Generating Functions of Switched Linear Systems - Purdue Engineering

IEEE-TAC FP-10-015

1

Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications Jianghai Hu, Member, IEEE, Jinglai Shen, Member, IEEE, and Wei Zhang, Member, IEEE

Abstract—In this paper, a unified framework is proposed to study the exponential stability of discrete-time switched linear systems, and more generally, the exponential growth rates of their trajectories, under three types of switching rules: arbitrary switching, optimal switching, and random switching. It is shown that the maximum exponential growth rates of system trajectories over all initial states under these three switching rules are completely characterized by the radii of convergence of three suitably defined families of functions called the strong, the weak, and the mean generating functions, respectively. In particular, necessary and sufficient conditions for the exponential stability of the switched linear systems are derived based on these radii of convergence. Various properties of the generating functions are established and their relations are discussed. Algorithms for computing the generating functions and their radii of convergence are also developed and illustrated through examples. Index Terms—Switched systems, stability, optimal control.

I. I NTRODUCTION WITCHED linear systems (SLSs) as a natural extension of linear systems are finding increasing applications in a diverse range of engineering systems [1]. A fundamental problem in the study of SLSs is to determine their stability. See [2], [3] for some recent reviews of the vast amount of existing results on this subject. These results can be roughly classified into two main categories: absolute (or uniform) stability where the switchings can be arbitrary; and stability under restricted switching rules such as switching rate constraints [4] and statedependent switchings [5], [6]. A predominant approach to the study of stability in both cases is through the construction of common or multiple Lyapunov functions [5], [7], [8]. Other approaches include Lie algebraic conditions [9]–[11] and the LMI methods [12]–[14], etc. The purpose of this paper is to characterize not only the stability of SLSs, but also the maximum exponential rates at which their trajectories can grow starting from all possible initial states under three switching rules: arbitrary switching, optimal switching, and random switching. Such rates give quantitative measures on the degree of the SLSs’ exponential stability/stabilizability. Most existing results in this direction

S

Manuscript received Jan. 13, 2010; revised May 28 and July 11, 2010. This work was supported in part by the National Science Foundation under Grant CNS-0643805 and Grant ECCS-0900960. J. Hu is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA (phone: 765-496-2395; fax: 765494-3371; e-mail: [email protected]). J. Shen is with the Department of Mathematics and Statistics, University of Maryland, Baltimore County, MD 21250 USA (e-mail: [email protected]). W. Zhang is with the Department of EECS, University of California, Berkeley, CA 94720 USA (e-mail: [email protected]).

focus on the arbitrary switching case. The maximum exponential growth rate of system trajectories in this case is called the joint spectral radius (JSR) of subsystem matrices, and has been studied extensively (see, e.g., [15]–[19]). In comparison, the maximum exponential growth rate under optimal switching remains much less studied. The smallest such rate under all open-loop switching policies is given by the joint spectral subradius (JSS) of subsystem matrices [16], [17], [20], [21]; whereas in this paper, we study the smallest such rate under the more general closed-loop, state-dependent switching policies (see Section IV-B for an example showing their difference). Finally, under random switching, the SLSs become instances of random dynamical systems, for which the concept of Lyapunov exponents [22], [23] can be used to characterize the expected exponential growth rate of their trajectories. It is well known that finding the maximum exponential growth rates of SLSs are difficult problems. For example, approximating the JSR within arbitrary precision has been proved to be an NP-hard problem [24]; and determining whether the JSR is less than or equal to one is algorithmically undecidable [20]. Computing the JSS is even more difficult [20], [24]. Despite these negative results, many approximation algorithms have been proposed in the literature, e.g. [25]–[29] for the JSR and [17], [21], [29] for the JSS, some with prescribed accuracy. In this paper, we propose a unified method to characterize the maximum exponential growth rates of the SLSs’ trajectories under the above three switching rules. The method is based on the novel concept of generating functions of SLSs, which are suitably defined power series with coefficients determined from the trajectories of the SLSs. The importance of these generating functions is twofold: (i) their radii of convergence characterize precisely the maximum exponential growth rates of the system trajectories; and (ii) they possess many amenable properties that make their efficient computation possible. Thus, generating functions provide both a theoretical framework and the computational tools for characterizing the maximum exponential growth rates of interest. In particular, they provide valid tests for the exponential stability/stabilizability of SLSs. Compared with the existing methods, the proposed approach studies the stability of SLSs from the perspective of their optimal control: the generating functions are the value functions of certain optimal control problems for the SLSs with a varying discount factor, and automatically become Lyapunov functions for stable SLSs. This perspective enables us to uncover some common properties of the exponential growth rates of the SLSs under the different switching rules (see, e.g., Propositions 3, 9 and 14). Moreover, it makes our approach easily extendable to more general classes of systems, such

IEEE-TAC FP-10-015

2

as conewise linear inclusions [6], switched positive systems [30], [31], and controlled SLSs. A similar perspective has been adopted in [15], and by the variational approach [32], [33], which studies the stability of SLSs under arbitrary switching by finding their most divergent trajectories. This paper is organized as follows. In Section II, the relevant stability notions of SLSs are introduced. In Section III (resp. Section IV), the strong (resp. weak) generating functions are defined, analyzed, and used to characterize the exponential stability of the SLSs under arbitrary (resp. optimal) switching. Their numerical computation algorithms are also presented. Section V discusses extensions to randomly switching linear systems. Finally, concluding remarks are given in Section VI. II. S TABILITY

OF

S WITCHED L INEAR S YSTEMS

A discrete-time (autonomous) SLS is defined as follows: its state x(t) ∈ Rn evolves by switching among a set of linear dynamics indexed by the finite index set M := {1, . . . , m}: x(t + 1) = Aσ(t) x(t),

t = 0, 1, . . . .

(1)

Here, σ(t) ∈ M for all t, or simply σ, is called the switching sequence; and Ai ∈ Rn×n , i ∈ M, are the subsystem (dynamics) matrices. Starting from the initial state x(0) = z and under the switching sequence σ, the trajectory of the SLS is denoted by x(t; z, σ). In this paper, unless otherwise stated, k · k denotes both the Euclidean norm on Rn and its induced matrix norm on Rn×n . Definition 1: The SLS (1) is called • exponentially stable under arbitrary switching (with the parameters κ and r) if there exist κ ≥ 0 and r ∈ [0, 1) such that starting from any initial state z and under any switching sequence σ, the trajectory x(t; z, σ) satisfies kx(t; z, σ)k ≤ κrt kzk, for all t = 0, 1, . . .. • exponentially stable under optimal switching (with the parameters κ and r) if there exist κ ≥ 1 and r ∈ [0, 1) such that starting from any initial state z, there exists a switching sequence σ for which the trajectory x(t; z, σ) satisfies kx(t; z, σ)k ≤ κrt kzk, for all t = 0, 1, . . .. As for linear systems, we can also define stability (in the sense of Lyapunov) and asymptotic stability of SLSs under arbitrary (resp. optimal) switching. By homogeneity, local and global stability notions are equivalent for SLSs. Moreover, the asymptotic stability and the exponential stability of SLSs under arbitrary switching are equivalent [6], [34], [35]. We show next that this is also the case under optimal switching. Theorem 1: Under optimal switching, the asymptotic stability and the exponential stability of the SLS (1) are equivalent. Proof: It suffices to show that asymptotic stability implies exponential stability. Assume that the SLS (1) is asymptotically stable under optimal switching. Then, for any initial state z on the unit sphere Sn−1 := {z ∈ Rn | kzk = 1}, there is a switching sequence σz such that x(t; z, σz ) → 0 as t → ∞; hence kx(Tz ; z, σz )k ≤ 41 for a time Tz large enough. As x(Tz ; z, σz ) is continuous in z for fixed σz and Tz , kx(Tz ; y, σz )k ≤ 12 for y in a neighborhood Uz of z in Sn−1 . The union of all such Uz is an open cover of the compact set Sn−1 ; hence Sn−1 ⊆ ∪ℓi=1 Uzi for some

z1 , . . . , zℓ ∈ Sn−1 with ℓ < ∞. Starting from any initial x(0) = z ∈ Sn−1 , we have z ∈ Uzi for some 1 ≤ i ≤ ℓ. By our construction, x(τ1 ) := x(Tzi ; z, σzi ) with τ1 := Tzi satisfies kx(τ1 )k ≤ 12 . Assume without loss of generality that x(τ1 ) 6= 0. Then x(τ1 )/kx(τ1 )k ∈ Uzj for some 1 ≤ j ≤ ℓ, and as a result, x(τ2 ) := x(Tzj ; x(τ1 ), σzj ) with τ2 := τ1 +Tzj satisfies kx(τ2 )k ≤ 12 kx(τ1 )k. By induction, we obtain a switching sequence σz by concatenating σzi , σzj , . . . and a sequence of times 0 = τ0 < τ1 < τ2 < · · · at most τ∗ := maxi Tzi apart such that the resulting trajectory x(t; z, σz ) satisfies kx(τk+1 ; z, σz )k ≤ 21 kx(τk ; z, σz )k Pτ∗ j for all k ≥ 0. Let κ := j=0 (maxi∈M kAi k) . Then t/τ∗ −1 kx(t; z, σz )k ≤ κ(0.5) kzk for all t. Thus, the SLS (1) is exponentially stable under optimal switching. Remark 1: The above proof implies that to prove the exponential stability of the SLS (1) under optimal switching, it suffices to show that for any z ∈ Rn , x(t; z, σz ) → 0 as t → ∞ for at least one σz . This fact will be used in Section IV. Another switching rule we consider is random switching. Let P p := {pi }i∈M be a probability distribution with pi ≥ 0 and i∈M pi = 1. The SLS (1) under the (stationary) random switching probability p has the dynamics x(t + 1) = A(t)x(t),

t = 0, 1, . . . .

(2)

Here, at each time t, A(t) is drawn independently randomly from {Ai }i∈M with the probability P{A(t) = Ai } = pi . Denote by x(t; z, p) the stochastic trajectory of the system (2) from a deterministic initial state x(0) = z, and denote by P and E the probability and expectation operators, respectively. Definition 2: The SLS (2) under the random switching probability p is called • mean square exponentially stable (with the parameters κ and r) if there exist κ ≥ 0 and r ∈ [0, 1) such that for any z ∈ Rn , E[kx(t; z, p)k2 ] ≤ κrt kzk2, for all t = 0, 1, . . .. Similarly, the SLS (2) is called mean square asymptotically stable if E[kx(t; z, p)k2 ] → 0 as t → ∞ for all z ∈ Rn , and almost sure asymptotically stable if P{limt→∞ kx(t; z, p)k = 0} = 1 for all z ∈ Rn . From results on random jumped linear systems [36, Theorem 4.1.1], mean square asymptotic stability and mean square exponential stability of the SLS (2) are equivalent; and each of them implies almost sure asymptotic stability. We shall focus on mean square exponential stability. III. S TRONG G ENERATING F UNCTIONS Central to the stability analysis of SLSs is the task of determining the exponential rate at which kx(t; z, σ)k grows as t → ∞ for trajectories x(t; z, σ) of the SLSs. The following lemma, adopted from [37, Corollary 1.1.10], hints at an indirect way of characterizing this growth rate. Lemma 1: Given P∞a scalar sequence {wt }t=0,1,... , suppose the power series t=0 wt λt has the radius of convergence R. Then for any r > R1 , there exists a constant Cr such that |wt | ≤ Cr rt for all t = 0, 1, . . .. As a result, for any trajectory x(t; z, σ), an (asymptotically) tight bound on the exponential growth rate of kx(t; z, σ)k2 as t → ∞ is given by the reciprocal of the radius of convergence P∞ of the power series t=0 λt kx(t; z, σ)k2 .

IEEE-TAC FP-10-015

3

Motivated by this, we define the strong generating function G(·, ·) : R+ × Rn → R+ ∪ {∞} of the SLS (1) as G(λ, z) := sup σ

∞ X t=0

λt kx(t; z, σ)k2 ,

∀λ ≥ 0, z ∈ Rn , (3)

where the supremum is taken over all switching sequences σ. For a fixed z, G(·, z) is nondecreasing in λ ≥ 0. Indeed, due to the supremum in (3), G(·, z) can be viewed intuitively as the power series in λ corresponding to the “most divergent” trajectories of the SLS starting from z. Thus, by Lemma 1, its radius of convergence defined by λ∗ (z) := sup{λ ≥ 0 | G(λ, z) < ∞}

(4)

is expected to characterize the fastest exponential growth rate of the SLS trajectories starting from z. We call λ∗ (z) the radius of strong convergence of the SLS at z, For each fixed λ ≥ 0, G(λ, z) is a function of z only: Gλ (z) := G(λ, z),

∀z ∈ Rn .

(5)

By definition (3), Gλ (·) is nonnegative and homogeneous of degree two, with G0 (z) = kzk2 . Since Gλ (·) is nondecreasing in λ, we have Gλ (z) ≥ kzk2, ∀z, for λ ≥ 0. We will also refer to Gλ (z) as the strong generating function of the SLS (1). A. Properties of General Gλ (z) We first prove some useful properties of the function Gλ (z). Proposition 1: Gλ (z) has the following properties: 1. (Bellman Equation): Let λ ≥ 0 be arbitrary. Then Gλ (z) = kzk2 + λ · maxi∈M Gλ (Ai z), ∀z ∈ Rn . 2. p (Sub-Additivity): p Let λ ≥ p0 be arbitrary. Then n Gλ (z1 + z2 ) ≤ Gλ (z1 ) + Gλ (z2 ), ∀z1 , zp 2 ∈R . 3. (Convexity): For each λ ≥ 0, both Gλ (z) and Gλ (z) are convex functions of z on Rn . 4. (Invariant Subspace): For each λ ≥ 0, the set Gλ := {z ∈ Rn | Gλ (z) < ∞} is a subspace of Rn invariant under {Ai }i∈M , i.e., Ai Gλ ⊆ Gλ for all i ∈ M. 5. For each λ ≥ 0, Gλ (z) < ∞ for all z ∈ Rn implies that Gλ (z) ≤ ckzk2 for all z ∈ Rn for some finite constant c. 6. For 0 ≤ λ < (maxi∈M kAi k2 )−1 , Gλ (z) < ∞, ∀z. Proof: 1. Note that Gλ (·) is the value function of an infinite P horizon optimal control problem maximizing the t 2 functional ∞ t=0 λ kx(t; z, σ)k . Property 1 is a direct consequence of the dynamic programming principle. 2. For a fixed λ ≥ 0, since x(t; z, σ) is linear in z, we have Gλ (z1 + z2 ) = sup σ

≤ Gλ (z1 ) + 2 sup σ

∞ X t=0

∞ X

λt kx(t; z1 , σ) + x(t; z2 , σ)k2

λt kx(t; z1 , σ)k kx(t; z2 , σ)k + Gλ (z2 )

p t=0 p ≤ Gλ (z1 ) + 2 Gλ (z1 ) Gλ (z2 ) + Gλ (z2 ), ∀z1 , z2 ∈ Rn .

The Cauchy-Schwartz inequality is used in the last step. Taking the square root yields the desired conclusion. 3. For a fixed λ ≥ 0, Gλ (z) is convex in z as by (3) it is the pointwise supremum of a family of convex (indeed, quadratic) p functions of z indexed by σ. The convexity of Gλ (z) follows

p p from sub-additivitypas Gλ (α1 z1p + α2 z2 ) ≤ Gλ (α1 z1 ) + p Gλ (α2 z2 ) = α1 Gλ (z1 ) + α2 Gλ (z2 ), for any z1 , z2 ∈ Rn and α1 , α2 ≥ 0 with α1 + α2 = 1. 4. This follows directly from Properties 1 and 2. 5. Assume λ is such that Gλ (z) < ∞, ∀z ∈ Rn . Write n an arbitrary z ∈ Sn−1 Pnin a 2standard basis {ei } of R as z = P n we i=1 αi ei , where pαi = 1. In light ofPsub-additivity, Pn i=1 n have Gλ (z) ≤ [ i=1 Gλ (αi ei )]2 ≤ n i=1 α2i Gλ (ei ) ≤ c, where c := n · max1≤i≤n Gλ (ei ) < ∞ by our assumption on λ. By homogeneity, we have Gλ (z) ≤ ckzk2 , ∀z ∈ Rn . 6. We simply note that any trajectory x(t; z, σ) of the SLS satisfies kx(t; z, σ)k2 ≤ (maxi∈M kAi k2 )t kzk2, ∀t. From Proposition 1, Gλ is a subspace of Rn that decreases monotonically from G0 = Rn at λ = 0 to G∞ := ∩λ≥0 Gλ as λ → ∞. Let λ1 < λ2 < · · · < λd for some integer d < n be the exact values of λ at which Gλ shrinks. Then the set of all distinct Gλ forms a filtration of subspaces of Rn as: G0 = Gλ− ) Gλ+ = Gλ− ) Gλ+ = · · · = Gλ− ) Gλ+ = G∞ , 1 1 2 2 d d where Gλ− := limλ↑λj Gλj and Gλ+ := limλ↓λj Gλj for j j each j. Since each subspace Gλ− (or Gλ+ ) is invariant under j j {Ai }i∈M , the SLS (1) restricted on it defines a sub-SLS. Intuitively, the restricted SLS on G∞ is “the most exponentially stable” as its trajectories have the slowest exponential growth rate. On bigger subspaces, the restricted SLSs will be “less exponentially stable” as they contain faster growing trajectories. Equivalently, a suitable change of coordinates can simultaneously transform {Ai }i∈M into the same row block upper echelon form, with their last row blocks corresponding to the restricted SLS on G∞ ; their last two row blocks corresponding to the restricted SLS on Gλd − , and so on. From the above discussion, the radius of strong convergence λ∗ (z) at z ∈ Rn can have at most d + 1 ≤ n distinct values: {λ1 , . . . , λd , ∞}. In particular, if the SLS is irreducible, namely, it has no nontrivial invariant subspaces other than Rn and {0} (which occurs with probability one for randomly generated SLSs), then d = 1, and Gλ (z) is either finite everywhere or infinite everywhere for any λ ≥ 0. Remark 2: In the Multiplicative Ergodic Theorem√for non√ switched dynamical systems, {− log λ1 , . . . , − log λd , ∞} are called the Lyapunov exponents of the systems [22]. Example 1: Consider a SLS on R2 with two subsystems:  7  5 4 − 65 3 3 A1 = 65 , A = (6) 2 7 4 5 . −6 6 3 3 Starting from any initial z = (z1 , z2 )T , let x(t; z, σ1 ) and x(t; z, σ2 ) be the state trajectories under the switching sequences σ1 = (1, 1, . . .) and σ2 = (2, 2, . . .), respectively. Then it can be mean trivially) that Gλ (z) Pprovedt (though by no 2 is maxi=1,2 ∞ t=0 λ kx(t; z, σi )k , or more explicitly,  n 2 9(z1 +z2 )2 1 −z2 )  max + (z ,  2(9−λ) 2(1−4λ)   o   9(z1 −z2 )2 (z1 +z2 )2 , if 0 ≤ λ < 91 2(1−9λ) + 2(9−λ)  (z1 −z2 )2 · 1 1 1   {z1 +z2 =0} + ∞ · 1{z1 +z2 6=0} , if 9 ≤ λ < 4   2(1−4λ)  ∞, if λ ≥ 41 .

Here, 1{z1 +z2 =0} denotes the indicator function for the set {(z1 , z2 ) ∈ R2 | z1 + z2 = 0}. Similarly for 1{z1 +z2 6=0} .

IEEE-TAC FP-10-015

4

Thus, Gλ is R2 for 0 ≤ λ < 91 ; the 1D subspace V := {(α, −α)T | α ∈ R} for 19 ≤ λ < 41 ; and {0} for λ ≥ 14 . Each of these is an invariant subspace of R2 for {A1 , A2 }. For example, V is a common eigenspace of A1 and A2 . In fact, A1 and A2 commute and can be simultaneously diago1 1 −1 −1 nalized as h Q A1 Q =diag( i 3 , 2) and Q A2 Q =diag(3, 3 ) cos(π/4) sin(π/4) by Q = − sin(π/4) cos(π/4) . Under the transformation by Q, V becomes the vertical axis. See also [38] for results on the nice reachability of such SLSs. B. Radius of Strong Convergence We next define a quantity that characterizes the stability of the SLS under arbitrary switching. Definition 3: The radius of strong convergence of the SLS (1), denoted by λ∗ ∈ (0, ∞], is defined as  λ∗ := sup λ ≥ 0 | Gλ (z) < ∞, ∀z ∈ Rn }.

By Property 5 of Proposition 1, λ∗ can also be defined as λ∗ = sup{λ ≥ 0 | Gλ (z) < ckzk2, ∀z ∈ Rn , for some finite c}. By Property 6, λ∗ ≥ (maxi∈M kAi k2 )−1 > 0. It is possible that λ∗ = ∞. This is the case, for example, if all solutions x(t; z, σ) of the SLS converge to the origin within a finite time uniformly in z and σ. For the SLS in Example 1, λ∗ = 19 . The following theorem shows that the radius of strong convergence is sufficient for determining the exponential stability of the SLS (1) under arbitrary switching. Theorem 2: The following statements are equivalent: 1) The SLS (1) is exponentially stable under arbitrary switching. 2) Its radius of strong convergence λ∗ > 1. 3) The strong generating function at λ = 1, G1 (z), is finite for all z ∈ Rn . Proof: To show 1) ⇒ 2), suppose there exist constants κ ≥ 1 and r ∈ [0, 1) such that kx(t; z, σ)k ≤ κrt kzk, ∀t, for all trajectory x(t; z, σ) of the SLS. Then for any λ < r−2 , Gλ (z) = sup σ

≤

∞ X t=0

∞ X t=0

λt kx(t; z, σ)k2

λt κ2 r2t kzk2 =

κ2 kzk2 < ∞, ∀z ∈ Rn . 1 − λr2

It follows that λ∗ ≥ r−2 > 1. The implication 2) ⇒ 3) follows directly from the definition of λ∗ . Finally, to show 3) ⇒ 1), suppose G1 (z) < ∞, ∀z. By Proposition 1, G1 (z) ≤ ckzk2, ∀z, for some finitePconstant c. Thus, for any trajectory ∞ 2 x(t; z, σ) of the SLS, ≤ ckzk2 . This t=0 kx(t; z, σ)k √ implies that kx(t; z, σ)k ≤ ckzk, ∀t; and that x(t; z, σ) → 0 as t → ∞. Consequently, the SLS is asymptotically stable, hence exponentially stable [6], under arbitrary switching. Theorem 2 implies the following stronger conclusions. Corollary 1: Given a SLS with a radius of strong convergence λ∗ , for any r > (λ∗ )−1/2 , there exists a constant κr such that kx(t; z, σ)k ≤ κr rt kzk, ∀t, for all trajectories x(t; z, σ) of the SLS. Furthermore, (λ∗ )−1/2 is also the smallest value for the previous statement to hold. Proof: Suppose r > (λ∗ )−1/2 . The scaled SLS with subsystem dynamics matrices {Ai /r}i∈M is easily seen to

have its strong generating function to be G(λ/r2 , z); hence, its radius of strong convergence is r2 λ∗ > 1. By Theorem 2, the scaled SLS is exponentially stable under arbitrary switching. In particular, all its trajectories x ˜(t; z, σ) satisfy k˜ x(t; z, σ)k ≤ κr kzk, ∀t, for some κr > 0. For all trajectories x(t; z, σ) of the original SLS, since x ˜(t; z, σ) = r−t x(t; z, σ), we must t have kx(t; z, σ)k = r k˜ x(t; z, σ)k ≤ κr rt kzk, ∀t. The second conclusion is a direct consequence of Theorem 2. In other words, the maximum exponential growth rate of all the trajectories of the SLS is (λ∗ )−1/2 . Later on in Theorem 3, we will study how to infer the constant κr from Gλ (z). Remark 3: For the SLS (1), the joint spectral radius [17], [18] of the subsystem dynamics matrices {Ai }i∈M is defined  by ρ∗ := limk→∞ sup kAi1 · · · Aik k1/k , i1 , . . . , ik ∈ M ; and the Lyapunov exponent of the corresponding linear difference inclusion is γ ∗ := supσ,z6=0 lim supt→∞ 1t ln kx(t; z, σ)k (see [15]). These two quantities also characterize the maximal exponential growth rate of the SLS trajectories, and are related ∗ to λ∗ by: (λ∗ )−1/2 = ρ∗ = eγ . In this sense, Corollary 1 is equivalent to [17, Prop. 1.4] and to [18, Prop. 4.17]. C. Smoothness of Finite Gλ (z) When λ is in the range of [0, λ∗ ), the function Gλ (z) is finite everywhere. We shall focus on such finite Gλ (z), and establish some smoothness properties of them in this section. We first introduce a few notions. A function f : Rn → R is called directionally differentiable at z0 ∈ Rn if its (onesided) directional derivative at z0 along any direction v ∈ Rn (z0 ) exists. If f is defined as f ′ (z0 ; v) := limτ ↓0 f (z0 +τ v)−f τ both directionally differentiable at z0 and locally Lipschitz continuous in a neighborhood of z0 , it is called B(ouligand)differentiable at z0 . Finally, f is semismooth at z0 if it is Bdifferentiable in a′ neighborhood of z0 and the following limit ′ 0 )−f (z0 ;z−z0 )| holds: limz→z0 | f (z;z−zkz−z = 0. 0k z6=z0 p Proposition 2: For λ ∈ [0, λ∗ ), both Gλ (z) and Gλ (z) are convex, locally p Lipschitz continuous, and semismooth on Rn . Moreover, Gλ (z) is globally Lipschitz p continuous. Proof: The convexity of Gλ (z) and Gλ (z) has been proved in Proposition 1. Being convex, they must also be semismooth according to [39, Prop. 7.4.5]. Finally, using the sub-additivity in Proposition 1, we obtain, ∀z, ∆z ∈ Rn , p p p p − Gλ (−∆z) ≤ Gλ (z + ∆z) − Gλ (z) ≤ Gλ (∆z). p p p √ Thus, Gλ (z + ∆z)− Gλ (z) ≤ Gλp (±∆z) ≤ ck∆zk for some finite constant c as λ < λ∗ , i.e., Gλ (z) is globally √ Lipschitz continuous on Rn with the Lipschitz constant c. p As a result, Gλ (z), hence Gλ (z), is also locally Lipschitz continuous on Rn . Note that for λ > λ∗ , Gλ (z) can not be continuous on Rn . Indeed, in this case, Gλ (z0 ) = ∞ at some z0 ∈ Rn . Thus as 6 0. k → ∞, the sequence zk0 → 0, but Gλ ( zk0 ) → ∞ = D. Quadratic Bounds of Finite Gλ (z) For λ ∈ [0, λ∗ ), Gλ (z) is finite everywhere, hence quadratically bounded by Proposition 1. Define gλ := sup{Gλ (z) | kzk = 1},

λ ∈ [0, λ∗ ).

(7)

IEEE-TAC FP-10-015

5

By homogeneity, gλ can be equivalently defined as the smallest constant c such that Gλ (z) ≤ ckzk2, ∀z ∈ Rn . It is easy to see that gλ is finite and strictly increasing on [0, λ∗ ) (we exclude the trivial case where all Ai are zero), with g0 = 1. We next prove an affine lower bound of 1/gλ . Lemma 2: g1λ ≥ 1−λ·maxi∈M kAi k2 , ∀λ ∈ [0, λ∗ ). Thus, 1 λ ≥ , 1 − 1/gλ maxi∈M kAi k2

∀λ ∈ (0, λ∗ ).

1 λ kx(t; z, σ)k ≤ kzk2, ∀σ. 2 1 − λ · max i∈M kAi k t=0 t

t=0

P∞ t Then the power series t=0 wt λ has its radius of convergence at least λ1 := λ0 /(1 − 1/β). Moreover, t=0

wt λt ≤

βw0 < ∞, ∀λ ∈ [λ0 , λ1 ). 1 − (β − 1)(λ/λ0 − 1)

Using Lemma 3, we obtain the following estimate of gλ . Proposition 3: The function λ/(1 − 1/gλ) is nondecreasing for λ ∈ (0, λ∗ ), and is upper bounded by λ ≤ λ∗ , 1 − 1/gλ

∀λ ∈ (0, λ∗ ).

(10)

Proof: Consider a fixed λ0 ∈ (0, λ∗ ). Let x(t; z, σ) be an arbitrary trajectory of the SLS. For each s = 0, 1, . . ., by the definition of gλ0 , the trajectory x(t + s; z, σ), t = 0, 1, . . ., of the SLS starting from the initial state x(s; z, σ) satisfies ∞ X t=0

λt0 kx(t + s; z, σ)k2 ≤ Gλ0 (x(s; z, σ)) ≤ gλ0 kx(s; z, σ)k2 .

Hence, the sequence {wt := kx(t; z, σ)k2 }t=0,1,... satisfies the condition (9) with β = gλ0 . By Lemma 3, we have ∞ X t=0

λt kx(t; z, σ)k2 ≤

gλ0 kzk2 , 1 − (gλ0 − 1)(λ/λ0 − 1)

for λ ∈ [λ0 , λ1 ), where λ1 := λ0 /(1 − 1/gλ0 ). As the trajectory x(t; z, σ) is arbitrary, we conclude that gλ0 Gλ (z) ≤ kzk2 < ∞, ∀λ ∈ [λ0 , λ1 ). 1 − (gλ0 − 1)(λ/λ0 − 1)

This implies λ∗ ≥ λ1 , i.e., the desired conclusion (10); and gλ ≤

(λ, 1/gλ )

0

1 − λ · maxi∈M kAi k2

Fig. 1.

λ/(1 − 1/gλ )

λ∗

λ

Plot of the function 1/gλ (in solid line).

2

By definition, we have gλ ≤ (1−λ·maxi∈M kAi k2 )−1 , which is the desired conclusion for 0 ≤ λ < (maxi∈M kAi k2 )−1 . When (maxi∈M kAi k2 )−1 ≤ λ < λ∗ , the desired conclusion is trivial: g1λ ≥ 0 ≥ 1 − λ · maxi∈M kAi k2 . The following auxiliary result on general power series is proved in Appendix A. Lemma 3: Let {wt }t=0,1,... be a sequence of nonnegative scalars such that for some λ0 > 0 and β ≥ 1, ∞ X wt+s λt0 ≤ βws , s = 0, 1, . . . . (9)

∞ X

1 − λ/λ∗

(8)

Proof: Let λ ∈ [0, λ∗ ). For an arbitrary trajectory x(t; z, σ) of the SLS, kx(t; z, σ)k2 ≤ (maxi∈M kAi k2 )t kzk2 for all t. Therefore, for 0 ≤ λ < (maxi∈M kAi k2 )−1 , ∞ X

1/gλ 1

1 − (gλ0

λ0 gλ0 λ ≥ ⇒ , − 1)(λ/λ0 − 1) 1 − 1/gλ 1 − 1/gλ0

for λ ∈ [λ0 , λ1 ). Since λ0 ∈ (0, λ∗ ) is arbitrary and λ1 > λ0 , this proves the monotonicity of λ/(1 − 1/gλ ) on (0, λ∗ ). In Appendix B, the following generic properties of the functions gλ and 1/gλ are proved. Proposition 4: The function 1/gλ is strictly decreasing, semismooth, and Lipschitz continuous with Lipschitz constant maxi∈M kAi k2 on [0, λ∗ ). Moreover, 1/gλ → 0 as λ ↑ λ∗ . Correspondingly, gλ is strictly increasing, convex, semismooth, and locally Lipschitz continuous on [0, λ∗ ), with gλ → ∞ as λ ↑ λ∗ . Remark 4: Since gλ → ∞ as λ ↑ λ∗ , the generating function Gλ∗ (z) must have infinite value at some z ∈ Rn , according to Property 5 of Proposition 1. This implies that, as λ increases, λ∗ is precisely the first value at which Gλ (·) starts to have infinite values. Fig. 1 plots the graph of a generic 1/gλ as a function of λ ∈ [0, λ∗ ). According to (8) and (10), the graph of 1/gλ is sandwiched between those of two affine functions: 1−λ maxi∈M kAi k2 from the left and 1−λ/λ∗ from the right. In addition, by Proposition 3, the ray (middle dashed line) emitting from the point (0, 1) and passing through (λ, 1/gλ ) λ , 0) that moves intersects the λ-axis at the point ( 1−1/g λ ∗ monotonically to the right towards (λ , 0) as λ increases in [0, λ∗ ), i.e., the ray rotates around its starting point (1, 0) counterclockwise monotonically. It is conjectured that the function 1/gλ is indeed convex on [0, λ∗ ) for any SLS. It is easy to show that the directional derivative of gλ at λ = 0 is gλ′ (0+ ) = maxi∈M kAi k2 . Hence the directional derivative of 1/gλ at λ = 0 is: (1/gλ )′ (0+ ) = − maxi∈M kAi k2 . In Fig. 1, this means that the graph of 1/gλ is tangential to the leftmost dashed ray emitting from (0, 1). E. Norms Induced by Finite Gλ (z) ∗ pAs an immediate result of Proposition 1, for λ ∈ [0, λ ), Gλ (z) is finite, sub-additive, and homogeneous of degree one; thus it defines a norm on the vector space Rn : p (11) kzkGλ := Gλ (z), ∀z ∈ Rn .

As λ increases, the norm k · kGλ increases, hence its unit ball shrinks. See Fig. 2 for the plots of such unit balls for the SLS (6) in Example 1. The vector norm k·kGλ induces a matrix norm for A ∈ Rn×n by: kAkGλ := supz6=0 {kAzkGλ /kzkGλ }.

IEEE-TAC FP-10-015

6

1

for all trajectories x(t; z, σ) of the SLS (1), where 1/2  dλ ≤ (λ∗ )−1/2 + ε. rλ = 1 + λdλ

λ=0.05 λ=0.09 λ=0.11

0.8

0.6

0.4

0.2

Remark 5: Once Gλ (z) is computed at any λ ∈ [0, λ∗ ), (13) gives a lower bound of λ∗ . An upper bound of λ∗ can be derived by using a result in [27, Lemma 3.3] as −1  Gλ (Ai z) ∗ . (14) λ ≤ inf max z6=0 i∈M Gλ (z)

0

−0.2

−0.4

−0.6

−0.8

−1 −1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

This upper bound, together with the lower bound in (13), gives an interval for the possible values of λ∗ . As λ ↑ λ∗ , it can be shown that the two bounds converge towards each other; and the corresponding norm k · kGλ approaches asymptotically a Barabanov norm [23], [27] of {Ai }i∈M .

1

Unit balls of k · kGλ for the SLS (6).

Fig. 2.

For λ ∈ [0, λ∗ ), define the constant:

dλ := sup max kAi zk2Gλ = sup max Gλ (Ai z). kzk=1 i∈M

kzk=1 i∈M

(12)

Lemma 4: For each λ ∈ [0, λ∗ ), the norm k · kGλ satisfies: p max kAi kGλ = dλ /(1 + λdλ ). i∈M

Proof: Using the Bellman equation, we write

Gλ (Ai z) maxi∈M Gλ (Ai z) = sup i∈M kzk=1 Gλ (z) Gλ (z) kzk=1

maxkAi k2Gλ = max sup i∈M

maxi∈M Gλ (Ai z) = sup kzk=1 1 + λ · maxi∈M Gλ (Ai z) =

Note that the second last step follows as x/(1 + λx) is continuous and increasing in x ∈ R+ for any λ ≥ 0. The above results yields bounds on both the exponential growth rate of the SLS trajectories and λ∗ as follows. Corollary 2: Suppose λ ∈ [0, λ∗ ). Then  t/2 dλ √ kx(t; z, σ)k ≤ gλ kzk, t = 0, 1, . . . , 1 + λdλ holds for all trajectories of the SLS (1). As a result, we have 1 , dλ

∀λ ∈ [0, λ∗ ).

We next present some algorithms for computing the finite strong generating functions. The idea is that Gλ (z) as the value function of an infinite horizon optimal control problem can be approximated by those of a sequence of finite horizon problems. Specifically, for each k = 0, 1, . . ., define Gkλ (z) := max σ

k X t=0

λt kx(t; z, σ)k2 , ∀z ∈ Rn .

(15)

Maximum is used here instead of supremum as only the first k steps of σ affect the summation. Proposition 5: For any λ ≥ 0 and k = 0, 1, . . . , Gkλ (z) is a convex function of z on Rn satisfying

supkzk=1 maxi∈M Gλ (Ai z) dλ . = 1 + λ · supkzk=1 maxi∈M Gλ (Ai z) 1 + λdλ

λ∗ > λ +

F. Algorithms for Computing Gλ (z)

G0λ (z) ≤ G1λ (z) ≤ G2λ (z) ≤ · · · ≤ Gλ (z), ∀z ∈ Rn . Moreover, for λ ∈ [0, λ∗ ), Gkλ (z) as k → ∞ converges exponentially fast to Gλ (z): ∀k = 0, 1, . . ., |Gkλ (z) − Gλ (z)| ≤ gλ (1 − 1/gλ )k+1 kzk2 , ∀z ∈ Rn . Proof: The convexity proof is identical to that of Proposition 1, hence omitted. Fix λ ≥ 0 and z ∈ Rn , and let σk be a switching sequence achieving the maximum in (15). Then,

(13)

√ Proof: We note that kzk ≤ kzkGλ ≤ gλ kzk, ∀z ∈ Rn , i.e., the norm k · kGλ is equivalent to the Euclidean norm k · k. For any trajectory x(t; z, σ) of the SLS, we then have kx(t; z, σ)k ≤ kx(t; z, σ)kGλ = kAσ(t−1) · · · Aσ(0) zkGλ ≤ kAσ(t−1) kGλ · · · kAσ(0) kGλ · kzkGλ , ∀t. √ Applying Lemma 4 and noting that kzkGλ ≤ gλ kzk, we obtain the first conclusion. This in turn implies that λ∗ > (1 + λdλ )/dλ , which is the desired conclusion (13). As λ ↑ λ∗ , by (13), we must have dλ → ∞; hence dλ /(1 + λdλ ) → 1/λ∗ . The following then holds. Theorem 3: For any ε > 0, there exists a λ ∈ (0, λ∗ ) sufficiently close to λ∗ such that √ kx(t; z, σ)k ≤ gλ (rλ )t kzk, t = 0, 1, . . . ,

Gkλ (z) =

k k+1 X X λt kx(t; z, σk )k2 ≤ λt kx(t; z, σk )k2 ≤ Gk+1 λ (z). t=0

t=0

Similarly, we can show Gkλ (z) ≤ tonicity. When λ ∈ [0, λ∗ ), Gλ (z)

Gλ (z), hence the monois finite for each z ∈ Rn . Let x ˆ(t) := x(t; z, σ) be a trajectory under an optimal switchingP sequence σ that achieves the supremum in (3), i.e., ∞ x(t)k2 . By the Bellman equation, Gλ (z) = t=0 λt kˆ

Gλ (ˆ x(k − 1))−λ · Gλ (ˆ x(k)) = kˆ x(k − 1)k2 ≥

Gλ (ˆ x(k − 1)) , gλ

for k = 1, 2, . . ., where the last step follows from the definition of gλ . Rearranging and by induction, we obtain Gλ (ˆ x(k)) ≤ λ−1 (1 − 1/gλ ) Gλ (ˆ x(k − 1)) ≤ · · ·

≤ λ−k (1 − 1/gλ )k Gλ (z) ≤ λ−k gλ (1 − 1/gλ )k kzk2 ,

IEEE-TAC FP-10-015

7

Algorithm 1 Computing Gλ (z) on Grid Points of Sn−1

0.8

n−1

Let S = be a set of grid points of S ; 0 b Initialize k = 0, and Gλ (zj ) = 1 for all zj ∈ S; repeat k ← k + 1; for each zj ∈ S do for each i ∈ M do Find a minimal subset Sij of S whose elements span a convex cone containing Ai zj ; P Express Ai zj as zℓ ∈Sij αℓij zℓ with αℓij > 0; q P b k−1 (zℓ ); Set gij = zℓ ∈Sij αℓij G λ end for bk (zj ) = 1 + λ · maxi∈M g 2 ; Set G ij λ end for until k is large enough b k (zj ) for all zj ∈ S return G λ

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1

−0.5

0

0.5

1

1

0.8

0.6 1/gλ

{zj }N j=1

0.4

0.2

0

0

0.05

0.1

0.15

0.2 λ

0.25

0.3

0.35

0.4

Fig. 3. Top: Unit balls of k · kGλ for the SLS in Example 2 at λ = 0.1, 0.2, 0.3, 0.37, 0.38 (inward). Bottom: Plot of 1/gλ .

for k = 0, 1, . . .. The optimality of x ˆ(t) then implies that ∞ X t=k

λt kˆ x(t)k2 = λk

∞ X t=0

λt kˆ x(t + k)k2 = λk Gλ (ˆ x(k))

≤ gλ (1 − 1/gλ )k kzk2 , ∀k = 0, 1, . . . . Pk Consequently, Gλ (z) ≥ Gkλ (z) ≥ t=0 λt kˆ x(t)k2 = Gλ (z)− P ∞ t 2 x(t)k ≥ Gλ (z) − gλ (1 − 1/gλ )k+1 kzk2. t=k+1 λ kˆ Therefore, Gkλ (z) for k large enough provide approximations of Gλ (z) within arbitrary precision. A recursive procedure to compute the functions Gkλ (z) is as follows: G0λ (z) = kzk2 ,

Gkλ (z) = kzk2 + λ · max Gλk−1 (Ai z), i∈M

k = 1, 2, . . . . (16)

To implement the recursion numerically, one first represents each Gλk−1 (z) by its values on some fine grid points of the unit sphere, and then carries out the recursion (16) by estimating conservatively the values of Gλk−1 (Ai z) at those Ai z not aligned with the direction of any q grid point, using convexity and homogeneity of the function Gλk−1 (z). The above idea is summarized in Algorithm 1. Similar ray gridding techniques have also been used in the previous studies [40], [41]. b k : S → R+ , Algorithm 1 returns a sequence of mappings G λ k = 0, 1, . . ., where S is a set of grid points of the unit sphere. They provide upperbounds of Gkλ (z) on S as follows. b k (zj ), ∀zj ∈ S, ∀k = 0, 1, . . . Proposition 6: Gkλ (zj ) ≤ G λ Proof: We prove by induction. At k = 0, we have b 0 (zj ) = 1 for all zj ∈ S. Suppose the conclusion G0λ (zj ) = G λ is true for 0, 1, . . . , k − 1. For any zj ∈ S, let Sij and αℓij be b k (zj ) = 1 + λ · maxi∈M g 2 , as given in Algorithm 1. Then G ij λ where by the induction hypothesis and sub-additivity, q X q X Gλk−1 (αℓij zℓ ) gij ≥ αℓij Gλk−1 (zℓ ) = zℓ ∈Sij

zℓ ∈Sij

≥

s  X  q Gλk−1 αℓij zℓ = Gλk−1 (Ai zj ). zℓ ∈Sij

Using the Bellman equation in Proposition 1, we then have b k (zj ) ≥ 1 + λ · maxi∈M Gk−1 (Ai zj ) = Gk (zj ). Thus, the G λ λ λ conclusion also holds for k. This completes the proof. Combining Proposition 6 and Theorem 2, we have the following stability test. Corollary 3: A sufficient condition for the SLS (1) to be exponentially stable under arbitrary switching is that the mapb k : S → R+ , k = 0, 1, . . ., obtained by Algorithm 1 pings G 1 are uniformly bounded. By repeatedly applying Algorithm 1 to a sequence of λ whose values increase according to λnext = λ/(1 − 1/gλ ) or by (13), increasingly accurate underestimates of λ∗ can be obtained. In view of Section III-E, this procedure is somewhat similar to the norm iteration proposed in [27], although the iterations in [27] are performed through a max-relaxation scheme and in this paper through the Bellman equation. As λ approaches λ∗ , however, the computation time of Algorithm 1 will get exponentially longer for two reasons. First, the convergence of Gkλ (z) to Gλ (z) is much slower by ˆ k (z) over-approximating Proposition 5. Second, errors of G λ k Gλ (z) will accumulate quickly in time. Thus, a denser grid and more iterations are generally needed to ensure a given accuracy. This is not surprising given that the problem of finding λ∗ (or the JSR) is known to be NP-hard [24]. Hence the complexity of Algorithm 1, like other approximation algorithms, will grow exponentially with respect to the state dimension and the required accuracy. Some recently developed algorithms for estimating the JSR (e.g., [29]) have the desirable feature of providing prescribed performance guarantee of the computed estimates. Besides (13) and (14), we are currently working on developing further performance assurance for Algorithm 1. G. Examples Example 2: The following example is taken from [25]:     1 1 1 0 A1 = , A2 = . 0 1 1 1

IEEE-TAC FP-10-015

8

H. Generalized Strong Generating Functions The definition of strong generating functions can be generalized. Let q be a positive integer, and let k · k now denote an arbitrary norm on Rn . Define a generalized strong generating ∞ X function as Gλ, q (z) := sup λt kx(t; z, σ)kq . When q = 2 σ

t=0

and k · k is the Euclidean norm, Gλ, q (z) reduces to Gλ (z) defined in (3). The function Gλ, q (z) retains most of the properties of Gλ (z) in Proposition 1. For instance, for any λ ≥ 0, [Gλ,q (z)]1/q is subadditive, positively homogeneous, and convex in z; it is further finite, globally Lipschitz continuous, and semismooth whenever λ is smaller than λq,∗ := sup{λ ≥ 0 | Gλ, q (z) < ∞, ∀z ∈ Rn }.

1 0.8

1/g

λ

0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

λ

Fig. 4. Top: Unit ball of k · kG1 for the SLS in Example 3. Bottom: Plot of 1/gλ .

Algorithm 1 is used to compute the functions Gλ (z) of this SLS for different values of λ: λ =0.1, 0.2, 0.3, 0.37, and 0.38. The results are shown in Fig. 3, where the top and bottom figures plot respectively the unit balls of the norm k · kGλ and the graph of the function 1/gλ . From the plot, the graph of 1/gλ is very close to a straight line. Thus, by computing 1/gλ at two different λ and extrapolating, one can obtain an accurate estimate of λ∗ . This gives some justification to the fact that the joint spectral radius in this case is available analytically [18]: √ ρ∗ = 1+2 5 . Thus λ∗ = 1/(ρ∗ )2 ≃ 0.3820. Example 3: Consider the following SLS in R3 : 

 0.5 0 −0.7 A1 =  0 0.3 0 , 0 −0.4 −0.6   0 −1 0 A3 =  0.9 0.2 0.3  . −0.2 0.3 −0.5



 0.5 0 0 A2 = 0.4 0.2 0.3 , 0 0 0.3

Overestimates of the function Gλ (z) are computed by applying Algorithm 1 on 752 grid points of the unit p sphere. The unit ball corresponding to the estimated norm G1 (z) is shown at the top of Fig. 4; the bottom figure depicts the computed 1/gλ for λ = 0.2, 0.4, 0.6, 0.8, 1, and 1.1. Since gλ at λ = 1.1 is finite, λ∗ > 1.1, hence the given SLS is exponentially stable under arbitrary switching. An extrapolation of the function 1/gλ provides an estimate of λ∗ at around 1.1064.

We call λq,∗ the radius of strong convergence corresponding to Gλ, q (z), and note that its definition does not depend on the choice of the norm k · k. Similar to Theorem 2, we can show that the SLS is exponentially stable under arbitrary switching if and only if λq,∗ > 1. Indeed, λq,∗ is related to λ∗ and the JSR ρ∗ as: (λq,∗ )1/q = (λ∗ )1/2 = (ρ∗ )−1 , for all q. Although different choices of q and k · k lead to equivalent stability tests, the numerical robustness of such tests may vary. Noting satisfies
sequence {wt }t=0,1,... P∞ thatqr any nonnegative P∞ q r qr,∗ (w ) ≤ (w ) , we have λ ≥ (λq,∗ )r , t t t=0 t=0 ∀q, r = 1, 2, . . .. For a barely exponentially stable SLS with a convergence radius λ2,∗ only slightly above 1, by choosing r > 1, the new radius λ2r,∗ ≥ (λ2,∗ )r has a larger gap with 1; hence it may lead to a more robust stability test. IV. W EAK G ENERATING F UNCTIONS A. Definition and Properties The weak generating function H : R+ × Rn → R+ ∪ {∞} of the SLS (1) is defined as H(λ, z) := inf σ

∞ X t=0

λt kx(t; z, σ)k2 ,

∀λ ≥ 0, z ∈ Rn , (17)

where the infimum is over all switching sequences σ of the SLS. Then H(λ, z) is monotonically increasing in λ, with H(0, z) = kzk2 when λ = 0. The threshold λ∗ (z) := sup{λ ≥ 0 | H(λ, z) < ∞} is called the radius of weak convergence of the SLS at z. For each fixed λ ≥ 0, write Hλ (z) := H(λ, z), ∀z ∈ Rn . Then Hλ (·) is homogeneous of degree two, with H0 (·) = k · k2 . Some properties of the function Hλ (z) are listed below. It is noted that many properties (e.g. convexity) of the strong generating function Gλ (z) are not valid for Hλ (z). Proposition 7: Hλ (z) has the following properties. 1. (Bellman Equation): Let λ ≥ 0 be arbitrary. Then Hλ (z) = kzk2 + λ · mini∈M Hλ (Ai z), ∀z ∈ Rn . 2. (Invariant Subset): For each λ ≥ 0, the set Hλ := {z ∈ Rn | Hλ (z) = ∞} is a subset of Rn invariant under {Ai }i∈M , i.e., Ai Hλ ⊆ Hλ for all i ∈ M. 3. For 0 ≤ λ < (mini∈M kAi k2 )−1 , Hλ (z) ≤ ckzk2 for some finite constant c > 1.

IEEE-TAC FP-10-015

9

Proof: Property 1 is proved by applying the dynamic programming P∞ principle to the optimal control problem of minimizing t=0 λt kx(t; z, σ)k2 . Property 2 follows directly from Property 1. For Property 3, by choosing σ0 := (i0 , i0 , i0 , . . .) with no switching where i0 = argmini∈MkAi k, we have kx(t; z, σ0 )k2 ≤ (mini∈M kAi k2 )t kzk2 , ∀t. Therefore, for 0 ≤ λ < (mini∈M kAi k2 )−1 , Hλ (z) ≤

∞ X t=0

λt kx(t; z, σ0 )k2 ≤

1 kzk2, 1 − λ · mini∈M kAi k2

which is finite and bounded by a quadratic function. Note that Hλ , unlike Gλ , cannot be a subspace of Rn as it does not contain the origin. B. Radius of Weak Convergence Definition 4: The radius of weak convergence of the SLS (1), denoted by λ∗ ∈ (0, ∞], is defined by λ∗ := sup{λ ≥ 0 | Hλ (z) < ∞, ∀z ∈ Rn }.

By Proposition 7, we must have λ∗ ≥ (mini∈M kAi k2 )−1 . The value of λ∗ could reach ∞ if starting from any z, a switching sequence σz exists so that x(t; z, σz ) reaches the origin within a uniform time T < ∞. The next result shows that a function Hλ (z) finite everywhere on Rn must be bounded by a quadratic function. Thus, λ∗ can also be defined to be sup{λ ≥ 0 | Hλ (z) ≤ ckzk2, ∀z ∈ Rn , for some constant c}. Proposition 8: For each λ ≥ 0, the following statements are equivalent: 1) Hλ (z) < ∞, ∀z ∈ Rn ; 2) Hλ (z) ≤ ckzk2 , ∀z ∈ Rn , for some positive√constant c; 3) The scaled SLS with subsystem matrices { λAi }i∈M is exponentially stable under optimal switching. Proof: It is obvious that 2) ⇒ 1). We show in the following that 1) ⇒ 3), and 3) ⇒ 2). To prove 1) ⇒ 3), assume λ ≥ 0 is such that 1) holds. Then n for any a switching sequence σz such P∞z ∈ t R , there exists 2 the scaled SLS that t=0 λ kx(t; z, σz )k < ∞. Consider √ with subsystem dynamics matrices { λAi }i∈M , and denote by x e(t; z, σz ) = P λt/2 x(t; z, σz ) its solution starting from z ∞ under σz . Since t=0 ke x(t; z, σz )k2 < ∞, x e(t; z, σz ) → 0 as t → ∞ for each z. By Remark 1, the scaled SLS is exponentially stable under optimal switching. To show 3) ⇒ 2), assume the scaled SLS is exponentially stable under optimal switching. Then there exist constants κ > 0 and r ∈ (0, 1) such that for each z, the trajectory x e(t; z, σz ) of the scaled SLS under at least one switching sequence σz satisfies ke x(t; z, σz )k2 ≤ κrt kzk2, ∀t. As a result, ∀z, Hλ (z) ≤

∞ X t=0

λt kx(t; z, σz )k2 =

∞ X t=0

ke x(t; z, σz )k2 ≤

2

κkzk , 1−r

which is exactly the conclusion of 2) with c = κ/(1 − r). We next show that the radius of weak convergence λ∗ characterizes the exponential stability of the SLS under optimal switching just as λ∗ does for the exponential stability under arbitrary switching.

Theorem 4: The SLS (1) is exponentially stable under optimal switching if and only if λ∗ > 1. Proof: To prove necessity, we observe that for a SLS exponentially stable under optimal switching with the parameters κ > 0 and r ∈ [0, 1), by a similar argument as in the κ2 2 proof of Theorem 2, we must have Hλ (z) ≤ 1−λr 2 kzk , ∀z, −2 −2 for λ < r ; hence λ∗ ≥ r > 1. To show sufficiency, suppose λ∗ > 1. Then at λ = 1, H1 (z) is finite for all z. By Proposition√8, this implies that the scaled SLS with subsystem matrices { λAi }i∈M , which is also the original SLS (1), is exponentially stable under optimal switching. Following similar steps as in the proof of Corollary 1, we can prove the subsequent result. Corollary 4: Consider the SLS (1). For any r > (λ∗ )−1/2 , there is a constant κr such that starting from each z ∈ Rn , kx(t; z, σz )k ≤ κr rt kzk, ∀t, for at least one switching sequence σz . Furthermore, (λ∗ )−1/2 is the smallest possible value for the previous statement to be true. To sum up, the maximum exponential growth rate over all initial states of the trajectories of the SLS (1) under optimal switching is given by precisely (λ∗ )−1/2 . Another related (but generally larger) quantity characterizing such a rate is the joint spectral subradius of {Ai }i∈M defined by [17]: o n ρˇ := lim inf kAi1 · · · Aik k1/k , i1 , . . . , ik ∈ M . (18) k→∞

Difference between these two rates is shown via the following SLS (inspired by [42, pp. 1135]) with subsystem matrices:  2 0 3 , A2 = QA1 QT , A3 = QA2 QT , A1 = 0 32

where Q ∈ R2×2 is the rotation matrix of 60◦ counterclockwise. Let C be the symmetric cone consisting of all those x ∈ R2 ≃ C with phase angle within the range of either [−30◦ , 30◦ ] or [150◦ , 210◦ ]. It is easy q q to check that kA1 xk ≤ 43 43 x ∈ C; kA2 xk ≤ 48 kxk for 48 kxk for x ∈ QC; and q 2 2 2 kA3 xk ≤ 43 48 kxk for x ∈ Q C. Since R = C ∪ QC ∪ Q C, a state feedback switching policy can be designed as follows: σ(x) = 1, 2, and 3, for x in C, QC, and Q2 C, respectively (if x is in more than one set, any tie breaking rule can be applied). Then starting from any z, the q resulting closed-loop trajectory satisfies kx(t + 1; z, σ)k ≤ 43 48 · kx(t; z, σ)k for all t. Thus, the SLS isqexponentially stable under σ(x), and we 43 48 have (λ∗ )−1/2 ≤ 48 , i.e., λ∗ > 43 . In comparison, since each of the subsystem matrices has determinant one, any finite product Ai1 · · · Aik also has determinant one, hence norm at least one. By (18), we must have ρˇ ≥ 1, hence ρˇ > (λ∗ )−1/2 . This gap can be explained as ρˇ defined by (18) satisfies ρˇ = lim

inf

sup kAi1 · · · Aik xk1/k

k→∞ i1 ,...,ik ∈M kxk=1

≥ lim sup

inf

k→∞ kxk=1 i1 ,...,ik ∈M

kAi1 · · · Aik xk1/k ,

where the right hand side is exactly (λ∗ )−1/2 . In other words, the JSS ρˇ and the rate (λ∗ )−1/2 studied in this paper characterize the smallest possible worst-case exponential growth rate over all initial states of the SLS trajectories achievable by open-loop switching policies of the form

IEEE-TAC FP-10-015

10

(i1 , . . . , ik , i1 , . . . , ik , . . .) and by closed-loop switching policies with state feedback switching laws, respectively.

E1

C. Quadratic Bounds of Finite Hλ (z)

hλ := sup{Hλ (z) | kzk = 1}.

∀λ ∈ (0, λ∗ ).

(20)

t=0

λt kx(t; z, σz )k2 ≤

hλ0 kzk2 , 1 − (hλ0 − 1)(λ − λ0 )/λ0

for λ ∈ [λ0 , λ1 ), where λ1 := λ0 /(1 − 1/hλ0 ). Therefore, Hλ (z) ≤

∞ X t=0

λt kx(t; z, σz )k2 ≤

hλ0 kzk2 1 − (hλ0 − 1)(λ − λ0 )/λ0

is finite for all z ∈ Rn and λ ∈ [λ0 , λ1 ). This implies that λ∗ ≥ λ1 , which is the desired (20); and that for λ ∈ [λ0 , λ1 ), hλ ≤

hλ0 λ λ0 ⇒ ≥ . 1 − (hλ0 − 1)(λ − λ0 )/λ0 1 − 1/hλ 1 − 1/hλ0

This proves the monotonicity of λ/(1 − 1/hλ ) on (0, λ∗ ). The following result follows directly from Proposition 9. Corollary 5: For each λ ∈ [0, λ∗ ), 1/hλ ≤ 1 − λ/λ∗ . As a result, 1/hλ → 0 and hλ → ∞ as λ ↑ λ∗ . The directional derivative of hλ at λ = 0 is computed in Appendix C as follows. Lemma 5: The directional derivative of hλ at 0 exists and hλ − h0 is given by h′λ (0+ ) := lim = sup min kAi zk2 . λ↓0 λ kzk=1 i∈M As a result of Lemma 5, lim λ↓0

E3 Fig. 5. Geometric interpretation of the Lipschitz constants in Propositions 4 2. and 10: maxi∈M kAi k2 = 1/rg2 and supkzk=1 mini∈M kAi zk2 = 1/rh

1/hλ 1 1 − λ · supkzk=1 mini∈M kAi zk2

n

Proof: Let λ0 ∈ (0, λ∗ ). For any z ∈ R , let σz be a switching sequence so that the resulting trajectory z, σz ) achieves the infimum in (17): Hλ0 (z) = P∞ x(t; t λ kx(t; z, σz )k2 < ∞. By the optimality of σz , for 0 t=0 each s = 0, 1, . . ., the time shifted trajectory of the SLS, x(t + s; z, σz ),P∀t, starting from the initial state x(s; z, σz ) is ∞ also optimal: t=0 λt0 kx(t + s; z, σz )k2 = Hλ0 (x(s; z, σz )) ≤ hλ0 kx(s; z, σz )k2 . In other words, the sequence defined by {wt := kx(t; z, σz )k2 }t=0,1,... satisfies the condition (9) with β = hλ0 . By Lemma 3, we then have ∞ X

rg

(19)

Obviously, hλ is strictly increasing in λ, with h0 = 1. Similar to Proposition 3 for gλ , we have the following estimates of hλ . Proposition 9: The function λ/(1−1/hλ) is nondecreasing for λ ∈ (0, λ∗ ), and is upper bounded by λ ≤ λ∗ , 1 − 1/hλ

rh

E2

For each λ ∈ [0, λ∗ ), define the constant hλ as the smallest constant c such that Hλ (z) ≤ ckzk2 for all z:

λ 1 1 = ′ . = 1 − 1/hλ hλ (0+ ) supkzk=1 mini∈M kAi zk2

Since by Proposition 9, λ/(1 − 1/hλ ) is nondecreasing in λ on (0, λ∗ ), the above limit implies that, for λ ∈ (0, λ∗ ), 1 λ λ ≥ lim = . λ↓0 1 − 1/hλ 1 − 1/hλ supkzk=1 mini∈M kAi zk2

1 − λ/λ∗ (λ, 1/hλ )

0

λ/(1 − 1/hλ )

Fig. 6.

λ∗

λ

Plot of the function 1/hλ (in solid line).

This leads to the following estimate of hλ , which is the counterpart of Lemma 2 for gλ . 1 Corollary 6: ≥ 1 − λ · sup min kAi zk2 , ∀λ ∈ [0, λ∗ ). hλ kzk=1 i∈M Similar to the proof of Proposition 4 in Appendix B, we can use Proposition 9 and Corollary 6 to prove the following. Proposition 10: The function 1/hλ defined on [0, λ∗ ) is strictly decreasing and Lipschitz continuous with Lipschitz constant supkzk=1 mini∈M kAi zk2 . The function hλ is strictly increasing and locally Lipschitz continuous on [0, λ∗ ). Remark 6: The Lipschitz constants of 1/gλ in Proposition 4 and of 1/hλ in Proposition 10 are given as supkzk=1 maxi∈M kAi zk2 and supkzk=1 mini∈M kAi zk2 , respectively. An overestimate of the latter is given by mini∈M supkzk=1 kAi zk2 = mini∈M kAi k2 . For a geometric interpretation, associate each matrix Ai with an ellipsoid Ei := {z ∈ Rn | kAi zk2 ≤ 1} in Rn . The intersection of all such ellipsoids, ∩i∈M Ei , is a convex set that can embed a maximal ball centered at 0 with the radius rg ; whereas their union ∪i∈M Ei can embed a maximal ball centered at 0 with the radius rh (see Fig. 5). Further, let rℓ := maxi∈M {the length of the shortest principal axis of Ei }. The two Lipschitz constants and the overestimate can then be expressed as max kAi k2 = 1/rg2 , sup min kAi zk2 = 1/rh2 , i∈M

and min kAi k2 = 1/rℓ2 .

kzk=1 i∈M

i∈M

A generic plot of the function 1/hλ for λ ∈ [0, λ∗ ) is shown in Fig. 6. The function decreases strictly from 1 at λ = 0 to 0 as λ ↑ λ∗ . Its graph is sandwiched by

IEEE-TAC FP-10-015

11

those of two affine functions: 1 − λ/λ∗ from the right, and 1 − λ · supkzk=1 mini∈M kAi zk2 from the left. Moreover, as λ increases from 0 towards λ∗ , the ray emitting from the point (1, 0) and passing through the point (λ, 1/hλ ) rotates counterclockwise monotonically, and intersects the λaxis at a point whose λ-coordinate, λ/(1 − 1/hλ ), provides an asymptotically tight underestimate of λ∗ . D. Approximating Finite Hλ (z) For each λ ∈ [0, λ∗ ), the function Hλ (z) is finite everywhere on Rn . We next show that it is the limit of a sequence of functions Hλk (z), k = 0, 1, . . ., defined by Hλk (z) := min σ

k X t=0

λt kx(t; z, σ)k2 ,

∀z ∈ Rn .

(21)

As the value functions of an optimal control problem with variable finite horizons, Hλk (z) can be computed recursively as follows: Hλ0 (z) = kzk2, ∀z ∈ Rn ; and for k = 1, 2, . . ., Hλk (z) = kzk2 + λ · min Hλk−1 (Ai z), ∀z ∈ Rn . i∈M

(22)

Equivalently, we can write Hλk (z) = min{z T P z : P ∈ Pk },

∀z ∈ Rn ,

(23)

where Pk , k = 0, 1, . . ., is a sequence of sets of positive definite matrices defined by: P0 = {I}; and for k = 1, 2, . . ., Pk = {I + λATi P Ai | P ∈ Pk−1 , i ∈ M}.

(24)

Proposition 11: The sequence of functions Hλk (z) is monotonic: Hλ0 ≤ Hλ1 ≤ Hλ2 ≤ · · · ≤ Hλ ; and for λ ∈ [0, λ∗ ), it converges exponentially fast to Hλ (z): for k = 0, 1, . . ., |Hλk (z) − Hλ (z)| ≤ h2λ (1 − 1/hλ )k+1 kzk2 , ∀z ∈ Rn . (25) Proof: Fix k ≥ 1. For each z ∈ Rn , let σk be a switching sequence achieving the minimum in (21). Then, Hλk (z) = Pk Pk−1 t k−1 t 2 2 (z). t=0 λ kx(t; z, σk )k ≥ t=0 λ kx(t; z, σk )k ≥ Hλ k Similarly, we have Hλ (z) ≤ Hλ (z), proving the monotonicity. Next assume λ ∈ [0, λ∗ ). For any z ∈ Rn and k = 0, 1, . . ., we note that kzk2 ≤ Hλk (z) ≤ Hλ (z) ≤ hλ kzk2 . Let σk be such that x ˆ(t) := x(t; z, σk ) achieves the minimum in (21). For each s = 0, 1, . . . , k − 1, since x ˆ(t) is also optimal over the time horizon t = s, s + 1, . . . , k, we have Hλk−s (ˆ x(s)) = kˆ x(s)k2 + λ 2

= kˆ x(s)k +

k−s−1 X

λt kˆ x(t + s + 1)k2

t=0 λHλk−s−1 (ˆ x(s

+ 1)).

Since kˆ x(s)k2 ≥ Hλk−s (ˆ x(s))/hλ , the above implies that Hλk−s−1 (ˆ x(s + 1)) ≤ λ−1 (1 − 1/hλ )Hλk−s (ˆ x(s)). Applying this inequality for s = k − 1, . . . , 0, we have kˆ x(k)k2 = Hλ0 (ˆ x(k)) ≤ λ−1 (1 − 1/hλ )Hλ1 (ˆ x(k − 1))

≤ · · · ≤ λ−k (1 − 1/hλ )k Hλk (z) ≤ λ−k hλ (1 − 1/hλ )k kzk2 .

Algorithm 2 Computing Over Approximations of Hλk (z). Initialize k = 0, P˜0ε = {In }, and P0ε = P˜0ε ; repeat k ← k + 1; ε P˜kε = {I + λATi P Ai | i ∈ M, P ∈ Pk−1 }; ε ε ˜ Find an ε-equivalent subset Pk ⊂ Pk ; until k is large enough return Hλk,ε (z) = min{z T P z | P ∈ Pkε }.

Using this inequality, and adopting the switching sequence that first follows σk for k steps and thereafter follows an infinitehorizon optimal σ∗ starting from xˆ(k), we obtain Hλ (z) ≤ = ≤

≤

k X

λt kˆ x(t)k2 +

t=0 Hλk (z) Hλk (z) Hλk (z)

∞ X

t=k+1

λt kx(t − k; x ˆ(k), σ∗ )k2

  x(k)) − kˆ x(k)k2 + λ Hλ (ˆ k

+ (hλ − 1)λk kˆ x(k)k2

+ h2λ (1 − 1/hλ )k+1 kzk2.

This, together with Hλk (z) ≤ Hλ (z), proves (25). By (25), for any λ0 ∈ [0, λ∗ ), Hλk (z), k = 0, 1, . . ., is a sequence of functions of (λ, z) converging uniformly to Hλ (z) on [0, λ0 ] × Sn−1 . As each Hλk (z) is continuous in (λ, z) on [0, λ0 ] × Sn−1 , so is Hλ (z). Using homogeneity and the arbitrariness of λ0 ∈ [0, λ∗ ), we obtain the following result. Corollary 7: The function Hλ (z) = H(λ, z) is continuous in (λ, z) on [0, λ∗ ) × Rn . E. Relaxation Algorithm for Computing Hλ (z) According to (25), Hλk (z) for large k provide increasingly accurate estimates of Hλ (z). By (23), to characterize Hλk (z), it suffices to compute the set Pk . To deal with the rapidly increasing size of Pk as k increases, we introduce the following complexity reduction technique, inspired by [43], [44]. A subset Pkε ⊆ Pk is called ε-equivalent to Pk for some ε > 0 if Hλk,ε (z) := minε z T P z ≤ min z T P z + εkzk2, ∀z ∈ Rn . P ∈Pk

P ∈Pk

A sufficient (though not necessary) condition for this to hold is that, for each P ∈ Pk , P + εIn is bounded from below by a convex combination of matrices in Pkε , i.e., there exist ε constants αP Q ≥ 0, ∀Q ∈ Pk , adding up to 1 such that P + εIn  Q∈P ε αQ · Q. This leads to a natural procedure k of removing matrices from Pk iteratively until a minimal εequivalent subset Pkε is achieved. By applying this procedure at each step of the iteration (24), we obtain Algorithm 2, which returns approximations of Hλk (z) for all k with uniformly bounded approximation errors as follows. Proposition 12: Assume λ ∈ [0, λ∗ ). Then for k = 0, 1, . . ., Hλk (z) ≤ Hλk,ε (z) ≤ (1 + ε)Hλk (z),

∀z ∈ Rn .

(26)

Proof: Obviously, (26) holds for k = 0. Assume it holds for some k − 1 ≥ 0. Define, for z ∈ Rn , ˜ k,ε (z) := min z T P z = kzk2 + λ · min H k−1,ε (Ai z). H λ λ ˜ε P ∈P k

i∈M

IEEE-TAC FP-10-015

12

1

its mean generating function F : R+ × Rn → R+ ∪ {∞} as # ∞ "∞ X X   t 2 λt E kx(t; z, p)k2 , F (λ, z) := E λ kx(t; z, p)k =

0.9 0.8 0.7

1/h

λ

(27)

0.5 0.4

n

for z ∈ R , λ ≥ 0. For each λ ≥ 0, Fλ (z) := F (λ, z) is the averaged sum of the power series along the random system trajectory; thus, its value lies between two extremes:

0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Hλ (z) ≤ Fλ (z) ≤ Gλ (z),

0.9

λ 1

∀z ∈ Rn .

(28)

The absence of maximum or minimum in the definition of Fλ (z) makes its characterization much easier compared to Gλ (z) and Hλ (z). For example, Fλ (z) is quadratic in z ∈ Rn :   ∞ t−1 0 X Y Y T t T Fλ (z) = z E I + λ A(s) A(s) z. (29)

0.8

1/h

λ

0.6

0.4

0.2

0

t=0

t=0

0.6

t=1

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

λ

Fig. 7. Plots of 1/hλ for SLSs in Example 2 (top) and Example 3 (bottom).

By our hypothesis, Hλk−1,ε (Ai z) ≤ (1 + ε)Hλk−1 (Ai z). Thus, ˜ k,ε (z) ≤ kzk2 + λ(1 + ε)H k−1 (Ai z), ∀z ∈ Rn , i ∈ M. H λ λ Since Pkε is ε-equivalent to P˜kε , we then have,   ˜ k,ε (z)+εkzk2 ≤ (1 + ε) kzk2 + λH k−1 (Ai z) , Hλk,ε (z) ≤ H λ λ

for each i ∈ M. By (22), we have Hλk,ε (z) ≤ (1 + ε)Hλk (z), ∀z. That Hλk,ε (z) ≥ Hλk (z) can be trivially proved. The estimated 1/hλ obtained by Algorithm 2 are plotted in Fig. 7 for the SLSs in Example 2 (top) and Example 3 (bottom). In both cases, 1/hλ decreases from 1 at λ = 0 to 0 at λ = λ∗ . (For Example 3, high computational complexity of Algorithm 2 prevents us from getting accurate estimates for λ close to λ∗ ). We observe that in each case, 1/hλ is convex and “more curved” than the plots of 1/gλ . It is conjectured that the function 1/hλ is convex on [0, λ∗ ) for all SLSs. According to Propositions 11 and 12, by choosing k large enough and ε small enough, Algorithm 2 can return estimates of Hλ (z), hence λ∗ , with an error as small as possible. See [44] for the expressions of the approximation error in terms of k, ε, and matrices Ai ’s. On the other hand, to attain a higher accuracy, the computation time of Algorithm 2 still grows exponentially, as is reflected by the rapidly increasing size of the set Pkε , despite our relaxation effort. Our numerical experiments suggest that computing λ∗ is no less challenging than computing the joint spectral subradius, which in itself is an NP-hard problem [20], [24]. A future direction of our research is to prove this formally and to see if the algorithms for computing the joint spectral subradius (e.g. [29]) can be adapted to compute the radius of weak convergence λ∗ . V. M EAN G ENERATING F UNCTIONS The notion of generating functions can also be extended to the SLS (2) under the random switching probability p. Define

s=0

s=t−1

We write Fλ (z) = z T Qλ z, where Qλ ∈ Rn×n is positive definite (with possibly infinite entries) and increasing in λ. The function Fλ (z) has a similar set of properties as Gλ (z). Proposition 13: Fλ (z) has the following properties. 1. (Bellman Equation): P Let λ ≥ 0 be arbitrary. Then Fλ (z) = kzk2 + λ i∈M pi Fλ (Ai z), ∀z ∈ Rn . 2. p (Sub-Additivity and p Convexity): p For any λ ≥ 0, Fλ (z1 + zp Fλ (z1 ) + Fλ (z1 ), ∀z1 , z2 ∈ Rn . 2) ≤ As a result, Fλ (z) is a convex function of z on Rn . 3. (Invariant Subspace): For each λ ≥ 0, the set Fλ := {z ∈ Rn | Fλ (z) < ∞} is a subspace of Rn invariant under {Ai }i∈M′ , where M′ := {i ∈ M | pi > 0}. 4. For any λ ≥ 0, if Fλ (z) is finite for all z, then Fλ (z) ≤ ckzk2 for some finite constant c. The proof of Proposition 13 is straightforward, hence omitted. The radius of convergence of Fλ (z) is defined as λ∗p := sup{λ ≥ 0 | Fλ (z) < ∞, ∀z ∈ Rn },

which depends on the probability distribution p. By (27), λ∗p is the  minimal radius  of convergence of the deterministic series E kx(t; z, p)k2 t=0,1,... over all z ∈ Rn . Theorem 5: The following statements are equivalent: 1) The SLS (2) under the random switching probability p is mean square exponentially stable; 2) Its mean generating function Fλ (z) has a radius of convergence λ∗p > 1; 3) The mean generating function at λ = 1, F1 (z), is finite everywhere on Rn . Proof: To show 1) ⇒ 2), suppose the SLS (2) is mean square exponentially  stable, i.e., there exist κ > 0 and r ∈ [0, 1) such that E kx(t; z, p)k2 ≤ κrt kzk2 , ∀t, ∀z. Then, Fλ (z) ≤

∞ X t=0

λt κrt kzk2 ≤

κ kzk2 < ∞, ∀z ∈ Rn , 1 − λr

whenever 0 ≤ λ < r−1 . Consequently, λ∗p ≥ r−1 > 1. That 2) ⇒ 3) is P obvious. Finally, to show 3) ⇒ 1), we  ∞ 2 note that F (z) = E kx(t; z, p)k < ∞ implies that 1 t=0   E kx(t; z, p)k2 → 0 as t → ∞, for all z. It follows that the SLS (2) is mean square asymptotically stable, hence mean square exponentially stable by [36, Theorem 4.1.1].

IEEE-TAC FP-10-015

13

Lg : 1 − λ · maxi∈M kAik2

For λ ∈ [0, λ∗p ), Fλ (·) is finite everywhere. Define fλ := sup{Fλ (z) | kzk = 1} < ∞,

λ∈

[0, λ∗p ).

1

(30)

Then, fλ = σmax (Qλ ), the largest eigenvalue of Qλ in (29). Proposition 14: For λ ∈ (0, λ∗p ), the function λ/(1 − 1/fλ ) is nondecreasing and upper bounded by λ/(1 − 1/fλ ) ≤ λ∗p . n Proof: Let λ0 ∈ (0, λ∗p ). For  each z ∈ R , define the 2 sequence wt := E kx(t; z, p)k , t = 0, 1, . . ., which satisfies "∞ # ∞ X X t t 2 wt+s λ0 = E λ0 kx(s + t; z, p)k = E[Fλ0 (x(s; z, p))] t=0

 t=0 ≤ E fλ0 kx(s; z, p)k2 = fλ0 ws , ∀s = 0, 1, . . . .

Applying Lemma 3 with β = fλ0 , we conclude that Fλ (z) = P ∞ t t=0 wt λ is finite for λ ∈ [λ0 , λ1 ), where λ1 := λ0 /(1 − 1/fλ0 ). Therefore, λ∗p ≥ λ1 , which is the second conclusion. That λ/(1 − 1/fλ ) is nondecreasing is proved in exactly the same way as in Proposition 9 (with hλ replaced by fλ ). As a result, we have the following estimate. Corollary 8: For each λ ∈ [0, λ∗p ), 1/fλ ≤ 1 − λ/λ∗p . Hence, 1/fλ → 0 and fλ → ∞ as λ ↑ λ∗p . We next compute the directional derivative of fλ at λ = 0. Lemma 6: The directional derivative of fλ at 0 exists and ! X f − f λ 0 is given by fλ′ (0+ ) := lim = σmax pi ATi Ai . λ↓0 λ i∈M Proof: Let z ∈ Sn−1 be arbitrary. For small λ
> 0, we can P T use (27) to write Fλ (z)−1 = zT i∈M pi Ai Ai z + O(λ), λ where O(λ) ≥ 0 is uniform in z. Therefore, we can exchange the order of the limit and supremum below to obtain Fλ (z) − 1 Fλ (z) − 1 = sup lim , λ↓0 kzk=1 λ λ kzk=1 λ↓0
P T which is exactly supkzk=1 z T i∈M pi Ai Ai z. Similar to Corollary 6 and Proposition 10, we can show the following two results. ! X 1 T Corollary 9: ≥ 1−λ σmax pi Ai Ai , ∀λ ∈ [0, λ∗p ). fλ i∈M Proposition 15: The function 1/fλ defined on [0, λ∗p ) is strictly decreasing
continuous with Lipschitz P and Lipschitz T . Hence, the function fλ is p A A constant σmax i i i i∈M strictly increasing and locally Lipschitz continuous on [0, λ∗p ). From (28), we have hλ ≤ fλ ≤ gλ , thus 1/gλ ≤ 1/fλ ≤ 1/hλ . The Lipschitz constants of the functions 1/hλ , 1/fλ , and 1/gλ in Propositions 10, 15, and 4, respectively, satisfy ! X sup min kAi zk2 ≤ σmax pi ATi Ai ≤ max kAi k2 . fλ′ (0+ ) = lim sup

kzk=1 i∈M

i∈M

Lf : 1 − λ · inf p σmax(

i∈M

By setting the probability distribution p to be pi∗ = 1 for i∗ = arg maxi∈M kAi k2 and pi = 0 for i 6= i∗ , the second inequality becomes equality. On the other hand, the first inequality is in general strict regardless of the choice of p, due to the generally lossy nature of the S-procedure [45]. Indeed, when the cardinality |M| ≥ 3 and the state dimension n ≥ 2, there exist matrices {Ai }i∈M and a constant γ > 0 T T such that: (i) mini∈M zP Ai Ai z ≤ γkzk2, ∀z ∈ Rn ; (ii) there exists no p such that i∈M pi ATi Ai PγI. Then for
all p, T supkzk=1 mini∈M kAi zk2 ≤ γ < σmax i∈M pi Ai Ai .

P T i∈M pi Ai Ai )

Lh: 1 − λ · supkzk=1 mini∈M kAi zk2

1/hλ 1/fλ

1/gλ 0

λ∗p

λ∗

Fig. 8.

λ

λ∗

Plot of the function 1/fλ .

A general plot of the function 1/fλ is shown in Fig. 8. The graph of 1/fλ is sandwiched between those of 1/gλ and 1/hλ . By Proposition 14, the ray emitting from (0, 1) and passing through (λ, 1/fλ ) rotates counterclockwise monotonically as λ increases. The discussions in the preceding paragraph further imply that the graph of 1/fλ leaves (0, 1) along a direction within the shaded conic region bounded by the lines Lg and Lf ; whereas generally a gap exists between Lf and the asymptotic direction Lh in which 1/hλ leaves (0, 1). Algorithms based on the Bellman equation can also be devised to compute Fλ (z), hence λ∗p . The details are omitted here. VI. C ONCLUSION Generating functions (more precisely their radii of convergence) of switched linear systems provide effective characterizations of the growth rates of the system trajectories, and in particular their exponential stability, under various switching rules. Numerical algorithms for their computation have also been developed based on their many properties derived here. A PPENDIX A P ROOF OF L EMMA 3 ∞ ∞ X X t Proof: Write wt+s λ0 = ws +λ0 wt+s+1 λt0 . Since t=0

t=0

by assumption the left hand side is at most βws , we have ∞ X t=0

wt+s+1 λt0 ≤

β−1 ws , λ0

s = 0, 1, . . . .

Define the power series: W (λ) :=

∞ X t=0

(31)

wt λt , λ ∈ R.

Denote by RW its radius of convergence. Since W (λ0 ) = P ∞ t t=0 wt λ0 ≤ βw0 < ∞, we have RW > λ0 . We cannot have RW = λ0 since the power series W (λ) with nonnegative coefficients must have its radius of convergence as a singular point, at which W (λ) diverges (see [47, Theorem 5.7.1]). As a power series defines an analytic function within its radius of convergence [47], W (λ) is analytic hence infinite time differentiable at λ0 . Its first order derivative at λ0 is ∞ ∞ X X W ′ (λ0 ) = twt λt−1 = (t + 1)wt+1 λt0 0 t=1

=

∞ X t X t=0 s=0

t=0

wt+1 λt0 =

∞ X s=0

λs0

∞ X t=0

wt+s+1 λt0 .

IEEE-TAC FP-10-015

14

Applying (31), we obtain the estimate ∞ β−1X β−1 W ′ (λ0 ) ≤ w0 . ws λs0 ≤ β λ0 s=0 λ0

(32)

Similarly, the second order derivative W ′′ (λ0 ) is W ′′ (λ0 ) =

∞ X

By Lemma 2, the above inequality also holds for λ0 = 0. This proves the Lipschitz continuity of 1/gλ , hence the local Lipschitz continuity of gλ , on [0, λ∗ ). Finally, by (10), 0 ≤ 1/gλ ≤ 1 − λ/λ∗ for λ ∈ (0, λ∗ ). Thus, limλ↑λ∗ 1/gλ = 0. Consequently, limλ↑λ∗ gλ = ∞.

(t + 2)(t + 1)wt+2 λt0

A PPENDIX C P ROOF OF L EMMA 5

t=0

=2

∞ X t X

(s + 1)wt+2 λt0 = 2

t=0 s=0

∞ X

(s + 1)λs0

s=0

∞ X

wt+s+2 λt0 .

t=0

Using (31) and (32), we can derive the estimateW ′′ (λ 20 ) ≤ β−1 β−1 β−1 P∞ s ′ w0 . 2 λ0 s=0 (s + 1)ws+1 λ0 = 2 λ0 W (λ0 ) ≤ 2β λ0 By induction, we can show, for k = 0, 1, 2, . . ., k  β−1 (k) w0 . (33) W (λ0 ) ≤ k! β λ0 ∞ X 1 (k) ¯ (λ) := W (λ0 )(λ−λ0 )k be the Taylor series Let W k! k=0 expansion of W (λ) at λ0 . For λ ≥ λ0 and close to λ0 , ¯ (λ) = W (λ) as W (λ) is analytic at λ0 ; and by (33), W  k ∞ X β−1 ¯ W (λ) ≤ β w0 (λ − λ0 )k λ0 k=0

≤

βw0 , 1 − (β − 1)(λ/λ0 − 1)

where the last inequality holds if (β − 1)(λ/λ0 − 1) < 1, or equivalently, if λ ∈ [λ0 , λ1 ) with λ1 defined in the lemma ¯ (λ) centered at λ0 is statement. Therefore, the power series W convergent, and thus defines an analytic function, on [λ0 , λ1 ). ¯ (λ) yields an analytic The concatenation of W (λ) and W continuation of W (λ) defined at least on [0, λ1 ). Being a power series of nonnegative coefficients, W (λ) does not admit any analytic continuation beyond its radius of convergence RW [47, Definition 5.7.1]. Hence, RW ≥ λ1 . This implies in particular that any λ ∈ [λ0 , λ1 ) is within the radius of ¯ (λ) and as such we must convergence of both W (λ) and W ¯ have W (λ) = W (λ), ∀λ ∈ [λ0 , λ1 ). This together with the last inequality above proves the desired conclusion. A PPENDIX B P ROOF OF P ROPOSITION 4 Proof: The monotonicity of gλ , hence of 1/gλ , is obvious as Gλ (z) is nondecreasing in λ. For convexity, let λ = αλ1 + (1 − α)λ2 for some λ1 , λ2 ∈ [0, λ∗ ), α ∈ [0, 1]; and let z ∈ Sn−1 be such that Gλ (z) = gλ . Since Gλ (z) is a convex function of λ ∈ [0, λ∗ ) for any fixed z (it is the maximum of a family of convex functions of λ ≥ 0), we have gλ = Gλ (z) ≤ αGλ1 (z) + (1 − α)Gλ2 (z) ≤ αgλ1 + (1 − α)gλ2 . This proves the convexity of gλ and hence its semismoothness [39, Prop. 7.4.5]. Being the composition of two semismooth functions gλ and x 7→ 1/x, 1/gλ is also semismooth [39, Prop. 7.4.4]. Pick any λ0 , λ ∈ (0, λ∗ ) with λ0 < λ. Proposition 3 implies that λ/(1 − 1/gλ ) ≥ λ0 /(1 − 1/gλ0 ). Thus by Lemma 2, 1 1 − 1/gλ0 1 − ≥ −(λ − λ0 ) max kAi k2 , ≥ −(λ − λ0 ) i∈M gλ gλ0 λ0

Proof: Fix an arbitrary z ∈ Sn−1 and let λ > 0 be small. Recalling the definition of Hλ (z), we write P∞ inf σ t=0 λt kx(t; z, σ)k2 − 1 Hλ (z) − 1 = λ λ ∞ ∞ X X = inf λt−1 kx(t; z, σ)k2 = inf λt kx(t + 1; z, σ)k2 . σ

σ

t=1

t=0

2

Since for all σ, kx(t; z, σ)k ≤ (maxi∈M kAi k2 )t , ∀t; and inf σ kx(1; z, σ)k2 = mini∈M kAi zk2 , the above implies min kAi zk2 ≤

i∈M

Hλ (z) − 1 ≤ min kAi zk2 + O(λ), i∈M λ

for some term O(λ) uniform in σ and z. Thus, we can exchange the order of limit and supremum below to obtain: h′λ (0+ ) = lim λ↓0

Hλ (z) − 1 hλ − 1 = lim sup λ↓0 kzk=1 λ λ

= sup lim kzk=1 λ↓0

Hλ (z) − 1 = sup min kAi zk2 . λ kzk=1 i∈M

This completes the proof of Lemma 5. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their constructive suggestions and for pointing out many of the references of this paper. R EFERENCES [1] D. Liberzon, Switching in Systems and Control. Birkh¨auser, 2003. [2] H. Lin and P. Antsaklis, “Stability and stabilizability of switched linear systems: A survey of recent results,” IEEE Trans. Autom. Control, vol. 54, no. 2, pp. 308–322, 2009. [3] R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King, “Stability criteria for switched and hybrid systems,” SIAM Rev., vol. 49, no. 4, pp. 545– 592, 2007. [4] J. Hespanha and A. S. Morse, “Stability of switched systems with average dwell time,” in Proc. 38th IEEE Int. Conf. Decision and Control, Phoenix, AZ, 1999, pp. 2655–2660. [5] M. Johansson and A. Rantzer, “Computation of piecewise quadratic Lyapunov functions for hybrid systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 555–559, 1998. [6] J. Shen and J. Hu, “Stability of discrete-time conewise linear inclusions and switched linear systems,” in Proc. Amer. Control Conf., Baltimore, MD, 2010, pp. 4034–4039. [7] M. Branicky, “Multiple Lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 475–482, 1998. [8] T. Hu, L. Ma, and Z. Lin, “Stabilization of switched systems via composite quadratic functions,” IEEE Trans. Autom. Control, vol. 53, no. 11, pp. 2571–2585, 2008. [9] A. Agrachev and D. Liberzon, “Lie-algebraic conditions for exponential stability of switched systems,” in Proc. 38th IEEE Int. Conf. Decision and Control, Phoenix, AZ, 1999, pp. 2679–2684. [10] D. Liberzon, J. Hespanha, and A. S. Morse, “Stability of switched linear systems: A Lie-algebraic condition,” Syst. Control Lett., vol. 37, no. 3, pp. 117–122, 1999.

IEEE-TAC FP-10-015

[11] Y. Sharon and M. Margaliot, “Third-order nilpotent, nice reachability and asymptotic stability,” J. Differ. Equations, vol. 233, pp. 136–150, 2007. [12] S. Pettersson and B. Lennartson, “LMI for stability and robustness of hybrid systems,” in Proc. Amer. Control Conf., Albuquerque, NM, 1997, pp. 1714–1718. [13] D. Mignone, G. Ferrari-Trecate, and M. Morari, “Stability and stabilization of piecewise affine and hybrid systems: an LMI approach,” in Proc. 39th IEEE Int. Conf. Decision and Control, Sydney, Australia, 2000, pp. 504–509. [14] A. Rantzer and M. Johansson, “Piecewise linear quadratic optimal control,” IEEE Trans. Autom. Control, vol. 45, no. 4, pp. 629–637, 2000. [15] N. Barabanov, “Lyapunov exponent and joint spectral radius: Some known and new results,” in Proc. 44th IEEE Int. Conf. Decision and Control, Seville, Spain, 2005, pp. 2332–2337. [16] L. Gurvits, “Stability of discrete linear inclusions,” Linear Algebra Appl., vol. 231, pp. 47–85, 1995. [17] R. Jungers, The Joint Spectral Radius, Theory and Applications. Berlin: Springer-Verlag, 2009. [18] J. Theys, “Joint spectral radius: Theory and approximations,” Ph.D. dissertation, Universite Catholique de Louvain, 2005. [19] F. Wirth, “The generalized spectral radius and extremal norms,” Linear Algebra Appl., vol. 342, no. 1-3, pp. 17–40, 2002. [20] V. Blondel and J. Tsitsiklis, “The boundedness of all products of a pair of matrices is undecidable,” Syst. Control Lett., vol. 41, pp. 135–140, 2000. [21] R. Jungers, V. Protasov, and V. Blondel, “Overlap-free words and spectra of matrices,” Theor. Comput. Sci., pp. 3670–3684, 2009. [22] L. Arnold, Random Dynamical Systems. Berlin: Springer-Verlag, 1998. [23] N. Barabanov, “A method for calculating the Lyapunov exponent of a differential inclusion,” Automat. Rem. Contr., vol. 50, pp. 475–479, 1989. [24] J. Tsitsiklis and V. Blondel, “The Lyapunov exponent and joint spectral radius of pairs of matrices are hard - when not impossible - to compute and to approximate,” Math. Control, Signals, and Systems, vol. 10, pp. 31–41, 1997. [25] V. Blondel, J. Theys, and Y. Nesterov, “Approximations of the rate of growth of switched linear systems,” in Hybrid Systems: Comput. and Control, Philadelphia, PA. Springer-Verlag, 2004, pp. 173–186. [26] V. Blondel, Y. Nesterov, and J. Theys, “On the accuracy of the ellipsoid norm approximation of the joint spectral radius,” Linear Algebra Appl., vol. 394, pp. 91–107, 2005. [27] V. Kozyakin, “Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets,” Discret. Contin. Dynamical Sys. Series B, vol. 14, pp. 143–158, 2010. [28] P. Parrilo and A. Jadbabaie, “Approximation of the joint spectral radius using sum of squares,” Linear Algebra Appl., vol. 428, no. 10, pp. 2385– 2402, 2008. [29] V. Protasov, R. Jungers, and V. Blondel, “Joint spectral characteristics of matrices: a conic programming approach,” SIAM J. Matrix Anal. Appl., 2010, to appear. [30] L. Fainshil, M. Margaliot, and P. Chigansky, “On the stability of positive linear switched systems under arbitrary switching laws,” IEEE Trans. Autom. Control, vol. 54, pp. 897–899, 2009. [31] J. Shen and J. Hu, “Stability of switched linear systems on cones: A generating function approach,” to appear in Proc. 49th IEEE Int. Conf. Decision and Control, Atlanta, GA, 2010. [32] M. Margaliot, “Stability analysis of switched systems using variational principles: An introduction,” Automatica, vol. 42, pp. 2059–2077, 2006. [33] E. Pyatnitsky, “Absolute stability of nonstationary nonlinear systems,” Automat. Rem. Contr, no. 1, pp. 5–15, 1970. [34] A. Molchanov and Y. S. Pyatnitsky, “Criteria of asymptotic stability of differential and difference inclusions encountered in control theory,” Syst. Control Lett., vol. 13, no. 1, pp. 59–64, 1989. [35] D. Angeli, “A note on stability of arbitrarily switched homogeneous systems,” 1999, preprint. [36] X. Feng, “Lyapunov exponents and stability of linear stochastic systems,” Ph.D. dissertation, Case Western Reserve Univ., 1990. [37] S. Krantz and H. Parks, A Primer of Real Analytic Functions, 2nd ed. Boston: Birkh¨auser-Verlag, 2002. [38] M. Margaliot and M. Branicky, “Nice reachability for planar bilinear control systems with applications to planar linear switched systems,” IEEE Trans. Autom. Control, vol. 54, pp. 1430–1435, 2009. [39] F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems. New York: Springer-Verlag, 2003.

15

[40] C. Seatzu, D. Corona, A. Giua, and A. Bemporad, “Optimal control of continuous-time switched affine systems,” IEEE Trans. Autom. Control, vol. 51, pp. 726–741, 2006. [41] C. Yfoulis and R. Shorten, “A numerical technique for stability analysis of linear switched systems,” in Hybrid Systems: Comput. and Control, Philadelphia, PA. Springer-Verlag, 2004, pp. 631–645. [42] D. Stanford and J. Urbano, “Some convergence properties of matrix sets,” SIAM J. Matrix Anal. Appl., vol. 15, no. 4, pp. 1132–1140, 1994. [43] W. Zhang, A. Abate, and J. Hu, “Efficient suboptimal solutions of switched LQR problems,” in Proc. Amer. Control Conf., St Louis, MO, 2009, pp. 1084–1091. [44] W. Zhang and J. Hu, “Switched LQR problem in discrete time: Theory and algorithm,” 2010, Purdue Technical Report TR-ECE 09-03. Submitted to IEEE Trans. Autom. Control, 2010. [45] I. Polik and T. Terlaky, “A survey of the S-lemma,” SIAM Rev., vol. 49, no. 3, pp. 371–418, 2007. [46] J. Hu, J. Shen, and W. Zhang, “A generating function approach to the stability analysis of switched linear systems,” in Proc. 13th ACM Int. Conf. Hybrid Systems: Comput. and Control. Stockholm, Sweden: ACM, 2010, pp. 273–282. [47] E. Hille, Analytic Function Theory. Providence, Rhodes Island: AMS Chelsea Publishing, 1959, vol. I.

Jianghai Hu (S’99-M’04) received the B.E. degree in automatic control from Xi’an Jiaotong University, P.R. China, in 1994, and the M.A. degree in Mathematics and the Ph.D. degree in Electrical Engineering from the University of California, Berkeley, in 2002 and 2003, respectively. He is currently an assistant professor at the School of Electrical and Computer Engineering, Purdue University. His research interests include hybrid systems, multiagent coordinated control, control of systems with uncertainty, and applied mathematics.

Jinglai Shen (S’01-A’03-M’05) Jinglai Shen received his B.S.E. and M.S.E. degrees in automatic control from Beijing University of Aeronautics and Astronautics, China, in 1994 and 1997, respectively, and his Ph.D. degree in aerospace engineering from the University of Michigan, Ann Arbor, in 2002. Currently, he is an Assistant Professor with the Department of Mathematics and Statistics, University of Maryland Baltimore County (UMBC). Before joining UMBC, he was a Postdoctoral Research Associate at Rensselaer Polytechnic Institute, Troy, NY. His research interests include hybrid and switching systems, particularly complementarity systems, optimization, multibody dynamics and nonlinear control with applications to engineering, biology and statistics.

Wei Zhang (S’07-M’10) Wei Zhang received the B.E. degree in Automatic Control from the University of Science and Technology of China, Hefei, China, in 2003, the M.S.E.E degree in Electrical Engineering from the University of Kentucky, Lexington, KY, in 2005, and the PhD degree in Electrical Engineering from Purdue University, West Lafayette, IN, in 2009. He is currently a postdoctoral researcher in the Department of Electrical Engineering and Computer Sciences at the University of California, Berkeley. His research interests are in the control and estimation of hybrid and stochastic dynamical systems, and in their applications in various engineering fields, especially robotics and air traffic control.