Stability analysis of periodically switched linear systems using Floquet theory

Stability of a switched system that consists of a set of linear 
time invariant subsystems and a periodic switching rule is 
investigated. Based on the Floquet theory, necessary and 
sufficient conditions are given for exponential stability. It is 
shown that there exists a slow switching rule that achieves 
exponential stability if at least one of these subsystems is 
asymptotically stable. It is also shown that there exists a fast 
switching rule that achieves exponential stability if the average 
of these subsystems is asymptotically stable. The results are 
illustrated by examples.


Introduction
A switched system consists of a set of subsystems and a rule that describes switching among them.The subsystems may be continuous-time or discrete-time systems and the switching rule may depend on time or states of individual subsystems.
Switched systems arise when dynamics of systems undergo abrupt changes due to component failures, parameter changes, switching elements, or switching controllers.Such systems have been studied extensively in the literature.A recent survey of switched systems can be found in [4] and various applications of switched systems are discussed in [5].
In this paper, a special class of switched systems is considered.Specifically, we consider a periodically switched linear system of the form A σ x(t), t σ−1 + lT ≤ t < t σ + lT, l = 0,1,2,..., t ≥ t 0 , with t σ = t 0 + T, where x(t) ∈ R n and A 1 ,A 2 ,...,A σ ∈ R n×n are (not necessarily different) constant matrices.Defining it can be easily seen that each subsystem ẋ(t) = A k x(t) is active for ∆t k seconds within each period.Systems of this type can be used to model sampled data control systems, switch mode power supplies, switched capacitor filters, and switching amplifiers.The goal of this paper is to investigate stability of periodically switched linear systems of the above form.The main tool used for this purpose is the Floquet theory [7,8].It should be mentioned that although Floquet theory has been used in the literature to investigate stability of periodically time varying linear systems, there seems to be no explicit study of stability of periodically switched linear systems using this theory.
This paper is organized as follows.In Section 2, the Floquet theory is reviewed.In Section 3, the stability theorems are presented.In Section 4, the results are illustrated.In Section 5, conclusions are given.

Floquet theory
Floquet theory transforms a linear periodically time varying system into a linear time invariant system through a Lyapunov transformation.Hence, the stability of the former system can be inferred from that of the latter.Below is a brief review of the Floquet theory.
Consider the linear time varying system ẋ(t) = A(t)x(t), t ≥ t 0 , where x(t) ∈ R n , the matrix A(t) ∈ R n×n is piecewise continuous, bounded, and periodic with period T. Letting Φ(t,t 0 ) be the state transition matrix of system (2.1), the Floquet theorem [7,8] states that (a) Φ(t + T,t 0 + T) = Φ(t,t 0 ), (b) there exist a nonsingular matrix P(t,t 0 ), which satisfies P(t + T,t 0 ) = P(t,t 0 ), and a constant matrix Q such that (c) the Lyapunov transformation transforms system (2.1) into the linear time invariant system ż(t) = Qz(t), t ≥ t 0 , the matrix Q can be expressed as The eigenvalues λ k , k = 1,...,n, of the matrix Q are called the characteristic exponents and the eigenvalues µ k , k = 1,...,n, of the matrix R are called the characteristic multipliers.
The characteristic exponents are related to the characteristic multipliers through Hence, it follows from the Floquet theorem that system (2.1) is exponentially stable if and only if the matrix Q is Hurwitz, that is, all eigenvalues of Q have negative real parts.Equivalently, system (2.1) is exponentially stable if and only if the matrix R is Schur, that is, all eigenvalues of R have magnitudes less than one.Note that the matrix Q defined above is not necessarily real.If it turns out to be complex, then a real Q can be obtained from This follows from the fact that A(t) is also periodic with period 2T.

Stability results
Consider the periodically switched linear system discussed above.Application of Floquet theory to this system yields the following result.

1) is exponentially stable if and only if the matrix
is Schur.Equivalently, system (1.1) is exponentially stable if and only if the matrix is Hurwitz.

Stability analysis of periodically switched systems
Proof.For each k = 1,...,σ, define p k (t) as system (1.1) can be compactly written as Clearly, A(t) is piecewise continuous, bounded, and periodic with period T. Thus, the hypothesis of the Floquet theorem is satisfied.
Thus, it follows that Hence, the conclusion follows from the Floquet theorem.
Next, the above result is applied to the case where the matrices A 1 ,A 2 ,...,A σ are pairwise commutative.

System (1.1) with
is exponentially stable if and only if the matrix

1) is exponentially stable if and only if the matrix
is Hurwitz.
Given the subsystem matrices A 1 ,A 2 ,...,A σ and the activation durations ∆t 1 ,∆t 2 ,..., ∆t σ for these subsystems, the matrices Q and R defined above can be easily calculated.Hence, checking whether a given system is stable or not is relatively straightforward.However, the results of Theorem 3.1, except those of Corollary 3.2, are not very useful for designing periodically switched systems.This is because, for a given set of matrices A 1 ,A 2 ,...,A σ , it is not easy to determine the activation durations ∆t 1 ,∆t 2 ,...,∆t σ such that R is Schur or Q is Hurwitz.Below, two extreme cases, slow and fast switching, are considered.In each case, it is shown that exponential stability can be achieved under a very mild assumption.Theorem 3.3.Consider system (1.1) and assume that at least one of A 1 ,A 2 ,...,A σ is Hurwitz.Then, there exist a sufficiently large T > 0 and activation durations ∆t 1 ,∆t 2 ,...,∆t σ such that system (1.1) is exponentially stable.
This theorem basically states that making the activation durations of the asymptotically stable subsystems sufficiently large compared to those of unstable ones results in an overall exponentially stable switched system.It should be pointed out that using Lyapunov theory, similar results have been obtained in the literature for general switched systems [2,6].
Proof.Let ∆t 1 = η 1 T, ∆t 2 = η 2 T,..., ∆t σ = η σ T.Then, for a sufficiently small T, the matrix can be written as Cevat Gökc ¸ek 7 where Since by assumption Hurwitz and since the eigenvalues of a matrix depend continuously on its elements, it follows that there exists a sufficiently small T such that Q is Hurwitz.Hence, by Theorem 3.1, the system is exponentially stable.
This theorem basically states that making the activation durations of every subsystem sufficiently small results in an overall exponentially stable switched system provided that the average of those subsystems is asymptotically stable.It should be pointed out that using averaging theory, similar results have been obtained in the literature for general periodic systems [1,3].

Examples
Two examples are given below to illustrate the salient futures of the developed results.Since the purpose is to show the usefulness of the developed theory, the chosen examples are fairly simple.
The first example considers the case where all subsystems of a switched system are asymptotically stable and demonstrates applications of Theorems 3.1 and 3.3.It also illustrates that for some switching rule, the switched system can be unstable even if all of its subsystems are asymptotically stable.Assume that ∆t 1 = ∆t 2 = T/2.The eigenvalues of A 1 are −1 ± j2 and the eigenvalues of A 2 are −3 ± j4.Note that although both A 1 and A 2 are Hurwitz, A 1 + A 2 is not Hurwitz.Thus, it follows from Theorem 3.3 that for sufficiently large T, the system is exponentially stable.The magnitudes of the eigenvalues of R as functions of T are plotted by the dashed and dashdot curves on Figure 4.1.From this figure, it follows that this system is unstable when T = 1 whereas it is exponentially stable when T = 2.A possible trajectory for each case is shown on Figure 4.2.
The second example, on the other hand, considers the case where all subsystems of a switched system are unstable and demonstrates applications of Theorems 3.1 and 3.4.It also illustrates that for some switching rule, the switched system can be stable even if all of its subsystems are unstable.
Example 4.2.Consider system (1.1) with σ = 2 and Assume that ∆t 1 = ∆t 2 = T/2.The eigenvalues of A 1 are 1 ± j2 and the eigenvalues of A 2 are 0.5, −4.Note that although neither A 1 nor A 2 is Hurwitz, A 1 + A 2 is Hurwitz.Thus, it follows from Theorem 3.4 that for sufficiently small T, the system is exponentially stable.The magnitudes of the eigenvalues of R as functions of T are plotted by the dashed and dashdot curves on Figure 4.3.From this figure, it follows that this system is exponentially stable when T = 1, whereas it is unstable when T = 2.A possible trajectory for each case is shown on

Conclusions
In this paper, stability of periodically switched linear systems is considered.Using the Floquet theory, necessary and sufficient conditions are derived for exponential stability.It is shown that there exists a slow switching rule that achieves exponential stability if at least one of the subsystems is asymptotically stable.It is also shown that there exists a fast switching rule that achieves exponential stability if the average of the subsystems is stable.