Lower bounds and dense discontinuity phenomena for the stabilizability radius of linear switched systems
aa r X i v : . [ m a t h . O C ] J un Lower bounds and dense discontinuity phenomena forthe stabilizability radius of linear switched systems
Carl P. Dettmann a , R. M. Jungers b , P. Mason c a School of Mathematics, University of Bristol, Fry Building, Woodland Road, BristolBS81UG, UK. b ICTEAM Institute, Universit´e catholique de Louvain, 4 avenue Georges Lemaitre, B-1348Louvain-la-Neuve, Belgium. R. J. is an F.R.S.-FNRS honorary research associate. c Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Laboratoire des signaux et syst`emes,91190, Gif-sur-Yvette, France.
Abstract
We investigate the stabilizability of discrete-time linear switched systems, whenthe sole control action of the controller is the switching signal, and when thecontroller has access to the state of the system in real time. Despite theirapparent simplicity, determining if such systems are stabilizable appears to bea very challenging problem, and basic examples have been known for long, forwhich the stabilizability question is open.We provide new results allowing us to bound the so-called stabilizabilityradius, which characterizes the stabilizability property of discrete-time linearswitched systems. These results allow us to compute significantly improvedexplicit lower bounds on the stabilizability radius for the above-mentioned ex-amples. As a by-product, we exhibit a discontinuity property for this problem,which brings theoretical understanding of its complexity.
Keywords:
Switched systems, stabilizability, joint spectral characteristics
1. Introduction
Joint spectral characteristics are numerical quantities that describe the asymp-totic behaviour of matrix semigroups. They have found many applications, inparticular in Systems and Control.Consider a finite set of m matrices M ∈ R n × n , and the corresponding lineardiscrete time switching systems, which is a system whose behaviour follows thefollowing law: x ( k + 1) = A σ k x ( k ) σ k ∈ { , . . . , m } . (1) Email addresses: [email protected] (Carl P. Dettmann), [email protected] (R. M. Jungers), [email protected] (P.Mason)
Preprint submitted to Elsevier June 17, 2020 hese systems are not uniquely defined, but any ‘switching signal’ σ implies awell defined law of evolution for the system. The joint spectral characteristicshave emerged quite independently during the second half of the 20th century,with the goal of characterizing the rate of growth of System (1) for some possibleswitching signal. These quantities have attracted a lot of attention, not onlybecause of their applications, but probably also because, despite the apparentsimplicity of their definition, they turn out to be extremely hard to compute.See for instance [1, 2] for typical complexity results on the topic.The first quantity, and perhaps the most well-known, was introduced in thecontext of robust control, and represents the worst case rate of growth of aswitching system: ρ ∞ ( M ) = lim k →∞ max A ∈M k {|| A || /k } . It is commonly referred to as the
Joint Spectral Radius (JSR in short) of the set M . It has been introduced by Rota and Strang [3]. See [4] for a monograph onthe topic. Since then, several other quantities were proposed, in order to describeother possible rates of growth of the system. Let us mention the p-radius, [5, 6]with motivations in mathematical analysis; the
Lyapunov exponent (see [7, 8])with motivations in randomly switching systems; or the
Joint Spectral Subradius ,which represents the minimal rate of growth for the evolution operator of System(1) (see [9, 10]).In this paper, we are concerned with the stabilizability radius , which, simi-larly to the subradius, is also related to the smallest possible rate of growth overall switching signals, but now it is assumed that one can choose the matrix se-quence depending on the initial condition x (0) . The stabilizability radius is thussmaller than the previously introduced subradius. It has only been introducedformally recently [11], but the reader can find earlier implicit studies of it in[12 ? , 13].Following [11], we introduce the following definition, which is the main topicof study of the present work: Definition 1.
The stabilizability radius of M is defined as ˜ ρ ( M ) = sup x ∈ R n ˜ ρ x ( M ) , where ˜ ρ x ( M ) , inf (cid:26) λ ≥ (cid:12)(cid:12)(cid:12) ∃ x ( · ) solution of (1) with initial point x and M > s.t. | x ( k ) | ≤ M λ k | x | for any t ≥ (cid:27) minimizes the exponential growth rate of the trajectories of (1) starting at x .As shown in [11] we may equivalently write ˜ ρ ( M ) = inf (cid:26) λ ≥ (cid:12)(cid:12)(cid:12) ∃ M > s.t. | x ( k ) | ≤ M λ k | x | for any x ∈ R n ,t ≥ and for some solution x ( · ) of (1) starting at x (cid:27) . The stabilizability radius is related to the possibility of stabilizing (1) byappropriately choosing the switching law, either in open loop form or in feedback2orm, see [11, Proposition 2.5 and Corollary 3.4]. Note that the systems ofthe form (1) represent a special class of the systems considered in nonlinearcontrol [14] and of variable structure systems [ ? ], where stabilization issuesplay a crucial role. Hence the study of the stabilizability radius is an importantstep to understand the complexity of the stabilization problem in a more generalcontext than the one considered in this paper.We recall the following basic properties of the stabilizability radius: Proposition 1.
The stabilizability radius satisfies the following basic properties:(i) Homogeneity: For any compact set of matrices M , ∀ γ > , ˜ ρ ( γ M ) = γ ˜ ρ ( M ) ,(ii) For any compact set of matrices M , ∀ k ∈ N , ˜ ρ ( M k ) = ˜ ρ ( M ) k , where M k denotes the set of k products of matrices in M . We illustrate the above concept with an example, which we will use as arunning example throughout this paper:
Example 1. [Based on an example by Stanford and Urbano [13]] Let us consider M = { A , A } , where A = (cid:18) cos π sin π − sin π cos π (cid:19) = √ (cid:18) − (cid:19) , A = (cid:18)
00 2 (cid:19) . It is easy to see that the norm of any product of matrices in M is larger thanor equal to one. Indeed, det( A σ ( k ) . . . A σ (0) ) = 1 independently on the switchingsequence, which implies that k A σ ( k ) . . . A σ (0) k ≥ . However, the definition of thestabilizability radius allows the switching sequence to depend on the value of x (0) , so that the stabilizability radius can be smaller than one, and it is the case in ourexample. Indeed, for any value of x ∈ R there always exists a natural number n x ≤ such that the absolute value of the angle formed by the vector A n x x andthe x axis is smaller than or equal to π/ . As a consequence it is easy to obtainthe estimate | A A n x x | < . | x | . Hence, starting from any initial condition x (0) we can easily construct recursively a switching sequence in such a way that thecorresponding solution x ( k ) of System (1) satisfies | x ( k ) | ≤ × . k/ | x (0) | ,which implies ˜ ρ ( M ) < . / ∼ . . In [11, Theorem 4.7] it was shown that the minimum singular value computedamong all matrices of M provides a lower bound for ˜ ρ ( M ) : Theorem 1. [11, Theorem 4.7] One has ˜ ρ ( M ) ≥ min A ∈M σ m ( A ) where σ m ( A ) is the smallest singular value of the matrix A . However, the study of Example 1 seems to suggest that such a lower bounddoes not represent a good approximation of the actual value of the stabilizabilityradius:
Example 2. (Example 1, continued.) A simple application of Theorem 1 tothe matrices from Example 1 gives ˜ ρ ( M ) ≥ / , and applying the same resultto M k , together with Item ( ii ) in Proposition 1, does not improve the bound. Is it possibleto improve Theorem 1 and provide a generally better formula for a lower bound?In particular, how can one compute a better lower bound on ˜ ρ ( M ) in Example1? Some techniques have been proposed in the control literature, which allow toderive an upper bound on the stabilizability radius, mainly based on semidefiniteprogramming [15, 12, 16]. However, it seems much harder to provide a tightlower bound. In this manuscript we tackle the above question. By a closerinspection at all the singular values of a matrix product, we provide a muchbetter lower bound, which can be improved by increasing the length of theproducts. In particular, we improve the lower bound previously obtained byapplying Theorem 1 to Example 1. Then, in Section 3, we show that the presentresult is actually general and has rather nonintuitive consequences concerningthe regularity of the radius ˜ ρ x in terms of the initial condition x . Notation:
For a matrix M ∈ R n × n we denote by s ( M ) ≤ s ( M ) · · · ≤ s n ( M ) the corresponding singular values. The sphere in R n is S n − = { x ∈ R n | k x k = 1 } . Finally, we denote by clos( A ) the closure of a subset A of atopological space.
2. Lower bound for the stabilizability radius
In this section we first provide a simple and actionable lower bound on thestabilizability radius, based on the determinants of the matrices in M . We thenprovide a more powerful bound, which is in some sense less actionable becauseit relies on more involved computations. Theorem 1 provides a simple lowerbound on the stabilizability radius in terms of the smallest singular value of thematrices. We start with a simple lemma that pushes this reasoning further, bypointing out a geometric property of a given matrix related both to its smallestsingular value and to its determinant.
Lemma 1.
Let n ≥ . Given a matrix A ∈ R n × n and r > we define S r,A = { x ∈ S n − | k Ax k ≤ r } . Then • S r,A is empty if s ( A ) > r , • for every nonsingular matrix A ∈ R n × n , and if s ( A ) ≤ r , the measure of S r,A is bounded by m n − min { r n | det A | − , } , where m n − is the surfacearea of the unit sphere.Proof. The fact that the set S r,A is empty if s ( A ) > r follows from the defini-tions. In order to prove the second part of the lemma we define the cone C r,A = { x ∈ R n | k x k ≤ , x/ k x k ∈ S r,A } and consider the image of C r,A by the matrix A . We know that the correspond-ing volumes scale by a factor | det( A ) | and that the image of C r,A is completelycontained inside the closed ball of radius r . We deduce that V r,A ≤ M n min { r n | det( A ) | − , } V r,A is the volume of C r,A and M n denotes the volume of the unit ball.The desired bound is then obtained by observing that the surface area of S r,A is equal to V r,A m n − /M n . Remark 1.
It is possible to show that the surface area of S r,A is bounded by πr ) n − s ( A ) | det A | − whenever A is nonsingular and s ( A ) ≤ r . This im-proves the estimate provided by Lemma 1 in the case s ( A ) ≪ r ≤ | det( A ) | /n .We omit the proof of this result as it is more involved than that of Lemma 1and the improved estimate does not allow to enhance the later results. Below we exploit the previous lemma in order to provide a first lower boundto ˜ ρ ( M ). Roughly speaking, the idea is that for any λ > ˜ ρ ( M ) and any x ∈ S n − there should exist a matrix A in M k , for k large enough, such that Ax belongs to the ball of radius λ k . In other words, in the notations of the previouslemma, the union of all the sets S λ k ,A for A ∈ M k must cover the whole sphere S n − . Theorem 2.
Consider System (1) and assume that M only contains nonsin-gular matrices. Then, the stabilizing radius satisfies ˜ ρ ( M ) ≥ ˜ ρ − , (cid:16) m X h =1 | det A h | − (cid:17) − /n . (2) Proof.
Let us fix λ = ρ + ǫ for some ǫ > , and fix some T ∈ N . By Lemma 1, forany product A ∈ M T , the set S r,A of unit vectors which are mapped inside theball of radius r by A has measure bounded by m n − r n | det( A ) | − , where m n − is the surface area of the unit sphere. This implies that the set ∪ A ∈M T S r,A hasmeasure bounded by m n − r n X A ∈M T | det( A ) | − = m n − r n X σ ∈{ ,...,m } T Y i =1 ,...,T | det( A σ i ) | − = m n − r n (cid:16) m X h =1 | det( A h ) | − (cid:17) T . Now, by definition, for any λ > ˜ ρ ( M ) there exists a positive constant C suchthat any x ∈ S n − may be mapped to a ball of radius Cλ T by at least one prod-uct of T matrices in M . Setting r = Cλ T , we deduce that the set ∪ A ∈M T S r,A must cover the whole sphere S n − , so that m n − C n λ nT (cid:16) m X h =1 | det( A h ) | − (cid:17) T ≥ m n − . Letting T tend to infinity we get λ ≥ (cid:16) m X h =1 | det( A h ) | − (cid:17) − /n , and the thesis follows since, by definition, λ = ˜ ρ + ǫ for an arbitrary small ǫ. xample 3. (Example 1, continued.) Despite the strikingly simple idea leadingto it, the previous result allows us to provide an answer to [11, Open Question 2].Indeed, let us consider again the set M in Example 1. One has that ˜ ρ − = √ / ,considerably improving the lower bound min { s ( A ) , s ( A ) } = 1 / obtainedapplying [11, Theorem 4.7]. It is worth noticing that, for an arbitrary given set M , Theorem 2 mightnot necessarily improve on the simple lower bound from Theorem 1: ˜ ρ ( M ) ≥ min A ∈M s ( A ) . (Indeed consider for instance a trivial example with two rotationmatrices. For such an example we have min A s ( A ) = 1 and ˜ ρ − = √ / . )Intuitively, on the one hand Theorem 1 estimates the maximal norm contractionat each step, among all available matrices and all initial conditions, but it doesnot take into account the fact that trajectories may move away from the mostcontracting directions; on the other hand, Theorem 2 is essentially based onthe assumption of a homogeneous occupation measure, and it does not exploitthe possible presence of privileged directions or modes which may be used foroptimizing the contraction rate. The result below mingles the two approaches. Theorem 3.
Consider System (1) and assume that M only contains nonsin-gular matrices. Let us denote, for simplicity, δ i = s ( A i ) and ∆ i = | det A i | .Consider the simplex Σ m defined as Σ m = { ν ∈ [0 , m | m X h =1 ν h = 1 } , the map Ψ : Σ m → R , Ψ( ν ) = m X h =1 ν h log (cid:18) ν h ∆ h δ nh (cid:19) and the element ¯ ν ∈ Σ m whose components are defined by ¯ ν h = ∆ − h P mj =1 ∆ − j . Thenwe have the following alternative:(a) If Ψ(¯ ν ) ≥ , then ˜ ρ ( M ) ≥ ˜ ρ − ≥ min h =1 ,...,m δ h .(b) If Ψ(¯ ν ) < , then the set Z = { ν ∈ Σ m | Ψ( ν ) = 0 } is nonempty and,setting ˜ ρ ∗− , min ν ∈Z m Y h =1 δ ν h h , (3) we have ˜ ρ ∗− ≥ min h =1 ,...,m δ h and ˜ ρ ( M ) ≥ ˜ ρ ∗− > ˜ ρ − . (4) Moreover, if δ i = δ j for every i = j , the argument of the minimum in (3) takes the form ˆ ν h ( β ) = ( if h / ∈ S δ βh ∆ − h P j ∈ S δ βj ∆ − j if h ∈ S (5)6 or some real value β and S ⊆ { , . . . , m } . As a consequence, ˜ ρ ∗− may becalculated numerically by solving the scalar equations Ψ(ˆ ν ( β ) , . . . , ˆ ν m ( β )) =0 obtained for all possible S ⊂ { , . . . , m } .Proof. If A is a product of length T containing n h copies of A h then | det A | = Q mh =1 ∆ n h h . By Lemma 1, the portion of the unit sphere that is mapped into aball of radius r has measure bounded by m n − r n | det A | − = m n − r n Q mh =1 ∆ − n h h .Moreover one has that P mh =1 n h log δ h = log( Q mh =1 δ n h h ) ≤ log( s ( A )) so that,applying the first item in Lemma 1, it follows that S r,A is empty whenever P mh =1 n h log δ h > log r . Thus, we obtain the following upper bound on the mea-sure of the union of all sets S r,A among all possible products of length T ofmatrices of M U ( T ) = m n − r n X T ! Q h n h ! 1 Q h ∆ n h h = m n − r n X T ! Q h ( n h !∆ n h h ) , (6)where in the last two equalities the sum is taken over all m -tuples of positiveintegers satisfying m X h =1 n h log δ h ≤ log r, m X h =1 n h = T. (7)Noticing that Q h ( n h !∆ n h h ) = Q h,n h =0 ( n h !∆ n h h ) , we can apply Stirling approxi-mation N ! ≈ N N + e − N , N > c ˜ U ( T ) ≤ U ( T ) ≤ c ˜ U ( T ) for some positive numbers c , c only depending on m, n , where˜ U ( T ) = r n X T Q h,n h =0 n h ! T T Q h,n h =0 ( n h ∆ h ) n h = r n X T Q h,n h =0 n h ! Y h,n h =0 (cid:16) n h T ∆ h (cid:17) − n h . Since, for any
T > T Q h,nh =0 n h ≤ T max h n h ≤ m , the expression on the right isbounded by √ mr n X Y h (cid:16) n h T ∆ h (cid:17) − n h ≤ √ mr n ( T + 1) m − max Y h (cid:16) n h T ∆ h (cid:17) − n h where we estimate the number of elements in the summation by (cid:0) T + m − m − (cid:1) ≤ ( T + 1) m − , and the maximum is taken over all m -tuples of positive integerssatisfying (7).In particular, by replacing each n h T with a continuous variable ν h , the subsetof the unit sphere which can be mapped into the ball of radius ˆ Cρ T has measure7ounded by E ( ρ, ˆ C, T ) , C ˆ C n ρ nT ( T + 1) m − max ν Y h ( ν h ∆ h ) − ν h ! T (8)for some C >
0, where the ν h ’s satisfy m X h =1 ν h = 1 , where ν h ≥ ∀ h, (9a) m X h =1 ν h log δ h ≤ log ρ. (9b)Note that the constraint (9a) corresponds to ν ∈ Σ m . Whenever ρ > ˜ ρ ( M ) andfor some ˆ C large enough, E ( ρ, ˆ C, T ) must necessarily be larger than the measureof the ( n − T >
0. In particular,setting u ( ρ ) , lim T →∞ T log E ( ρ, ˆ C, T ) = n log ρ − min ν X h ν h log( ν h ∆ h ) ,ρ > ˜ ρ ( M ) implies u ( ρ ) ≥
0. Since u is strictly increasing, we actually havethat ρ > ˜ ρ ( M ) implies u ( ρ ) >
0. As a consequence, if u ( ρ ) ≤ ρ ≤ ˜ ρ ( M ) and the problem of finding the maximum ρ satisfying u ( ρ ) ≤ ρ ( M ). Note that u ( ρ ) ≤ ν h among which such a maximummust be seek satisfy m X h =1 ν h log δ h ≤ log ρ ≤ n m X h =1 ν h log( ν h ∆ h ) , and, as a consequence,Ψ( ν ) = m X h =1 ν h log( ν h ∆ h ) − n m X h =1 ν h log δ h ≥ . (10)Note that (10) is always satisfied at the vertices of the simplex Σ m , with equalityif the corresponding matrix in M is proportional to an orthogonal matrix, thatis, ∆ h = δ nh .From what precedes a lower bound ρ ∗ for ˜ ρ ( M ) should satisfy the followingminimization problem:find ρ ∗ , min ν e n Φ( ν ) subject to (9a)-(10) (11)where Φ( ν ) , m X h =1 ν h log( ν h ∆ h ) .
8n particular Φ is convex, as it is the sum of convex functions of a singlereal variable. Also, Φ( ν ) = Ψ( ν ) + n P mh =1 ν h log δ h ≥ n P mh =1 ν h log δ h ≥ n min h =1 ,...,m log δ h , which implies ρ ∗ ≥ min h =1 ,...,m δ h . Consider now the problem of minimizing Φ under the sole condition (9a),i.e. on the simplex Σ m . Since ∂ Φ ∂ν h tends to −∞ if ν h goes to 0, the minimum ofΦ is not attained at the boundary of Σ m and can therefore be computed usingLagrange multipliers. In particular the value ¯ ν ∈ Σ m minimizing Φ is given by¯ ν h = ∆ − h P j ∆ − j , and it is easy to see that e n Φ(¯ ν ) = ˜ ρ − . Therefore, if Ψ(¯ ν ) ≥ ρ ∗ = ˜ ρ − in (11), concluding the proof of Item (a).Assume now that Ψ(¯ ν ) <
0. We claim that the minimum in the definitionof ρ ∗ is attained when Ψ is equal to 0. Indeed, by continuity of Ψ, for any ν satisfying (10) there exists a convex combination ν λ = λν + (1 − λ )¯ ν suchthat Ψ( ν λ ) = 0. Moreover Φ( ν λ ) ≤ Φ( ν ) (with equality only if ν = ν λ ) byconvexity of Φ and since Φ(¯ ν ) < Φ( ν ). Thus, without loss of generality, inthe problem (11) one may replace the constraint (10) with Ψ = 0, that is wecan minimize Φ restricted to the subset Z . We observe that Φ( ν ) = Ψ( ν ) + n P mh =1 ν h log δ h = n P mh =1 ν h log δ h for ν ∈ Z , from which we deduce that ρ ∗ = ˜ ρ ∗− = min ν ∈Z Q mh =1 δ ν h h . This proves the first inequality in (4). The laststrict inequality in (4) is a consequence of the uniqueness of the minimizer ¯ ν obtained with the sole constraint (9a).Concerning the last part of the theorem, if the minimum of Φ restricted to Z is attained in the interior of Σ m then it can be computed by using Lagrangemultipliers. In particular, if the values δ i are all different, one finds that ν h = αδ βh ∆ − h , where α, β depend on the parameters δ i , ∆ i , i = 1 , . . . , m . Since ν ∈ Σ m , weget α = α ( β ) = ( P h δ βh ∆ − h ) − , so that, setting ˆ ν h ( β ) = α ( β ) δ βh ∆ − h the value β may be found numerically by solving the equationΨ(ˆ ν ( β )) = 0 . If the minimum of Φ restricted to Z is attained at the boundary of Σ m theneither it is attained at one vertex of Σ m , or in the interior of a subsimplex { ν ∈ Σ m | ν h = 0 , ∀ h / ∈ S } , for some S ⊂ { , . . . , m } . In the latter case itminimizes the restriction of Φ to that subsimplex under the constraint Ψ = 0 andone can again find the minimizer by means of Lagrange multipliers, obtainingthat min ν ∈Z Φ( ν ) is attained at a point ˆ ν of the form (5). This concludes theproof of Item (b). 9 emark 2. Theorem 3 shows that the inequality ˜ ρ ( M ) ≥ min h =1 ,...,m δ h (firstprovided in [11, Theorem 4.7]) is actually strict, except for very special cases.In particular the strict inequality holds if min h =1 ,...,m δ h is attained only for asingle index h = ¯ h and A ¯ h is not proportional to an orthogonal matrix (that is,if ∆ ¯ h > δ n ¯ h ). On the other hand, the lower bound obtained in Theorem 3 in thecase ( a ) coincides with the one obtained in Theorem 2. Remark 3.
There are at least two simple ways to possibly improve the lowerbound in Theorem 3: • Unlike ˜ ρ ( M ) , the value ˜ ρ ∗− in Theorem 3 may actually vary if one per-forms a linear coordinate transformation (common to each A ∈ M ), as thesingular values are not invariant with respect to linear coordinate transfor-mations. Therefore one can consider the problem of optimizing the lowerbound by coordinate changes. • We have ˜ ρ ( M ) = ˜ ρ ( M k ) /k (see Proposition 1). In particular, computinga lower bound for ˜ ρ ( M k ) for k > by means of Theorem 3 may lead tobetter estimate of ˜ ρ ( M ) compared to a direct application of Theorem 3 tothe set M .We show below through a simple example that Theorem 3 may strictly increasethe lower bound on ˜ ρ ( M ) . Whether the iteration of such a method leads asymp-totically to the actual value ˜ ρ ( M ) remains an open problem. Example 4.
To illustrate the previous result we consider a set of matrices M = { A , A } where A = diag( c, c − ) with c ∈ (0 , , and A is a two-by-two orthogonal matrix. In particular the matrices in Example 1 satisfy suchassumptions with c = 1 / . In the notation of Theorem 3 we have δ = c, δ = 1 and ∆ = ∆ = 1 . Moreover, ¯ ν = ¯ ν = 1 / and Ψ(¯ ν ) = log − log c .Thus, for c ∈ (0 , / we fall into case ( a ) of the theorem; the lower bound ˜ ρ − = 1 / √ provided by both Theorem 2 and Theorem 3 improves the value δ min , min { δ , δ } = c of [11, Theorem 4.7]. On the other hand, if c ∈ (1 / , we fall into case ( b ) of Theorem 3, and the lower bound ˜ ρ ∗− is strictly largerthan both ˜ ρ − and δ min . In this case it is easy to see that the minimum isattained at the interior of Σ and that it is associated with the unique solutionof the equation Ψ(ˆ ν ( β ) , . . . , ˆ ν m ( β )) = 0 , obtained for S = { , } , which maybe easily found numerically. In Table 1 we collect the lower bounds δ min , ˜ ρ − , ˜ ρ ∗− for different values of c ∈ (0 , .
3. Dependence of the stabilizability radius on the initial condition
We consider now an application of Theorem 3. We are interested in studyingthe dependence of ˜ ρ x ( M ) on the initial condition x . In general, one cannotexpect this function to be everywhere continuous. For instance, if M is made ofa single matrix A = diag { λ , . . . , λ n } , then the image of such a function is equalto {| λ | , . . . , | λ n |} and ˜ ρ x ( M ) = | λ i | if ( x ) i = 0 and ( x ) j = 0 for all j such10 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 δ min ρ − ρ ∗− - - - Table 1: Computation of δ min , ˜ ρ − and ˜ ρ ∗− in terms of the parameter c appearing in the set M of Example 4. Italic numbers represent the best bound. that | λ i | ≤ | λ j | . In particular ˜ ρ x ( M ) is discontinuous at any point x with zerocomponent along the eigenspaces corresponding to the eigenvalues of maximumabsolute value. We show below a much more surprising result, which entails theexistence of linear switched systems such that ˜ ρ x ( M ) is nowhere continuous.Namely, we provide general conditions on the linear switched system ensuringthe existence of an open set A ⊂ R n and two disjoint subsets U, S both densein A , such that ˜ ρ x ( M ) ≥ c for x ∈ U and ˜ ρ x ( M ) ≤ c for x ∈ S for some c > c > M by a common constant,from each point of S it is possible to stabilize exponentially the system and fromeach point of U it is impossible to stabilize the system.We start with the following definitions which adapt classical notions fromcontinuous-time controlled dynamical systems. Definition 2.
Consider the discrete-time switched system on the projectivespace P n − (for simplicity, we identify here a point of P n − with a pair of op-posite points z, − z in S n − ) z ( k + 1) = B σ k z ( k ) | B σ k z ( k ) | , B σ k ∈ N ⊂ R n × n , (12) and denote the attainable set in positive time O + ( z (0)) from z (0) as the set of allpoints that can be reached from z (0) for all k ≥ and switching signals σ , that is O + ( z (0)) = (cid:8) Bz (0) | Bz (0) | | B ∈ N k , k ≥ (cid:9) . If N is made by nonsingular matrices,we also define the attainable set in negative time from z (0) as O − ( z (0)) = (cid:8) B − z (0) | B − z (0) | | B ∈ N k , k ≥ (cid:9) . We have the following general result concerning the stabilizability rate ˜ ρ x ( M ). Proposition 2.
Consider System (1) and assume that M = { A , . . . , A m } ⊂ R n × n only contains nonsingular matrices. Consider the projected switched sys-tem on P n − z ( k + 1) = A σ k z ( k ) | A σ k z ( k ) | , A σ k ∈ M . (13) Assume that there exist two points z (1) , z (2) ∈ P n − ⊂ R n such that ˜ ρ z (1) ( M ) < ˜ ρ z (2) ( M ) . Then the function x ˜ ρ x ( M ) is discontinuous at every point of thecone D = { x ∈ R n : x = λ (clos( O − ( z (1) )) ∩ clos( O + ( z (2) ))) , λ > } . roof. Clearly ˜ ρ z ( M ) ≤ ˜ ρ z (1) ( M ) if z ∈ O − ( z (1) ) and ˜ ρ z ( M ) ≥ ˜ ρ z (2) ( M ) if z ∈ O + ( z (2) ). The conclusion follows since both O − ( z (1) ) and O + ( z (2) ) aredense in D .Of course the previous result is of some interest only if the set D is nonempty.This is the case under some (approximate) controllability property of the system.For instance, if n = 2 and if there exists k ∈ N and A ∈ M k with nonrealeigenvalues such that A h is not proportional to the identity for every integer h = 0 then clos( O + ( z )) = clos( O − ( z )) = P for any z ∈ P . Indeed, in this case A is similar to a multiple of a rotation matrix whose corresponding attainablesets are dense in P . Furthermore, Theorem 3 allows to determine some setsof matrices for which the existence of z (1) , z (2) such that ˜ ρ z (1) ( M ) < ˜ ρ z (2) ( M ),as required in Proposition 2, is satisfied. An example of application, whichimmediately follows from the discussion above and from Remark 2, is given asfollows. Proposition 3.
Let n = 2 and assume that there exists A ∈ ∪ k ∈ N M k withnonreal eigenvalues such that A l is not proportional to the identity for everynonzero integer l and that there is a single index ¯ h satisfying min h =1 ,...,m δ h = δ ¯ h , with A ¯ h diagonal and not proportional to the identity. Then ˜ ρ x ( M ) isdiscontinuous at each point of R . It is easy to see that the matrices in Example 1 satisfy the assumptions ofProposition 3. Indeed, the matrix A A is similar to a rotation matrix of angle θ , with θ incommensurable with π (see [11] for more details). An even simplerapplication of Proposition 3 is obtained if one directly replaces the rotationangle π in the matrix A of Example 1 with any angle incommensurable with π . A further numerical example is given below. Example 5.
We consider the set of matrices M = { A , A , A } where A = (cid:18) − − (cid:19) , A = (cid:18) − . (cid:19) , A = (cid:18) − − − (cid:19) . The assumptions of Proposition 3 are satisfied since A has nonreal eigenvalueswhich are not proportional to roots of the unit while the minimum singular valueis equal to . and is associated with the diagonal matrix A . An application ofTheorem 3 gives the lower bounds ˜ ρ − = 1 . and ˜ ρ ∗− = 1 . for ˜ ρ ( M ) . As aconsequence, there exists a dense subset of R starting from which it is possibleto stabilize exponentially the system (with the exponential rate ˜ ρ x ( M ) = 0 . )and another dense subset starting from which it is not possible to stabilize thesystem.
4. Conclusion
In this paper, we have studied the stabilizability radius of linear switchedsystems. Even though such systems are well known to be extremely complex to12nalyse, we believe that the stabilizability radius exhibits particularly complexphenomena (see for instance Example 5), and on the other hand it has been thetopic of very little study in the literature. As an example of this, no methodwas available in order to provide a nontrivial lower bound on the stabilizabilityradius of the matrices in Example 1, even though they were introduced morethan 25 years ago.Our lower bounds provide a useful complement to previously available meth-ods, which offer upper bounds (that is, sufficient conditions for stabilizability).Indeed, when the lower bound is larger than one, one can directly deduce in-feasability of the sufficient conditions.We have provided two results allowing to improve these lower bounds. Inparticular, Theorem 3 provides a lower bound that can be refined by simplyiteratively computing longer products of the matrices in the studied set. Weleave open the question of whether this procedure leads to the true value of thestabilizability radius (as is the case for other classical algorithms allowing tocompute other joint spectral characteristics). In Section 3, we provide a moretheoretical analysis, showing that complex discontinuity phenomena occur, evenfor quite simple examples.From a control-theoretic perspective, we believe that the problem studiedhere is of high importance in the context of formal methods and cyber-physicalsystems control, where the set of control actions available to the controller isoften made of a discrete set. We hope that the present research sheds some lighton the complexity of the phenomena at stake, and that it will motivate furtherresearch in that direction.
Acknowledgement
This work was supported by the Engineering and Physical Sciences ResearchCouncil, grant EP/N002458/1, by the FNRS, the Innoviris Foundation and theWalloon Region, and by the iCODE institute, research project of the Idex Paris-Saclay. No new data were created in this study.The authors would like to thank J. Ouaknine for organising the Bellairsworkshop Algorithmic Aspects of Dynamical Systems, (Barbados, March 2017)where this work was initiated.
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