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Automatica 44 (2008) 2857–2862

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Growth rate of switched homogeneous systemsI S. Emre Tuna ∗ Electrical and Electronics Engineering Department, Middle East Technical University, Ankara 06531, Turkey

article

info

Article history: Received 26 June 2007 Received in revised form 30 November 2007 Accepted 19 March 2008 Available online 26 September 2008

a b s t r a c t We consider discrete-time homogeneous systems under arbitrary switching and study their growth rate, the analogue of joint spectral radius for switched linear systems. We show that a system is asymptotically stable if and only if its growth rate is less than unity. We also provide an approximation algorithm to compute growth rate with arbitrary accuracy. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Homogeneity Switched system Growth rate Joint spectral radius Approximation algorithm

1. Introduction Stability analysis of systems under arbitrary switching has for a decade been a major direction of research in systems theory. The main reason, as indicated in an early survey (Liberzon & Morse, 1999), is that many recent engineering systems are of switched nature. The area is still in its infancy and even linear systems are yet not fully explored under switching. Hence much of the work now is concentrated on switched linear systems, see the survey (Shorten, Wirth, Mason, Wulff, & King, 2007). One way to determine the stability of a discrete-time switched linear system is to examine the joint spectral radius (JSR) of its system matrices: if the JSR is less than unity then (and only then) the system is asymptotically stable. In fact, JSR is more than just a sharp indicator of stability; it also tells how fast the trajectories will converge to (or diverge from) the origin. As a consequence, JSR has been extensively studied (Barabanov, 1988; Gurvits, 1995; Wirth, 2002) and active research is going on to devise efficient algorithms to approximate its value (Blondel & Nesterov, 2005; Protasov, 2005). In this paper we consider switched homogeneous systems, a superclass of switched linear systems, which have been the subject of Filippov (1980) and later Holcman and Margaliot (2003)

I This paper was presented at the 7th IFAC Symposium on Nonlinear Control Systems, Pretoria, South Africa. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor Hassan K. Khalil. ∗ Tel.: +90 312 210 2368; fax: +90 312 210 2304. E-mail address: [email protected]

0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.03.017

and Tuna and Teel (2005). In particular, we consider a discretetime homogeneous system under arbitrary switching and study its growth rate, the analogue of JSR for linear systems. We first establish the result that the norm of the trajectories of a discrete-time switched homogeneous system can uniformly (in the norm of the initial condition) be upperbounded by an exponential envelope. The infimum of growth rates of such envelopes is the growth rate of the system. Then we show that asymptotic stability of the system is equivalent to the growth rate’s being less than unity. Hence, the first contribution of the paper is the generalization of the result of applying JSR to the case of homogeneous systems. As a second contribution, we provide an algorithm to approximate growth rate to an arbitrary accuracy. For certain applications this novel algorithm (cf. Tuna (2005)) may stand as an alternative for the known JSR approximation algorithms. The algorithm we provide exploits homogeneity such that the computations are performed on a grid of the unit sphere in Rn . As the dimension of the system increases, the number of grid points necessary to approximate the growth rate, and hence the computation time, should be expected to increase exponentially. That should prevent our algorithm from being applicable to systems with high order unless further tricks, other than exploiting homogeneity, can be applied to the system at hand. Despite this not unexpected vulnerability to curse of dimensionality, the computation time depends on the number of subsystems only linearly, provided that the number of grid points do not increase with increasing number of subsystems. This feature should possibly make the algorithm agreeable for applications with many subsystems of low dimension. The algorithm provides an explicit upperbound on the growth rate, which is shown to

2858

S.E. Tuna / Automatica 44 (2008) 2857–2862

converge to the actual growth rate as one chooses finer and finer grids of the unit sphere. However, in our analysis we have not been able to come up with a lowerbound. Having given an overview of what this paper is about, we also would like to take a few words to make clear what it is not. When one talks about an algorithm that runs behind certain computations, interesting from both theoretical and practical points of view is the computational complexity of the algorithm. Indeed this is a fairly well-studied problem for JSR computing algorithms, where one works with matrices representing linear subsystems and therefore has at his discretion versatile tools such as eigenvalues, singular values, Kronecker products, and invariant subspaces. Related to JSR computations, we refer the reader to Blondel and Nesterov (2005) for a thorough review of existing algorithms, their complexities, along with theoretical limitations to how good (fast) an algorithm can get. When, however, one attempts to study homogeneous systems and therefore one relinquishes the above mentioned tools which, generally, do not exist for nonlinear systems, it becomes insuperably difficult to make precise statements about the computational complexity of an algorithm which approximates the growth rate. Therefore, although in this paper we provide an algorithm, we do not study its computational complexity nor do we compare it with the existing algorithms for switched linear systems. Such an undertaking is beyond our scope, simply because we have no idea how to deal with it. The rest of the paper is organized as follows. The next section gives definitions and notation. In Section 3 we extend the results of JSR of switched linear systems and obtain their counterparts on growth rate of switched homogeneous systems. Section 4 is where we establish the equivalence of asymptotic stability and growth rates being less than unity. In Section 5 we present a numerical algorithm to approximate the growth rate. In Section 6 we provide further generalization of the results of the previous sections by adopting a more general definition of homogeneity. Section 7 presents simplifications to the analysis when the system is linear.

System (1) is asymptotically stable if there exists β ∈ KL such that φ(k, x, q) ≤ β( x , k) for all k, x, and q. A continuous map V : Rn → R≥0 is a Lyapunov function for system (1) if there exist α1 , α2 ∈ K∞ and α3 ∈ K such that α1 ( x ) ≤ V (x) ≤ α2 ( x ) and maxq V (fq x) − V (x) ≤ −α3 ( x ) for all x. Below, we make our first assumption which will henceforth hold.

2. Definitions and notation

In this section we study the growth rate of switched homogeneous systems. The results we establish are the generalizations of what has been known for switched linear systems. The following result is obvious and therefore the proof is omitted.

Nonnegative integers are denoted by N. A function α : R≥0 → R≥0 belongs to class-K (α ∈ K ) if it is zero at zero, continuous, and strictly increasing. It belongs to class-K∞ (α ∈ K∞ ) if it is also unbounded. A function β : R≥0 × R≥0 → R≥0 belongs to classKL (β ∈ KL) if it satisfies: for all t ≥ 0, β(·, t ) is nondecreasing and lims→0+ β(s, t ) = 0, and for all s ≥ 0, β(s, ·) is nondecreasing and limt →∞ β(s, t ) = 0. · stands for some p-norm. Let u (and sometimes ui ) denote a unit vector in Rn , i.e. u = 1. An operator1 f : Rn → Rn is homogeneous (with respect to standard dilation) if for all x ∈ Rn and λ ≥ 0 it satisfies f (λx) = λf (x). Consider the (discrete-time) switched system x+ = fq (x)

Assumption 1. System (1) is homogeneous, i.e. for each index, fq is homogeneous. Observe that, due to homogeneity, the solution satisfies

φ(k, λx, q) = λφ(k, x, q) for all k, λ, x, and q. Given x, q, and k let

Φk (x) := max φ(k, x, q) . q

Then let the (maximum) growth rate of system (1) be

0012 00131/k σ := lim sup sup Φk (u) . k→∞

Remark 2. The growth rate is independent of the choice of the norm, for all p-norms are equivalent. For a homogeneous operator f : Rn → Rn we define the norm of f as

f := sup fu .

(3)

u

Note that max fqk fqk−1 · · · fq1 = sup Φk (u) q

for all k. Finally, let f := max fq and q

µ := inf fq u . u ,q

3. Growth rate under homogeneity

Lemma 3. For homogeneous operators f , g, we have f g ≤ f g . Claim 4. Growth rate σ is finite. Proof. By (4) and Lemma 3 one can write

00131/k 0012 00131/k lim sup sup Φk (u) = lim sup max fqk fqk−1 · · · fq1 0012

u

k→∞

q

k→∞

0013 0001 1/k ≤ lim sup max fqk fqk−1 · · · fq1 0012

k→∞

q

k 1/k

n

≤ lim sup(f ) k→∞

= f. The result hence follows.

0004

Lemma 5. In (2) the limit exists, i.e.

00131/k

0012 lim

k→∞

1 In this paper, we use the word operator out of its classical meaning. We use it to refer to a function describing the righthand side of a system in Rn .

(4)

u

(1)

where x ∈ R is the state and q ∈ {1, 2, . . . , q¯ } (q¯ ∈ N) is the index that determines the righthand side by selecting a transition map from a parametrized family of locally bounded operators {f1 , f2 , . . . , fq¯ }, and x+ is the state at the next time instant. The solution of system (1) at time k ∈ N, having started at the initial condition x and evolved under the influence of an index sequence q := (q1 , q2 , . . .) with qi ∈ {1, 2, . . . , q¯ } is denoted by φ(k, x, q). For convenience we will sometimes write fq x instead of fq (x).

(2)

u

sup Φk (u) u

= σ.

Proof. For compactness let us adopt the symbol Φk instead of supu Φk (u). By Lemma 3 we can write

S.E. Tuna / Automatica 44 (2008) 2857–2862

Proof. Suppose not. Then there exists ω0 > σ such that for all M ≥ 1 there exist k0 , x0 , and q0 such that

Φk = max fqk fqk−1 · · · fq1 q 0013 0012 ≤ max fqk fqk−1 · · · fqm+1 {qk ,qk−1 ,...,qm+1 } 0013 0012 × max fqm fqm−1 · · · fq1

k

φ(k0 , x0 , q0 ) > M ω00 x0 .

{qm ,qm−1 ,...,q1 }

= Φk−m Φm .

(5)

Let us first show that for each m ∈ N≥1 and real ε > 0 there exists 1/k

1/m

m0 ∈ N≥1 such that for all k ≥ m0 we have Φk ≤ Φm + ε . The result follows trivially if Φ1 = f = 0. Hence we suppose otherwise. Let us be given a pair (m, ε). First choose ν > 0 such that (1 + ε/Φ1 )/(1 + ν) > 1. Then choose l0 ∈ N≥1 large enough l /(m(l +1))

1/l0

0 ≤ (1 + ε/Φ1 )/(1 + ν) and Φm0 ≤ (1 + ν) hold. Let m0 := l0 m. Given k ≥ m0 there exist nonnegative integers l ≥ l0 and d ≤ m such that k = lm + d.

so that both Φ1 1/m Φm

By (5) we write 1/k

Φk

1/(lm+d)

≤ (Φlm Φd )1/k = Φlm

1/(lm+d)

Φd

= ≤

(1 + ε/Φ1 ) =

+ ε(

Φm1/m

1/k0

/Φ1 )

(6)

Due to what we have shown at the beginning of the proof, there exists m0 such that for all k ≥ m0 we have

πk ≤ πlm + δ/2 ≤ π1 − (m − 1)δ/2 from which one infers, due to our assumption that (πk ) is not Cauchy, that there exists lm+1 ≥ m0 such that

πlm+1 ≤ π1 − (m − 1)δ/2 − δ = π1 − (m + 1)δ/2. Now recall that πk ≤ π1 for all k. Hence, by assumption, there must exist l1 such that πl1 ≤ π1 − δ , which implies

πl1 ≤ π1 − δ/2. Therefore by induction there exists an index sequence (l1 , l2 , . . .) such that (6) holds. However the sequence (πk ) is nonnegative, i.e. bounded from below, which poses a contradiction to our assumption that it is not Cauchy. Hence (πk ) is Cauchy. Recalling that every Cauchy sequence is convergent, the result follows. 0004

u

ω0

(9)

1/k0

Φk0 (x0 )1/k0 > M0

ω0 ≥ f

which is a contradiction. Otherwise, if k0 > m0 , we can write 1/k0

ω0 − ε = σ + ε ≥ Φk0 (x0 )1/k0 > M0

which would be another contradiction.

ω0 ≥ ω0 0004

4. Stability For switched linear systems, growth rate (known as JSR) is a keen indicator of stability. A switched linear system is asymptotically stable if and only if its JSR is strictly less than unity. This result extends to the more general case of switched homogeneous systems, as indicated by the result below. Theorem 8. If system (1) is asymptotically stable then and only then σ < 1. Proof. Suppose σ < 1. Then by Lemma 7 there exist M ≥ 1 and ω ∈ (σ , 1) such that φ(k, x, q) ≤ M ωk x for all k, x, and q. Define β0 (s, t ) := M ωt s. Note that β0 ∈ KL. Hence the asymptotic stability. Let us now show the other direction. Suppose that there exists β ∈ KL such that φ(k, x, q) ≤ β( x , k) for all k, x and q. Let the integer k0 ≥ 1 be such that β(1, k0 ) ≤ 1/2. Let σ0 := (1/2)1/k0 . Note that σ0 < 1. Then sup Φk0 (x)1/k0 ≤ β(1, k0 )1/k0

x =1

All that is left now is to combine (10) and Remark 6.

≥ σ.

Lemma 7. For each ω > σ there exists M φ(k, x, q) ≤ M ωk x for all k, x, q.

.

≤ σ0 .

Remark 6. Observe that for all k

00131/k

0001m0

is satisfied? The answer is negative. For if k0 ≤ m0 then we can write

+ε

πlm ≤ π1 − mδ/2.

sup Φk (u)

(8)

The result follows by contradiction if f = 0 since f ≥ (sup x =1 Φk (x))1/k for all k, which, in turn, implies no (k0 , x0 ) pair can exist to satisfy (8). Let us consider otherwise, i.e. sup x =1 Φ1 (x) > 0. Let us let ε := (ω0 − σ )/2. By Lemma 5 there exists m0 ≥ 1 such that Φk (x)1/k ≤ σ + ε for all k ≥ m0 and x = 1. Let us set

Φk0 (x0 )1/k0 > M0

where we used the fact, an implication of (5), that Φk ≤ Φ1k for all k. We now move to the second part of the proof where we show that 1/k the (nonnegative) sequence (πk ) := (Φk ) is a Cauchy sequence. Recall that a sequence (ak ) is Cauchy if for each δ > 0 there exists n0 such that al − am < δ for all l, m ≥ n0 . Suppose that (πk ) is not Cauchy. Then there exists δ > 0 such that for each n0 there exist l, m ≥ n0 such that πl − πm ≥ δ . Suppose there exist integers m, lm ≥ 1 such that

0012

≤ Φk0 (x0 )1/k0 .

Now we ask the following question. Is there a pair (k0 , x0 ) such that (8) with M = M0 , i.e.

≤ Φm1/m (1 + ν) · (1 + ε/Φ1 )/(1 + ν) Φm1/m

M 1/k0 ω0 < φ(k0 , x0 , q0 ) 1/k0

d/(lm+d)

(7)

Note that in (7) it must be that x0 > 0. Otherwise φ(k0 , x0 , q0 ) would have to be zero (due to homogeneity) and the inequality would break down. Since x0 > 0, without loss of generality, we can take it to equal unity (again due to homogeneity.) Hence we can rewrite the inequality and then move one step forward as

M0 := max{f, f−1 } · max ω0 , ω0−1

≤ Φml/(lm+d) Φ1 n o 1/l ≤ max Φm1/m , Φml0 /(m(l0 +1)) · max Φ1 0 , 1 Φm1/m Φm1/m

2859

(10) 0004

We will need the following classic result for later use.

≥ 1 such that

Theorem 9. System (1) is asymptotically stable if there exists a Lyapunov function for it.

2860

S.E. Tuna / Automatica 44 (2008) 2857–2862

5. Approximating growth rate Due to its importance in determining the stability of a system, it may be essential to compute σ for certain applications. For that reason, we provide an algorithm to calculate the growth rate to an arbitrary accuracy. We begin this section with the following assumptions on system (1) to hold throughout the rest of the section.

Proof. Suppose that there exists k such that for all u1 and m satisfying u1 − xm ≤ h we have

Φk (u1 ) ≤ (1 + ρ(h)/µ)k Ψm,k .

(13)

Now let us be given some u2 . Let i, δ1 , and u3 be such that xi +δ1 u3 = u2 and δ1 ∈ [0, h]. Let q be such that Φk+1 (u2 ) = Φk (fq u2 ). There exist δ2 ∈ [0, h], j, and u4 such that

Assumption 10. For each index, fq is continuous.

fq u2 = fq u2 (xj + δ2 u4 ).

Assumption 11. We have that µ > 0.

By Lemma 13 we have that j ∈ Liq . Therefore λiq Ψj,k ≤ Ψi,k+1 . We can write

Define ρ : R≥0 → R≥0 as

Φk+1 (u2 ) = fq (xi + δ1 u3 ) Φk (xj + δ2 u4 )

ρ(s) := sup fq (u1 + su2 ) − fq u1

≤ ( fq xi + ρ(δ1 ))Φk (xj + δ2 u4 )

u1 ,u2 ,q

which we observe is continuous and zero at zero.

≤ (λiq + ρ(h))Φk (xj + δ2 u4 )

Claim 12. Function ρ is monotonic.

≤ λiq (1 + ρ(h)/µ) Φk (xj + δ2 u4 ) ≤ (1 + ρ(h)/µ)k+1 λiq Ψj,k

We omit the proof of Claim 12 for lack of space. However, for the reader who would be dissatisfied with this absence of demonstration, we would like to mention that in the analysis that follows ρ can be substituted by a suitable upperbounding class-K function without any problem. See Remark 20.

Eq. (13) trivially holds for k = 0. Hence the result by induction. 0004

5.1. Gridding unit sphere

5.2. Approximate growth rate

Let us be given G := {xi : i = 1, 2, . . . , ¯ı}, a finite grid (¯ı ∈ N) with xi = 1 for all i. Then let

Let us now define the approximate growth rate γ , which depends on system (1) and our choice of grid G, as

h := sup min u − xi and u

i

λiq := fq xi .

≤ (1 + ρ(h)/µ)k+1 Ψi,k+1 .

0012 00131/k γ := lim sup max Ψi,k .

The following two results, namely Claim 15 and Lemma 16, are the analogues of Claim 4 and Lemma 5, respectively. These are not surprising results. The proofs are omitted.

Liq := j : fq xi − λiq xj ≤ fh + (2 + h)ρ(h)

which is never empty. Then define for k ∈ N

Ψi,k+1 := max max λiq Ψj,k q

(11)

j∈Liq

with Ψi,0 = 1 for all i.

Claim 15. Approximate growth rate γ is finite. Lemma 16. In (14) the limit exists, i.e.

Lemma 13. Given i, j, q, and u, if u − xi ≤ h and fq u −1 fq u − xj ≤ h then j ∈ Liq . i

Proof. Let us be given i, j, q, and u satisfying u − x ≤ h and fq u −1 fq u − xj ≤ h. Note that there exist u1 and ε ∈ [0, h] such that u = xi + ε u1 . Also note that there exist δ1 ∈ [0, ρ(ε)] and u2 such that fq xi = fq u + δ1 u2 . It follows that fq u − λiq ≤ δ1 . Let u3 and δ2 ∈ [0, h] be such that fq u −1 fq u = xj + δ2 u3 . We can now write fq xi − λiq xj = fq u + δ1 u2 − λiq xj

= fq u (xj + δ2 u3 ) + δ1 u2 − λiq xj = ( fq u − λiq + λiq )(xj + δ2 u3 ) + δ1 u2 − λiq xj = ( fq u − λiq )(xj + δ2 u3 ) + λiq δ2 u3 + δ1 u2 whence, since fq u − λiq ≤ δ1 ,

fq xi − λiq xj ≤ δ1 + δ1 δ2 + λiq δ2 + δ1 = λiq δ2 + (2 + δ2 )δ1 ≤ fh + (2 + h)ρ(h) which implies j ∈ Liq .

(14)

i

k→∞

Also, for each i let

k→∞

max Ψi,k

00131/k

i

= γ.

The result below is a direct implication of Lemma 14. Theorem 17. We have that σ ≤ (1 + ρ(h)/µ)γ . Theorem 17 is practically important. It implies that if the computed approximate growth rate satisfies γ < (1 + ρ(h)/µ)−1 then one can be sure, by Theorem 8, that system (1) is asymptotically stable. Of course, it would not be of much worth if γ did not converge to σ as the grid gets finer. Although we have not been able to come up with an explicit lowerbound on σ in terms of γ and h, it is still the case that γ converges to σ as the grid fineness h gets smaller. The result below formalizes that observation. Theorem 18. For each ε > 0 there exists δ > 0 such that if h ≤ δ then σ − γ ≤ ε . Proof. Let us be given ε > 0. We let r := σ + ε and define

0004

Lemma 14. For all u and i, satisfying u − xi ≤ h, and k we have

Φk (u) ≤ (1 + ρ(h)/µ)k Ψi,k .

0012 lim

(12)

V (x) := sup q

∞ X

r −k φ(k, x, q) .

k=0

Due to homogeneity, V satisfies V (λx) = λV (x). Observe that V (x) ≥ x . By Lemma 7 there exists M ≥ 1 such that φ(k, x, q) ≤

S.E. Tuna / Automatica 44 (2008) 2857–2862

M (σ +ε/2)k x for all k, x, and q. If we let c := (σ +ε/2)/(σ +ε) < 1, then it follows that V (x) ≤ M (1−c )−1 x . Therefore we can write

x ≤ V (x) ≤ M (1 − c )−1 x .

max V (r q

x

=r

−1

(20)

Recall Lemma 14. Combining (12) with Lemma 16, and since γ ≤ f, one obtains

fq x) − V (x) ≤ − x .

Finally, V is continuous due to Assumption 10. As a result, V is a Lyapunov function for the system +

and by Theorem 9 system (19) is stable. Note that since the growth rate of system (17) is γ , the growth rate of system (19) has to be γ /r. Stability, by Theorem 8, implies that γ /r < 1. Therefore

γ ≤r = σ + ε.

Then we observe that V satisfies −1

2861

f q x.

σ ≤ (1 + ρ(h)/µ)γ ≤ γ + fρ(δ)/µ ≤ γ +ε thanks to (16). Finally, we put together (20) and (21).

Let α be a class-K function satisfying

α(s) ≥ sup V (x + su) − V (x) . x ≤f,u

Such an α exists since V is continuous. Now let us pick δ > 0 such that the following inequalities hold:

(21) 0004

Remark 19. Assumption 11 can be removed but is preferred to be present for the ease of analysis. To be precise, Theorem 18 would still hold without it, however, with it the proof is simpler. Moreover, the upperbound in Lemma 14 can be bettered in a way that obviates Assumption 11. It may also be worth pointing out that if the fq are obtained via sample and hold from continuous-time homogeneous (with degree zero) systems, then the assumption comes for free.

r −1 α(fδ + (2 + δ)ρ(δ)) + α(δ) ≤ 1/2

(15)

fρ(δ)/µ ≤ ε.

(16)

The following remark is for the case when ρ is not known exactly but an upperbound is known instead.

We are able to choose such a δ as ρ is monotonic by Claim 12, continuous, and zero at zero. Now let grid G be such that h ≤ δ . Let us define an auxiliary system

Remark 20. Let α be any class-K function satisfying α(s) ≥ fs + (2 + s)ρ(s) and

ξ + = gt ξ

(17)

for t ∈ {1, 2, . . . , t¯} such that (i) system (17) is homogeneous and (ii) for each ξ = 1 we have {gt ξ : t ∈ {1, 2, . . . , t¯}} = {λiq xj : q ∈ {1, 2, . . . , q¯ }, ξ − xi ≤ h, j ∈ Liq }. Observe that gt ≤ f for all t. Hence the local boundedness. Our second observation is that the growth rate of system (17) equals γ . Given ξ , satisfying ξ = 1, let i be such that ξ −xi ≤ h. Then let j and q0 be such that j ∈ Liq0 and maxt V (r −1 gt ξ ) = r −1 V (λiq0 xj ). Since j ∈ Liq0 there exist ω1 ≤ fh + (2 + h)ρ(h) and u1 such that λiq0 xj = fq0 xi + ω1 u1 . Moreover, there exist ω2 ≤ h and u2 be such that ξ = xi + ω2 u2 . We can write by (15) max V (r −1 gt ξ ) − V (ξ ) t

≤ max V (r −1 fq xi ) − V (xi ) + r −1 α(ω1 ) + α(ω2 ) q

≤ max V (r −1 fq xi ) − V (xi ) + 1/2 = −1/2.

(18)

Now let us be given an arbitrary ξ . If ξ = 0 then we can trivially write maxt V (r −1 gt ξ ) − V (ξ ) ≤ − ξ /2 since each term in the inequality is zero. If ξ > 0 then we can write, by homogeneity and (18), that max V (r −1 gt ξ ) − V (ξ ) t

ξ =r

−1

gt ξ

Remark 21. For k ∈ N consider the recursion 1

k

γi,k+1 = max max λiqk+1 γj,kk+1 q

(22)

j∈Liq

with γi,0 = 1 for all i. Note that γik,k = Ψi,k and therefore limk→∞ maxi γi,k = γ . Also note that µ ≤ γi,k ≤ f for all i, k whereas Ψi,k for some i may either blow up or vanish as k → ∞. Hence, for numerical calculations, (22) should be preferable to (11).

(23)

that is homogeneous with respect to ∆, i.e. Γq ∆λ ξ = ∆λ Γq ξ for all q, λ, and ξ . Assume that Γq is a locally bounded operator for each q. Let ψ denote the solution of system (23). The homogeneous (with respect to ∆) norm is defined as

00011/p kξ k := ξ1 p/r1 + . . . + ξn p/rn for some p ≥ 1. Let us define for k ∈ N ∆ Φk (ξ ) := max kψ(k, ξ , q)k. q

0013

0012 −V

ξ ξ

00130013

We then define the growth rate of system (23) as

≤ − ξ /2

∆

since ξ / ξ is a unit vector. Therefore V is a Lyapunov function for the system +

Then Lemma 14 and Theorem 18 would still hold if Liq were replaced by L∗iq . However, then, a finer grid (i.e. a smaller h) would possibly be required to attain a desired closeness of the approximate growth rate γ to the actual growth rate σ .

ξ + = Γq (ξ )

q

ξ = ξ max V r −1 gt t ξ

In this section we consider homogeneity in a more general sense and extend our analysis on growth rate. First, we need the following definition. A dilation ∆ is such that for each λ (nonnegative) ∆λ = diag(λr1 , . . . , λrn ) (i.e. a diagonal matrix with entries λr1 , . . . , λrn ) with fixed ri > 0 for i = 1, . . . , n. Consider the following switched system in Rn

= r −1 V (Λq0 xi + ω1 u1 ) − V (xi + ω2 u2 ) 0001 ≤ r −1 V (Λq0 xi ) + α(ω1 ) − V (xi ) + α(ω2 )

0012

6. Homogeneity considered in general

= r −1 V (λiq0 xj ) − V (ξ )

0012

L∗iq := j : fq xi − λiq xj ≤ α(h) .

(19)

σ := lim sup k→∞

0012

00131/k

sup ∆Φk (ξ )

.

kξ k=1

Remark 22. The growth rate ∆σ is independent of the choice of the (homogeneous) norm.

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S.E. Tuna / Automatica 44 (2008) 2857–2862

For b ∈ R and r > 0, let b[r ] := sgn(b) b r . Now let us define a transformation H : Rn → Rn as

0010

0011

[r ]

Hx := x1 1 , . . . , x[nrn ] . Note that H −1 exists and is H

−1

0012

−1

[r1 ]

ξ := ξ1

[rn−1 ]

, . . . , ξn

7. Switched linear systems An important special case of switched homogeneous systems are switched linear systems. System (1) is a switched linear system if fq ∈ Rn×n for each q. Let us denote the system as x+ = Aq x

0013

.

We observe that H λx = ∆λ Hx for all x and H −1 ∆λ ξ = λH −1 ξ for all ξ . We also observe that H −1 Γq H λx = λH −1 Γq Hx for all x and that H −1 Γq H is locally bounded for each q. Therefore, for our purposes, we may henceforth and without loss of generality assume that system (23) and system (1) satisfy the following relation H −1 Γq H = fq . The following result then accrues. Theorem 23. We have that ∆σ = σ .

(24)

if it is linear. As we mentioned earlier, the growth rate of a switched linear system has been named the joint spectral radius in the literature. Hence our approximation algorithm presented in Section 5 calculates the JSR of system (24). Linearity brings some simplifications to our analysis. We point them out below. Remark 27. For system (24) Assumption 10 comes for free. Let σi (Aq ) denote the ith largest singular value of Aq and let us define σmax := maxq σ1 (Aq ) and σmin := minq σn (Aq ). Then we realize that for system (24) we have ρ(s) = σmax s and µ = σmin provided that we are working with the Euclidean norm. The following result comes as a corollary of Theorem 17. Corollary 28. For system (24) we have

Remark 24. The practical importance of Theorem 23 is that the growth rate of an arbitrary homogeneous system ξ + = Γq ξ can be computed by the approximation algorithm presented in Section 5 applied to the auxiliary system x+ = fq x which is but homogeneous with respect to standard dilation. The following result is a byproduct of Theorem 23 and Lemma 7. ∆

Corollary 25. For each ω > σ there exists M ≥ 1 such that

kψ(k, ξ , q)k ≤ M ωk kξ k for all k, ξ , and q. System (23) is considered to be asymptotically stable if there exists β ∈ KL such that ψ(k, ξ , q) ≤ β( ξ , k) for all k, ξ , and q. Then we have the following result. Theorem 26. If system (23) is asymptotically stable then and only then ∆σ < 1. Proof. Let α1 and α2 be two class-K∞ functions such that

α1 (kξ k) ≤ ξ ≤ α2 (kξ k) for all ξ . Suppose that system (23) is asymptotically stable with some β ∈ KL characterizing stability. That is, ψ(k, ξ , q) ≤ β( ξ , k) for all k, ξ , and q. That implies

kψ(k, ξ , q)k ≤ β1 (kξ k, k) for all k, ξ , and q; and for β1 (s, t ) := α1−1 (β(α2 (s), t )). Note that β1 ∈ KL. Then we have

φ(k, x, q) = = ≤ =

kH φ(k, x, q)k kψ(k, Hx, q)k β1 (kHxk, k) β1 ( x , k)

for all k, x, and q. Hence the asymptotic stability of system (1). By Theorem 8 therefore we have that σ < 1. Then from Theorem 23 it follows that ∆σ < 1. The other direction is nothing but going backwards since all the above implications (⇒) are indeed equivalences (⇔). Hence the result. 0004

0012 0013 σmax σ ≤ 1+ ·h γ σmin if we work with 2-norm. 8. Conclusion We have generalized the results on the joint spectral radius for switched linear systems to the growth rate for switched homogeneous systems. We have shown that the growth rate of a (switched) homogeneous system is less than unity if and only if the system is asymptotically stable. We also have introduced an algorithm to numerically approximate growth rate to an arbitrary accuracy. This algorithm is shown to yield an upperbound on growth rate. References Barabanov, N. E. (1988). On the Lyapunov exponent of discrete inclusions I-III. Automation and Remote Control, 49, 152–157, 283–287, 558–565. Blondel, V. D., & Nesterov, Y. (2005). Computationally efficient approximations of the joint spectral radius. SIAM Journal of Matrix Analysis, 27, 256–272. Filippov, A. F. (1980). Stability conditions in homogeneous systems with arbitrary regime switching. Automation and Remote Control, 41, 1078–1085. Gurvits, L. (1995). Stability of discrete linear inclusion. Linear Algebra and its Applications, 231, 47–85. Holcman, D., & Margaliot, M. (2003). Stability analysis of second-order switched homogeneous systems. SIAM Journal on Control and Optimization, 41, 1609–1625. Liberzon, D., & Morse, A. S. (1999). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19, 59–70. Protasov, V. (2005). The geometric approach for computing the joint spectral radius. In Proc. of the 44th IEEE conference on decision and control (pp. 3001–3006). Shorten, R., Wirth, F., Mason, O., Wulff, K., & King, C. (2007). Stability criteria for switched and hybrid systems. SIAM Review, 49, 545–592. Tuna, S. E. (2005). Optimal regulation of homogeneous systems. Automatica, 41, 1879–1890. Tuna, S. E., & Teel, A. R. (2005). Regulating discrete-time homogeneous systems under arbitrary switching. In Proc. of the 44th IEEE conference on decision and control (pp. 2586–2591). Wirth, F. (2002). The generalized spectral radius and extremal norms. Linear Algebra and its Applications, 342, 17–40. S. Emre Tuna was born in 1979, in Iskenderun, Turkey. He received a B.S. degree in electrical and electronics engineering from Middle East Technical University, Ankara, in 2000. He received a Ph.D. degree in electrical and computer engineering from University of California, Santa Barbara, in 2005. He is currently an assistant professor at Middle East Technical University.

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Brief paper

Growth rate of switched homogeneous systemsI S. Emre Tuna ∗ Electrical and Electronics Engineering Department, Middle East Technical University, Ankara 06531, Turkey

article

info

Article history: Received 26 June 2007 Received in revised form 30 November 2007 Accepted 19 March 2008 Available online 26 September 2008

a b s t r a c t We consider discrete-time homogeneous systems under arbitrary switching and study their growth rate, the analogue of joint spectral radius for switched linear systems. We show that a system is asymptotically stable if and only if its growth rate is less than unity. We also provide an approximation algorithm to compute growth rate with arbitrary accuracy. © 2008 Elsevier Ltd. All rights reserved.

Keywords: Homogeneity Switched system Growth rate Joint spectral radius Approximation algorithm

1. Introduction Stability analysis of systems under arbitrary switching has for a decade been a major direction of research in systems theory. The main reason, as indicated in an early survey (Liberzon & Morse, 1999), is that many recent engineering systems are of switched nature. The area is still in its infancy and even linear systems are yet not fully explored under switching. Hence much of the work now is concentrated on switched linear systems, see the survey (Shorten, Wirth, Mason, Wulff, & King, 2007). One way to determine the stability of a discrete-time switched linear system is to examine the joint spectral radius (JSR) of its system matrices: if the JSR is less than unity then (and only then) the system is asymptotically stable. In fact, JSR is more than just a sharp indicator of stability; it also tells how fast the trajectories will converge to (or diverge from) the origin. As a consequence, JSR has been extensively studied (Barabanov, 1988; Gurvits, 1995; Wirth, 2002) and active research is going on to devise efficient algorithms to approximate its value (Blondel & Nesterov, 2005; Protasov, 2005). In this paper we consider switched homogeneous systems, a superclass of switched linear systems, which have been the subject of Filippov (1980) and later Holcman and Margaliot (2003)

I This paper was presented at the 7th IFAC Symposium on Nonlinear Control Systems, Pretoria, South Africa. This paper was recommended for publication in revised form by Associate Editor Zongli Lin under the direction of Editor Hassan K. Khalil. ∗ Tel.: +90 312 210 2368; fax: +90 312 210 2304. E-mail address: [email protected]

0005-1098/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2008.03.017

and Tuna and Teel (2005). In particular, we consider a discretetime homogeneous system under arbitrary switching and study its growth rate, the analogue of JSR for linear systems. We first establish the result that the norm of the trajectories of a discrete-time switched homogeneous system can uniformly (in the norm of the initial condition) be upperbounded by an exponential envelope. The infimum of growth rates of such envelopes is the growth rate of the system. Then we show that asymptotic stability of the system is equivalent to the growth rate’s being less than unity. Hence, the first contribution of the paper is the generalization of the result of applying JSR to the case of homogeneous systems. As a second contribution, we provide an algorithm to approximate growth rate to an arbitrary accuracy. For certain applications this novel algorithm (cf. Tuna (2005)) may stand as an alternative for the known JSR approximation algorithms. The algorithm we provide exploits homogeneity such that the computations are performed on a grid of the unit sphere in Rn . As the dimension of the system increases, the number of grid points necessary to approximate the growth rate, and hence the computation time, should be expected to increase exponentially. That should prevent our algorithm from being applicable to systems with high order unless further tricks, other than exploiting homogeneity, can be applied to the system at hand. Despite this not unexpected vulnerability to curse of dimensionality, the computation time depends on the number of subsystems only linearly, provided that the number of grid points do not increase with increasing number of subsystems. This feature should possibly make the algorithm agreeable for applications with many subsystems of low dimension. The algorithm provides an explicit upperbound on the growth rate, which is shown to

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S.E. Tuna / Automatica 44 (2008) 2857–2862

converge to the actual growth rate as one chooses finer and finer grids of the unit sphere. However, in our analysis we have not been able to come up with a lowerbound. Having given an overview of what this paper is about, we also would like to take a few words to make clear what it is not. When one talks about an algorithm that runs behind certain computations, interesting from both theoretical and practical points of view is the computational complexity of the algorithm. Indeed this is a fairly well-studied problem for JSR computing algorithms, where one works with matrices representing linear subsystems and therefore has at his discretion versatile tools such as eigenvalues, singular values, Kronecker products, and invariant subspaces. Related to JSR computations, we refer the reader to Blondel and Nesterov (2005) for a thorough review of existing algorithms, their complexities, along with theoretical limitations to how good (fast) an algorithm can get. When, however, one attempts to study homogeneous systems and therefore one relinquishes the above mentioned tools which, generally, do not exist for nonlinear systems, it becomes insuperably difficult to make precise statements about the computational complexity of an algorithm which approximates the growth rate. Therefore, although in this paper we provide an algorithm, we do not study its computational complexity nor do we compare it with the existing algorithms for switched linear systems. Such an undertaking is beyond our scope, simply because we have no idea how to deal with it. The rest of the paper is organized as follows. The next section gives definitions and notation. In Section 3 we extend the results of JSR of switched linear systems and obtain their counterparts on growth rate of switched homogeneous systems. Section 4 is where we establish the equivalence of asymptotic stability and growth rates being less than unity. In Section 5 we present a numerical algorithm to approximate the growth rate. In Section 6 we provide further generalization of the results of the previous sections by adopting a more general definition of homogeneity. Section 7 presents simplifications to the analysis when the system is linear.

System (1) is asymptotically stable if there exists β ∈ KL such that φ(k, x, q) ≤ β( x , k) for all k, x, and q. A continuous map V : Rn → R≥0 is a Lyapunov function for system (1) if there exist α1 , α2 ∈ K∞ and α3 ∈ K such that α1 ( x ) ≤ V (x) ≤ α2 ( x ) and maxq V (fq x) − V (x) ≤ −α3 ( x ) for all x. Below, we make our first assumption which will henceforth hold.

2. Definitions and notation

In this section we study the growth rate of switched homogeneous systems. The results we establish are the generalizations of what has been known for switched linear systems. The following result is obvious and therefore the proof is omitted.

Nonnegative integers are denoted by N. A function α : R≥0 → R≥0 belongs to class-K (α ∈ K ) if it is zero at zero, continuous, and strictly increasing. It belongs to class-K∞ (α ∈ K∞ ) if it is also unbounded. A function β : R≥0 × R≥0 → R≥0 belongs to classKL (β ∈ KL) if it satisfies: for all t ≥ 0, β(·, t ) is nondecreasing and lims→0+ β(s, t ) = 0, and for all s ≥ 0, β(s, ·) is nondecreasing and limt →∞ β(s, t ) = 0. · stands for some p-norm. Let u (and sometimes ui ) denote a unit vector in Rn , i.e. u = 1. An operator1 f : Rn → Rn is homogeneous (with respect to standard dilation) if for all x ∈ Rn and λ ≥ 0 it satisfies f (λx) = λf (x). Consider the (discrete-time) switched system x+ = fq (x)

Assumption 1. System (1) is homogeneous, i.e. for each index, fq is homogeneous. Observe that, due to homogeneity, the solution satisfies

φ(k, λx, q) = λφ(k, x, q) for all k, λ, x, and q. Given x, q, and k let

Φk (x) := max φ(k, x, q) . q

Then let the (maximum) growth rate of system (1) be

0012 00131/k σ := lim sup sup Φk (u) . k→∞

Remark 2. The growth rate is independent of the choice of the norm, for all p-norms are equivalent. For a homogeneous operator f : Rn → Rn we define the norm of f as

f := sup fu .

(3)

u

Note that max fqk fqk−1 · · · fq1 = sup Φk (u) q

for all k. Finally, let f := max fq and q

µ := inf fq u . u ,q

3. Growth rate under homogeneity

Lemma 3. For homogeneous operators f , g, we have f g ≤ f g . Claim 4. Growth rate σ is finite. Proof. By (4) and Lemma 3 one can write

00131/k 0012 00131/k lim sup sup Φk (u) = lim sup max fqk fqk−1 · · · fq1 0012

u

k→∞

q

k→∞

0013 0001 1/k ≤ lim sup max fqk fqk−1 · · · fq1 0012

k→∞

q

k 1/k

n

≤ lim sup(f ) k→∞

= f. The result hence follows.

0004

Lemma 5. In (2) the limit exists, i.e.

00131/k

0012 lim

k→∞

1 In this paper, we use the word operator out of its classical meaning. We use it to refer to a function describing the righthand side of a system in Rn .

(4)

u

(1)

where x ∈ R is the state and q ∈ {1, 2, . . . , q¯ } (q¯ ∈ N) is the index that determines the righthand side by selecting a transition map from a parametrized family of locally bounded operators {f1 , f2 , . . . , fq¯ }, and x+ is the state at the next time instant. The solution of system (1) at time k ∈ N, having started at the initial condition x and evolved under the influence of an index sequence q := (q1 , q2 , . . .) with qi ∈ {1, 2, . . . , q¯ } is denoted by φ(k, x, q). For convenience we will sometimes write fq x instead of fq (x).

(2)

u

sup Φk (u) u

= σ.

Proof. For compactness let us adopt the symbol Φk instead of supu Φk (u). By Lemma 3 we can write

S.E. Tuna / Automatica 44 (2008) 2857–2862

Proof. Suppose not. Then there exists ω0 > σ such that for all M ≥ 1 there exist k0 , x0 , and q0 such that

Φk = max fqk fqk−1 · · · fq1 q 0013 0012 ≤ max fqk fqk−1 · · · fqm+1 {qk ,qk−1 ,...,qm+1 } 0013 0012 × max fqm fqm−1 · · · fq1

k

φ(k0 , x0 , q0 ) > M ω00 x0 .

{qm ,qm−1 ,...,q1 }

= Φk−m Φm .

(5)

Let us first show that for each m ∈ N≥1 and real ε > 0 there exists 1/k

1/m

m0 ∈ N≥1 such that for all k ≥ m0 we have Φk ≤ Φm + ε . The result follows trivially if Φ1 = f = 0. Hence we suppose otherwise. Let us be given a pair (m, ε). First choose ν > 0 such that (1 + ε/Φ1 )/(1 + ν) > 1. Then choose l0 ∈ N≥1 large enough l /(m(l +1))

1/l0

0 ≤ (1 + ε/Φ1 )/(1 + ν) and Φm0 ≤ (1 + ν) hold. Let m0 := l0 m. Given k ≥ m0 there exist nonnegative integers l ≥ l0 and d ≤ m such that k = lm + d.

so that both Φ1 1/m Φm

By (5) we write 1/k

Φk

1/(lm+d)

≤ (Φlm Φd )1/k = Φlm

1/(lm+d)

Φd

= ≤

(1 + ε/Φ1 ) =

+ ε(

Φm1/m

1/k0

/Φ1 )

(6)

Due to what we have shown at the beginning of the proof, there exists m0 such that for all k ≥ m0 we have

πk ≤ πlm + δ/2 ≤ π1 − (m − 1)δ/2 from which one infers, due to our assumption that (πk ) is not Cauchy, that there exists lm+1 ≥ m0 such that

πlm+1 ≤ π1 − (m − 1)δ/2 − δ = π1 − (m + 1)δ/2. Now recall that πk ≤ π1 for all k. Hence, by assumption, there must exist l1 such that πl1 ≤ π1 − δ , which implies

πl1 ≤ π1 − δ/2. Therefore by induction there exists an index sequence (l1 , l2 , . . .) such that (6) holds. However the sequence (πk ) is nonnegative, i.e. bounded from below, which poses a contradiction to our assumption that it is not Cauchy. Hence (πk ) is Cauchy. Recalling that every Cauchy sequence is convergent, the result follows. 0004

u

ω0

(9)

1/k0

Φk0 (x0 )1/k0 > M0

ω0 ≥ f

which is a contradiction. Otherwise, if k0 > m0 , we can write 1/k0

ω0 − ε = σ + ε ≥ Φk0 (x0 )1/k0 > M0

which would be another contradiction.

ω0 ≥ ω0 0004

4. Stability For switched linear systems, growth rate (known as JSR) is a keen indicator of stability. A switched linear system is asymptotically stable if and only if its JSR is strictly less than unity. This result extends to the more general case of switched homogeneous systems, as indicated by the result below. Theorem 8. If system (1) is asymptotically stable then and only then σ < 1. Proof. Suppose σ < 1. Then by Lemma 7 there exist M ≥ 1 and ω ∈ (σ , 1) such that φ(k, x, q) ≤ M ωk x for all k, x, and q. Define β0 (s, t ) := M ωt s. Note that β0 ∈ KL. Hence the asymptotic stability. Let us now show the other direction. Suppose that there exists β ∈ KL such that φ(k, x, q) ≤ β( x , k) for all k, x and q. Let the integer k0 ≥ 1 be such that β(1, k0 ) ≤ 1/2. Let σ0 := (1/2)1/k0 . Note that σ0 < 1. Then sup Φk0 (x)1/k0 ≤ β(1, k0 )1/k0

x =1

All that is left now is to combine (10) and Remark 6.

≥ σ.

Lemma 7. For each ω > σ there exists M φ(k, x, q) ≤ M ωk x for all k, x, q.

.

≤ σ0 .

Remark 6. Observe that for all k

00131/k

0001m0

is satisfied? The answer is negative. For if k0 ≤ m0 then we can write

+ε

πlm ≤ π1 − mδ/2.

sup Φk (u)

(8)

The result follows by contradiction if f = 0 since f ≥ (sup x =1 Φk (x))1/k for all k, which, in turn, implies no (k0 , x0 ) pair can exist to satisfy (8). Let us consider otherwise, i.e. sup x =1 Φ1 (x) > 0. Let us let ε := (ω0 − σ )/2. By Lemma 5 there exists m0 ≥ 1 such that Φk (x)1/k ≤ σ + ε for all k ≥ m0 and x = 1. Let us set

Φk0 (x0 )1/k0 > M0

where we used the fact, an implication of (5), that Φk ≤ Φ1k for all k. We now move to the second part of the proof where we show that 1/k the (nonnegative) sequence (πk ) := (Φk ) is a Cauchy sequence. Recall that a sequence (ak ) is Cauchy if for each δ > 0 there exists n0 such that al − am < δ for all l, m ≥ n0 . Suppose that (πk ) is not Cauchy. Then there exists δ > 0 such that for each n0 there exist l, m ≥ n0 such that πl − πm ≥ δ . Suppose there exist integers m, lm ≥ 1 such that

0012

≤ Φk0 (x0 )1/k0 .

Now we ask the following question. Is there a pair (k0 , x0 ) such that (8) with M = M0 , i.e.

≤ Φm1/m (1 + ν) · (1 + ε/Φ1 )/(1 + ν) Φm1/m

M 1/k0 ω0 < φ(k0 , x0 , q0 ) 1/k0

d/(lm+d)

(7)

Note that in (7) it must be that x0 > 0. Otherwise φ(k0 , x0 , q0 ) would have to be zero (due to homogeneity) and the inequality would break down. Since x0 > 0, without loss of generality, we can take it to equal unity (again due to homogeneity.) Hence we can rewrite the inequality and then move one step forward as

M0 := max{f, f−1 } · max ω0 , ω0−1

≤ Φml/(lm+d) Φ1 n o 1/l ≤ max Φm1/m , Φml0 /(m(l0 +1)) · max Φ1 0 , 1 Φm1/m Φm1/m

2859

(10) 0004

We will need the following classic result for later use.

≥ 1 such that

Theorem 9. System (1) is asymptotically stable if there exists a Lyapunov function for it.

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S.E. Tuna / Automatica 44 (2008) 2857–2862

5. Approximating growth rate Due to its importance in determining the stability of a system, it may be essential to compute σ for certain applications. For that reason, we provide an algorithm to calculate the growth rate to an arbitrary accuracy. We begin this section with the following assumptions on system (1) to hold throughout the rest of the section.

Proof. Suppose that there exists k such that for all u1 and m satisfying u1 − xm ≤ h we have

Φk (u1 ) ≤ (1 + ρ(h)/µ)k Ψm,k .

(13)

Now let us be given some u2 . Let i, δ1 , and u3 be such that xi +δ1 u3 = u2 and δ1 ∈ [0, h]. Let q be such that Φk+1 (u2 ) = Φk (fq u2 ). There exist δ2 ∈ [0, h], j, and u4 such that

Assumption 10. For each index, fq is continuous.

fq u2 = fq u2 (xj + δ2 u4 ).

Assumption 11. We have that µ > 0.

By Lemma 13 we have that j ∈ Liq . Therefore λiq Ψj,k ≤ Ψi,k+1 . We can write

Define ρ : R≥0 → R≥0 as

Φk+1 (u2 ) = fq (xi + δ1 u3 ) Φk (xj + δ2 u4 )

ρ(s) := sup fq (u1 + su2 ) − fq u1

≤ ( fq xi + ρ(δ1 ))Φk (xj + δ2 u4 )

u1 ,u2 ,q

which we observe is continuous and zero at zero.

≤ (λiq + ρ(h))Φk (xj + δ2 u4 )

Claim 12. Function ρ is monotonic.

≤ λiq (1 + ρ(h)/µ) Φk (xj + δ2 u4 ) ≤ (1 + ρ(h)/µ)k+1 λiq Ψj,k

We omit the proof of Claim 12 for lack of space. However, for the reader who would be dissatisfied with this absence of demonstration, we would like to mention that in the analysis that follows ρ can be substituted by a suitable upperbounding class-K function without any problem. See Remark 20.

Eq. (13) trivially holds for k = 0. Hence the result by induction. 0004

5.1. Gridding unit sphere

5.2. Approximate growth rate

Let us be given G := {xi : i = 1, 2, . . . , ¯ı}, a finite grid (¯ı ∈ N) with xi = 1 for all i. Then let

Let us now define the approximate growth rate γ , which depends on system (1) and our choice of grid G, as

h := sup min u − xi and u

i

λiq := fq xi .

≤ (1 + ρ(h)/µ)k+1 Ψi,k+1 .

0012 00131/k γ := lim sup max Ψi,k .

The following two results, namely Claim 15 and Lemma 16, are the analogues of Claim 4 and Lemma 5, respectively. These are not surprising results. The proofs are omitted.

Liq := j : fq xi − λiq xj ≤ fh + (2 + h)ρ(h)

which is never empty. Then define for k ∈ N

Ψi,k+1 := max max λiq Ψj,k q

(11)

j∈Liq

with Ψi,0 = 1 for all i.

Claim 15. Approximate growth rate γ is finite. Lemma 16. In (14) the limit exists, i.e.

Lemma 13. Given i, j, q, and u, if u − xi ≤ h and fq u −1 fq u − xj ≤ h then j ∈ Liq . i

Proof. Let us be given i, j, q, and u satisfying u − x ≤ h and fq u −1 fq u − xj ≤ h. Note that there exist u1 and ε ∈ [0, h] such that u = xi + ε u1 . Also note that there exist δ1 ∈ [0, ρ(ε)] and u2 such that fq xi = fq u + δ1 u2 . It follows that fq u − λiq ≤ δ1 . Let u3 and δ2 ∈ [0, h] be such that fq u −1 fq u = xj + δ2 u3 . We can now write fq xi − λiq xj = fq u + δ1 u2 − λiq xj

= fq u (xj + δ2 u3 ) + δ1 u2 − λiq xj = ( fq u − λiq + λiq )(xj + δ2 u3 ) + δ1 u2 − λiq xj = ( fq u − λiq )(xj + δ2 u3 ) + λiq δ2 u3 + δ1 u2 whence, since fq u − λiq ≤ δ1 ,

fq xi − λiq xj ≤ δ1 + δ1 δ2 + λiq δ2 + δ1 = λiq δ2 + (2 + δ2 )δ1 ≤ fh + (2 + h)ρ(h) which implies j ∈ Liq .

(14)

i

k→∞

Also, for each i let

k→∞

max Ψi,k

00131/k

i

= γ.

The result below is a direct implication of Lemma 14. Theorem 17. We have that σ ≤ (1 + ρ(h)/µ)γ . Theorem 17 is practically important. It implies that if the computed approximate growth rate satisfies γ < (1 + ρ(h)/µ)−1 then one can be sure, by Theorem 8, that system (1) is asymptotically stable. Of course, it would not be of much worth if γ did not converge to σ as the grid gets finer. Although we have not been able to come up with an explicit lowerbound on σ in terms of γ and h, it is still the case that γ converges to σ as the grid fineness h gets smaller. The result below formalizes that observation. Theorem 18. For each ε > 0 there exists δ > 0 such that if h ≤ δ then σ − γ ≤ ε . Proof. Let us be given ε > 0. We let r := σ + ε and define

0004

Lemma 14. For all u and i, satisfying u − xi ≤ h, and k we have

Φk (u) ≤ (1 + ρ(h)/µ)k Ψi,k .

0012 lim

(12)

V (x) := sup q

∞ X

r −k φ(k, x, q) .

k=0

Due to homogeneity, V satisfies V (λx) = λV (x). Observe that V (x) ≥ x . By Lemma 7 there exists M ≥ 1 such that φ(k, x, q) ≤

S.E. Tuna / Automatica 44 (2008) 2857–2862

M (σ +ε/2)k x for all k, x, and q. If we let c := (σ +ε/2)/(σ +ε) < 1, then it follows that V (x) ≤ M (1−c )−1 x . Therefore we can write

x ≤ V (x) ≤ M (1 − c )−1 x .

max V (r q

x

=r

−1

(20)

Recall Lemma 14. Combining (12) with Lemma 16, and since γ ≤ f, one obtains

fq x) − V (x) ≤ − x .

Finally, V is continuous due to Assumption 10. As a result, V is a Lyapunov function for the system +

and by Theorem 9 system (19) is stable. Note that since the growth rate of system (17) is γ , the growth rate of system (19) has to be γ /r. Stability, by Theorem 8, implies that γ /r < 1. Therefore

γ ≤r = σ + ε.

Then we observe that V satisfies −1

2861

f q x.

σ ≤ (1 + ρ(h)/µ)γ ≤ γ + fρ(δ)/µ ≤ γ +ε thanks to (16). Finally, we put together (20) and (21).

Let α be a class-K function satisfying

α(s) ≥ sup V (x + su) − V (x) . x ≤f,u

Such an α exists since V is continuous. Now let us pick δ > 0 such that the following inequalities hold:

(21) 0004

Remark 19. Assumption 11 can be removed but is preferred to be present for the ease of analysis. To be precise, Theorem 18 would still hold without it, however, with it the proof is simpler. Moreover, the upperbound in Lemma 14 can be bettered in a way that obviates Assumption 11. It may also be worth pointing out that if the fq are obtained via sample and hold from continuous-time homogeneous (with degree zero) systems, then the assumption comes for free.

r −1 α(fδ + (2 + δ)ρ(δ)) + α(δ) ≤ 1/2

(15)

fρ(δ)/µ ≤ ε.

(16)

The following remark is for the case when ρ is not known exactly but an upperbound is known instead.

We are able to choose such a δ as ρ is monotonic by Claim 12, continuous, and zero at zero. Now let grid G be such that h ≤ δ . Let us define an auxiliary system

Remark 20. Let α be any class-K function satisfying α(s) ≥ fs + (2 + s)ρ(s) and

ξ + = gt ξ

(17)

for t ∈ {1, 2, . . . , t¯} such that (i) system (17) is homogeneous and (ii) for each ξ = 1 we have {gt ξ : t ∈ {1, 2, . . . , t¯}} = {λiq xj : q ∈ {1, 2, . . . , q¯ }, ξ − xi ≤ h, j ∈ Liq }. Observe that gt ≤ f for all t. Hence the local boundedness. Our second observation is that the growth rate of system (17) equals γ . Given ξ , satisfying ξ = 1, let i be such that ξ −xi ≤ h. Then let j and q0 be such that j ∈ Liq0 and maxt V (r −1 gt ξ ) = r −1 V (λiq0 xj ). Since j ∈ Liq0 there exist ω1 ≤ fh + (2 + h)ρ(h) and u1 such that λiq0 xj = fq0 xi + ω1 u1 . Moreover, there exist ω2 ≤ h and u2 be such that ξ = xi + ω2 u2 . We can write by (15) max V (r −1 gt ξ ) − V (ξ ) t

≤ max V (r −1 fq xi ) − V (xi ) + r −1 α(ω1 ) + α(ω2 ) q

≤ max V (r −1 fq xi ) − V (xi ) + 1/2 = −1/2.

(18)

Now let us be given an arbitrary ξ . If ξ = 0 then we can trivially write maxt V (r −1 gt ξ ) − V (ξ ) ≤ − ξ /2 since each term in the inequality is zero. If ξ > 0 then we can write, by homogeneity and (18), that max V (r −1 gt ξ ) − V (ξ ) t

ξ =r

−1

gt ξ

Remark 21. For k ∈ N consider the recursion 1

k

γi,k+1 = max max λiqk+1 γj,kk+1 q

(22)

j∈Liq

with γi,0 = 1 for all i. Note that γik,k = Ψi,k and therefore limk→∞ maxi γi,k = γ . Also note that µ ≤ γi,k ≤ f for all i, k whereas Ψi,k for some i may either blow up or vanish as k → ∞. Hence, for numerical calculations, (22) should be preferable to (11).

(23)

that is homogeneous with respect to ∆, i.e. Γq ∆λ ξ = ∆λ Γq ξ for all q, λ, and ξ . Assume that Γq is a locally bounded operator for each q. Let ψ denote the solution of system (23). The homogeneous (with respect to ∆) norm is defined as

00011/p kξ k := ξ1 p/r1 + . . . + ξn p/rn for some p ≥ 1. Let us define for k ∈ N ∆ Φk (ξ ) := max kψ(k, ξ , q)k. q

0013

0012 −V

ξ ξ

00130013

We then define the growth rate of system (23) as

≤ − ξ /2

∆

since ξ / ξ is a unit vector. Therefore V is a Lyapunov function for the system +

Then Lemma 14 and Theorem 18 would still hold if Liq were replaced by L∗iq . However, then, a finer grid (i.e. a smaller h) would possibly be required to attain a desired closeness of the approximate growth rate γ to the actual growth rate σ .

ξ + = Γq (ξ )

q

ξ = ξ max V r −1 gt t ξ

In this section we consider homogeneity in a more general sense and extend our analysis on growth rate. First, we need the following definition. A dilation ∆ is such that for each λ (nonnegative) ∆λ = diag(λr1 , . . . , λrn ) (i.e. a diagonal matrix with entries λr1 , . . . , λrn ) with fixed ri > 0 for i = 1, . . . , n. Consider the following switched system in Rn

= r −1 V (Λq0 xi + ω1 u1 ) − V (xi + ω2 u2 ) 0001 ≤ r −1 V (Λq0 xi ) + α(ω1 ) − V (xi ) + α(ω2 )

0012

6. Homogeneity considered in general

= r −1 V (λiq0 xj ) − V (ξ )

0012

L∗iq := j : fq xi − λiq xj ≤ α(h) .

(19)

σ := lim sup k→∞

0012

00131/k

sup ∆Φk (ξ )

.

kξ k=1

Remark 22. The growth rate ∆σ is independent of the choice of the (homogeneous) norm.

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S.E. Tuna / Automatica 44 (2008) 2857–2862

For b ∈ R and r > 0, let b[r ] := sgn(b) b r . Now let us define a transformation H : Rn → Rn as

0010

0011

[r ]

Hx := x1 1 , . . . , x[nrn ] . Note that H −1 exists and is H

−1

0012

−1

[r1 ]

ξ := ξ1

[rn−1 ]

, . . . , ξn

7. Switched linear systems An important special case of switched homogeneous systems are switched linear systems. System (1) is a switched linear system if fq ∈ Rn×n for each q. Let us denote the system as x+ = Aq x

0013

.

We observe that H λx = ∆λ Hx for all x and H −1 ∆λ ξ = λH −1 ξ for all ξ . We also observe that H −1 Γq H λx = λH −1 Γq Hx for all x and that H −1 Γq H is locally bounded for each q. Therefore, for our purposes, we may henceforth and without loss of generality assume that system (23) and system (1) satisfy the following relation H −1 Γq H = fq . The following result then accrues. Theorem 23. We have that ∆σ = σ .

(24)

if it is linear. As we mentioned earlier, the growth rate of a switched linear system has been named the joint spectral radius in the literature. Hence our approximation algorithm presented in Section 5 calculates the JSR of system (24). Linearity brings some simplifications to our analysis. We point them out below. Remark 27. For system (24) Assumption 10 comes for free. Let σi (Aq ) denote the ith largest singular value of Aq and let us define σmax := maxq σ1 (Aq ) and σmin := minq σn (Aq ). Then we realize that for system (24) we have ρ(s) = σmax s and µ = σmin provided that we are working with the Euclidean norm. The following result comes as a corollary of Theorem 17. Corollary 28. For system (24) we have

Remark 24. The practical importance of Theorem 23 is that the growth rate of an arbitrary homogeneous system ξ + = Γq ξ can be computed by the approximation algorithm presented in Section 5 applied to the auxiliary system x+ = fq x which is but homogeneous with respect to standard dilation. The following result is a byproduct of Theorem 23 and Lemma 7. ∆

Corollary 25. For each ω > σ there exists M ≥ 1 such that

kψ(k, ξ , q)k ≤ M ωk kξ k for all k, ξ , and q. System (23) is considered to be asymptotically stable if there exists β ∈ KL such that ψ(k, ξ , q) ≤ β( ξ , k) for all k, ξ , and q. Then we have the following result. Theorem 26. If system (23) is asymptotically stable then and only then ∆σ < 1. Proof. Let α1 and α2 be two class-K∞ functions such that

α1 (kξ k) ≤ ξ ≤ α2 (kξ k) for all ξ . Suppose that system (23) is asymptotically stable with some β ∈ KL characterizing stability. That is, ψ(k, ξ , q) ≤ β( ξ , k) for all k, ξ , and q. That implies

kψ(k, ξ , q)k ≤ β1 (kξ k, k) for all k, ξ , and q; and for β1 (s, t ) := α1−1 (β(α2 (s), t )). Note that β1 ∈ KL. Then we have

φ(k, x, q) = = ≤ =

kH φ(k, x, q)k kψ(k, Hx, q)k β1 (kHxk, k) β1 ( x , k)

for all k, x, and q. Hence the asymptotic stability of system (1). By Theorem 8 therefore we have that σ < 1. Then from Theorem 23 it follows that ∆σ < 1. The other direction is nothing but going backwards since all the above implications (⇒) are indeed equivalences (⇔). Hence the result. 0004

0012 0013 σmax σ ≤ 1+ ·h γ σmin if we work with 2-norm. 8. Conclusion We have generalized the results on the joint spectral radius for switched linear systems to the growth rate for switched homogeneous systems. We have shown that the growth rate of a (switched) homogeneous system is less than unity if and only if the system is asymptotically stable. We also have introduced an algorithm to numerically approximate growth rate to an arbitrary accuracy. This algorithm is shown to yield an upperbound on growth rate. References Barabanov, N. E. (1988). On the Lyapunov exponent of discrete inclusions I-III. Automation and Remote Control, 49, 152–157, 283–287, 558–565. Blondel, V. D., & Nesterov, Y. (2005). Computationally efficient approximations of the joint spectral radius. SIAM Journal of Matrix Analysis, 27, 256–272. Filippov, A. F. (1980). Stability conditions in homogeneous systems with arbitrary regime switching. Automation and Remote Control, 41, 1078–1085. Gurvits, L. (1995). Stability of discrete linear inclusion. Linear Algebra and its Applications, 231, 47–85. Holcman, D., & Margaliot, M. (2003). Stability analysis of second-order switched homogeneous systems. SIAM Journal on Control and Optimization, 41, 1609–1625. Liberzon, D., & Morse, A. S. (1999). Basic problems in stability and design of switched systems. IEEE Control Systems Magazine, 19, 59–70. Protasov, V. (2005). The geometric approach for computing the joint spectral radius. In Proc. of the 44th IEEE conference on decision and control (pp. 3001–3006). Shorten, R., Wirth, F., Mason, O., Wulff, K., & King, C. (2007). Stability criteria for switched and hybrid systems. SIAM Review, 49, 545–592. Tuna, S. E. (2005). Optimal regulation of homogeneous systems. Automatica, 41, 1879–1890. Tuna, S. E., & Teel, A. R. (2005). Regulating discrete-time homogeneous systems under arbitrary switching. In Proc. of the 44th IEEE conference on decision and control (pp. 2586–2591). Wirth, F. (2002). The generalized spectral radius and extremal norms. Linear Algebra and its Applications, 342, 17–40. S. Emre Tuna was born in 1979, in Iskenderun, Turkey. He received a B.S. degree in electrical and electronics engineering from Middle East Technical University, Ankara, in 2000. He received a Ph.D. degree in electrical and computer engineering from University of California, Santa Barbara, in 2005. He is currently an assistant professor at Middle East Technical University.