7
SMC of Switched State-Delayed Hybrid Systems: Continuous-Time Case
7.1 Introduction
SMC theory and methodologies have been developed for many kinds of systems such as uncertain systems, time-delay systems, and stochastic systems. Unfortunately, little progress has been made toward solving SMC of switched hybrid systems. The research in this area has not been fully investigated and still remains challenging. In this chapter, we will investigate the SMC design problem for continuous-time switched hybrid systems with time-varying delay. First, the original system is transformed into a regular form through model transformation, and then by designing a linear sliding surface, the dynamical equation for the sliding mode dynamics is derived. By utilizing the average dwell time approach and the piecewise Lyapunov function technique, a delay-dependent sufficient condition for the existence of a desired sliding mode is proposed, and an explicit parametrization of the desired sliding surface is also given. Since the obtained conditions are not all expressed in terms of strict LMIs (some matrix equality constraints exist), the CCL method is exploited to cast them into a sequential minimization problem subject to LMI constraints, which can be easily solved numerically. Then, a discontinuous SMC law is synthesized, by which the system state trajectories can be driven onto the prescribed sliding surface in a finite time and maintained there for all subsequent time. Since the designed SMC law contains state-delay terms, it requires the time-varying delay to be explicitly known a priori in the practical implementation of the controller. However, in some practical situations, the information about time delay is unavailable, or difficult to measure. In such a case, the designed SMC law is not applicable. To overcome this, we suppose the the state-delay terms in controller are unknown and unmeasurable, but that they are norm-bounded with an unknown upper bound. We will design an adaptive law to estimate the unknown upper bound, and thus an adaptive SMC law is synthesized, which can also guarantee that the system state trajectories reach onto the the prescribed sliding surface in a finite time.
7.2 System Description and Preliminaries
Consider the continuous-time state-delayed hybrid systems described by
where x(t) ∈ Rn is the system state vector; u(t) ∈ Rm is the control input; f(t) ∈ Rp is the nonlinearity representing the external disturbance or unmodeled dynamics; 
is a family of matrices parameterized by an index set 
; and 
(denoted by α for simplicity) is a switching signal defined as the same in Chapter 5. Also, φ(t) ∈ Cn, d is a differentiable vector-valued initial function on [ − d, 0] for a known constant d > 0, and d(t) denotes the time-varying delays which satisfy 0 ≤ d(t) ≤ d and 
.
For each possible value α(t) = i, 
, we denote the system matrices associated with mode i by A(i) = A(α), Ad(i) = Ad(α), and F(i) = F(α), where A(i), Ad(i), and F(i) are constant matrices. In addition, B is assumed to be of full column rank, and for the nonlinearity f(t), we suppose that
where 
are real constants.
Introduce the following definitions for the autonomous system of (7.1a):
Definition 7.2.1 The switched state-delayed hybrid system in (7.2) is said to be exponentially stable under α(t) if the solution x(t) of the system satisfies
where η ≥ 1 and λ > 0 are two real constants, and
7.3 Main Results
7.3.1 Sliding Mode Dynamics Analysis
In this section, we will consider the SMC problem for system (7.1a)–(7.1b). First of all, we design the switching function and analyze the stability of sliding mode dynamics. Since B is of full column rank by assumption, there exists a nonsingular matrix T such that
where B1 ∈ Rm × m is nonsingular. Taking a singular value decomposition of B, we have
where 
and W ∈ Rm × m are unitary matrices with U1 ∈ Rn × (n − m), U2 ∈ Rn × m, and Γ ∈ Rm × m is a diagonal positive-definite matrix. For convenience, choose T = UT, then by the transformation z(t) = Tx(t), system (7.1a)–(7.1b) becomes
Let 
with z1(t) ∈ Rn − m, z2(t) ∈ Rm, and
then (7.5) can be written in the following regular form:
where
Obviously, the first subsystem of (7.6) represents the sliding mode dynamics. We design the following switching function:
where C ∈ Rm × (n − m) is the parametric matrix to be designed.
Remark 7.1 Note that the switching function defined in (7.7) does not switch with the switching signal α (i.e. we design C not C(α) in (7.7)), that is, there is a unique non-switched sliding surface. The reason for this is to avoid repetitive jumps of the trajectories of the state components of the closed-loop system between sliding surfaces and hence the possible instability. ♦
When the system state trajectories reach onto the sliding surface s(t) = 0, that is, z2(t) = −Cz1(t), the sliding mode dynamics is attained. Substituting z2(t) = −Cz1(t) into the first subsystem of (7.6) yields the sliding mode dynamics:
Now, we will analyze the stability of the sliding mode dynamics in (7.8) based on the result obtained in Theorem 5.2.3, and give the following theorem.
Theorem 7.3.1 For a given constant β > 0, there exist matrices P > 0, 
, R(i) > 0, 
, 
, 
, 
, 
, 
, and 
such that for 
,
where
Then the sliding mode dynamics in (7.8) is exponentially stable for any switching signal with average dwell time satisfying 
, where μ ≥ 1 and satisfies
Moreover, if the conditions above are feasible, the matrix C in (7.7) is given by 
, that is, the switching function can be designed as
Proof. By Theorem 5.2.3, we know that if there exist matrices P > 0, Q(i) > 0, R(i) > 0, X(i), and Y(i) such that for 
,
where
then the sliding mode dynamics in (7.8) is exponentially stable for any switching signal with average dwell time satisfying 
, where μ ≥ 1 and satisfies
Define the following matrices:
Performing a congruence transformation on (7.12) with 
, we have
where 
, 
, and 
are defined in (7.9a).
Notice that (7.15) is not of LMI form because of the term of 
. Now, replacing 
in (7.15) with 
, it follows that (7.15) holds if (7.9a) holds and for 
,
By Schur complement, (7.16) is equivalent to
which implies (7.9b) by (7.14) and letting 
.
Moreover, considering (7.13)–(7.14), we have (7.10). This completes the proof. ▀
Remark 7.2 It should be pointed out that the matrix variables P and 
in Theorem 7.3.1 do not depend on the switching set and are fixed. As the designed switching function in (7.7) is a parameter-independent function, the parameter 
in (7.7) is guaranteed to be fixed if the matrix variable 
is fixed. ♦
Note that the conditions in Theorem 7.3.1 are not all of strict LMI form due to (7.9c), so we can not solve them by LMI procedures directly. Now, by using CCL method [66], we suggest the following minimization problem involving LMI conditions instead of the original nonconvex feasibility problem formulated in Theorem 7.3.1.
Problem SMDA (Sliding mode dynamics analysis)
subject to (7.9a)–(7.9b), (7.10) and for i 
,
According to the CCL method [66], if the solution of the above minimization problem is (1 + 2N)(n − m), then the conditions in Theorem NaN are solvable. We give the following algorithm to solve Problem SMDA.
Algorithm SMDA
- Step 1. Find a feasible set
satisfying (7.9a)–(7.9b), (7.10), and (7.18). Set κ = 0. - Step 2. Solve the following optimization problem:
subject to (7.9a)–(7.9b), (7.10), and (7.18) and denote f* as the optimized value.
- Step 3. Substitute the obtained matrices
into (7.17). If (7.17) is satisfied, with
for a sufficiently small scalar ϵ > 0, then output the feasible solutions
, so EXIT. - Step 4. If
where 
is the maximum number of iterations allowed, so EXIT. - Step 5. Set κ = κ + 1,
, and go to Step 2.
7.3.2 SMC Law Design
In the following, we are in a position to synthesize an SMC law to drive the system state trajectories onto the predefined sliding surface s(t) = 0, and give the following result.
Theorem 7.3.2 Suppose that the conditions in (7.9a)–(7.10) have a set of feasible solutions P > 0, 
, R(i) > 0, 
, 
, 
, 
, 
, 
, and 
, and the switching function is given by (7.11). Then the state trajectories of the closed-loop system (7.6) can be driven onto the sliding surface s(t) = 0 in a finite time by the control of
where ρ(i) > 0, 
are adjustable parameters.
Proof. We will show that the control law (7.19) can not only drive the system state trajectories onto the sliding surface, but also keep it there for all subsequent time. Consider the switching function as
and choose the following Lyapunov function:
Then as with the solution of the system in (7.6) for a fixed α, we have
Substituting the following control law into (7.22):
and noting that ‖sT(t)B1‖ ≤ |sT(t)B1|, we have
where 
.
As in the proof of Theorem 5.2.3, for an arbitrary piecewise constant switching signal α, and for any t > 0, we let 0 = t0 < t1 < ⋅⋅⋅ < tk < ⋅⋅⋅, k = 0, 1, …, denote the switching points of α over the interval (0, t). The ikth subsystem is activated when t ∈ [tk, tk + 1). Integrating 
from tk to t and tk − 1 to tk, k = 1, 2, …, we have
Summing the terms on both sides of (7.24) gives
It can be seen from (7.25) that there exists a time t* ≤ 2W1/2(0)/ρ such that W(t) = 0, and consequently s(t) = 0, for t ≥ t*, which means that the system state trajectories can reach onto the predefined sliding surface s(t) = 0 in a finite time. Since the reaching condition 
holds, the system state trajectories can be driven onto the predefined sliding surface and maintained there for all subsequent time. This completes the proof. ▀
Notice that the SMC law in (7.19) is applicable only when the time-varying delay d(t) is explicitly known a priori, since there exist z1(t − d(t)) and z2(t − d(t)) in (7.19). However, in some practical situations, the information for delay d(t) is unavailable, or difficult to measure. To overcome this, in what follows, we provide another kind of SMC law.
We assume that there exists a constant r > 0 such that
where the constant r is not known a priori, which is often the case in practical situations. Therefore, to obtain the value of r, we should design an adaptive law first to estimate it, and thus give an adaptive SMC law for system (7.6). Let r(t) represent the estimate of r. The corresponding estimation error is 
.
Theorem 7.3.3 Suppose the conditions in (7.9a)–(7.10) have a set of feasible solutions P > 0, 
, R(i) > 0, 
, 
, 
, 
, 
, 
, and 
, and the switching function is given by (7.11). Then the state trajectories of the closed-loop system (7.6) can be driven onto the sliding surface s(t) = 0 with the control of
where δ(i) > 0, 
are constants, and the adaptive law is given as
with r(0) = 0, where l > 0 is a given scalar.
Proof. Choose a Lyapunov function of the following form:
Then as with the solution of the system in (7.6) for a fixed α and by noting (7.26), we have
Substituting the control law (7.27) into (7.29) yields
Note from (7.28) that 
, which implies 
. Therefore, there exists a time instant 
such that 
for 
, and consequently 
for 
. Substituting the adaptive law (7.28) (with i replaced by α) into (7.30), when 
we have
where 
. By (7.31) and noting 
for 
, we have 
< 0 for 
, thus the reaching condition is satisfied. This completes the proof. ▀
7.4 Illustrative Example
Example7.4.1 Consider the switched state-delayed hybrid system in (7.1a)–(7.1b) with N = 2 (that is, there are two subsystems) and the following parameters:
and d = 2.0, β = 0.6, τ = 0.5, and f(t) = 0.5exp ( − t)sin (t). It can be verified that system (7.1a)–(7.1b) with u(t) = 0 and the above parameters is unstable for a switching signal given in Figure 7.1 (which is generated randomly; here, ‘1’ and ‘2’ represent the first and second subsystems, respectively), the states of the open-loop system are shown in Figure 7.2 with the initial condition given by 
, θ ∈ [ − 2, 0]. Therefore, our aim is to design an SMC law u(t) such that the closed-loop system is stable with arbitrary switching. To check the stability of (7.8) with arbitrary switching, we solve conditions (7.9a)–(7.9c) in Theorem 7.3.1 with R(i) = R(j) = R, 
, 
, 
by Algorithm SMDA, which gives
According to (7.11), we have
The existence of a feasible solution shows that there exists a mode-independent Lyapunov function for checking the exponential stability of the sliding mode dynamics in (7.8), that is, we can find a desired switching function in (7.32) such that the resulting sliding mode dynamics in (7.8) is exponentially stable for arbitrary switching. The remaining task is to design an SMC law such that the system state trajectories can be driven onto the predefined sliding surface s(t) = 0 and maintained there for all subsequent time. When delay d(t) in (7.1a)–(7.1b) is explicitly given as d(t) = 1.5 + 0.5sin t, the SMC law in (7.19) can be computed as
When delay d(t) in (7.1a)–(7.1b) is unknown, the SMC law designed in (7.27)–(7.28) can be applied, and given by
Set δ(1) = δ(2) = 2 and l = 1. The adaptive law in (7.28) is computed as
To reduce the chattering, we replace sign(s(t)) with s(t)/(0.01 + ‖s(t)‖). Figure 7.3 shows the state response of the closed-loop switched system with (7.33). The switching function and the control input are given in Figures 7.4 and 7.5, respectively. The corresponding simulation results with (7.34) are given in Figures 7.6–7.9.
Figure 7.1 Switching signal
Figure 7.2 States of the open-loop system
Figure 7.3 States of the closed-loop system with (7.33)
Figure 7.4 Sliding function with (7.33)
Figure 7.5 Control input (7.33)
Figure 7.6 States of the closed-loop system with (7.34)
Figure 7.7 Sliding function with (7.34)
Figure 7.8 Control input (7.34)
Figure 7.9 Adaptive estimate r(t)
7.5 Conclusion
In this chapter, the SMC design problem has been investigated for continuous-time switched systems with time-varying delay. By model transformation, the system has first been transformed into the regular form, and then the sliding mode dynamics has been derived by designing a linear switching function. The corresponding sufficient condition for the existence of resulting sliding mode dynamics has been derived, and an explicit parametrization of the desired sliding surface has also been given. In addition, an adaptive SMC law for reaching motion has been designed such that the system state trajectories can be driven onto the prescribed sliding surface in a finite time and maintained there for all subsequent time. Finally, a numerical example has been provided to illustrate the effectiveness of the design scheme.