11
State Estimation and SMC of Switched Stochastic Hybrid Systems
11.1 Introduction
In Chapter 9, we studied the problems of stability, stabilization, and 
control for the continuous-time switched stochastic hybrid system. In this chapter, we are interested in investigating the SMC design problem for such hybrid systems, and some results developed in Chapter 9 will be used. First, by designing an integral switching function, we obtain the sliding mode dynamics, which is a switched stochastic hybrid system with the same order as the original systems. Based on the stability analysis result in Chapter 9, a sufficient condition for the existence of the sliding mode is proposed in terms of LMIs, and an explicit parametrization of the desired switching function is also given. Then, a discontinuous SMC law for reaching motion is synthesized, such that the state trajectories of the SMC system can be driven onto a prescribed sliding surface and maintained there for all subsequent time. Moreover, considering that some system state components may not be available in practical applications, we further consider the state estimation problem by designing an observer. Sufficient conditions are also established for the existence and the solvability of the desired observer, and then the observer-based SMC law is synthesized.
11.2 System Description and Preliminaries
Consider the switched stochastic hybrid systems which are established on the probability space 
and are described by
where x(t) ∈ Rn is the system state vector; u(t) ∈ Rm is the control input; ϖ(t) is a one-dimensional Brownian motion satisfying E{dϖ(t)} = 0, and E{dϖ2(t)} = dt. 
is a family of matrices parameterized by an index set 
and 
is switching signal (denoted by α for simplicity), which is defined as in Chapter 5. For each possible value α = i, 
, we denote the system matrices associated with mode i by A(i) = A(α), B(i) = B(α), D(i) = D(α), and F(i) = F(α), where A(i), B(i), D(i), and F(i) are constant matrices. The pairs (A(i), B(i)) are controllable for 
, and matrices B(i) are assumed to be of full column rank. For scalars φ(α) > 0, 
, the unknown nonlinear function f(x, t) satisfies
The autonomous system of (11.1) can be formulated as
Definition 11.2.1 The switched stochastic hybrid system in (11.3) is said to be mean-square exponentially stable under α if its solution x(t) satisfies
where η ≥ 1 and λ > 0 are two real constants.
According to Theorem 9.3.2, we have the following result for the mean-square exponential stability of system (11.3).
Theorem 11.2.2 Given a scalar β > 0, suppose that there exist matrices P(i) > 0 such that for 
,
Then the switched stochastic hybrid system in (11.3) is mean-square exponentially stable for any switching signal with average dwell time satisfying 
with μ ≥ 1 and satisfying
Moreover, an estimate of the mean-square of the state decay is given by
where
11.3 Main Results
11.3.1 Sliding Mode Dynamics Analysis
We design the following integral switching function:
where G(i) ∈ Rm × n and K(i) ∈ Rm × n are real matrices to be designed. In particular, the matrices G(i) are to be chosen such that G(i)B(i) are nonsingular and G(i)D(i) = 0 for all 
.
Then, the solution of x(t) can be given by
It follows from (11.4) and (11.5) that
As 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
By substituting (11.6) into (11.1), the sliding mode dynamics can be obtained as
We will analyze the stability of the sliding mode dynamics in (11.7) based on Theorem 11.2.2, by which an explicit parametrization of the desired switching function designed in (11.4) is given.
Theorem 11.3.1 For a given constant β > 0, suppose that there exist matrices 
and 
such that for 
,
Then the sliding mode dynamics in (11.7) is mean-square exponentially stable for any switching signal with average dwell time satisfying 
with μ ≥ 1 and satisfying
Moreover, if the conditions above are feasible, the matrices K(i) in (11.4) can be solved by
Proof. From Theorem 11.2.2, if there exist matrices P(i) > 0 such that the following conditions hold for 
:
then the sliding mode dynamics in (11.7) is mean-square exponentially stable for any switching signal with average dwell time satisfying 
with μ ≥ 1 and satisfying
Letting 
, and performing a congruence transformation on (11.11) with 
, we have that for 
,
Let 
, and we have (11.8) from (11.13). Furthermore, considering (11.12) and noting 
yields (11.9). This completes the proof. ▀
11.3.2 SMC Law Design
In this section, we will synthesize an SMC law to drive the system state trajectories onto the predefined sliding surface s(t) = 0.
Theorem 11.3.2 Consider the switched stochastic hybrid system (11.1). Suppose that the switching function is designed as (11.4) with K(i) being solved by (11.10) in Theorem 11.3.1. Then the state trajectories of system (11.1) can be driven onto the sliding surface s(t) = 0 in a finite time by the following SMC law:
with
where δ > 0 is a real constant and G(i) are adjustable parameters to be chosen such that G(i)B(i) are nonsingular for 
.
Proof. Choose a Lyapunov function of the following form:
By (11.4), we have
Thus, taking the derivative of V(t) and considering the above equation, we have
Substituting (11.14) into (11.15) and noting ‖s(t)‖ ≤ |s(t)|, we have
It can be shown from (11.16) that there exists an instant 
such that V(t) = 0 (equivalently, s(t) = 0) when t ≥ t*. Thus, it is concluded that the system trajectories can be driven onto the predefined sliding surface in a finite time. ▀
11.4 Observer-Based SMC Design
In this section, we will study the SMC problem under the assumption that some of the system state components are not available. We will first utilize a state observer to generate an estimate of the unmeasured states, and then synthesize an SMC law based on the state estimates. To begin with, we give the following measured output:
where y(t) ∈ Rp is the measured output. We design the following sliding mode observer to estimate the states of the switched stochastic hybrid system in (11.1):
where 
represents the estimate of the system states x(t); L(i) ∈ Rn × p are the observer gains to be designed; and the control term v(t) is chosen to eliminate the effect of the nonlinear function f(x, t).
Let 
denote the estimation error. According to (11.1) and (11.17)–(11.18), the estimation error dynamics is obtained as
Remark 11.1 Notice from (11.19) that the estimation error dynamics corresponds to a switched stochastic hybrid system, and is dependent on the observer feedback matrix L(i) and state estimates 
. This means that the stability analysis of the error dynamics (11.19) is not independent of the observer dynamics (11.18). ♦
Define the following switching functions in the state estimation space and in the state estimation error space, respectively,
where 
are adjustable matrices which are chosen such that 
are Hurwitz for 
. In addition, X(i) > 0 are matrices to be designed such that there always exist appropriately dimensioned matrices N(i) for BT(i)X(i) = N(i)C(i), thus
The state-estimate-based SMC laws are designed as
where ϱ > 0 and κ > 0 are two real constants, and
We will show in the following that the sliding motion will be driven onto the specified sliding surface sx(t) = 0 in a finite time.
Theorem 11.4.1 The state trajectories of systems (11.18) can be driven onto the sliding surface sx(t) = 0 in a finite time by the observer-based SMC law in (11.22a)–(11.22b).
Proof. Select the following Lyapunov function:
Noting ‖sx(t)‖ ≤ |sx(t)| and sTx(t)sign(se(t)) ≤ |sx(t)|, we have
Substituting SMC law (11.22a)–(11.22b) into (11.24), we have
where 
. It can be shown from (11.25) that there exists an instant 
such that 
(equivalently, sx(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. This completes the proof. ▀
According to the SMC theory, it follows from 
that the following equivalent control law can be obtained:
Substituting ueq(t) above into (11.18) yields the sliding mode dynamics in the state estimation space, which can be formulated as
In Theorem 11.4.1, our intention is to design an SMC law based on the estimated system states, such that the system state trajectories can be driven onto the predefined sliding surface sx(t) = 0 in a finite time, and the sliding mode dynamics in the state estimation space then results – see (11.26). In the following, we will propose a sufficient stability condition for overall closed-loop system composed of the estimation error dynamics (11.19) and the sliding mode dynamics in the state estimation space (11.26).
Theorem 11.4.2 Consider the switched stochastic hybrid system in (11.1) with (11.17). Its unmeasured states are estimated by the observer (11.18). The switching functions in the state estimation space and in the state estimation error space are chosen as (11.20a)–(11.20b), and the observer-based SMC law is synthesized by (11.22a)–(11.22b). If there exist matrices X(i) > 0, N(i) > 0 and 
such that for 
,
where
then the overall closed-loop switched stochastic hybrid system is globally asymptotically stable. Moreover, the observer gain is given by
Proof. Select the following Lyapunov functions:
Then, as with the solution of systems (11.19) and (11.26), we have
Thus, we have
Notice (11.22) and ‖se(t)‖ ≤ |se(t)|. Thus,
However, the following inequalities hold:
Considering (11.31)–(11.34), we have
where 
and 
with
Let 
and by Schur complement, (11.27) implies Ω(i) < 0. Thus,
We know that the overall closed-loop switched stochastic hybrid system composed of the estimation error dynamics (11.19) and the sliding mode dynamics in the state estimation space (11.26) is globally asymptotically stable. This completes the proof. ▀
Note that the conditions in Theorem 11.4.2 are not all expressed in LMI form due to (11.28), thus they can not be solved directly by an LMI procedure. In fact, (11.28) can be equivalently converted to
We consider the following matrix inequalities for scalar ℏ > 0,
By Schur complement, (11.36) is equivalent to
Therefore, when ℏ > 0 is chosen as a sufficiently small scalar, (11.28) can be solved through LMI (11.37).
11.5 Illustrative Example
Example 11.5.1 (SMC problem) Consider system (11.1) with N = 2 and the following parameters:
Set β = 0.5. It can be shown that the system in (11.1) with u(t) = 0 and the above parametric matrices is unstable for a switching signal given in Figure 11.1 (which is generated randomly; here, ‘1’ and ‘2’ represent the first and second subsystems, respectively). Thus, our aim is to design the SMC law u(t) in (11.14) such that the closed-loop system is mean-square exponentially stable for Ta > T*a = 0.1 (in this case, the allowable minimum of μ is μmin = 1.0513). To check the stability of the sliding mode dynamics in (11.7) with Ta > T*a = 0.1 (that is, set μ = 1.0513), we solve the conditions (11.8)–(11.9) in Theorem 11.3.1, and by (11.10), we obtain
We choose
Thus, the switching function defined in (11.4) is given by
and the SMC law designed in (11.14) can be computed as
where δ > 0 is an adjustable constant.
To prevent the control signals from chattering, we replace sign(s(t)) in the SMC law with s(t)/(0.01 + ‖s(t)‖). Set δ = 0.5 and suppose that the initial condition is 
. 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, step size Δt = ρδt with ρ = 2, and the number of discretized Brownian paths p = 10. The simulation results are given in Figures 11.2–11.6. Specifically, Figures 11.2–11.4 are the simulation results along an individual discretized Brownian path, with Figure 11.2 showing the states of the closed-loop system under the designed SMC law. The switching function and the SMC input are given in Figures 11.3 and 11.4, respectively. Figures 11.5–11.6 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.
Example 11.5.2 (Observer-based SMC problem) Consider system (11.1) with N = 2 and the following parameters:
In this example, we will consider the SMC design in the case where some of the system state components are not available. We design a sliding mode observer in the form of (11.18) to estimate the system states, and then synthesize the observer-based SMC laws in (11.22a)–(11.22b) for the reaching motion. First, we select matrices 
and 
as follows:
which guarantee that both 
and 
are Hurwitz. Then, solving (11.27) and (11.37), and by (11.29), we obtain
According to (11.20a), we have
and by (11.21), we have
Thus, the state estimate-based SMC laws designed in (11.22a)–(11.22b) are computed with φ(1) = 1, φ(2) = 2 and
Figure 11.1 Switching signal
Figure 11.2 States of the closed-loop system
Figure 11.3 Switching function
Figure 11.4 Control input
Figure 11.5 Individual paths and the average of the states of the closed-loop system
Figure 11.6 Individual paths and the average of the switching function
11.6 Conclusion
The problem of the SMC of a continuous-time switched stochastic hybrid system has been investigated in this chapter. An integral switching function has been designed, and a sufficient condition for the existence of sliding mode has been established in terms of LMIs, and an explicit parametrization of the desired switching function has also been given. Then, a discontinuous SMC law for reaching motion has been synthesized to drive the system state trajectories onto the predefined sliding surface in a finite time. Moreover, we have further studied the observer design and observer-based SMC problems for the case that some system state components are not accessible. Sufficient conditions have also been proposed for the existence of the desired sliding mode, and the observer-based SMC law has been designed for the reaching motion. Two numerical examples have been provided to illustrate the effectiveness of the proposed design scheme.