square exponentially stable.
2. control. Both the state feedback control and the DOF control are considered according to the availability of the system states. Specifically,

1. When system states are fully available, design a state feedback controller such that the closed-loop system is mean-square exponentially stable with a weighted performance.
2. When system states are not all available, design a DOF controller such that the closed-loop system is mean-square exponentially stable with a weighted performance.

9.3 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 system (9.1a)–(9.1b) with ω(t) = 0.

Before proceeding, we cite the following result of Itô’s formula, which plays an important role in the stability analysis for stochastic systems (see [144] for a detailed account of Itô stochastic systems).

Lemma 9.3.1 [144] (Itô’s formula) Let x(t) be an n-dimensional Itô’s process on t ≥ 0 with the stochastic differential

where f(t) ∈ Rⁿ and g(t) ∈ R^{n × m}. Let V(x(t), t) ∈ C^{2, 1}(Rⁿ × R⁺; R⁺). Then, V(x(t), t) is a real-valued Itô process with its stochastic differential given by

where C^{2, 1}(Rⁿ × R⁺; R⁺) denotes the family of all real-valued functions V(x(t), t) such that they are continuously twice differentiable in x and t. If V(x(t), t) ∈ C^{2, 1}(Rⁿ × R⁺; R⁺), we set

Firstly, we present the following stability analysis result for the switched stochastic hybrid system in (9.1a)–(9.1b) with u(t) = 0 and ω(t) = 0.

Theorem 9.3.2 Given a scalar β > 0, suppose there exist matrices P(i) > 0 such that for ,

(9.9)

Then the switched stochastic hybrid system in (9.1a)–(9.1b) with u(t) = 0 and ω(t) = 0 is mean-square exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies

(9.10)

Moreover, an estimate of the state decay is given by

(9.11)

where

(9.12)

Proof. Choose a Lyapunov function as

(9.13)

where P(α) > 0, are to be determined. Then, as with the solution of the system (9.1a)–(9.1b) with u(t) = 0 and ω(t) = 0 for a fixed α, by Itô’s formula, we have

where

(9.14)

By Schur complement, LMI (9.9) implies

which implies from (9.14) that

Thus, we have

Observe that

(9.15)

Integrate both sides of (9.15) from T > 0 to t and take expectations. Then, by some mathematical operations, we have

(9.16)

Now, for an arbitrary piecewise constant switching signal α, and for any t > 0, we let 0 = t₀ < t₁ < ⋅⋅⋅ < t_k < ⋅⋅⋅, k = 0, 1, …, denote the switching points of α over the interval (0, t). As mentioned earlier, the i_kth subsystem is activated when t ∈ [t_k, t_{k + 1}). Letting T = t_k in (9.16) gives

(9.17)

Using (9.10) and (9.13), at switching instant t_k, we have

(9.18)

where t⁻_k denotes the left limit of t_k.

Therefore, it follows from (9.17)–(9.18) and the relationship ϑ = N_α(0, t) ≤ (t − 0)/T_a that

(9.19)

Notice from (9.13) that

(9.20)

where a and b are defined in (9.12). Combining (9.19) and (9.20) yields

(9.21)

which implies (9.11). By Definition 9.2.1 with t₀ = 0, the system in (9.1a)–(9.1b) with u(t) = 0 and ω(t) = 0 is mean-square exponentially stable. This completes the proof. ▀

Remark 9.1 Note that the scalar β is introduced in the stability analysis of Theorem 9.3.2, this is the characteristic of the exponential stability for the switched system by using the average dwell time approach. Here, β plays a key role in controlling the low bound of the average dwell time due to . From we can see that when β is given a bigger 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 9.2 When μ = 1 in we have T_a > T*_a = 0, which means that the switching signal α can be arbitrary. In this case, (9.10) turns out to be P(i) ≤ P(j), . The only possibility for that is P(i) = P(j) = P, , and this implies that it requires a common (that is, mode-independent) Lyapunov function for all subsystems. However, when μ > 1 and β → 0 in , we have T_a → ∞, that is, there is no switching. In such a case, switched stochastic hybrid system (9.1a)–(9.1b) is effectively operating at one of the subsystems all the time. We have the following result. ♦

Corollary 9.3.3 Suppose there is no switching in system (9.1a)–(9.1b) with u(t) = 0 and ω(t) = 0 (when β → 0 as discussed in Remark 9.2), that is, system (9.1a)–(9.1b) with u(t) = 0 and ω(t) = 0 is transformed to a common stochastic system (thus, the parameters become as (A, E)). If there exists a matrix P > 0 such that

(9.22)

then the common stochastic system is mean-square asymptotically stable.

Remark 9.3 The mean-square asymptotic stability for the common stochastic system in Corollary 9.3.3 is consistent with the result in [240], which shows that Theorem 9.3.2 has extended some results in [240] to the switched hybrid systems. ♦

Now, we present a solution to the stabilization problem, and give the following result.

Theorem 9.3.4 Given a scalar β > 0, suppose there exist matrices R(i) > 0 and L(i) > 0 such that for ,

(9.23)

where

Then, the closed-loop system in (9.3) is mean-square exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies

(9.24)

Moreover, the gain matrices K(i) of the stabilization controller in (9.2) can be chosen by

(9.25)

Proof. Replacing A(i) in (9.9) with A(i) + B(i)K (i) in Theorem 9.3.2, we have that the closed-loop stabilization system in (9.3) is mean-square exponentially stable if there exist matrices P(i) > 0 such that for ,

(9.26)

where

Letting R(i)≜P^{− 1}(i) and performing a congruence transformation on (9.26) by diag(R(i), R(i)) yields

(9.27)

where

Set L(i) = K(i)R(i), thus (9.27) is equal to (9.23). This completes the proof. ▀

9.4 Control

9.4.1 Performance Analysis

First, we will investigate the weighted performance for the switched stochastic hybrid system in (9.1a)–(9.1b) with u(t) = 0.

Theorem 9.4.1 Given scalars β > 0 and γ > 0, suppose there exist matrices P(i) > 0 such that for ,

(9.28)

where

Then the switched stochastic hybrid system in (9.1a)–(9.1b) with u(t) = 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 (9.10).

Proof. The proof of mean-square exponential stability can be carried out along the same lines as that in the proof of Theorem 9.3.2. Now, we will establish the weighted performance defined in (9.8); to this end, we introduce the following index:

where the Lyapunov function V(x, α) is given in (9.13) and

Thus,

where and

By Schur complement, LMI (9.28) is equal to Π(α) < 0, thus . Let Γ(t)≜z^T(t)z(t) − γ²ω^T(t)ω(t), then we have

Thus, we have

Observe that

(9.29)

Integrate both sides of (9.29) from T > 0 to t and take expectations. Then, by some mathematical operations, we have

(9.30)

Let 0 = t₀ < t₁ < ⋅⋅⋅ < t_k < ⋅⋅⋅, k = 1, …, denote the switching points of α over the interval (0, t), and suppose that the i_kth subsystem is activated when t ∈ [t_k, t_{k + 1}). Setting T = t_k in (9.30), we have

(9.31)

Using (9.10) and (9.13), at switching instant t_k, we have

(9.32)

Therefore, it follows from (9.31)–(9.32) and the relationship ϑ = N_α(0, t) ≤ (t − 0)/T_a that

(9.33)

Under zero initial condition, that is, x(0) = 0, (9.33) implies

(9.34)

Multiplying both sides of (9.34) by yields

(9.35)

Notice that as N_α(0, s) ≤ s/T_a and T_a > T*_a = ln μ/β, we have N_α(0, s)ln μ ≤ βs. Thus, (9.35) implies

Integrating the above inequality from t = 0 to ∞ yields (9.8). This completes the proof. ▀

Remark 9.4 Note that Theorem 9.4.1 presents a weighted performance for the switched stochastic hybrid system in (9.1a)–(9.1b) with u(t) = 0. The term ‘weighted’ refers to the weighting function e^{− βt} in the left-hand side of (9.8). This is also the characteristic of the exponential stability result to the switched system by using the average dwell time approach combining with the piecewise Lyapunov function technique. When β = 0, we know from Remark 9.2 that there is no switching. Thus, the result in Theorem 9.4.1 becomes an asymptotic stability condition with an performance for the deterministic stochastic system, which is also consistent with the results in [240].

9.4.2 State Feedback Control

In this sequel, we will present a solution to the state feedback control problem.

Theorem 9.4.2 Given scalars β > 0 and γ > 0, suppose there exist matrices R(i) > 0 and L(i) > 0, such that for ,

(9.36)

where

Then the closed-loop system in (9.4a)–(9.4b) is mean-square exponentially stable with a weighted performance level γ for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies

(9.37)

Moreover, if the above LMIs have feasible solutions, then the gain matrices K(i) of the controller in (9.2) can be chosen by

(9.38)

The result can be carried out by employing the same techniques as used with Theorems 9.3.4 and 9.4.1.

9.4.3 DOF Controller Design

In the following, we will study the DOF control problem for the switched stochastic hybrid system (9.1a)–(9.1b). First, we present the following results, and its proof can be worked out along the same line of reasoning as in the derivation of Theorems 9.3.2 and 9.4.1.

Theorem 9.4.3 Given scalars β > 0 and γ > 0, suppose there exist matrices P(i) > 0 such that for ,

(9.39)

where

Then the closed-loop switched stochastic hybrid system in (9.7a)–(9.7b) is mean-square exponentially stable with a weighted performance level γ for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies

(9.40)

Moreover, an estimate of the mean-square of the state decay is given by

where

Now, we present a solution to the DOF control problem.

Theorem 9.4.4 Consider the switched stochastic hybrid system in (9.1a)–(9.1b). For given constants β > 0 and γ > 0, suppose there exist matrices , , , , and and a scalar ϵ > 0 such that for ,

(9.41a)

(9.41b)

where

Then the closed-loop system in (9.7a)–(9.7b) is mean-square exponentially stable with a weighted performance level γ for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies

(9.42)

Moreover, if the above conditions have feasible solutions, then a desired weighted DOF controller realization is given by

(9.43)

Proof. From Theorem 9.4.3, we know that the closed-loop system in (9.7a)–(9.7b) is mean-square exponentially stable with a weighted performance level γ > 0, if there exist matrices P(i) > 0 satisfying (9.39). It is not difficult to see that these conditions are satisfied if there exist matrices P(i) > 0 and a scalar ϵ > 0 such that for ,

(9.44)

(9.45)

where .

Let P(i) be partitioned as

(9.46)

Without loss of generality, we assume P₂(i) and Q₂(i) are nonsingular (if not, P₂(i) and Q₂(i) may be perturbed by matrices ΔP₂(i) and ΔQ₂(i) with sufficiently small norms respectively such that P₂(i) + ΔP₂(i) and Q₂(i) + ΔQ₂(i) are nonsingular and satisfy (9.44)–(9.45)). Then, define the following nonsingular matrices:

(9.47)

Notice that , , and P₁(i)Q₁(i) + P₂(i)Q^T₂(i) = I, .

Performing congruence transformations on (9.44) and (9.45) by matrices and , respectively, we obtain

(9.48)

(9.49)

Define , and

(9.50)

Then by (9.46)–(9.47) and (9.50), it follows from (9.48) and (9.49) that

(9.51)

and (9.41), respectively, where

and Ψ₁₁(i) are defined in (9.41a). By Schur complement, (9.41a) is equivalent to (9.51). Moreover, considering the conditions in (9.40) yields (9.42). This completes the proof. ▀

Remark 9.5 It should be pointed out that to solve the parameters of output feedback controller in (9.43), matrices P₂(i) and Q₂(i) should be available in advance, which can be obtained by taking any full rank factorization of (derived from ). ♦

Remark 9.6 Note that Theorem 9.4.4 provides a sufficient condition for solvability of the weighted DOF control problem and, since the resulting condition is in LMI form, a desired DOF controller which minimizes the weighted performance level (i.e. maximize the level of noise removal) can be determined by solving the following convex optimization problem:

with matrix variables , , , , and and a scalar ϵ > 0. ♦

9.5 Illustrative Example

Example 9.5.1 (Stabilization problem) Consider the switched stochastic hybrid system in (9.1a)–(9.1b) with N = 2 and the following parameters:

Given β = 0.5. By simulation, when setting μ = 1.78, thus , the conditions in (9.9) hold with

As analyzed above, the open-loop system is mean-square exponentially stable for T_a > T*_a = 1.1532. Moreover, taking T_a = 1.2 > T*_a = 1.1532 and according to (9.11)–(9.12) we have a = 1739.4, b = 5557.4, , 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 = 1.1532. Here, to show the effectiveness, we will design an appropriate stabilization controller in (9.2) such that the closed-loop system in (9.3) is mean-square exponentially stable for T_a ≥ T*_a = 0.1 (in this case, the allowable minimum of μ is μ_min  = 1.0513). By solving conditions (9.23) in Theorem 9.3.4, we have

Thus, by (9.25) we have

Therefore, by the stabilization controller in (9.2) with the above control gains, the closed-loop system is mean-square exponentially stable for T_a ≥ T*_a = 0.1.

Example 9.5.2 ( DOF control problem) Consider the switched stochastic hybrid system in (9.1a)–(9.1b) with N = 2 and

Given β = 0.5, we checked that the considered switched stochastic hybrid system with the above two subsystems is not stable for a switching signal given in Figure 9.1 (which is generated randomly; here, ‘1’ and ‘2’ represent the first and second subsystems, respectively). Here, our aim is to design a DOF 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  = 1.05). Letting ϵ = 0.18 and solving (9.41a)–(9.41b) in Theorem 9.4.4, then setting Q₂(i) = I and according to (9.43), we have γ = 3.3166 and

Given the initial conditions as and , suppose the disturbance input ω(t) be ω(t) = 0.5e^{− t}sin (t). By using the discretization approach [96], we simulate standard Brownian motion. Some initial parameters are given as follows: the simulation time t ∈ [0, T^⋆] with T^⋆ = 20, the normally distributed variance with N^⋆ = 2¹¹, step size Δt = ρδt with ρ = 2, and the number of discretized Brownian paths p = 10. The simulation results are given in Figures 9.2–9.6. Among them, Figures 9.2–9.4 are the simulation results along an individual discretized Brownian path. Figures 9.2 and 9.3 give respectively the states of the closed-loop system and the DOF controller. The control input is shown in Figure 9.4. Figures 9.5 and 9.6 are the simulation results on x(t) and along 10 individual paths (dotted lines) and the average over 10 paths (solid line), respectively.

9.6 Conclusion

In this chapter, the problems of stabilization and the control have been investigated for continuous-time switched stochastic hybrid systems. By applying the average dwell time method and the piecewise Lyapunov function technique, sufficient conditions have been proposed for the mean-square exponential stability and the weighted performance for the switched stochastic hybrid system. Then, the stabilization and the control including the state feedback and DOF control problems have been solved. Finally, two numerical examples have been provided to illustrate the effectiveness of the proposed theories.