12.1 Introduction

Dissipativity theory has played a critical part in analysis and control design of linear and nonlinear systems, especially for high-order systems, since from the practical application point of view, many systems need to be dissipative for achieving effective noise attenuation. It has been recognized that for more abstract systems one can still associate with them an energy-like function (called the storage function) and an input-power-like function (called the supply rate). Dissipativity is then characterized by storage functions and supply rates, which represent the energy stored inside the system and energy supplied from outside the system, respectively. Roughly speaking, dissipative systems are those for which the increase in stored energy is never larger than the amount of energy supplied by the environment, that is, dissipative systems can only dissipate but also not generate energy. The dissipative systems theory is closely related to the dynamic properties of a process and, in particular, to its stability properties.

In this chapter, we will study dissipativity analysis and SMC design for switched stochastic hybrid systems. A more general supply rate is proposed, and a strict -dissipativity is defined, which includes , positive realness, and passivity as its special cases. The main idea is to introduce the strict -dissipativity into the analysis of sliding mode dynamics so as to improve the transient performance of the SMC system. The objective is to conduct dissipativity analysis and investigate the dissipativity-based SMC design scheme, with a view to contributing to the development of SMC design and the dissipativity analysis methods for the switched stochastic hybrid system. Specifically, an integral sliding surface is designed such that the sliding mode exists with the same order as the original system.

Then, by using the average dwell time approach and the piecewise Lyapunov function technique, a sufficient condition is established in terms of LMIs, which guarantees the sliding mode dynamics to be mean-square exponentially stable with a strict dissipativity performance. In addition, a solution to the dissipativity synthesis is provided by designing a discontinuous SMC law such that the system state trajectories can be driven onto the predefined sliding surface in a finite time and maintained there for all subsequent time.

12.2 Problem Formulation and Preliminaries

12.2.1 System Description

Consider the continuous-time switched stochastic hybrid systems, which are established on the probability space , and are described by

(12.1a)

(12.1b)

where x(t) ∈ Rⁿ is the system state vector; u(t) ∈ R^m is the control input; z(t) ∈ R^q is the controlled output; ω(t) ∈ R^p which belongs to , is either a disturbance input or a reference signal; ϖ(t) is a one-dimensional Brownian motion satisfying E{dϖ(t)} = 0; and E{dϖ²(t)} = dt. In addition, f(x(t), t, α(t)) ∈ R^m is an unknown nonlinear function satisfying

where φ(α(t)) > 0, and φ(α(t))‖x(t)‖ is an upper bound of the norm of the nonlinear function.

Here, in system (12.1a)–(12.1b) is a family of matrices parameterized by an index set and is a piecewise constant function of time t called a switching signal (denoted by α for simplicity), which is defined as in Chapter 5. 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), and F(i) = F(α), where A(i), B(i), C(i), D(i), E(i), and F(i) are constant matrices.

Assumption 12.1 For each , the pair (A(α), B(α)) in system (12.1a) is controllable and the matrix B(α) has full column rank.

First, we consider the following switched stochastic hybrid systems:

(12.2a)

(12.2b)

where x(t) ∈ X ∈ Rⁿ is the state vector; ω(t) ∈ Ω ∈ R^p is the input; and z(t) ∈ Z ∈ R^q is the controlled output.

Definition 12.2.1 The switched stochastic hybrid system (12.2a) with ω(t) = 0 is said to be mean-square exponentially stable under α if its solution x(t) satisfies

where η ⩾ 1 and λ > 0 are real constants.

12.2.2 Dissipativity

In this section, we will give a brief introduction to dissipative systems. Dissipative systems can be regarded as a generalization of passive systems with more general internal and supplied energies [269]. A system is called ‘dissipative’ if there is ‘power dissipation’ in the system. Dissipative systems are those that cannot store more energy than that supplied by the environment and/or by other systems connected to them, that is, dissipative systems can only dissipate but not generate energy [169].

According to [97], associated with the switched stochastic hybrid system (12.2a)–(12.2b) is a real-valued function Φ(ω, z) called the supply rate, which is formally defined as follows.

Definition 12.2.2 (Supply Rate) The supply rate is a real-valued function: Φ(ω, z): , which is assumed to be locally Lebesgue integrable independently of the input and the initial conditions, that is, for any ω ∈ Ω, z ∈ Z, and t* ⩾ 0, it holds that ∫^t*₀|Φ(ω(t), z(t))|dt < +∞.

The classical form of dissipativity in [97] is obviously applicable to the switched stochastic hybrid system in (12.1a)–(12.1b).

Definition 12.2.3 (Dissipative system) The switched stochastic hybrid system (12.2a)–(12.2b) with supply rate Φ(ω, z) is said to be dissipative if there exists a nonnegative function V(x): X → R, called the storage function, such that the following dissipation inequality holds:

(12.3)

for all initial condition x₀ ∈ X, input ω ∈ Ω, and t* ⩾ 0 (or said differently: for all admissible inputs ω(t) that drive the state from x(0) to x(t*) on the interval [0, t*], where x(t*) is the state variable at time t = t*).

Remark 12.1 Inequality (12.3) is called the dissipation inequality and it formalizes the property that the increase in stored energy is never greater than the amount of energy supplied by the environment. Passive systems are a special class of dissipative systems that have a bilinear supply rate, that is, Φ(ω, z) = z^Tω. If a system with a constant positive feedforward of is passive, then the process is dissipative with respect to the supply rate , where . Similarly, if a system with a constant negative feedback of is passive, then the process is dissipative with respect to the supply rate , where . ♦

Motivated by the above facts, a more general supply rate is proposed in the following definition.

Definition 12.2.4 Given matrices , , and with and being symmetric, the switched stochastic hybrid system (12.2a)–(12.2b) is called -dissipative if for some real function γ( · ) with γ(0) = 0,

(12.4)

Furthermore, if for some scalar δ > 0 and ∀t* ⩾ 0,

(12.5)

then the switched stochastic hybrid system (12.2a)–(12.2b) is called strictly -δ-dissipative.

Remark 12.2 In Definition 12.2.4, we assume that , thus the performance defined in Definition 12.2.4 includes , positive realness, and passivity as special cases. Specifically,

• Case 1. When , , and , (12.5) reduces to an performance requirement.
• Case 2. When , , , and γ(x₀) = 0, (12.5) corresponds to an extended strict positive real problem.
• Case 3. When , , and , θ ∈ [0, 1], (12.5) corresponds to the mixed and positive real performances.
• Case 4. When , , and , (12.5) corresponds to a passivity problem. ♦

12.3 Dissipativity Analysis

In this section, we will apply the average dwell time method combined with the piecewise Lyapunov function technique to investigate the dissipativity and the mean-square exponential stability for the switched stochastic hybrid system in (12.2a)–(12.2b).

Theorem 12.3.1 Given matrices , , and , with and being symmetric, and scalars β > 0, δ > 0, suppose that there exist matrices 0 < P(i) ∈ R^{n × n} such that for ,

(12.6)

where

then the switched stochastic hybrid system in (12.2a)–(12.2b) is strictly -δ-dissipative in the sense of Definition 12.2.4 for any switching signal with the average dwell time satisfying (where μ ⩾ 1) and satisfying

(12.7)

Proof. Choose the following Lyapunov function:

(12.8)

where P(α) > 0, are to be determined. Then, as with the solution of system (12.2a)–(12.2b), by Itô’s formula, we obtain the stochastic differential as

(12.9)

To show the strict dissipativity of system (12.2a)–(12.2b), we consider (12.9), and for any nonzero , it follows that

where Π(α) is defined in (12.6). By (12.6) we have Γ(x, α) < 0, that is,

where

Thus, we have

Observe that

(12.10)

Integrating both sides of (12.10) from t^⋆ > 0 to t and then taking expectations results in

which is equivalent to

(12.11)

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}).

According to (12.11) and letting t^⋆ = t_k, we have

(12.12)

Using (12.7) and (12.8), at switching instant t_k, we have

(12.13)

Therefore, it follows from (12.12)–(12.13) and the relationship ϑ = N_α(0, t) ⩽ (t − 0)/T_a that

(12.14)

Under the zero initial condition, that is, x(0) = 0, (12.14) implies

(12.15)

Multiplying both sides of (12.15) by yields

(12.16)

Noting N_α(0, t) ⩽ t/T_a and T_a > T*_a = ln μ/β, we have N_α(0, t)ln μ ⩽ βt. Thus, (12.16) implies

It is true that for arbitrary t* ⩾ 0,

which satisfies (12.5). Hence, the proof is completed. ▀

From the proof of Theorem 12.3.1, we also have the following result.

Theorem 12.3.2 Given a scalar β > 0, suppose that there exist matrices P(i) > 0 such that for ,

(12.17)

Then the switched stochastic hybrid system (12.2a)–(12.2b) with ω(t) = 0 is mean-square exponentially stable for any switching signal with the average dwell time satisfying (where μ ⩾ 1) and satisfying (12.7). Moreover, an estimate of the mean square of the state decay is given by

(12.18)

where

(12.19)

Proof. Choose the Lyapunov function as (12.8). Inequality (12.17) implies

Considering (12.9) for ω(t) = 0, we have

Thus,

Observe that

(12.20)

Integrating both sides of (12.20) from T > 0 to t and taking expectations, with some mathematical operations, we have

(12.21)

As the analysis made in the proof of Theorem 12.3.1, we let 0 = t₀ < t₁ < ⋅⋅⋅ < t_k < ⋅⋅⋅ (k = 1, 2, …) 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}). Letting T = t_k in (12.21) gives

(12.22)

Therefore, it follows from (12.13) and (12.22), and the relationship ϑ = N_α(0, t) ⩽ (t − 0)/T_a that

(12.23)

Note from (12.8) that

(12.24)

where a and b are defined in (12.19). Combining (12.23) and (12.24) together yields

which implies (12.18). By Definition 12.2.1 with t₀ = 0, system (12.2a)–(12.2b) with u(t) = 0 is mean-square exponentially stable. ▀

12.4 Sliding Mode Control

12.4.1 Sliding Mode Dynamics

We design the following integral switching function:

(12.25)

where K(i) ∈ R^{m × n} are real matrices to be designed, and matrices G(i) are to be chosen such that G(i)B(i) are nonsingular and G(i)F(i) = 0 for .

The solution of x(t) can be given by

(12.26)

It follows from (12.25) and (12.26) that

It is well known that 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

(12.27)

By substituting (12.27) into (12.1a), the sliding mode dynamics can be obtained as

(12.28)

For notational simplicity, we define

Thus, the sliding mode dynamics in (12.28) combined with the controlled output equation in (12.1b) can be formulated as

(12.29a)

(12.29b)

In this chapter, we choose G(i) = B^T(i)X(i), where 0 < X(i) ∈ R^{n × n} is to be designed later. Thus,

(12.30)

Note that the positive definiteness of matrix X(i) guarantees the nonsingularity of G(i)B(i) due to Assumption 12.1.

The above analysis gives the first step of the SMC for the switched stochastic hybrid system (12.1a)–(12.1b). Specifically, we design an integral-type sliding surface as given in (12.25) so that the dynamics restricted to the sliding surface (i.e. the sliding mode dynamics) has the form of (12.29a)–(12.29b). The remaining problems to be addressed in this chapter can be stated as follows.

1. Dissipativity analysis. Given all the system matrices in (12.1a)–(12.1b), determine the matrices G(i) and K(i) in the switching function (12.25) such that the sliding mode dynamics in (12.29a)–(12.29b) is mean-square exponentially stable and strictly -δ-dissipative in the sense of Definitions 12.2.1 and 12.2.4, respectively.
2. SMC law design. Synthesize an SMC law to globally 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. This is the second step of SMC.

12.4.2 Sliding Mode Dynamics Analysis

First, we give the following result for the dissipativity of the sliding mode dynamics in (12.29a)–(12.29b).

Theorem 12.4.1 Given matrices , , and , with and being symmetric, and scalars β > 0, δ > 0, suppose that there exist matrices X(i) > 0 such that for ,

(12.31)

where

Then the sliding mode dynamics in (12.29a)–(12.29b) is mean-square exponentially stable and strictly -δ-dissipative for any switching signal with the average dwell time satisfying (where μ ⩾ 1) and satisfying

Proof. The result can be obtained by employing the same techniques as used in the proof of Theorem 12.3.1 and noticing (12.30) and

Thus, the detailed proof is omitted for brevity. ▀

In the following, based on the result in Theorem 12.4.1, we are in a position to present a solution to the dissipativity synthesis problem for the sliding mode dynamics in (12.29a)–(12.29b).

Theorem 12.4.2 Given matrices , , and , with and being symmetric, and scalars β > 0, δ > 0, suppose that there exist matrices X(i) > 0, , and such that for ,

(12.32a)

(12.32b)

(12.32c)

where is defined in Theorem 12.4.1 and

Then the sliding mode dynamics in (12.29a)–(12.29b) is mean-square exponentially stable and strictly -δ-dissipative for any switching signal with the average dwell time satisfying (where μ ⩾ 1) and satisfying

(12.33)

Moreover, if the above conditions are feasible, then the matrix variable K(i) in (12.25) can be computed by

(12.34)

Proof. Let and . Then by performing a congruence transformation on (12.31) with and by Schur complement, the result can be obtained. ▀

Remark 12.3 Note that there exist two matrix equalities of (12.32b) and (12.32c) in Theorem 12.4.2, which cannot be solved directly by applying the LMI Toolbox in the Matlab environment. In the following, we will propose some algorithms to solve them. First, to solve (12.32b), for a scalar ϵ > 0, we consider the following matrix inequalities:

(12.35)

By Schur complement, (12.35) is equivalent to

(12.36)

Therefore, when ϵ > 0 is chosen as a sufficiently small scalar, matrix equality (12.32b) can be solved through LMI (12.36). Next, we use the CCL method to solve (12.32c) by formulating it into a sequential optimization problem subject to LMI constraints. ♦

Now, combining the methods for solving (12.32b)–(12.32c) together, we introduce the following minimization problem involving LMI conditions instead of the original nonconvex feasibility problem formulated in Theorem 12.4.2.

Problem SMA (Sliding mode analysis):

subject to (12.32a), (12.33), (12.36) and

(12.37)

Remark 12.4 By CCL method [66], if the solution of the above minimization problem is Nn, that is, , then the conditions in Theorem 12.4.2 are solvable. Although it is still not possible to always find the global optimal solution, the proposed minimization problem is easier to solve than the original nonconvex feasibility problem. ♦

12.4.3 SMC Law Design

In this section, we will synthesize a discontinuous SMC law, by which the state trajectories of the switched stochastic hybrid system (12.1a)–(12.1b) can be driven onto the pre-specified sliding surface s(t) = 0 in a finite time and then are maintained there for all subsequent time.

Theorem 12.4.3 Consider the switched stochastic hybrid system (12.1a)–(12.1b). Suppose that the switching function is designed as (12.25) with K(i) being solved by (12.34), and G(i) is chosen as G(i) = B^T(i)X(i) with X(i) > 0 being solved in Theorem 12.4.2. Then the state trajectories of system (12.1a)–(12.1b) can be driven onto the sliding surface s(t) = 0 in a finite time by the following SMC law:

(12.38)

where

with ϱ being a positive constant.

Proof. Choose a Lyapunov function of the following form:

According to (12.27), we have

Thus, taking the derivative of W(t) and considering the above equation, we have

(12.39)

Substituting (12.38) into (12.39) and noting that ‖s(t)‖ ⩽ |s(t)|, we have

(12.40)

where

It can be shown from (12.40) that there exists an instant such that W(t) = 0 (equivalently, s(t) = 0) when t ⩾ t*. Thus, we can say that the system state trajectories can be driven onto the predefined sliding surface in a finite time. ▀

12.5 Illustrative Example

Example 12.5.1  Consider system (12.1a)–(12.1b) with N = 2 and the following parameters

Suppose β = 0.5 and

(thus φ(1) and φ(2) can be chosen as φ(1) = φ(2) = 0.5), and the exogenous input ω(t) is given by ω(t) = 1/(1 + t²).

It is found that the system in (12.1a)–(12.1b) with u(t) = 0 and ω(t) = 0 and above parametric matrices is unstable for a switching signal given in Figure .1 (which is generated randomly; here, ‘1’ and ‘2’ represent the first and second subsystems, respectively). Therefore, our aim is to design the SMC law u(t) in (12.38) such that the resulting closed-loop system is mean-square exponentially stable and strictly -δ-dissipative for T_a > T*_a = 0.1 (in this case, the allowable minimum of μ is μ_min  = 1.0513). Firstly, we need to check the stability and the strict -δ-dissipativity of the sliding mode dynamics in (12.29a)–(12.29b) with T_a > T*_a = 0.1 (that is, set μ = 1.0513). To this end, we choose , , , and δ as , , , and δ = 0.1, respectively. Solve the conditions (12.32a)–(12.33) in Theorem 12.4.2 according to Remark 12.3 and applying the CCL method, and by (12.34), we have

Thus, the switching function defined in (12.25) is given by

and the SMC law designed in (12.38) can be computed as

To prevent the SMC system from chattering, we replace sign(s(t)) by

Set ϱ = 0.5 and the initial condition . 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* = 10, the normally distributed variance with N* = 2¹¹, the step size Δt = ρδt with ρ = 2, and the number of discretized Brownian paths p = 10. The simulation results are presented in Figures .2–.5. Specifically, Figures .2–.3 display the simulation results along an individual discretized Brownian path, with Figure .2 showing the states of the closed-loop system, and the switching function is given in Figure .3. Figures .4–.5 are the simulation results on x(t) and s(t) along 10 individual paths (dotted lines) and the average over 10 paths (solid line), respectively.

12.6 Conclusion

In this chapter, the problems of dissipativity analysis and SMC design have been studied for a class of continuous-time switched stochastic hybrid systems. The average dwell time approach and the piecewise Lyapunov function technique have been utilized to establish the LMI-type sufficient condition for guaranteeing the mean-square exponential stability and the strict dissipativity of the sliding mode dynamics. This was followed by the derivation of the condition for achieving the dissipativity synthesis. Furthermore, it has been shown that a discontinuous SMC law can be synthesized to drive the system state trajectories onto the predefined sliding surface in a finite time. Finally, the developed theory was validated by a numerical example together with computer simulations.