In the previous three chapters, we have solved SMC design problems for systems with Markovian switching parameters. From this chapter onward, we shall study the analysis and synthesis problems for another kind of parameter-switching systems, namely switched hybrid systems. In Chapter 1, we presented an overview of the recent developments in switched hybrid systems. The stability analysis problem for switched hybrid systems can be classified into two major categories: stability analysis under arbitrary switching and under restricted switchings. It was shown that switched systems may fail to preserve stability under arbitrary switching, but may be stable under restricted switching signals [131]. That is to say, the stability results under restricted switchings may have less conservativeness than those under arbitrary switching. Among the restricted switchings, the most famous concept is the average dwell time. In this book, we shall assume that the switching signal in the considered switched hybrid systems is not arbitrary, but is a restricted one having an average dwell time.
In this chapter, we shall investigate the stability analysis and stabilization problems for continuous- and discrete-time switched hybrid systems with time-varying delays. For continuous-time system, the time-varying delay d(t) is assumed to satisfy either (A1) 0 ≤ d(t) ≤ d and or (A2) 0 ≤ d(t) ≤ d. By using the average dwell time approach and the piecewise Lyapunov function technique, two delay-dependent sufficient conditions are established for the exponential stability of the considered hybrid system with (A1) and (A2), respectively. Here, the slack matrix approach is applied to further reduce the conservativeness of the stability conditions caused by the time delay. For discrete-time system, the stability conditions are also derived by the average dwell time approach, and the results are all delay-dependent and thus less conservative. The stabilization problem is then solved by designing a memoryless state feedback controller, and an explicit expression for the desired controller is given. The work in this chapter is an important foundation for the development of the SMC methodologies for switched hybrid systems in subsequent chapters.
Consider the continuous-time switched state-delayed hybrid systems described by
where x(t) ∈ Rn is the state vector; u(t) ∈ Rm is the control input; φ(t) ∈ Cn, d is a differentiable vector-valued initial function on [ − d, 0] for a known constant d > 0; d(t) denotes the time-varying delay satisfying either (A1) 0 ≤ d(t) ≤ d ; and or (A2) 0 ≤ d(t) ≤ d.
In system (5.1a), is a family of matrices parameterized by an index set , and is a piecewise constant function of time t called a switching signal. At a given time t, the value of α(t), denoted by α for simplicity, might depend on t or x(t), or both, or may be generated by any other hybrid scheme. Therefore, the switched delayed hybrid system effectively switches among N subsystems with the switching sequence controlled by α. We assume that the value of α is unknown, but its instantaneous value is available in real time.
For each , we will denote the system matrices associated with mode i by A(i) = A(α), Ad(i) = Ad(α), and B(i) = B(α), where A(i), Ad(i), and B(i) are constant matrices. Corresponding to the switching signal α, we have the switching sequence with t0 = 0, which means that the ikth subsystem is activated when t ∈ [tk, tk + 1).
For the switching signal α, we revisit the average dwell time property from the following definition.
Definition 5.2.1 [129] For any T2 > T1 ≥ 0, let Nα(T1, T2) denote the number of switching of α over (T1, T2). If Nα(T1, T2) ≤ N0 + (T2 − T1)/Ta holds for Ta > 0, N0 ≥ 0, then Ta is called an average dwell time.
Assumption 5.1 The switching signal α(t) has an average dwell time.
Definition 5.2.2 The continuous-time switched state-delayed hybrid system in (5.1a)–(5.1b) with u(t) = 0 is said to be exponentially stable under α(t) if the solution x(t) of the system satisfies
for constants η ≥ 1 and λ > 0, and
Remark 5.1 By the average dwell time switching, we mean a class of switching signals such that the average time interval between consecutive switchings is at least Ta. Then, a basic problem for such systems is how to specify the minimal Ta and thereby get the admissible switching signals such that the system is stable and satisfies a prescribed performance if the system dynamics meets some conditions. As commonly used in the literature, we choose N0 = 0 in Definition 5.2.1. ♦
In this section, we will establish an exponential stability condition for system (5.1a)–(5.1b) with u(t) = 0 by applying the average dwell time approach and the piecewise Lyapunov function technique, and give the following result.
Theorem 5.2.3 For a given constant β > 0, suppose (A1) holds and there exist matrices P(i) > 0, Q(i) > 0, R(i) > 0, and X(i), Y(i) such that for ,
where
Then the switched system in (5.1a)–(5.1b) with u(t) = 0 is exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies
Moreover, an estimate of the state decay is given by
where
Proof Choose a Lyapunov function of the following form:
with
where P(α) > 0, Q(α) > 0, and R(α) > 0 are to be determined. Then, as with the solution of (5.1a)–(5.1b) for a fixed α, we have
However, the Newton–Leibniz formula gives
Then, for any appropriately dimensioned matrices , we have
where . Considering (5.7)–(5.10), it follows that
where
with
Notice that, in (5.11),
Performing a congruence transformation on (5.2) by diag{I, I, I, eβdI} and considering , , by Schur complement, (5.2) implies
Thus, it follows from (5.11)–(5.13) that
Now, for an arbitrary piecewise constant switching signal α, and for any t > 0, we let 0 < t1 < ⋅⋅⋅ < tk < ⋅⋅⋅, k = 0, 1, …, denote the switching points of α over the interval (0, t). As mentioned earlier, the ikth subsystem is activated when t ∈ [tk, tk + 1). Integrating (5.14) from tk to t gives
Using (5.3) and (5.6), at switching instant tk, we have
Therefore, it follows from (5.15)–(5.16) and the relationship ϑ = Nα(0, t) ≤ (t − 0)/Ta that
Notice from (5.6) that
where a and b are defined in (5.5). Combining (5.17)–(5.18) yields
which implies (5.4). By Definition 5.2.1 with t0 = 0, system (5.1a)–(5.1b) is exponentially stable. This completes the proof. ▀
Remark 5.2 Notice that Theorem 5.2.3 gives a delay-dependent sufficient condition for the exponential stability of system (5.1a)–(5.1b) with u(t) = 0. In the derivation of the delay-dependent result in Theorem 5.2.3, no model transformation was performed to system (5.1a)–(5.1b). Moreover, we introduced slack variables and , which helps avoid using bounding techniques and hence the possible conservativenes. ♦
Remark 5.3 Notice that there exist constraints P(i) ≤ μP(j), Q(i) ≤ μQ(j), and R(i) ≤ μR(j), in (5.3) of Theorem 5.2.3. So μ( > 1) is only dependent upon (5.3), and it is independent of (5.2). In fact, μ can be found to have very many solutions, for example,
and any value larger than μ* can also be considered as a solution of μ. ♦
Remark 5.4 When μ = 1 in , we have Ta > T*a = 0, which means that the switching signal α can be arbitrary. In this case, (5.3) turns out to be P(i) ≤ P(j), Q(i) ≤ Q(j), and R(i) ≤ R(j), . Thus the only possibility is P(i) = P(j) = P, Q(i) = Q(j) = Q, and R(i) = R(j) = R, , which implies that a common (that is, mode-independent) Lyapunov function is required for all subsystems. ♦
Remark 5.5 When μ > 1 and β → 0 in , we have Ta → ∞, that is, there is no switching. Switched system (5.1a)–(5.1b) is effectively operating at one of the subsystems all the time. In this case, according to the proof of Theorem 5.2.3, the asymptotic stability result of system (5.1a)–(5.1b) coincides with Theorem 1 in [243] when delay d(t) = d is constant. ♦
Remark 5.6 It should be pointed out that the methods used in this chapter for deriving the stability condition in Theorem 5.2.3 are different from that in [188], thus the obtained results are different. Since it introduced more slack matrices in Theorem 1 of [188], the condition becomes hard to apply to stabilization and controller synthesis problems. Our result in Theorem 5.2.3 overcomes the above difficulty, and this can be verified by the SMC problem presented in Chapter 7. ♦
The result in Theorem 5.2.3 is based on (A1), but when considering (A2), we have the following theorem. The result can be obtained by employing the same techniques used as in the proof of Theorem 5.2.3, thus we omit the proof.
Theorem 5.2.4 For a given constant β > 0, suppose (A2) holds and there exist matrices P(i) > 0, R(i) > 0, and X(i), Y(i) such that for ,
where
Then the switched system in (5.1a)–(5.1b) with u(t) = 0 is exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 and satisfies
Moreover, an estimate of state decay is given by
where
Remark 5.7 Comparing the results in Theorems 5.2.3 and 5.2.4, we found that the result in Theorem 5.2.4 requires a weaker condition on the time-varying delay when compared with Theorem 5.2.3. To obtain Theorem 5.2.4, a Lyapunov function is chosen as follows:
Notice that the derivative of the functional does not require bounding of the rate of delay d(t). In particular, the delay in Theorem 5.2.3 is required to be differentiable, but the one in Theorem 5.2.4 may be non-differentiable with arbitrarily fast time-varying behavior. ♦
Example 5.2.5 Consider the switched delay system in (5.1a)–(5.1b) with N = 2 and the following system parameters:
and d = 1.2, β = 0.5, τ = 0.3. It can be checked by using Theorem 1 of [243] that the above two subsystems are both asymptotically stable. We consider the average dwell time scheme, and set μ = 1.2 > 1, thus by (5.3). Solving LMIs (5.2)–(5.3), it follows that
which means that the above switched system is exponentially stable. Taking Ta = 1 > T*a, and considering (5.4)–(5.5) yield a = 2.6702, b = 6.6340, η = 1.5762 and λ = 0.1588, thus
Consider a discrete-time switched system with time delays, which can be described by the following dynamical equations:
where x(k) ∈ Rn is the system state vector; u(k) ∈ Rm represents the control input; φ(k) is the initial condition; is a family of matrices parameterized by an index set ; and is a piecewise constant function of time, called a switching signal, which takes its values in the finite set . At an arbitrary discrete time k, the value of α(k), denoted by α for simplicity, might depend on k or x(k), or both, or may be generated by any other hybrid scheme. We assume that the sequence of subsystems in switching signal α is unknown a priori, but its instantaneous value is available in real time. For the switching time sequence k0 < k1 < k2 < ⋅⋅⋅ of switching signal α, the holding time between [kl, kl + 1] is called the dwell time of the currently engaged subsystem, where . The delay d(k) satisfies 1 ≤ d1 ≤ d(k) ≤ d2, where d1 and d2 are constant positive scalars representing the minimum and maximum delays, respectively.
Remark 5.8 For each possible value α = i, , we will denote the system matrices associated with mode i by A(i) = A(α), Ad(i) = Ad(α), and B(i) = B(α), where A(i), Ad(i), and B(i) are constant matrices. Corresponding to the switching signal α, we have the switching sequence with k0 = 0, which means that the ilth subsystem is activated when k ∈ [kl, kl + 1). ♦
For the switching signal α, we introduce the following definition.
Definition 5.3.1 For a switching signal and any ki > kj > k0, let Nα(kj, ki) be the switching numbers of αk over the interval [kj, ki]. If for any given N0 > 0 and Ta > 0, we have Nα(kj, ki) ≤ N0 + (ki − kj)/Ta, then Ta and N0 are called average dwell time and the chatter bound, respectively.
Here, we assume N0 = 0 for simplicity as commonly used in the literature.
Assumption 5.2 The switching signal α(k) has an average dwell time.
Design a stabilization controller with the following general structure:
where K(α) ∈ Rm × n are parameter matrices to be designed.
Substituting the stabilization controller in (5.20) into system (5.19a)–(5.19b), we obtain the closed-loop system as
where
Definition 5.3.2 The discrete-time switched time-delay hybrid system in (5.19a)–(5.19b) with u(k) = 0 is said to be exponentially stable under α if the solution x(k) satisfies
for constants η ≥ 1 and 0 < ρ < 1, and
where ξ(θ)≜x(θ + 1) − x(θ).
Remark 5.9 Notice that the phrase ‘under α’ appears in Definition 5.3.2. This serves to emphasize that all results obtained subsequently in this chapter are dependent on the switching signal α, and α is not an arbitrary switching signal but a restricted one having an average dwell time. ♦
First, we will use the piecewise Lyapunov technique and the average dwell time approach to propose a sufficient condition for the exponential stability of the discrete-time switched time-delay system in (5.19a)–(5.19b) with u(k) = 0. We have the following theorem.
Theorem 5.3.3 Given a constant 0 < β < 1, suppose that there exist matrices P(i) > 0, Q(i) > 0, R(i) > 0, S1(i) > 0, and S2(i) > 0, and matrices L(i), M(i), and N(i) such that for ,
where
Then the discrete-time switched time-delay system in (5.19a)–(5.19b) with u(k) = 0 is exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies
Proof Choose a Lyapunov function of the following form:
where ξ(k)≜x(k + 1) − x(k), S(α)≜S1(α) + S2(α), and P(α) > 0, Q(α) > 0, R(α) > 0, S1(α) > 0, and S2(α) > 0 are real matrices. For k ∈ [kl, kl + 1), we define
thus with
Moreover, for and any appropriately dimensioned matrices L(α), M(α), and N(α), the following equations are true:
Considering (5.25)–(5.29) and (5.30), we have
where Φ(α) is defined in (5.22) and
Moreover, it can be seen from (5.22) that
Then we have
Now, for an arbitrary piecewise constant switching signal αk, and for any k > 0, we let k0 < k1 < ⋅⋅⋅ < kl < ⋅⋅⋅, l = 1, …, denote the switching points of αk over the interval (0, k). As mentioned earlier, the ilth subsystem is activated when k ∈ [kl, kl + 1). Therefore, for k ∈ [kl, kl + 1), it holds from (5.31) that
Using (5.23) and (5.24), at switching instant tk, we have
Therefore, it follows from (5.32)–(5.33) and the relationship ϑ = Nα(0, k) ≤ (k − k0)/Ta that
Note from (5.24) that there exist two positive constants a and b (a ≤ b) such that
Combining (5.34) and (5.35) yields
Furthermore, letting , it follows that
By Definition 5.3.2, we know that if 0 < ρ < 1, that is, , the discrete-time switched time-delay system in (5.19a)–(5.19b) with u(k) = 0 is exponentially stable, where function ceil(h) represents rounding real number h to the nearest integer greater than or equal to h. The proof is completed.
Remark 5.10 In Theorem 5.3.3, 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 given 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 5.11 Note that when μ = 1 in we have Ta > T*a = 0, which means that the switching signal α(k) can be arbitrary. In this case, (5.23) turns out to be P(i) = P(j) = P, Q(i) = Q(j) = P, R(i) = R(j) = P, S1(i) = S1(j) = S1, S2(i) = S2(j) = S2, , and the proposed approach becomes a quadratic one thus conservative. In this case, the system in (5.19a)–(5.19b) with u(k) = 0 turns out to be
and we have the following result for the system in (5.36a)–(5.36b). ♦
Corollary 5.3.4 The discrete-time time-delay system in (5.36a)–(5.36b) is asymptotically stable if there exist matrices P > 0, Q > 0, R > 0, S1 > 0, and S2 > 0, and matrices L, M, and N such that
where
Proof To prove the above result, we choose the following Lyapunov function:
where ξ(k)≜x(k + 1) − x(k), and P > 0, Q > 0, R > 0, S1 > 0, and S2 > 0 are real matrices to be determined. The remaining processes can be followed along the same lines as for the proof of Theorem 5.3.3, and we omit the details.
Notice that there exist two product terms between the Lyapunov matrices (i.e. P(i) and S1(i) + S2(i)) and the system matrices A(i) in the condition of Theorem 5.3.3, which will cause some problems with the solution of the stabilization control synthesis problem. In the following, a subsequent result is given in order to facilitate the control design procedure.
Corollary 5.3.5 Given a constant 0 < β < 1, suppose that there exist matrices P(i) > 0 and Q(i) > 0 such that for ,
where
Then the discrete-time switched time-delay system in (5.19a)–(5.19b) with u(k) = 0 is exponentially stable for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies
Now, based on the above corollary, we consider the stabilization problem for system (5.19a)–(5.19b).
Theorem 5.3.6 Given a constant 0 < β < 1, the system in (5.19a)–(5.19b) is stabilizable, that is, the closed-loop system in (5.21a)–(5.21b) is exponentially stable under the control input u(k) in (5.20), if there exist matrices , , and such that for ,
where
Then the discrete-time switched time-delay system in (5.19a)–(5.19b) is exponentially stabilizable for any switching signal with average dwell time satisfying , where μ ≥ 1 satisfies
In this case, a stabilizing state feedback controller can be chosen by
Proof By Schur complement, it can be seen from Corollary 5.3.5 that the closed-loop system in (5.21a)–(5.21b) is exponentially stable if there matrices P(i) > 0 and Q(i) > 0 such that for ,
where the switching signal has an average dwell time satisfying , where μ ≥ 1 satisfies
Performing a congruence transformation on (5.40) by (where ) and letting , we have
where is defined in (5.37). Moreover, we define , we have (5.37), and we know that . The proof is completed.▀
Example 5.3.8 (Stability analysis) Consider the system in (5.19a)–(5.19b) with N = 2, and its parameters are given as follows:
and d1 = 1, d2 = 2, β = 0.8. We consider the average dwell time approach proposed in this chapter, and set μ = 1.5 > 1, thus . Solving LMI (5.22) with (5.23), we can obtain a feasible solution of (P(1), P(2), Q(1), Q(2), R(1), R(2), S1(1), S1(2), . .S2(1), S2(2), L(1), L(2), M(1), M(2), N(1), N(2)). Therefore, we can conclude that the above discrete-time switched system is exponentially stable.
In addition, for d1 = 1, μ = 1.5, and τ = 0.6, considering different β, the upper bound of d2 for different cases are listed in Table 5.1.
5.3.8 Example (Stabilization problem) Consider the system in (5.19a)–(5.19b) with N = 2, and the system parameters are given as follows:
with d(k) = 2.5 + ( − 1)k/2 (thus d1 = 1, d2 = 3), and suppose that μ = 1.5, β = 0.9, and .
The switching signal is given in Figure 5.1 (which is generated randomly; here, ‘1’ and ‘2’ represent the first and second subsystems, respectively). The state trajectories of the open-loop system are shown in Figure 5.2, from which we can see that the open-loop system is not stable. In this situation, we will design a state feedback stabilization controller such that the closed-loop system is stable. To this end, by solving the LMI conditions in Theorem 5.3.6, we obtain
The state trajectories of the closed-loop system are shown in Figure 5.3.
Table 5.1 Upper bound of d2 (denoted by ) for different β
β | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
1.3344 | 1.7345 | 2.2247 | 2.9278 | 4.1299 |
The stability analysis and stabilization problems have been investigated for continuous- and discrete-time switched hybrid systems with time-varying delay. By using the average dwell time approach and the piecewise Lyapunov function technique, some delay-dependent sufficient conditions have been proposed to guarantee the exponential stability of the considered systems. To further reduce the conservativeness caused by the time-varying delays, the slack matrix variables technique has been applied to seek the relationship between the Newton–Leibniz formula, instead of applying the traditional model transformation. In addition, a stabilization controller design approach has been developed for discrete-time switched delayed hybrid systems, and an explicit expression for the desired state feedback controller has also been given. Finally, two numerical examples have been provided to illustrate the effectiveness of the theoretic results obtained.