Generating Functions of Switched Linear Systems: Analysis, Computation, and Stability Applications

Transcription

1 IEEE-TAC FP--5 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. INTRODUCTION SWITCHED linear systems (SLSs) as a natural extension of linear systems are finding increasing applications in a diverse range of engineering systems []. 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] [] and the LMI methods [2] [4], 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 Manuscript received Jan. 3, 2; revised May 28 and July, 2. This work was supported in part by the National Science Foundation under Grant CNS and Grant ECCS-996. J. Hu is with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 4797 USA (phone: ; fax: ; jianghai@purdue.edu). J. Shen is with the Department of Mathematics and Statistics, University of Maryland, Baltimore County, MD 225 USA ( shenj@umbc.edu). W. Zhang is with the Department of EECS, University of California, Berkeley, CA 9472 USA ( weizhang@eecs.berkeley.edu). 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., [5] [9]). 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 [6], [7], [2], [2]; 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 [2]. Computing the JSS is even more difficult [2], [24]. Despite these negative results, many approximation algorithms have been proposed in the literature, e.g. [25] [29] for the JSR and [7], [2], [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 4). Moreover, it makes our approach easily extendable to more general classes of systems, such

2 IEEE-TAC FP--5 2 as conewise linear inclusions [6], switched positive systems [3], [3], and controlled SLSs. A similar perspective has been adopted in [5], 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. STABILITY OF SWITCHED LINEAR SYSTEMS A discrete-time (autonomous) SLS is defined as follows: its state x(t) R n evolves by switching among a set of linear dynamics indexed by the finite index set M := {,...,m}: x(t + ) = A (t) x(t), t =,,.... () Here, (t) M for all t, or simply, is called the switching sequence; and A i R n n, i M, are the subsystem (dynamics) matrices. Starting from the initial state x() = z and under the switching sequence, the trajectory of the SLS is denoted by x(t; z, ). In this paper, unless otherwise stated, denotes both the Euclidean norm on R n and its induced matrix norm on R n n. Definition : The SLS () is called exponentially stable under arbitrary switching (with the parameters κ and r) if there exist κ and r [, ) such that starting from any initial state z and under any switching sequence, the trajectory x(t; z, ) satisfies x(t; z, ) κr t z, for all t =,,.... exponentially stable under optimal switching (with the parameters κ and r) if there exist κ and r [, ) such that starting from any initial state z, there exists a switching sequence for which the trajectory x(t; z, ) satisfies x(t; z, ) κr t z, for all t =,,.... 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 : Under optimal switching, the asymptotic stability and the exponential stability of the SLS () are equivalent. Proof: It suffices to show that asymptotic stability implies exponential stability. Assume that the SLS () is asymptotically stable under optimal switching. Then, for any initial state z on the unit sphere S n := {z R n z = }, there is a switching sequence z such that x(t; z, z ) as t ; hence x(t z ; z, z ) 4 for a time T z large enough. As x(t z ; z, z ) is continuous in z for fixed z and T z, x(t z ; y, z ) 2 for y in a neighborhood U z of z in S n. The union of all such U z is an open cover of the compact set S n ; hence S n l i= U z i for some z,...,z l S n with l <. Starting from any initial x() = z S n, we have z U zi for some i l. By our construction, x(τ ) := x(t zi ; z, zi ) with τ := T zi satisfies x(τ ) 2. Assume without loss of generality that x(τ ). Then x(τ )/ x(τ ) U zj for some j l, and as a result, x(τ 2 ) := x(t zj ; x(τ ), zj ) with τ 2 := τ +T zj satisfies x(τ 2 ) 2 x(τ ). By induction, we obtain a switching sequence z by concatenating zi, zj,... and a sequence of times = τ < τ < τ 2 < at most τ := max i T zi apart such that the resulting trajectory x(t; z, z ) satisfies x(τ k+ ; z, z ) 2 x(τ k; z, z ) for all k. Let κ := τ j= (max i M A i ) j. Then x(t; z, z ) κ(.5) t/τ z for all t. Thus, the SLS () is exponentially stable under optimal switching. Remark : The above proof implies that to prove the exponential stability of the SLS () under optimal switching, it suffices to show that for any z R n, x(t; z, z ) as t for at least one z. This fact will be used in Section IV. Another switching rule we consider is random switching. Let p := {p i } i M be a probability distribution with p i and i M p i =. The SLS () under the (stationary) random switching probability p has the dynamics x(t + ) = A(t)x(t), t =,,.... (2) Here, at each time t, A(t) is drawn independently randomly from {A i } i M with the probability P{A(t) = A i } = p i. Denote by x(t; z, p) the stochastic trajectory of the system (2) from a deterministic initial state x() = 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 and r [, ) such that for any z R n, E[ x(t; z, p) 2 ] κr t z 2, for all t =,,.... Similarly, the SLS (2) is called mean square asymptotically stable if E[ x(t; z, p) 2 ] as t for all z R n, and almost sure asymptotically stable if P{lim t x(t; z, p) = } = for all z R n. From results on random jumped linear systems [36, Theorem 4..], 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. STRONG GENERATING FUNCTIONS Central to the stability analysis of SLSs is the task of determining the exponential rate at which x(t; z, ) grows as t for trajectories x(t; z, ) of the SLSs. The following lemma, adopted from [37, Corollary..], hints at an indirect way of characterizing this growth rate. Lemma : Given a scalar sequence {w t },,..., suppose the power series w t t has the radius of convergence R. Then for any r > R, there exists a constant C r such that w t C r r t for all t =,,.... As a result, for any trajectory x(t; z, ), an (asymptotically) tight bound on the exponential growth rate of x(t; z, ) 2 as t is given by the reciprocal of the radius of convergence of the power series t x(t; z, ) 2.

3 IEEE-TAC FP--5 3 Motivated by this, we define the strong generating function G(, ) : R + R n R + { } of the SLS () as G(, z) := sup t x(t; z, ) 2,, z R n, (3) where the supremum is taken over all switching sequences. For a fixed z, G(, z) is nondecreasing in. Indeed, due to the supremum in (3), G(, z) can be viewed intuitively as the power series in corresponding to the most divergent trajectories of the SLS starting from z. Thus, by Lemma, its radius of convergence defined by (z) := sup{ G(, z) < } (4) is expected to characterize the fastest exponential growth rate of the SLS trajectories starting from z. We call (z) the radius of strong convergence of the SLS at z, For each fixed, G(, z) is a function of z only: G (z) := G(, z), z R n. (5) By definition (3), G ( ) is nonnegative and homogeneous of degree two, with G (z) = z 2. Since G ( ) is nondecreasing in, we have G (z) z 2, z, for. We will also refer to G (z) as the strong generating function of the SLS (). A. Properties of General G (z) We first prove some useful properties of the function G (z). Proposition : G (z) has the following properties:. (Bellman Equation): Let be arbitrary. Then G (z) = z 2 + max i M G (A i z), z R n. 2. (Sub-Additivity): Let be arbitrary. Then G (z + z 2 ) G (z )+ G (z 2 ), z, z 2 R n. 3. (Convexity): For each, both G (z) and G (z) are convex functions of z on R n. 4. (Invariant Subspace): For each, the set G := {z R n G (z) < } is a subspace of R n invariant under {A i } i M, i.e., A i G G for all i M. 5. For each, G (z) < for all z R n implies that G (z) c z 2 for all z R n for some finite constant c. 6. For < (max i M A i 2 ), G (z) <, z. Proof:. Note that G ( ) is the value function of an infinite horizon optimal control problem maximizing the functional t x(t; z, ) 2. Property is a direct consequence of the dynamic programming principle. 2. For a fixed, since x(t; z, ) is linear in z, we have G (z + z 2 ) = sup G (z ) + 2 sup t x(t; z, ) + x(t; z 2, ) 2 t x(t; z, ) x(t; z 2, ) + G (z 2 ) G (z ) + 2 G (z ) G (z 2 ) + G (z 2 ), z, z 2 R n. The Cauchy-Schwartz inequality is used in the last step. Taking the square root yields the desired conclusion. 3. For a fixed, G (z) is convex in z as by (3) it is the pointwise supremum of a family of convex (indeed, quadratic) functions of z indexed by. The convexity of G (z) follows from sub-additivity as G (α z + α 2 z 2 ) G (α z ) + G (α 2 z 2 ) = α G (z ) + α 2 G (z 2 ), for any z, z 2 R n and α, α 2 with α + α 2 =. 4. This follows directly from Properties and Assume is such that G (z) <, z R n. Write an arbitrary z S n in a standard basis {e i } of R n as z = n i= α i e i, where n i= α2 i =. In light of sub-additivity, we have G (z) [ n i= G (α i e i )] 2 n n i= α2 i G (e i ) c, where c := n max i n G (e i ) < by our assumption on. By homogeneity, we have G (z) c z 2, z R n. 6. We simply note that any trajectory x(t; z, ) of the SLS satisfies x(t; z, ) 2 (max i M A i 2 ) t z 2, t. From Proposition, G is a subspace of R n that decreases monotonically from G = R n at = to G := G as. Let < 2 < < d for some integer d < n be the exact values of at which G shrinks. Then the set of all distinct G forms a filtration of subspaces of R n as: G = G G + = G G 2 + = = G 2 G d + = G, d where G := lim j G j and G j + := lim j G j for j each j. Since each subspace G (or G j + ) is invariant under j {A i } i M, the SLS () 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 {A i } 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 G d, and so on. From the above discussion, the radius of strong convergence (z) at z R n can have at most d + n distinct values: {,..., d, }. In particular, if the SLS is irreducible, namely, it has no nontrivial invariant subspaces other than R n and {} (which occurs with probability one for randomly generated SLSs), then d =, and G (z) is either finite everywhere or infinite everywhere for any. Remark 2: In the Multiplicative Ergodic Theorem for nonswitched dynamical systems, { log,..., log d, } are called the Lyapunov exponents of the systems [22]. Example : Consider a SLS on R 2 with two subsystems: [ 7 A = ], A 2 = [ ]. (6) Starting from any initial z = (z, z 2 ) T, let x(t; z, ) and x(t; z, 2 ) be the state trajectories under the switching sequences = (,,...) and 2 = (2, 2,...), respectively. Then it can be proved (though by no mean trivially) that G (z) is max i=,2 t x(t; z, i ) 2, or more explicitly, max { 9(z+z 2) 2 + (z z2)2 2( 4), 2(9 ) } (z +z 2) 2 2( 9) + 9(z z2)2 2(9 ), if < 9 (z z 2) 2 2( 4) {z +z 2=} + {z+z 2 }, if 9 < 4, if 4. Here, {z+z 2=} denotes the indicator function for the set {(z, z 2 ) R 2 z + z 2 = }. Similarly for {z+z 2 }.

4 IEEE-TAC FP--5 4 Thus, G is R 2 for < 9 ; the D subspace V := {(α, α) T α R} for 9 < 4 ; and {} for 4. Each of these is an invariant subspace of R 2 for {A, A 2 }. For example, V is a common eigenspace of A and A 2. In fact, A and A 2 commute and can be simultaneously diagonalized as Q A Q =diag( 3, 2) and Q A 2 Q =diag(3, 3 ) [ cos(π/4) sin(π/4) sin(π/4) cos(π/4) ]. Under the transformation by Q, 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 (), denoted by (, ], is defined as := sup { G (z) <, z R n }. By Property 5 of Proposition, can also be defined as = sup{ G (z) < c z 2, z R n, for some finite c}. By Property 6, (max i M A i 2 ) >. It is possible that =. This is the case, for example, if all solutions x(t; z, ) of the SLS converge to the origin within a finite time uniformly in z and. For the SLS in Example, = 9. The following theorem shows that the radius of strong convergence is sufficient for determining the exponential stability of the SLS () under arbitrary switching. Theorem 2: The following statements are equivalent: ) The SLS () is exponentially stable under arbitrary switching. 2) Its radius of strong convergence >. 3) The strong generating function at =, G (z), is finite for all z R n. Proof: To show ) 2), suppose there exist constants κ and r [, ) such that x(t; z, ) κr t z, t, for all trajectory x(t; z, ) of the SLS. Then for any < r 2, G (z) = sup t x(t; z, ) 2 t κ 2 r 2t z 2 = κ2 r 2 z 2 <, z R n. It follows that r 2 >. The implication 2) 3) follows directly from the definition of. Finally, to show 3) ), suppose G (z) <, z. By Proposition, G (z) c z 2, z, for some finite constant c. Thus, for any trajectory x(t; z, ) of the SLS, x(t; z, ) 2 c z 2. This implies that x(t; z, ) c z, t; and that x(t; z, ) as t. Consequently, the SLS is asymptotically stable, hence exponentially stable [6], under arbitrary switching. Theorem 2 implies the following stronger conclusions. Corollary : Given a SLS with a radius of strong convergence, for any r > ( ) /2, there exists a constant κ r such that x(t; z, ) κ r r t z, t, for all trajectories x(t; z, ) of the SLS. Furthermore, ( ) /2 is also the smallest value for the previous statement to hold. Proof: Suppose r > ( ) /2. The scaled SLS with subsystem dynamics matrices {A i /r} i M is easily seen to have its strong generating function to be G(/r 2, z); hence, its radius of strong convergence is r 2 >. By Theorem 2, the scaled SLS is exponentially stable under arbitrary switching. In particular, all its trajectories x(t; z, ) satisfy x(t; z, ) κ r z, t, for some κ r >. For all trajectories x(t; z, ) of the original SLS, since x(t; z, ) = r t x(t; z, ), we must have x(t; z, ) = r t x(t; z, ) κ r r t z, 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 ( ) /2. Later on in Theorem 3, we will study how to infer the constant κ r from G (z). Remark 3: For the SLS (), the joint spectral radius [7], [8] of the subsystem dynamics matrices {A i } i M is defined by ρ := lim k sup { A i A ik /k, i,..., i k M } ; and the Lyapunov exponent of the corresponding linear difference inclusion is γ := sup,z limsup t t ln x(t; z, ) (see [5]). These two quantities also characterize the maximal exponential growth rate of the SLS trajectories, and are related to by: ( ) /2 = ρ = e γ. In this sense, Corollary is equivalent to [7, Prop..4] and to [8, Prop. 4.7]. C. Smoothness of Finite G (z) When is in the range of [, ), the function G (z) is finite everywhere. We shall focus on such finite G (z), and establish some smoothness properties of them in this section. We first introduce a few notions. A function f : R n R is called directionally differentiable at z R n if its (onesided) directional derivative at z along any direction v R n defined as f f(z (z ; v) := lim +τv) f(z ) τ τ exists. If f is both directionally differentiable at z and locally Lipschitz continuous in a neighborhood of z, it is called B(ouligand)- differentiable at z. Finally, f is semismooth at z if it is B- differentiable in a neighborhood of z and the following limit f holds: (z;z z ) f (z ;z z ) limz z z z z z =. Proposition 2: For [, ), both G (z) and G (z) are convex, locally Lipschitz continuous, and semismooth on R n. Moreover, G (z) is globally Lipschitz continuous. Proof: The convexity of G (z) and G (z) has been proved in Proposition. Being convex, they must also be semismooth according to [39, Prop ]. Finally, using the sub-additivity in Proposition, we obtain, z, z R n, G ( z) G (z + z) G (z) G ( z). Thus, G (z + z) G (z) G (± z) c z for some finite constant c as <, i.e., G (z) is globally Lipschitz continuous on R n with the Lipschitz constant c. As a result, G (z), hence G (z), is also locally Lipschitz continuous on R n. Note that for >, G (z) can not be continuous on R n. Indeed, in this case, G (z ) = at some z R n. Thus as k, the sequence z k, but G ( z k ). D. Quadratic Bounds of Finite G (z) For [, ), G (z) is finite everywhere, hence quadratically bounded by Proposition. Define g := sup{g (z) z = }, [, ). (7)

5 IEEE-TAC FP--5 5 By homogeneity, g can be equivalently defined as the smallest constant c such that G (z) c z 2, z R n. It is easy to see that g is finite and strictly increasing on [, ) (we exclude the trivial case where all A i are zero), with g =. We next prove an affine lower bound of /g. Lemma 2: g max i M A i 2, [, ). Thus, /g max i M A i 2, (, ). (8) Proof: Let [, ). For an arbitrary trajectory x(t; z, ) of the SLS, x(t; z, ) 2 (max i M A i 2 ) t z 2 for all t. Therefore, for < (max i M A i 2 ), t x(t; z, ) 2 max i M A i 2 z 2,. By definition, we have g ( max i M A i 2 ), which is the desired conclusion for < (max i M A i 2 ). When (max i M A i 2 ) <, the desired conclusion is trivial: g max i M A i 2. The following auxiliary result on general power series is proved in Appendix A. Lemma 3: Let {w t },,... be a sequence of nonnegative scalars such that for some > and β, w t+s t βw s, s =,,.... (9) Then the power series w t t has its radius of convergence at least := /( /β). Moreover, w t t βw (β )(/ ) <, [, ). Using Lemma 3, we obtain the following estimate of g. Proposition 3: The function /( /g ) is nondecreasing for (, ), and is upper bounded by /g, (, ). () Proof: Consider a fixed (, ). Let x(t; z, ) be an arbitrary trajectory of the SLS. For each s =,,..., by the definition of g, the trajectory x(t + s; z, ), t =,,..., of the SLS starting from the initial state x(s; z, ) satisfies t x(t + s; z, ) 2 G (x(s; z, )) g x(s; z, ) 2. Hence, the sequence {w t := x(t; z, ) 2 },,... satisfies the condition (9) with β = g. By Lemma 3, we have t x(t; z, ) 2 g (g )(/ ) z 2, for [, ), where := /( /g ). As the trajectory x(t; z, ) is arbitrary, we conclude that g G (z) (g )(/ ) z 2 <, [, ). This implies, i.e., the desired conclusion (); and g g (g )(/ ) /g /g, Fig.. /g max i M A i 2 (,/g ) / /( /g ) Plot of the function /g (in solid line). for [, ). Since (, ) is arbitrary and >, this proves the monotonicity of /( /g ) on (, ). In Appendix B, the following generic properties of the functions g and /g are proved. Proposition 4: The function /g is strictly decreasing, semismooth, and Lipschitz continuous with Lipschitz constant max i M A i 2 on [, ). Moreover, /g as. Correspondingly, g is strictly increasing, convex, semismooth, and locally Lipschitz continuous on [, ), with g as. Remark 4: Since g as, the generating function G (z) must have infinite value at some z R n, according to Property 5 of Proposition. This implies that, as increases, is precisely the first value at which G ( ) starts to have infinite values. Fig. plots the graph of a generic /g as a function of [, ). According to (8) and (), the graph of /g is sandwiched between those of two affine functions: max i M A i 2 from the left and / from the right. In addition, by Proposition 3, the ray (middle dashed line) emitting from the point (, ) and passing through (, /g ) intersects the -axis at the point ( /g, ) that moves monotonically to the right towards (, ) as increases in [, ), i.e., the ray rotates around its starting point (, ) counterclockwise monotonically. It is conjectured that the function /g is indeed convex on [, ) for any SLS. It is easy to show that the directional derivative of g at = is g ( +) = max i M A i 2. Hence the directional derivative of /g at = is: (/g ) ( + ) = max i M A i 2. In Fig., this means that the graph of /g is tangential to the leftmost dashed ray emitting from (, ). E. Norms Induced by Finite G (z) As an immediate result of Proposition, for [, ), G (z) is finite, sub-additive, and homogeneous of degree one; thus it defines a norm on the vector space R n : z G := G (z), z R n. () As increases, the norm G increases, hence its unit ball shrinks. See Fig. 2 for the plots of such unit balls for the SLS (6) in Example. The vector norm G induces a matrix norm for A R n n by: A G := sup z { Az G / z G }.

6 IEEE-TAC FP Fig. 2. Unit balls of G for the SLS (6). For [, ), define the constant: z = z = =.5 =.9 =. d := sup max A iz 2 G i M = sup max G (A i z). (2) i M Lemma 4: For each [, ), the norm G satisfies: max i M A i G = d /( + d ). Proof: Using the Bellman equation, we write max A i 2 G i M = max sup G (A i z) G (z) i M z = max i M G (A i z) = sup z = G (z) max i M G (A i z) = sup z = + max i M G (A i z) sup z = max i M G (A i z) = + sup z = max i M G (A i z) = d. + d Note that the second last step follows as x/( + x) is continuous and increasing in x R + for any. The above results yields bounds on both the exponential growth rate of the SLS trajectories and as follows. Corollary 2: Suppose [, ). Then x(t; z, ) ( ) t/2 d g z, t =,,..., + d holds for all trajectories of the SLS (). As a result, we have > + d, [, ). (3) Proof: We note that z z G g z, z R n, i.e., the norm G is equivalent to the Euclidean norm. For any trajectory x(t; z, ) of the SLS, we then have x(t; z, ) x(t; z, ) G = A (t ) A () z G A (t ) G A () G z G, t. Applying Lemma 4 and noting that z G g z, we obtain the first conclusion. This in turn implies that > ( + d )/d, which is the desired conclusion (3). As, by (3), we must have d ; hence d /(+ d ) /. The following then holds. Theorem 3: For any ε >, there exists a (, ) sufficiently close to such that x(t; z, ) g (r ) t z, t =,,..., for all trajectories x(t; z, ) of the SLS (), where ( ) /2 d r = ( ) /2 + ε. + d Remark 5: Once G (z) is computed at any [, ), (3) gives a lower bound of. An upper bound of can be derived by using a result in [27, Lemma 3.3] as [ inf max z i M G (A i z) G (z) ]. (4) This upper bound, together with the lower bound in (3), gives an interval for the possible values of. As, it can be shown that the two bounds converge towards each other; and the corresponding norm G approaches asymptotically a Barabanov norm [23], [27] of {A i } i M. F. Algorithms for Computing G (z) We next present some algorithms for computing the finite strong generating functions. The idea is that G (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 =,,..., define G k (z) := max k t x(t; z, ) 2, z R n. (5) Maximum is used here instead of supremum as only the first k steps of affect the summation. Proposition 5: For any and k =,,..., G k (z) is a convex function of z on R n satisfying G (z) G (z) G2 (z) G (z), z R n. Moreover, for [, ), G k (z) as k converges exponentially fast to G (z): k =,,..., G k (z) G (z) g ( /g ) k+ z 2, z R n. Proof: The convexity proof is identical to that of Proposition, hence omitted. Fix and z R n, and let k be a switching sequence achieving the maximum in (5). Then, k k+ G k (z)= t x(t; z, k ) 2 t x(t; z, k ) 2 G k+ (z). Similarly, we can show G k (z) G (z), hence the monotonicity. When [, ), G (z) is finite for each z R n. Let ˆx(t) := x(t; z, ) be a trajectory under an optimal switching sequence that achieves the supremum in (3), i.e., G (z) = t ˆx(t) 2. By the Bellman equation, G (ˆx(k )) G (ˆx(k))= ˆx(k ) 2 G (ˆx(k )) g, for k =, 2,..., where the last step follows from the definition of g. Rearranging and by induction, we obtain G (ˆx(k)) ( /g )G (ˆx(k )) k ( /g ) k G (z) k g ( /g ) k z 2,

7 IEEE-TAC FP--5 7 Algorithm Computing G (z) on Grid Points of S n Let S = {z j } N j= be a set of grid points of Sn ; Initialize k =, and Ĝ (z j) = for all z j S; repeat k k + ; for each z j S do for each i M do Find a minimal subset S ij of S whose elements span a convex cone containing A i z j ; Express A i z j as z l S ij α l ij z l with α l ij > ; Ĝk (z l ); Set g ij = z l S ij α l ij end for Set Ĝk (z j) = + max i M gij 2 ; end for until k is large enough return Ĝk (z j) for all z j S for k =,,.... The optimality of ˆx(t) then implies that t ˆx(t) 2 = k t ˆx(t + k) 2 = k G (ˆx(k)) t=k g ( /g ) k z 2, k =,,.... Consequently, G (z) G k (z) k t ˆx(t) 2 = G (z) t=k+ t ˆx(t) 2 G (z) g ( /g ) k+ z 2. Therefore, G k (z) for k large enough provide approximations of G (z) within arbitrary precision. A recursive procedure to compute the functions G k (z) is as follows: G (z) = z 2, G k (z) = z 2 + max i M Gk (A i z), k =, 2,.... (6) To implement the recursion numerically, one first represents each G k (z) by its values on some fine grid points of the unit sphere, and then carries out the recursion (6) by estimating conservatively the values of G k (A i z) at those A i z not aligned with the direction of anygrid point, using convexity and homogeneity of the function G k (z). The above idea is summarized in Algorithm. Similar ray gridding techniques have also been used in the previous studies [4], [4]. Algorithm returns a sequence of mappings Ĝk : S R +, k =,,..., where S is a set of grid points of the unit sphere. They provide upperbounds of G k (z) on S as follows. Proposition 6: G k (z j) Ĝk (z j), z j S, k =,,... Proof: We prove by induction. At k =, we have G (z j) = Ĝ (z j) = for all z j S. Suppose the conclusion is true for,,..., k. For any z j S, let S ij and α l ij be as given in Algorithm. Then Ĝk (z j) = + max i M gij 2, where by the induction hypothesis and sub-additivity, g ij G k (z l ) = G k (α l ij z l) z l S ij α l ij G k ( ) α l ij z l = z l S ij z l S ij G k (A i z j ). /g Fig. 3. Top: Unit balls of G for the SLS in Example 2 at =.,.2,.3,.37,.38 (inward). Bottom: Plot of /g. Using the Bellman equation in Proposition, we then have Ĝ k (z j) + max i M G k (A i z j ) = G k (z j). Thus, the 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 () to be exponentially stable under arbitrary switching is that the mappings Ĝk : S R +, k =,,..., obtained by Algorithm are uniformly bounded. By repeatedly applying Algorithm to a sequence of whose values increase according to next = /( /g ) or by (3), increasingly accurate underestimates of 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 approaches, however, the computation time of Algorithm will get exponentially longer for two reasons. First, the convergence of G k (z) to G (z) is much slower by Proposition 5. Second, errors of Ĝ k (z) over-approximating G k (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 (or the JSR) is known to be NP-hard [24]. Hence the complexity of Algorithm, 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 (3) and (4), we are currently working on developing further performance assurance for Algorithm. G. Examples Example 2: The following example is taken from [25]: [ ] [ ] A =, A 2 =.

8 IEEE-TAC FP--5 8 H. Generalized Strong Generating Functions The definition of strong generating functions can be generalized. Let q be a positive integer, and let now denote an arbitrary norm on R n. Define a generalized strong generating function as G, q (z) := sup t x(t; z, ) q. When q = 2 and is the Euclidean norm, G, q (z) reduces to G (z) defined in (3). The function G, q (z) retains most of the properties of G (z) in Proposition. For instance, for any, [G,q (z)] /q is subadditive, positively homogeneous, and convex in z; it is further finite, globally Lipschitz continuous, and semismooth whenever is smaller than q, := sup{ G, q (z) <, z R n }. /g Fig. 4. Top: Unit ball of G for the SLS in Example 3. Bottom: Plot of /g. Algorithm is used to compute the functions G (z) of this SLS for different values of : =.,.2,.3,.37, and.38. The results are shown in Fig. 3, where the top and bottom figures plot respectively the unit balls of the norm G and the graph of the function /g. From the plot, the graph of /g is very close to a straight line. Thus, by computing /g at two different and extrapolating, one can obtain an accurate estimate of. This gives some justification to the fact that the joint spectral radius in this case is available analytically [8]: ρ = Thus = /(ρ ) Example 3: Consider the following SLS in R 3 :.5.7 A =.3, A 2 = , A 3 = Overestimates of the function G (z) are computed by applying Algorithm on 75 2 grid points of the unit sphere. The unit ball corresponding to the estimated norm G (z) is shown at the top of Fig. 4; the bottom figure depicts the computed /g for =.2,.4,.6,.8,, and.. Since g at =. is finite, >., hence the given SLS is exponentially stable under arbitrary switching. An extrapolation of the function /g provides an estimate of at around.64. We call q, the radius of strong convergence corresponding to G, q (z), and note that its definition does not depend on the choice of the norm. Similar to Theorem 2, we can show that the SLS is exponentially stable under arbitrary switching if and only if q, >. Indeed, q, is related to and the JSR ρ as: ( q, ) /q = ( ) /2 = (ρ ), for all q. Although different choices of q and lead to equivalent stability tests, the numerical robustness of such tests may vary. Noting that any nonnegative sequence {w t },,... satisfies (w t) qr ( (w t) q) r, we have qr, ( q, ) r, q, r =, 2,.... For a barely exponentially stable SLS with a convergence radius 2, only slightly above, by choosing r >, the new radius 2r, ( 2, ) r has a larger gap with ; hence it may lead to a more robust stability test. IV. WEAK GENERATING FUNCTIONS A. Definition and Properties The weak generating function H : R + R n R + { } of the SLS () is defined as H(, z) := inf t x(t; z, ) 2,, z R n, (7) where the infimum is over all switching sequences of the SLS. Then H(, z) is monotonically increasing in, with H(, z) = z 2 when =. The threshold (z) := sup{ H(, z) < } is called the radius of weak convergence of the SLS at z. For each fixed, write H (z) := H(, z), z R n. Then H ( ) is homogeneous of degree two, with H ( ) = 2. Some properties of the function H (z) are listed below. It is noted that many properties (e.g. convexity) of the strong generating function G (z) are not valid for H (z). Proposition 7: H (z) has the following properties.. (Bellman Equation): Let be arbitrary. Then H (z) = z 2 + min i M H (A i z), z R n. 2. (Invariant Subset): For each, the set H := {z R n H (z) = } is a subset of R n invariant under {A i } i M, i.e., A i H H for all i M. 3. For < (min i M A i 2 ), H (z) c z 2 for some finite constant c >.

9 IEEE-TAC FP--5 9 Proof: Property is proved by applying the dynamic programming principle to the optimal control problem of minimizing t x(t; z, ) 2. Property 2 follows directly from Property. For Property 3, by choosing := (i, i, i,...) with no switching where i = argmin i M A i, we have x(t; z, ) 2 (min i M A i 2 ) t z 2, t. Therefore, for < (min i M A i 2 ), H (z) t x(t; z, ) 2 min i M A i 2 z 2, which is finite and bounded by a quadratic function. Note that H, unlike G, cannot be a subspace of R n as it does not contain the origin. B. Radius of Weak Convergence Definition 4: The radius of weak convergence of the SLS (), denoted by (, ], is defined by := sup{ H (z) <, z R n }. By Proposition 7, we must have (min i M A i 2 ). The value of could reach if starting from any z, a switching sequence z exists so that x(t; z, z ) reaches the origin within a uniform time T <. The next result shows that a function H (z) finite everywhere on R n must be bounded by a quadratic function. Thus, can also be defined to be sup{ H (z) c z 2, z R n, for some constant c}. Proposition 8: For each, the following statements are equivalent: ) H (z) <, z R n ; 2) H (z) c z 2, z R n, for some positive constant c; 3) The scaled SLS with subsystem matrices { A i } i M is exponentially stable under optimal switching. Proof: It is obvious that 2) ). We show in the following that ) 3), and 3) 2). To prove ) 3), assume is such that ) holds. Then for any z R n, there exists a switching sequence z such that t x(t; z, z ) 2 <. Consider the scaled SLS with subsystem dynamics matrices { A i } i M, and denote by x(t; z, z ) = t/2 x(t; z, z ) its solution starting from z under z. Since x(t; z, z) 2 <, x(t; z, z ) as t for each z. By Remark, 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 > and r (, ) such that for each z, the trajectory x(t; z, z ) of the scaled SLS under at least one switching sequence z satisfies x(t; z, z ) 2 κr t z 2, t. As a result, z, H (z) t x(t; z, z ) 2 = x(t; z, z ) 2 κ z 2 r, which is exactly the conclusion of 2) with c = κ/( r). We next show that the radius of weak convergence characterizes the exponential stability of the SLS under optimal switching just as does for the exponential stability under arbitrary switching. Theorem 4: The SLS () is exponentially stable under optimal switching if and only if >. Proof: To prove necessity, we observe that for a SLS exponentially stable under optimal switching with the parameters κ > and r [, ), by a similar argument as in the proof of Theorem 2, we must have H (z) κ2 r z 2, z, 2 for < r 2 ; hence r 2 >. To show sufficiency, suppose >. Then at =, H (z) is finite for all z. By Proposition 8, this implies that the scaled SLS with subsystem matrices { A i } i M, which is also the original SLS (), is exponentially stable under optimal switching. Following similar steps as in the proof of Corollary, we can prove the subsequent result. Corollary 4: Consider the SLS (). For any r > ( ) /2, there is a constant κ r such that starting from each z R n, x(t; z, z ) κ r r t z, t, for at least one switching sequence z. Furthermore, ( ) /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 () under optimal switching is given by precisely ( ) /2. Another related (but generally larger) quantity characterizing such a rate is the joint spectral subradius of {A i } i M defined by [7]: { } ˇρ := lim inf A i A ik /k, i,..., i k M. (8) k Difference between these two rates is shown via the following SLS (inspired by [42, pp. 35]) with subsystem matrices: [ 2 ] A = 3 3, A 2 = QA Q T, A 3 = QA 2 Q T, 2 where Q R 2 2 is the rotation matrix of 6 counterclockwise. Let C be the symmetric cone consisting of all those x R 2 C with phase angle within the range of either [ 3, 3 ] or [5, 2 ]. It is easy to check that A x x for x C; A 2x x for x QC; and A 3 x 48 x for x Q2 C. Since R 2 = C QC Q 2 C, a state feedback switching policy can be designed as follows: (x) =, 2, and 3, for x in C, QC, and Q 2 C, respectively (if x is in more than one set, any tie breaking rule can be applied). Then starting from any z, the resulting closed-loop 43 trajectory satisfies x(t+; z, ) 48 x(t; z, ) for all t. Thus, the SLS is exponentially stable under (x), and we 43 have ( ) /2 48, i.e., > In comparison, since each of the subsystem matrices has determinant one, any finite product A i A ik also has determinant one, hence norm at least one. By (8), we must have ˇρ, hence ˇρ > ( ) /2. This gap can be explained as ˇρ defined by (8) satisfies ˇρ = lim inf sup k i,...,i k M x = lim sup k x = A i A ik x /k inf A i A ik x /k, i,...,i k M where the right hand side is exactly ( ) /2. In other words, the JSS ˇρ and the rate ( ) /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

10 IEEE-TAC FP--5 (i,..., i k, i,...,i k,...) and by closed-loop switching policies with state feedback switching laws, respectively. C. Quadratic Bounds of Finite H (z) For each [, ), define the constant h as the smallest constant c such that H (z) c z 2 for all z: E 2 E r h r g h := sup{h (z) z = }. (9) Obviously, h is strictly increasing in, with h =. Similar to Proposition 3 for g, we have the following estimates of h. Proposition 9: The function /( /h ) is nondecreasing for (, ), and is upper bounded by /h, (, ). (2) Proof: Let (, ). For any z R n, let z be a switching sequence so that the resulting trajectory x(t; z, z ) achieves the infimum in (7): H (z) = t x(t; z, z) 2 <. By the optimality of z, for each s =,,..., the time shifted trajectory of the SLS, x(t + s; z, z ), t, starting from the initial state x(s; z, z ) is also optimal: t x(t + s; z, z) 2 =H (x(s; z, z )) h x(s; z, z ) 2. In other words, the sequence defined by {w t := x(t; z, z ) 2 },,... satisfies the condition (9) with β = h. By Lemma 3, we then have t x(t; z, z ) 2 h (h )( )/ z 2, for [, ), where := /( /h ). Therefore, H (z) t x(t; z, z ) 2 h z 2 (h )( )/ is finite for all z R n and [, ). This implies that, which is the desired (2); and that for [, ), h h (h )( )/. /h /h This proves the monotonicity of /( /h ) on (, ). The following result follows directly from Proposition 9. Corollary 5: For each [, ), /h /. As a result, /h and h as. The directional derivative of h at = is computed in Appendix C as follows. Lemma 5: The directional derivative of h at exists and is given by h h h ( + ) := lim = sup min A iz 2. z = i M As a result of Lemma 5, lim /h = h ( +) = sup z = min i M A i z 2. Since by Proposition 9, /( /h ) is nondecreasing in on (, ), the above limit implies that, for (, ), lim /h /h = sup z = min i M A i z 2. E 3 Fig. 5. Geometric interpretation of the Lipschitz constants in Propositions 4 and : max i M A i 2 = /r 2 g and sup z = min i M A i z 2 = /r 2 h. Fig. 6. /h sup z = min i M A i z 2 (,/h ) /( /h ) Plot of the function /h (in solid line). / This leads to the following estimate of h, which is the counterpart of Lemma 2 for g. Corollary 6: sup min h A iz 2, [, ). z = 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 : The function /h defined on [, ) is strictly decreasing and Lipschitz continuous with Lipschitz constant sup z = min i M A i z 2. The function h is strictly increasing and locally Lipschitz continuous on [, ). Remark 6: The Lipschitz constants of /g in Proposition 4 and of /h in Proposition are given as sup z = max i M A i z 2 and sup z = min i M A i z 2, respectively. An overestimate of the latter is given by min i M sup z = A i z 2 = min i M A i 2. For a geometric interpretation, associate each matrix A i with an ellipsoid E i := {z R n A i z 2 } in R n. The intersection of all such ellipsoids, i M E i, is a convex set that can embed a maximal ball centered at with the radius r g ; whereas their union i M E i can embed a maximal ball centered at with the radius r h (see Fig. 5). Further, let r l := max i M {the length of the shortest principal axis of E i }. The two Lipschitz constants and the overestimate can then be expressed as max A i 2 = /rg 2, sup min A iz 2 = /rh 2, i M z = i M and min A i 2 = /rl 2. i M A generic plot of the function /h for [, ) is shown in Fig. 6. The function decreases strictly from at = to as. Its graph is sandwiched by

11 IEEE-TAC FP--5 those of two affine functions: / from the right, and sup z = min i M A i z 2 from the left. Moreover, as increases from towards, the ray emitting from the point (, ) and passing through the point (, /h ) rotates counterclockwise monotonically, and intersects the - axis at a point whose -coordinate, /( /h ), provides an asymptotically tight underestimate of. D. Approximating Finite H (z) For each [, ), the function H (z) is finite everywhere on R n. We next show that it is the limit of a sequence of functions H k (z), k =,,..., defined by H k (z) := min k t x(t; z, ) 2, z R n. (2) As the value functions of an optimal control problem with variable finite horizons, H k (z) can be computed recursively as follows: H (z) = z 2, z R n ; and for k =, 2,..., H k (z) = z 2 + min i M Hk (A i z), z R n. (22) Equivalently, we can write H k (z) = min{zt Pz : P P k }, z R n, (23) where P k, k =,,..., is a sequence of sets of positive definite matrices defined by: P = {I}; and for k =, 2,..., P k = {I + A T i PA i P P k, i M}. (24) Proposition : The sequence of functions H k (z) is monotonic: H H H2 H ; and for [, ), it converges exponentially fast to H (z): for k =,,..., H k (z) H (z) h 2 ( /h ) k+ z 2, z R n. (25) Proof: Fix k. For each z R n, let k be a switching sequence achieving the minimum in (2). Then, H k(z) = k t x(t; z, k ) 2 k t x(t; z, k ) 2 H k (z). Similarly, we have H k(z) H (z), proving the monotonicity. Next assume [, ). For any z R n and k =,,..., we note that z 2 H k(z) H (z) h z 2. Let k be such that ˆx(t) := x(t; z, k ) achieves the minimum in (2). For each s =,,..., k, since ˆx(t) is also optimal over the time horizon t = s, s +,...,k, we have k s H k s (ˆx(s)) = ˆx(s) 2 + t ˆx(t + s + ) 2 = ˆx(s) 2 + H k s (ˆx(s + )). Since ˆx(s) 2 H k s (ˆx(s))/h, the above implies that H k s (ˆx(s + )) ( /h )H k s (ˆx(s)). Applying this inequality for s = k,...,, we have ˆx(k) 2 = H (ˆx(k)) ( /h )H (ˆx(k )) k ( /h ) k H k (z) k h ( /h ) k z 2. Algorithm 2 Computing Over Approximations of H k (z). Initialize k =, Pε = {I n }, and P ε = P ε ; repeat k k + ; P ε k = {I + AT i PA i i M, P P ε k }; Find an ε-equivalent subset P ε k P ε k ; until k is large enough return H k,ε (z) = min{zt Pz P P ε k }. Using this inequality, and adopting the switching sequence that first follows k for k steps and thereafter follows an infinitehorizon optimal starting from ˆx(k), we obtain H (z) k t ˆx(t) 2 + t=k+ t x(t k; ˆx(k), ) 2 = H k (z) + k [ H (ˆx(k)) ˆx(k) 2] H k (z) + (h ) k ˆx(k) 2 H k (z) + h2 ( /h ) k+ z 2. This, together with H k(z) H (z), proves (25). By (25), for any [, ), H k (z), k =,,..., is a sequence of functions of (, z) converging uniformly to H (z) on [, ] S n. As each H k (z) is continuous in (, z) on [, ] S n, so is H (z). Using homogeneity and the arbitrariness of [, ), we obtain the following result. Corollary 7: The function H (z) = H(, z) is continuous in (, z) on [, ) R n. E. Relaxation Algorithm for Computing H (z) According to (25), H k (z) for large k provide increasingly accurate estimates of H (z). By (23), to characterize H k(z), it suffices to compute the set P k. To deal with the rapidly increasing size of P k as k increases, we introduce the following complexity reduction technique, inspired by [43], [44]. A subset Pk ε P k is called ε-equivalent to P k for some ε > if H k,ε (z) := min P P ε k z T Pz min P P k z T Pz + ε z 2, z R n. A sufficient (though not necessary) condition for this to hold is that, for each P P k, P + εi n is bounded from below by a convex combination of matrices in Pk ε, i.e., there exist constants α Q, Q Pk ε, adding up to such that P + εi n Q P α k ε Q Q. This leads to a natural procedure of removing matrices from P k iteratively until a minimal ε- equivalent subset Pk ε is achieved. By applying this procedure at each step of the iteration (24), we obtain Algorithm 2, which returns approximations of H k (z) for all k with uniformly bounded approximation errors as follows. Proposition 2: Assume [, ). Then for k =,,..., H k (z) H k,ε (z) ( + ε)hk (z), z R n. (26) Proof: Obviously, (26) holds for k =. Assume it holds for some k. Define, for z R n, H k,ε (z) := min z T Pz = z 2 + min P P ε k i M Hk,ε (A i z).

12 IEEE-TAC FP--5 2 Fig. 7. /h /h Plots of /h for SLSs in Example 2 (top) and Example 3 (bottom). By our hypothesis, H k,ε (A i z) ( + ε)h k (A i z). Thus, H k,ε (z) z 2 + ( + ε)h k (A i z), z R n, i M. Since Pk ε is ε-equivalent to P k ε, we then have, H k,ε (z) H k,ε (z)+ε z 2 ( + ε) [ z 2 + H k (A i z) ], for each i M. By (22), we have H k,ε (z) ( + ε)hk (z), z. That H k,ε (z) Hk (z) can be trivially proved. The estimated /h obtained by Algorithm 2 are plotted in Fig. 7 for the SLSs in Example 2 (top) and Example 3 (bottom). In both cases, /h decreases from at = to at =. (For Example 3, high computational complexity of Algorithm 2 prevents us from getting accurate estimates for close to ). We observe that in each case, /h is convex and more curved than the plots of /g. It is conjectured that the function /h is convex on [, ) for all SLSs. According to Propositions and 2, by choosing k large enough and ε small enough, Algorithm 2 can return estimates of H (z), hence, with an error as small as possible. See [44] for the expressions of the approximation error in terms of k, ε, and matrices A i 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 Pk ε, despite our relaxation effort. Our numerical experiments suggest that computing is no less challenging than computing the joint spectral subradius, which in itself is an NP-hard problem [2], [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. V. MEAN GENERATING FUNCTIONS The notion of generating functions can also be extended to the SLS (2) under the random switching probability p. Define its mean generating function F : R + R n R + { } as [ F(, z):=e t x(t; z, p) ]= 2 t E [ x(t; z, p) 2], (27) for z R n,. For each, F (z) := F(, z) is the averaged sum of the power series along the random system trajectory; thus, its value lies between two extremes: H (z) F (z) G (z), z R n. (28) The absence of maximum or minimum in the definition of F (z) makes its characterization much easier compared to G (z) and H (z). For example, F (z) is quadratic in z R n : [ t ] F (z) = z T E I + t A(s) T A(s) z. (29) t= s= s=t We write F (z) = z T Q z, where Q R n n is positive definite (with possibly infinite entries) and increasing in. The function F (z) has a similar set of properties as G (z). Proposition 3: F (z) has the following properties.. (Bellman Equation): Let be arbitrary. Then F (z) = z 2 + i M p i F (A i z), z R n. 2. (Sub-Additivity and Convexity): For any, F (z + z 2 ) F (z ) + F (z ), z, z 2 R n. As a result, F (z) is a convex function of z on R n. 3. (Invariant Subspace): For each, the set F := {z R n F (z) < } is a subspace of R n invariant under {A i } i M, where M := {i M p i > }. 4. For any, if F (z) is finite for all z, then F (z) c z 2 for some finite constant c. The proof of Proposition 3 is straightforward, hence omitted. The radius of convergence of F (z) is defined as p := sup{ F (z) <, z R n }, which depends on the probability distribution p. By (27), p is the { [ minimal radius ]} of convergence of the deterministic series E x(t; z, p) 2,,... over all z Rn. Theorem 5: The following statements are equivalent: ) The SLS (2) under the random switching probability p is mean square exponentially stable; 2) Its mean generating function F (z) has a radius of convergence p > ; 3) The mean generating function at =, F (z), is finite everywhere on R n. Proof: To show ) 2), suppose the SLS (2) is mean square exponentially stable, i.e., there exist κ > and r [, ) such that E [ x(t; z, p) 2] κr t z 2, t, z. Then, F (z) t κr t z 2 κ r z 2 <, z R n, whenever < r. Consequently, p r >. That 2) 3) is obvious. Finally, to show 3) ), we note that F (z) = E [ x(t; z, p) 2] < implies that E [ x(t; z, p) 2] 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..].

13 IEEE-TAC FP--5 3 For [, p ), F ( ) is finite everywhere. Define f := sup{f (z) z = } <, [, p ). (3) Then, f = max (Q ), the largest eigenvalue of Q in (29). Proposition 4: For (, p), the function /( /f ) is nondecreasing and upper bounded by /( /f ) p. Proof: Let (, p ). For each z Rn, define the sequence w t := E [ x(t; z, p) 2], t =,,..., which satisfies [ w t+s t =E t x(s + t; z, p) ]=E[F 2 (x(s; z, p))] E [ f x(s; z, p) 2] = f w s, s =,,.... Applying Lemma 3 with β = f, we conclude that F (z) = w t t is finite for [, ), where := /( /f ). Therefore, p, which is the second conclusion. That /( /f ) is nondecreasing is proved in exactly the same way as in Proposition 9 (with h replaced by f ). As a result, we have the following estimate. Corollary 8: For each [, p ), /f / p. Hence, /f and f as p. We next compute the directional derivative of f at =. Lemma 6: The directional derivative of f( at exists and ) is given by f ( f f +) := lim = max p i A T i A i. i M Proof: Let z S n be arbitrary. For small >, we can use (27) to write F (z) = z ( T i M p ia T i A i) z + O(), where O() is uniform in z. Therefore, we can exchange the order of the limit and supremum below to obtain z = f ( F (z) +) = lim sup = sup z = z = F (z) lim, which is exactly sup z = z ( T i M p ia T i A i) z. Similar to Corollary 6 and Proposition, we can show the following two results. ( ) Corollary 9: max p i A T i f A i, [, p ). i M Proposition 5: The function /f defined on [, p) is strictly decreasing and Lipschitz continuous with Lipschitz constant max (i M p ia T i A i). Hence, the function f is strictly increasing and locally Lipschitz continuous on [, p). From (28), we have h f g, thus /g /f /h. The Lipschitz constants of the functions /h, /f, and /g in Propositions, ( 5, and 4, respectively, ) satisfy sup min A iz 2 max p i A T i A i max A i 2. i M i M i M By setting the probability distribution p to be p i = for i = arg max i M A i 2 and p i = for i 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 {A i } i M and a constant γ > such that: (i) min i M z T A T i A iz γ z 2, z R n ; (ii) there exists no p such that i M p ia T i A i γi. Then for all p, sup z = min i M A i z 2 γ < max (i M p ia T i A i). L g : max i M A i 2 L f : inf p max ( i M p i A T i A i ) L h : sup z = min i M A i z 2 /g /f Fig. 8. Plot of the function /f. p /h A general plot of the function /f is shown in Fig. 8. The graph of /f is sandwiched between those of /g and /h. By Proposition 4, the ray emitting from (, ) and passing through (, /f ) rotates counterclockwise monotonically as increases. The discussions in the preceding paragraph further imply that the graph of /f leaves (, ) along a direction within the shaded conic region bounded by the lines L g and L f ; whereas generally a gap exists between L f and the asymptotic direction L h in which /h leaves (, ). Algorithms based on the Bellman equation can also be devised to compute F (z), hence p. The details are omitted here. VI. CONCLUSION 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. APPENDIX A PROOF OF LEMMA 3 Proof: Write w t+s t = w s+ w t+s+ t. Since by assumption the left hand side is at most βw s, we have w t+s+ t β w s, s =,,.... (3) Define the power series: W() := w t t, R. Denote by R W its radius of convergence. Since W( ) = w t t βw <, we have R W >. We cannot have R W = since the power series W() with nonnegative coefficients must have its radius of convergence as a singular point, at which W() diverges (see [47, Theorem 5.7.]). As a power series defines an analytic function within its radius of convergence [47], W() is analytic hence infinite time differentiable at. Its first order derivative at is W ( ) = tw t t = (t + )w t+ t = t= s= t w t+ t = s s= w t+s+ t.

14 IEEE-TAC FP--5 4 Applying (3), we obtain the estimate W ( ) β w s s β β w. (32) s= Similarly, the second order derivative W ( ) is W ( ) = (t + 2)(t + )w t+2 t = 2 s= t (s + )w t+2 t = 2 (s + ) s w t+s+2 t. s= Using (3) and (32), we can derive the estimate W ( ) 2 β ( ) 2 s= (s+)w s+ s = 2 β W ( ) 2β β w. By induction, we can show, for k =,, 2,..., ( ) k β W (k) ( ) k! β w. (33) Let W() := k! W (k) ( )( ) k be the Taylor series k= expansion of W() at. For and close to, W() = W() as W() is analytic at ; and by (33), ( ) k β W() β w ( ) k k= βw (β )(/ ), where the last inequality holds if (β )(/ ) <, or equivalently, if [, ) with defined in the lemma statement. Therefore, the power series W() centered at is convergent, and thus defines an analytic function, on [, ). The concatenation of W() and W() yields an analytic continuation of W() defined at least on [, ). Being a power series of nonnegative coefficients, W() does not admit any analytic continuation beyond its radius of convergence R W [47, Definition 5.7.]. Hence, R W. This implies in particular that any [, ) is within the radius of convergence of both W() and W() and as such we must have W() = W(), [, ). This together with the last inequality above proves the desired conclusion. APPENDIX B PROOF OF PROPOSITION 4 Proof: The monotonicity of g, hence of /g, is obvious as G (z) is nondecreasing in. For convexity, let = α + ( α) 2 for some, 2 [, ), α [, ]; and let z S n be such that G (z) = g. Since G (z) is a convex function of [, ) for any fixed z (it is the maximum of a family of convex functions of ), we have g = G (z) αg (z) + ( α)g 2 (z) αg + ( α)g 2. This proves the convexity of g and hence its semismoothness [39, Prop ]. Being the composition of two semismooth functions g and x /x, /g is also semismooth [39, Prop ]. Pick any, (, ) with <. Proposition 3 implies that /( /g ) /( /g ). Thus by Lemma 2, ( ) /g g g ( ) max i M A i 2, By Lemma 2, the above inequality also holds for =. This proves the Lipschitz continuity of /g, hence the local Lipschitz continuity of g, on [, ). Finally, by (), /g / for (, ). Thus, lim /g =. Consequently, lim g =. APPENDIX C PROOF OF LEMMA 5 Proof: Fix an arbitrary z S n and let > be small. Recalling the definition of H (z), we write H (z) = inf t x(t; z, ) 2 = inf t x(t; z, ) 2 = inf t x(t + ; z, ) 2. t= Since for all, x(t; z, ) 2 (max i M A i 2 ) t, t; and inf x(; z, ) 2 = min i M A i z 2, the above implies min A iz 2 H (z) min i M A iz 2 + O(), i M for some term O() uniform in and z. Thus, we can exchange the order of limit and supremum below to obtain: h ( + ) = lim h z = H (z) = lim sup z = H (z) = sup lim = sup min A iz 2. i M This completes the proof of Lemma 5. ACKNOWLEDGMENT z = The authors would like to thank the anonymous reviewers for their constructive suggestions and for pointing out many of the references of this paper. REFERENCES [] D. Liberzon, Switching in Systems and Control. Birkhäuser, 23. [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 , 29. [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 , 27. [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, 999, pp [5] M. Johansson and A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Trans. Autom. Control, vol. 43, no. 4, pp , 998. [6] J. Shen and J. Hu, Stability of discrete-time conewise linear inclusions and switched linear systems, in Proc. Amer. Control Conf., Baltimore, MD, 2, pp [7] M. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Autom. Control, vol. 43, no. 4, pp , 998. [8] T. Hu, L. Ma, and Z. Lin, Stabilization of switched systems via composite quadratic functions, IEEE Trans. Autom. Control, vol. 53, no., pp , 28. [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, 999, pp [] 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. 7 22, 999.

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Hu, Switched LQR problem in discrete time: Theory and algorithm, 2, Purdue Technical Report TR-ECE 9-3. Submitted to IEEE Trans. Autom. Control, 2. [45] I. Polik and T. Terlaky, A survey of the S-lemma, SIAM Rev., vol. 49, no. 3, pp , 27. [46] J. Hu, J. Shen, and W. Zhang, A generating function approach to the stability analysis of switched linear systems, in Proc. 3th ACM Int. Conf. Hybrid Systems: Comput. and Control. Stockholm, Sweden: ACM, 2, pp [47] E. Hille, Analytic Function Theory. Providence, Rhodes Island: AMS Chelsea Publishing, 959, vol. I. Jianghai Hu (S 99-M 4) received the B.E. degree in automatic control from Xi an Jiaotong University, P.R. China, in 994, and the M.A. degree in Mathematics and the Ph.D. degree in Electrical Engineering from the University of California, Berkeley, in 22 and 23, 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 -A 3-M 5) Jinglai Shen received his B.S.E. and M.S.E. degrees in automatic control from Beijing University of Aeronautics and Astronautics, China, in 994 and 997, respectively, and his Ph.D. degree in aerospace engineering from the University of Michigan, Ann Arbor, in 22. 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 7-M ) Wei Zhang received the B.E. degree in Automatic Control from the University of Science and Technology of China, Hefei, China, in 23, the M.S.E.E degree in Electrical Engineering from the University of Kentucky, Lexington, KY, in 25, and the PhD degree in Electrical Engineering from Purdue University, West Lafayette, IN, in 29. 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.