Robust stability conditions for switched linear systemsunder restricted switching

Robust stability conditions for switched linear systems under restricted switching

Atreyee Kundu
July 26, 2019
Abstract.

We propose matrix commutator based stability characterization for discrete-time switched linear systems under restricted switching. Given an admissible minimum dwell time, we identify sufficient conditions on subsystems such that a switched system is stable under all switching signals that obey the given restriction. The primary tool for our analysis is commutation relations between the subsystem matrices. Our stability conditions are robust with respect to small perturbations in the elements of these matrices. In case of arbitrary switching (i.e., given minimum dwell time ), we recover the prior result [1, Proposition 1] as a special case of our result.

Key words and phrases:
switched linear systems, stability, restricted dwell times, matrix commutators
AK is with the Department of Electrical Engineering, Indian Institute of Science Bangalore, India. Email: atreyeek@iisc.ac.in.
She thanks Debasish Chatterjee for helpful discussions.

1. Introduction

A switched system has two ingredients — a family of systems and a switching signal. The switching signal selects an active subsystem at every instant of time, i.e, the system from the family that is currently being followed [10, §1.1.2]. Switched systems find wide applications in power systems and power electronics, automotive control, aircraft and air traffic control, network and congestion control, etc. [3, p. 5].

We consider a family of discrete-time linear systems

 (1) x(t+1)=Aix(t),x(0)=x0,i∈P,t∈N0,

where is the vector of states at time , is an index set, and , are constant matrices. Let be a switching signal. A discrete-time switched linear system generated by the family of systems (1) and a switching signal is described as

 (2) x(t+1)=Aσ(t)x(t),x(0)=x0,t∈N0.

The solution to (2) is given by

 x(t)=Aσ(t−1)⋯Aσ(1)Aσ(0)x0,t∈N,

where we have suppressed the dependence of on for notational convenience.

In this paper we will work with switching signals that obey a pre-specified minimum dwell time on every subsystem , i.e., whenever a subsystem is activated by , it remains active for at least units of time.

Remark 1.

In many engineering applications, a restriction on minimum dwell time on subsystems is natural. For instance, actuator saturations may prevent switching frequency beyond a certain limit, or in order to switch from one component to another, a system may undergo certain operations of non-negligible durations leading to a minimum dwell time requirement on each subsystem [4].

Let be the switching instants; these are the points in time when switches from one subsystem to another. Our switching signals satisfy: there exists such that the following condition holds:

 (3) τi+1−τi≥δ,i=0,1,2,….

Let denote the set of all switching signals that satisfy condition (3). Our focus is on global uniform exponential stability (GUES) of the switched system (2).

Definition 1.

[1, Section 1] The switched system (2) is globally uniformly exponentially stable (GUES) over the set of switching signals if there exist positive numbers and such that for arbitrary choices of the initial condition and the switching signal , the following condition holds:

 (4) ∥x(t)∥≤ce−λt∥x0∥for allt∈N,

where denotes the Euclidean norm of a vector .

The term ‘uniform’ in the above definition indicates that the numbers and can be selected independent of . Let denote the matrix product corresponding to a switching signal , and be the set of all products corresponding to the switching signals . Condition (4) can be written equivalently as [1, Section 2]: for arbitrary choice of , the following condition holds:

 (5) ∥Wδ∥≤ceλ|Wδ|for all|Wδ|,

where for a matrix given by a product of matrices , denotes its induced Euclidean norm and denotes its length, i.e., the number of matrices that appear in the product, counting repetitions. We will solve the following problem:

Problem 1.

Given a minimum dwell time , find conditions on the matrices , , such that the switched system (2) is GUES over the set of switching signals .

Remark 2.

It is well-known that the switched system (2) is GUES if all subsystems are individually stable and the dwell time is sufficiently large, A vast body of switched systems literature is devoted to estimating “how large” a dwell time is required for stability, see e.g., [8, 12, 9] and the references therein. In contrast, Problem 1 deals with guaranteeing stability under a “given” minimum dwell time. Stability and optimal control of switched systems under restrictions on minimum dwell time have been addressed earlier in [5, 4, 6, 7].

A switched linear system is stable under arbitrary switching if all subsystems are Schur stable and commute pairwise [11] or are “sufficiently close” to a set of matrices whose elements commute pairwise [1]. On the one hand, these conditions are only sufficient in nature, and their non-satisfaction does not guarantee that a switched system is not stable under all switching signals. On the other hand, a switched system that is not stable under arbitrary switching, may be stable under sets of switching signals that obey a certain minimum dwell time. In many practical applications stability under switching signals satisfying a pre-specified minimum dwell time is useful. This feature motivates us to study matrix commutator based conditions for stability of discrete-time switched linear systems under restricted switching. Our objective is to propose stability conditions that are also robust to small perturbations in the elements of the subsystem matrices. Towards this end, we follow the combinatorial analysis technique presented in [1].

Given an admissible minimum dwell time , we split matrix products into sums and apply counting arguments on them. Our characterization of stability involves upper bounds on the norms of the commutators of the subsystem matrices and a set of scalars relating to the individual matrices and the given minimum dwell time. The main features of our result are:

• Our stability conditions are robust in the sense that if the elements of the subsystem matrices are perturbed by a small margin such that the matrices are not “too far” from a set of matrices for which certain products commute, then stability of the switched system (2) remains preserved under switching signals obeying a minimum dwell time.

• Our stability conditions generalize the conditions for stability under arbitrary switching proposed in [1] in the following sense: if a set of matrices does not satisfy the conditions of [1, Proposition 1], we cannot conclude about (in)stability under all switching signals solely based on commutation relations between the subsystem matrices. However, our conditions may still hold guaranteeing stability under a set of switching signals obeying a certain minimum dwell time. In fact, when (i.e., the case of arbitrary switching), we recover [1, Proposition 1] as a special case of our result.

The remainder of this paper is organized as follows: in §2 we catalog assumptions and notations required for our analysis. Our results appear in §3. We also describe various features of our results in this section. We present numerical examples in §4 and conclude in §5.

2. Preliminaries

Since the set includes constant switching signals, a necessary condition for GUES over is that all subsystem matrices , , are Schur stable. This implies that there exists such that the following condition holds:

 (6) ∥∥Ami∥∥≤ρ<1,for alli∈P.

Of course, the choice of such is not unique; we use the smallest that satisfies (6). Schur stability of the matrices , is, however, not sufficient to guarantee stability of (2) under all elements of , given.

Example 1.

Consider with

 A1=(0.470.12−3.900.19)andA2=(−0.030.780.600.47).

Clearly, both the matrices are Schur stable. Let . We have

 ∥∥A21∥∥=2.62,∥∥A31∥∥=0.71,∥∥A22∥∥=0.90,∥∥A32∥∥=0.86.

Therefore, the smallest for which (6) holds is . Now, define a switching signal as follows:

 σ(0) =1, τi+1−τi ={2,ifσ(τi)=1,4,ifσ(τi)=2.

It is evident that . The switched system (2) generated under the above is unstable. The corresponding (with initial condition ) is plotted in Figure 1.

The above example motivates the search for sufficient conditions on , , such that the switched system (2) is GUES over . Let

 (7) M=maxi∈P∥Ai∥,

be the largest integer satisfying

 (8) K1δ≤m,
 (9) K2=⌊(N−1)(m−1)δ⌋,

and

 (10) K3=(N−1)(m−1)−K2δ.

We define the commutators of the matrix products and , as follows:

 (11) Ep,qij=ApiAqj−AqjApi,i,j∈P.
Remark 3.

The use of commutators of the matrix products and , , instead of commutators of the matrices and , as employed in [1], is motivated by the structure of our switching signals , see Remark 5 for a detailed discussion.

We are now in a position to present our results.

3. Results

The following theorem identifies sufficient conditions on the subsystem matrices , , such that the switched system (2) is GUES over .

Theorem 1.

Consider the family of systems (1). Let be given, the matrices , , satisfy (6) with , and be an arbitrary positive number satisfying

 (12) ρeλm<1.

Suppose that there exist , small enough such that the following conditions hold:

 (13) ∥∥Ep,qij∥∥≤εp,qfor alli,j∈P,

and

 ρeλm+(K1K2εδ,δM(N−1)(m−1)+m−2δ+K1K3εδ,1M(N−1)(m−1)+m−δ−1 +(m−K1δ)K2ε1,δM(N−1)(m−1)+m−δ−1+(m−K1δ)K3ε1,1M(N−1)(m−1)+m−2) (14) ×eλ(N(m−1)+1)≤1,

where , , , and , , are as defined in (7), (8), (9) and (10), respectively. Then the switched system (2) is GUES over the set of switching signals .

Proof.

It suffices to show that if the conditions of Theorem 1 hold, then there exists a positive number such that (5) holds for arbitrary choice of . We will employ mathematical induction on to establish (5).

A. Induction basis: Pick large enough so that (5) holds with all satisfying .

B. Induction hypothesis: Let and assume that (5) is proved for all products of length less than .

C. Induction step: Let , where .

Claim 1.

There exists an index such that contains at least -many ’s.

The above claim follows from the fact that and there are subsystems. Without loss of generality, let . We rewrite as

 L=Am1L1+L2,

where . (Consider, for example, , and . Let . It can be rewritten as

 L=A23A22A1––––––A21=A23A1––––––A22A21−A23E1,212A21 =A1A23A22A21––––––−E1,213A22A21−A23E1,112A21 =A1A23A21––––––A22−A1A23E2,212−E1,213A22A21−A23E1,212A21 =A31A23A22−A1E2,213A22−A1A23E2,212−E1,213A22A21−A23E1,212A21.)

The sum contains at most

• terms of length with ’s and ,

• terms of length with ’s and ,

• terms of length with ’s and , and

• terms of length with ’s and .

Now, from the sub-multiplicativity and sub-additivity properties of the induced norm, we have

 ∥Wδ∥ ≤ρce−λ(|Wδ|−m)+(K1K2εδ,δM(N−1)(m−1)+m−2δ+K1K3εδ,1M(N−1)(m−1)+m−δ−1 +(m−K1δ)K2ε1,δM(N−1)(m−1)+m−δ−1+(m−K1δ)K3ε1,1M(N−1)(m−1)+m−2) ×ce−λ(|Wδ|−(N(m−1)+1)) =ce−λ|Wδ|(ρeλm+(K1K2εδ,δM(N−1)(m−1)+m−2δ+K1K3εδ,1M(N−1)(m−1)+m−δ−1 (15) +(m−K1δ)K2ε1,δM(N−1)(m−1)+m−δ−1+(m−K1δ)K3ε1,1M(N−1)(m−1)+m−2)×eλ(N(m−1)+1)).

The upper bounds on and are obtained from the relations and , respectively. Applying (1) to (3) leads to (5). Consequently, (2) is GUES over .

This completes our proof of Theorem 1. ∎

Theorem 1 characterizes a subset of the set of all Schur stable matrices that preserves stability of a switched system under all switching signals that obey a pre-specified minimum dwell time on all subsystems. The characterization involves upper bounds on the matrix norms of the commutators of the matrix products and , , , and the scalars , , , , , which are related to the matrices , , the total number of subsystems , and the given minimum dwell time . Notice that given and , there always exists a positive scalar such that (12) holds. We rely on the existence of small enough , that satisfy (13)-(1). The scalars , give a measure of the “closeness” of the set of matrices , to a set of matrices for which the matrix products under consideration commute.

Remark 4.

In the simplest case when for all and all , our stability condition relies on commutativity of the matrix products and , , .111 denotes the -dimensional -matrix. This setting is not robust in the sense that if the elements of the subsystem matrices are perturbed by a margin such that the matrix products of our interest cease to commute, then our stability conditions are no longer useful. Stability being a robust property, may not, however, be violated by small perturbations in the elements of the matrices , . Theorem 1 caters to sets of matrices , , for which and , , do not necessarily commute, but are -close in the induced Euclidean norm, . As a result, if the elements of , , are perturbed in a manner so that conditions (6), (13)-(1) continue to hold, then GUES of (2) remains preserved.

Remark 5.

Our analysis relies on splitting matrix products into sums and applying counting arguments. This technique was employed earlier in [1, Proof of Proposition 1] to cater to arbitrary switching. In order to arrive at the form , the authors split a matrix product into sums by exchanging at every step two matrices and , , that appear consecutively in . In contrast, we utilize the minimum dwell time property of switching signals to arrive at the desired structure of . Our procedure involves exchanging products of length of with at most products of length and entries of matrix , , and entries of with at most products of length and entries of matrix , . This leads us to the usage of the commutators , , and unlike only employed in [1].

Our stability conditions generalize [1, Proposition 1] in the following sense: if for a set of subsystems , , the conditions of [1, Proposition 1] do not hold, then we cannot conclude about (in)stability of the switched system (2) under arbitrary switching solely based on commutation relations between the subsystems. However, conditions of Theorem 1 may still hold ensuring GUES under a minimum dwell time . For (i.e., the case of arbitrary switching), we recover [1, Proposition 1] as a special case of our result.

Corollary 1.

Consider the family of systems (1). Let , the matrices , , satisfy (6) with , and be an arbitrary positive number satisfying (12). Suppose that there exists small enough such that

 (16) ∥∥E1,1ij∥∥≤εfor% alli,j∈P,

and

 (17) ρeλm+m(N−1)(m−1)εMN(m−1)+1×eλ(N(m−1)+1)≤1.

Then the switched system (2) is GUES over the set of switching signals .

Proof.

From Theorem 1, we have that the switched system (2) is GUES if conditions (13)-(1) hold.

Given , we have

 K1=m,m−K1δ=0,K2=(N−1)(m−1),andK3=0.

Consequently,

 K1K2εδ,δM(N−1)(m−1)+m−2δ =m(N−1)(m−1)ε1,1M(N−1)(m−1)+m−2 =m(N−1)(m−1)ε1,1MN(m−1)−1, K1K3εδ,1M(N−1)(m−1)+m−δ−1 =0, (m−K1δ)K2ε1,δM(N−1)(m−1)+m−δ−1 =0, (m−K1δ)K3ε1,1M(N−1)(m−1)+m−2 =0.

We have that condition (13) becomes

 ∥∥E1,1ij∥∥≤ε1,1for alli,j∈P,

and condition (1) becomes

 ρeλm

Setting completes our proof of Corollary 1. ∎

Remark 6.

A widely used tool for analyzing stability of a switched system under dwell time switching is multiple Lyapunov functions [2]. Given an admissible minimum dwell time , stability of (2) over the set can be ensured in terms of existence of positive definite matrices , , such that the following conditions hold:

 A⊤iPiAi−Pi ≺0,0<λ<1,i∈P, Pj ⪯μPi,μ≥1,i,j∈P lnμ|lnλ| ≤δ.

In contrast, we propose stability conditions that rely on commutation relations between certain products of the subsystem matrices. Both Lyapunov functions and matrix commutators based conditions for stability under restricted switching are only sufficient in nature. Non-satisfaction of these conditions does not imply that the switched system (2) is not stable over the set of switching signals .

Remark 7.

In many real-world applications the maximum dwell time on subsystems is also restricted. Consider, for example, systems whose components need regular maintenance or replacements, e.g., aircraft carriers, MEMS systems, etc., and systems that are dependent on diurnal or seasonal changes, e.g., components of an electricity grid have inherent restrictions on admissible maximum dwell times. Identifying conditions on , for stability of (2) under a class of that satisfies both a minimum dwell time and a maximum dwell time on all subsystems (i.e., , ) falls as a special case of our results. In this setting the admissible matrix products admit at least and at most consecutive ’s for any , and consequently, our technique of splitting a matrix product into sums works. However, Schur stability of all matrices , is restrictive for constrained maximum dwell times. Indeed, constant switching signals are no longer allowed, and one may obtain GUES of (2) when some or even all ’s are unstable.

Remark 8.

Recently in [6, 7] we studied the algorithmic design of stabilizing switching signals under pre-specified restrictions on admissible dwell times. The stability conditions proposed in [6, 7] involves designing negative weight cycles on the underlying weighted directed graph of a switched system. While those techniques cater to the general setting of nonlinear systems, the existence of stabilizing cycles depends on the existence of Lyapunov-like functions that satisfy certain conditions individually and among themselves. The design of such functions is a numerically difficult problem. In contrast, in this paper we restrict our attention to linear subsystems and restricted minimum dwell time switching, and rely on thew properties of the subsystem matrices to provide stability guarantees under switching signals that satisfy the given restriction.

4. Numerical examples

Consider with

 A1=(0.020.93−0.53−0.92)andA2=(0.040.090.08−0.11).

Clearly, both the subsystems are Schur stable. Let .

We have

 ∥∥A21∥∥ =1.1204,∥∥A31∥∥=0.5404,∥∥A22∥∥=0.0220,∥∥A32∥∥=0.0033.

Therefore, the smallest integer for which (6) holds is . Also, . Let leading to

 ρeλm=0.5569<1.

Now,

 M K1 =⌊mδ⌋=1, K2 =⌊(N−1)(m−1)δ⌋=1, K3 =(N−1)(m−1)−K2δ=0, εδ,δ =0.0133,εδ,1=0.1897,ε1,δ=0.1897,ε1,1=0.2108.

The conditions of [1, Proposition 1] do not hold in the setting described above. Indeed,

 ρeλm+m(N−1)(m−1)ε1,1MN(m−1)+1×eλ(N(m−1)+1) = 0.5569e0.01×3+3×1×2×0.2108×1.36832×2+1×e0.01(2×2+1) = 6.9513>1.

Consequently, we cannot conclude about (in)stability of (2) under arbitrary switching solely based on commutation relations between the subsystem matrices. However,

 ρeλm+(K1K2εδ,δM(N−1)(m−1)+m−2δ+K1K3εδ,1M(N−1)(m−1)+m−δ−1 +(m−K1δ)K2ε1,δM(N−1)(m−1)+m−δ−1+(m−K1δ)K3ε1,1M(N−1)(m−1)+m−2)×eλ(N(m−1)+1) = 0.5569e0.01×3+(1×1×0.0133×1.36831×2+3−4+1×0×0.1897×1.36831×2+3−2−1 +1×1×0.1897×1.36831×2+3−2−1+1×0×0.2108×1.36831×2+3−2)×e0.01(2×2+1) = 0.9664<1

implying that the switched system (2) is GUES under all switching signals that obey a minimum dwell time .

We generate random switching signals that obey a minimum dwell time on both the subsystems and , and plot the corresponding in Figure 2. The initial conditions are chosen from the interval uniformly at random.

We now perturb the elements of the matrices and to generate

 ~A1 =A1+(0.030.02−0.070)=(0.050.95−0.6−0.92),and ~A2 =A2+(000.020)=(0.040.090.1−0.11).

We have

 ∥∥A21∥∥ =1.1384,∥∥A31∥∥=0.5180,∥∥A22∥∥=0.0243,∥∥A32∥∥=0.0038

leading to and . Choosing gives

 ρeλm=0.5182<1.

Also,

 M K1 =K2=1,K3=0, εδ,δ =0.0157,εδ,1=0.2244,ε1,δ=0.2244,ε1,1=0.2579.

Consequently,

 ρeλm+(K1K2εδ,δM(N−1)(m−1)+m−2δ+K1K3εδ,1M(N−1)(m−1)+m−δ−1 +(m−K1δ)K2ε1,δM(N−1)(m−1)+m−δ−1+(m−K1δ)K3ε1,1M(N−1)(m−1)+m−2)×eλ(N(m−1)+1) = 0.5180e0.0001×3+(1×1×0.0157×1.40431×2+3−4+1×0×0.2244×1.40431×2+3−2−1 +1×1×0.1897×1.40431×2+3−2−1+1×0×0.2579×1.40431×2+3−2)×e0.0001(2×2+1) = 0.9830<1

We generate random switching signals that obey a minimum dwell time , and plot the corresponding in Figure 3. The initial conditions are chosen from the interval uniformly at random.

5. Conclusion

In this paper we presented sufficient conditions on subsystems such that stability of a switched linear system is preserved under every switching signal that obeys a “given” minimum dwell time. Our characterization of stability involves commutation relations between the subsystem matrices. Since we dealt with stability of a switched system under all switching signals obeying a given minimum dwell time, the overarching assumption has been Schur stability of all the subsystem matrices. In the presence of unstable subsystems, however, certain subsets of the set of switching signals may be stabilizing. Characeterization of these signals based on commutation relations between subsystem matrices is currently under investigation, and will be reported elsewhere.

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