Chance-constrained quasi-convex optimization with application to data-driven switched systems control
PProceedings of Machine Learning Research vol xxx:1–12, 2021
Chance-constrained quasi-convex optimization with application todata-driven switched systems control
Guillaume O. Berger
GUILLAUME . BERGER @ UCLOUVAIN . BE Rapha¨el M. Jungers
RAPHAEL . JUNGERS @ UCLOUVAIN . BE Zheming Wang
ZHEMING . WANG @ UCLOUVAIN . BE Institute of Information and Communication Technologies, Electronics and Applied Mathematics, Depart-ment of Mathematical Engineering (ICTEAM/INMA) at UCLouvain, 1348 Louvain-la-Neuve, Belgium
Abstract
We study quasi-convex optimization problems, where only a subset of the constraints can be sam-pled, and yet one would like a probabilistic guarantee on the obtained solution with respect to theinitial (unknown) optimization problem. Even though our results are partly applicable to generalquasi-convex problems, in this work we introduce and study a particular subclass, which we call“quasi-linear problems”. We provide optimality conditions for these problems. Thriving on this, weextend the approach of chance-constrained convex optimization to quasi-linear optimization prob-lems. Finally, we show that this approach is useful for the stability analysis of black-box switchedlinear systems, from a finite set of sampled trajectories. It allows us to compute probabilistic upperbounds on the JSR of a large class of switched linear systems.
Keywords:
Data-driven control, chance-constrained optimization, quasi-convex programming,switched systems.
1. Introduction
Data-driven control has gained a lot of interest from the control community in recent years; see,e.g., Duggirala et al. (2013); Huang and Mitra (2014); Blanchini et al. (2016); Kozarev et al. (2016);Balkan et al. (2017); Boczar et al. (2018). In many modern applications of control systems, one can-not rely on having a model of the system, but rather has to design a controller in a blackbox , data-driven fashion. This is the case for instance for proprietary systems; more usually, this happensbecause the system is too complex to be modeled, or because the obtained model is too complicatedto be analyzed with classical control techniques. In these situations, the control engineer can onlyrely on data — which sometimes come in huge amounts —, but make the problem of very differ-ent nature than the classical, model-based control problems. Examples of such situations includeself-driving cars, where the input to the controller is made of huge heterogeneous data (harvestedfrom cameras, lidars, etc.); or smart grid applications, where the heterogeneous parts of the system(prosumers, smart buildings, etc.) are best described with data harvested from observing these partsthan with a rigid, closed-form model (Aswani et al., 2012; Zhou et al., 2017).Data collected from a control system can be seen as samples extracted from a large set of pos-sible behaviors. Controller design can then be approached by synthesizing controllers based onthe sampled set of behaviors; the challenge is then to provide guarantees on the correctness of thecontroller for the whole behavior of the system. In optimization, this approach is known as chance-constrained optimization , which consists in sampling a subset of the constraints of an optimization © 2021 G.O. Berger, R.M. Jungers & Z. Wang. a r X i v : . [ m a t h . O C ] J a n HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION problem and solving the problem with these constraints only. The solution obtained in this way willin general not satisfy all of the constraints of the problem; however, probabilistic guarantees can beobtained on the measure of the set of constraints that are compatible with this solution (Calafiore,2010; Margellos et al., 2014; Campi et al., 2018).The approach of chance-constrained optimization has already proved useful in several areas ofcontrol, like robust control design (Calafiore and Campi, 2006) or quantized control (Campi et al.,2018). Recently, it has been successfully applied to data-driven control problems, as a techniqueto bridge the gap between data and model-based control; see, e.g., applications in data-enabledpredictive control (Van Parys et al., 2015; Coulson et al., 2020) and stability analysis of black-boxdynamical systems (Kenanian et al., 2019; Wang and Jungers, 2020).In this work, we introduce a new class of optimization problems: quasi-linear problems . Thisclass forms a subclass of quasi-convex optimization problems (see, e.g., Eppstein, 2005). We extendthe results from Calafiore (2010) for chance-constrained optimization of convex problems to quasi-linear optimization problems. This is achieved by showing that for any such optimization problemthere is a subset of constraints, called the essential set , with bounded cardinality, that provides thesame optimal solution as the original problem. This result draws on an akin result for quasi-convexproblems in Eppstein (2005), and improves it in two ways: we get a better upper bound on thecardinality of essential sets, while removing the assumption that the constraints are “continuouslyshrinking” (Eppstein, 2005).We believe that chance-constrained quasi-linear optimization can find application in many ar-eas of data-driven control. For instance, by replacing LMIs with sampled linear inequalities, onecould transform SDP problems in control (Boyd et al., 1994) into linear or quasi-linear programs,and use chance-constrained optimization to bridge the gap between the original and the sampledformulations.As a proof of concept, we demonstrate here that the setting of chance-constrained quasi-linearoptimization can be useful for the stability analysis of black-box switched linear systems . SwitchedLinear Systems are systems described by a finite set of linear modes among which the system canswitch over time. They constitute a paradigmatic class of hybrid and cyber-physical systems, andappear naturally in many engineering applications, or as abstractions of more complicated systems(Alur et al., 2009; Jadbabaie et al., 2003). These systems turn out to be extremely challenging interms of control and analysis, even for basic questions like stability or stabilizability. In particular,the computation of the
Joint Spectral Radius (JSR), a measure of stability of switched linear sys-tems, has been used as a benchmark for testing new approaches in complex systems (Blondel andNesterov, 2005; Parrilo and Jadbabaie, 2008; Jungers et al., 2017).Recently, the problem of JSR approximation was introduced for black-box switched linear sys-tems. It is well known that bounds on the JSR of switched linear systems can be obtained from theresolution of adequate quasi-convex optimization problems built from the system (Jungers et al.,2017). In Kenanian et al. (2019), the authors extend this approach when the system is not knownbut only a few trajectories are observed, and apply chance-constrained optimization techniques toobtain probabilistic upper bounds and lower bounds on the JSR of the system. In this work, weshow that this approach fits in fact into the framework of chance-constrained quasi-linear optimiza-tion. From this, probabilistic upper and lower bounds on the JSR of the system can be obtainedstraightforwardly, by applying the results introduced in this paper; the bounds obtained in that wayare also better than the ones proposed in Kenanian et al. (2019). HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION
The paper is organized as follows. In Section 2, we introduce the class of quasi-linear optimiza-tion problems and discuss their properties. In Section 3, we state and prove the main theorem ofthis paper, which extends the results of chance-constrained convex optimization to quasi-linear opti-mization problems. Then, in Section 4, we apply the framework of chance-constrained quasi-linearoptimization to the problem of stability analysis of black-box switched linear systems, and we showhow this framework can be used to obtain probabilistic bounds on the JSR of the system. Finally, inSection 5, we demonstrate the applicability of our results with several numerical examples.
Notation. N denotes the set of nonnegative integers, and N ∗ the set of positive integers. For aset of vectors V ⊆ R d , conv( U ) denotes the convex hull of U , and cone( U ) its conic hull . For aconvex function f : R d → R and x ∈ R d , we let Sub x ( f ) be the subdifferential of f at x , i.e., theset of vectors g ∈ R d such that f ( y ) − f ( x ) ≥ g (cid:62) ( y − x ) for all y ∈ R d ; for a convex set C ⊆ R d ,we let Nor x ( C ) be the normal cone of C at x , i.e., the set of vectors g ∈ R d such that g (cid:62) ( y − x ) ≤ for all y ∈ C . If ∆ is a set, ¯ ω := ( δ , . . . , δ N ) ∈ ∆ N and δ N +1 ∈ ∆ , we use ¯ ω (cid:107) δ N +1 to denote their concatenation : ¯ ω (cid:107) δ N +1 := ( δ , . . . , δ N +1 ) ; in Section 3, for the sake of simplicity, we will slightlyabuse the notation and write ω to denote the set obtained from the elements of ¯ ω := ( δ , . . . , δ N ) ,i.e., ω = { δ , . . . , δ N } .
2. Quasi-linear optimization problems
In this section, we introduce a novel class of optimization problems, which are a particular caseof quasi-convex problems. We particularize and improve some classical results of quasi-convexprogramming to this class.Let X be a compact convex subset of R d , with nonempty interior and with / ∈ X . Let ∆ bea set, and { a δ } δ ∈ ∆ and { b δ } δ ∈ ∆ be two collections — indexed by δ ∈ ∆ — of vectors in R d andsuch that b (cid:62) δ x > for all x ∈ X and δ ∈ ∆ . Consider the following optimization problem: min x ∈ R d , λ ≥ ( λ, c ( x )) s.t. x ∈ X , and a (cid:62) δ x ≤ λb (cid:62) δ x, ∀ δ ∈ ∆ , (1)where c : X → R is a strongly convex function. The objective of (1) is to minimize ( λ, c ( x )) in the lexicographical order , while respecting the constraints defined by ∆ and x ∈ X . See Figure 1 foran illustration.Sometimes, it is not possible to solve (1) with all the constraints defined by ∆ , either becauseonly a subset of these constraints are known (as it is the case for instance in data-driven controlproblems), or because the set ∆ is so large (or even infinite) that it is algorithmically impracticableto enforce all of these constraints. In these cases, for a finite set ω ⊆ ∆ , we consider the following sampled optimization problem: P ( ω ) : min x ∈ R d , λ ≥ ( λ, c ( x )) s.t. x ∈ X , and a (cid:62) δ x ≤ λb (cid:62) δ x, ∀ δ ∈ ω. (2)We let Opt( ω ) be the optimal solution of P ( ω ) and we let Cost( ω ) be its optimal cost. Theconstraints of P ( ω ) defined by ω will be called the sampled constraints , while the constraint x ∈ X is the common constraint .
1. “First component first”: ( λ , c ) < ( λ , c ) if λ < λ , or else λ = λ and c < c .2. By the strong convexity of c , Opt( ω ) is unique. HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION
𝒳 𝑎 (2)⊤ 𝑥 ≤ 𝜆 𝑏 (2)⊤ 𝑥𝑥 𝑥 𝑎 𝑥 ≤ 𝜆 (1) 𝑏 𝑥 𝒳 𝑥 𝑥 𝒳 𝑥 𝑥 𝑥 ∗ 𝑐 𝑥 = 𝑐𝑡𝑒 𝑎 ⊤ 𝑥 ≤ 𝜆 (2) 𝑏 ⊤ 𝑥 𝑎 (2)⊤ 𝑥 ≤ 𝜆 𝑏 (2)⊤ 𝑥 𝑎 𝑥 ≤ 𝜆 ∗ 𝑏 𝑥 𝑎 (2)⊤ 𝑥 ≤ 𝜆 ∗ 𝑏 (2)⊤ 𝑥 𝜆 (1) > 𝜆 > 𝜆 ∗ 𝑐 𝑥 = 𝑐𝑡𝑒 𝑐 𝑥 = 𝑐𝑡𝑒 Figure 1: Set of feasible points x ∈ R d of a quasi-linear optimization problem (1), for three differentvalues of λ . The blue set X represents the fixed constraints, while the two quasi-linear constraintsare represented in black. The dotted curves are level-curves of the secondary cost function c ( x ) .The smallest λ for which there is a feasible point x is the optimal λ , denoted λ ∗ . For this value of λ , the feasible point x that minimizes c is the optimal point x , denoted x ∗ .For a fixed value of λ , the sampled constraints of P ( ω ) are linear in x . Therefore, we will saythat P ( ω ) is a quasi-linear optimization problem . Note that P ( ω ) is a particular instance of quasi-convex optimization problems, as defined in Eppstein (2005). It is shown there that, under sometechnical assumption on the continuity of the constraints, the cardinality of any essential set (seeDefinition 1 below) of a quasi-convex problem is upper bounded by d + 1 , where d is the dimensionof x . In this paper, we provide for quasi-linear problems a better upper bound on the cardinality oftheir essential sets, and without the technical assumption of “continuously shrinking” constraints,present in Eppstein (2005). Definition 1 (Calafiore, 2010, Definition 2.9) An essential set for P ( ω ) is a set β ⊆ ω , withminimal cardinality, satisfying Cost( β ) = Cost( ω ) . Theorem 2
The cardinality of any essential set β of P ( ω ) satisfies | β | ≤ d . To prove this theorem, we will need the following lemma.
Lemma 3 (Rockafellar, 1970, Theorem 27.4) Let f : R d → R be a convex function and C ⊆ R d anonempty convex set. Then, x ∈ C is a mininizer of f over C if and only if ∈ Sub x ( f )+Nor x ( C ) . Proof of Theorem 2
Let β be an essential set for P ( ω ) and let ( λ ∗ , x ∗ ) = Opt( ω ) . For each δ ∈ ω ,let h δ = a δ − λ ∗ b δ . Let γ ⊆ ω be the set of all δ ∈ ω such that h (cid:62) δ x ∗ = 0 . We divide the proof intwo cases. Case 1:
First, we consider the case when λ ∗ = 0 . Assume that x ∈ X is a support constraint ,meaning that the optimal cost of P ( ω ) without this constraint is strictly smaller than Cost( ω ) .Then, by the classical argument , there is a set of at most d constraints among those of P ( ω ) (i.e.,
3. I.e., f ( x ∗ ) = inf x ∈C f ( x ) .4. Indeed, P ( ω ) with λ fixed to zero is a convex optimization problem, and the cardinality of essential sets of feasibleconvex optimization problems is bounded by d (see, e.g., Calafiore and Campi, 2006, Theorem 3). HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION among the constraints defined by ω , and the constraint x ∈ X ) such that the optimal solution of theproblem with these constraints only is equal to Cost( ω ) . Because x ∈ X is a support constraint, itmust belong to this set of constraints. Hence, there is a set β (cid:48) ⊆ ω , with | β (cid:48) | ≤ d − , such that Cost( β (cid:48) ) = Cost( ω ) . This shows that | β | ≤ d − when x ∈ X is a support constraint.Now, assume that x ∈ X is not a support constraint, i.e., the optimal cost of P ( ω ) without thisconstraint is the same as Cost( ω ) . By Lemma 3, it holds that ∈ Sub x ∗ ( c ) + cone( { h δ } δ ∈ γ ) . Notethat, by definition of γ , the vectors { h δ } δ ∈ γ are all orthogonal to x ∗ , so that they belong to a ( d − -dimensional subspace. Hence, by Caratheodory theorem , there is a set γ (cid:48) ⊆ γ , with | γ (cid:48) | ≤ d − ,such that ∈ Sub x ∗ ( c ) + cone( { h δ } δ ∈ γ (cid:48) ) . By Lemma 3, it thus follows that Cost( γ (cid:48) ) = Cost( ω ) .This shows that | β | ≤ d − when x ∈ X is not a support constraint; concluding the proof for thefirst case. Case 2:
Now, we consider the case when λ ∗ > . By Lemma 3 applied on f ( x ) = sup δ ∈ γ h (cid:62) δ x and C = X , it follows that ∈ conv( { h δ } δ ∈ γ ) + Nor x ∗ ( X ) . Let γ (cid:48) ⊆ γ be a nonempty subset withminimal cardinality such that ∈ conv( { h δ } δ ∈ γ (cid:48) ) + Nor x ∗ ( X ) . By Caratheodory theorem, it holdsthat | γ (cid:48) | ≤ d − f + 1 where f is the dimension of the linear subspace orthogonal to { h δ } δ ∈ γ (cid:48) . Weconclude the proof by using the same argument as in case 1: since the problem is now restricted toan f -dimensional problem (because x is in the subspace orthogonal to { h δ } δ ∈ γ (cid:48) ), we may find a set β (cid:48) ⊆ ω , with | β (cid:48) | ≤ f − , such that Opt( γ (cid:48) ∪ β (cid:48) ) = ( λ ∗ , x ∗ ) . This shows that | β | ≤ | γ (cid:48) | + | β (cid:48) | ≤ d ;concluding the proof for the second case.
3. Chance-constrained quasi-linear optimization
Let P be a probability measure on ∆ . Suppose that the constraints δ , . . . , δ N are sampled from ∆ according to P , and that we solve the problem P ( ω N ) where ω N = { δ , . . . , δ N } . This ap-proach of solving the optimization problem for a few randomly sampled constraints is called chance-constrained optimization . Under certain assumptions, probabilistic guarantees can be obtained onthe measure of the set of constraints δ ∈ ∆ that are compatible with the optimal solution of P ( ω N ) .This is the case, for instance, for a large class of convex optimization problems (see, e.g., Calafiore,2010) and non-convex optimization problems (though with weaker probabilistic guarantees; see,e.g., Campi et al., 2018). In the section, we extend the results from chance-constrained convexoptimization (Calafiore, 2010) to chance-constrained quasi-linear problems.Therefore, we make the following standing assumption on the set ∆ and on its probabilitymeasure P . First, let us introduce the notion of non-degenerate vector of constraints. Definition 4 (Calafiore, 2010, Definition 2.11) Let N ∈ N ∗ . We say that ¯ ω N := ( δ , . . . , δ N ) ∈ ∆ N is non-degenerate if there is a unique set I ⊆ { , . . . , N } such that { δ i } i ∈ I is an essential setfor P ( ω N ) . Assumption 5 (Calafiore, 2010, Assumption 2) For every N ∈ N ∗ , the vector ¯ ω N ∈ ∆ N is non-degenerate with probability one. For any vector of constraints ¯ ω N ∈ ∆ N , we define the violating probability associated to ¯ ω N : V (¯ ω N ) = P ( { δ ∈ ∆ : Cost( ω N ∪ { δ } ) > Cost( ω N ) } ) .
5. See, e.g., Rockafellar (1970, Corollary 17.1.2). HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION
We are now able to present the extension of Calafiore (2010, Theorem 3.3) to chance-constrainedquasi-linear programs. Therefore, let ζ ∈ N ∗ be an upper bound on the cardinality of any essentialset of P ( ω ) , with finite ω ⊆ ∆ . From Theorem 2, it holds that ζ ≤ d . Theorem 6
Consider the sampled quasi-linear optimization problem (2) , and let V (¯ ω N ) and ζ beas above. Let Assumption 5 hold. Let N ∈ N , N ≥ ζ , and let ε ∈ (0 , . Then, P N ( { ¯ ω N ∈ ∆ N : V (¯ ω N ) > ε } ) ≤ Φ( ε, ζ − , N ) , where Φ( · , ζ − , N ) is the regularized incomplete beta function . Proof (Adapted from Calafiore, 2010) Fix N ∈ N , N ≥ ζ . By Assumption 5, we may assumewithout loss of generality that ¯ ω N is non-degenerate for all ¯ ω N ∈ ∆ N . Hence, for each ¯ ω N :=( δ , . . . , δ N ) ∈ ∆ N , we let J (¯ ω N ) be the unique set I ∈ { , . . . , N } such that { δ i } i ∈ I is anessential set for P ( ω N ) . Label the elements of ∆ with labels belonging to a totally order set. Let J ∗ (¯ ω N ) be a completion of J (¯ ω N ) with the ζ − | J (¯ ω N ) | elements of { , . . . , N } \ J (¯ ω N ) suchthat { δ i } i ∈ J ∗ (¯ ω N ) \ J (¯ ω N ) have the largest labels among the elements of ω N . From Assumption 5, itfollows that J ∗ (¯ ω N ) is well defined with probability one; hence, in the following, we will assumewithout loss of generality that J ∗ (¯ ω N ) is well defined for all ¯ ω N ∈ ∆ N .Let { I , . . . , I M } be the set of all subsets of { , . . . , N } with ζ elements; in particular, M = N ! / ( ζ !( N − ζ )!) (cid:44) C ( N, ζ ) . For each i = 1 , . . . , M , let S i = { ¯ ω N ∈ ∆ N : J ∗ (¯ ω N ) = I i } . Thesets { S i } ≤ i ≤ M are disjoint and their union is equal to ∆ N . Moreover, by the symmetry of theirdefinition, they have the same probability; hence P ( S i ) = 1 /C ( N, ζ ) .Now, for each ¯ ω ζ ∈ ∆ ζ , we let V ∗ (¯ ω ζ ) be the violating probability of ¯ ω ζ with respect to (2) andthe labelling of the constraints: that is, V ∗ (¯ ω ζ ) = P ( { δ ∈ ∆ : J ∗ (¯ ω ζ (cid:107) δ ) (cid:54) = { , . . . , ζ } ) . From theuniqueness of the optimal solution of the problems P ( ω ) , ω ⊆ ∆ , it follows that for every L ∈ N ∗ , ¯ ω L ∈ ∆ L and δ, η ∈ ∆ , if J ∗ (¯ ω L (cid:107) δ ) = J ∗ (¯ ω L (cid:107) η ) = J ∗ (¯ ω L ) , then J ∗ ((¯ ω L (cid:107) δ ) (cid:107) η ) = J ∗ (¯ ω L ) . Itfollows that, for any v ∈ [0 , , P [ S i | V ∗ (¯ ω N,i ) = v ] = (1 − v ) N − ζ , ∀ N ≥ ζ, i = 1 , . . . , C ( N, ζ ) , where ¯ ω N,i is the restriction of ¯ ω N to the indices in I i : ¯ ω N,i = ( δ i ) i ∈ I i . Hence, we get that P ( S i ) = (cid:119) (1 − v ) N − ζ d F i ( v ) = 1 /C ( N, ζ ) , ∀ N ≥ ζ, i = 1 , . . . , C ( N, ζ ) , (3)where F i ( v ) = P N ( { ¯ ω N ∈ ∆ N : V ∗ (¯ ω N,i ) ≤ v } ) . Equation (3) describes a Hausdorff momentproblem ; it is shown in Calafiore (2010, p. 3436) that (3) implies that F i ( v ) = v ζ .Finally, for each i = 1 , . . . , C ( N, ζ ) , we let B i = { ¯ ω N ∈ ∆ N : V ∗ (¯ ω N,i ) > ε } . Using theexpression of F i , it can be shown that P N ( B i ) = Φ( ε, ζ − , N ) /C ( N, ζ ) . By symmetry, we getthat P N ( (cid:83) ≤ i ≤ M B i ) = Φ( ε, ζ − , N ) . Since { ¯ ω N ∈ ∆ N : V (¯ ω N ) > ε } ⊆ (cid:83) ≤ i ≤ M B i , weobtain the desired result.
6. See, e.g., Kenanian et al. (2019, Definition 2).7. This approach, from Calafiore (2010), requires the axiom of choice when ∆ is a general set. However, it is not neededfor instance if ∆ ⊆ R n , as it is the case in our application (see Section 4).8. See, e.g., Calafiore (2010, §2.1) for details.9. See, e.g., Calafiore (2010, Theorem 3.3). HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION
4. Application to data-driven stability analysis of switched linear systems
Let A = { A , . . . , A m } be a fixed set of matrices in R n × n , and let S be the unit sphere (boundaryof the unit Euclidean ball) in R n . Let ∆ = A × S , and let P be the uniform distribution on ∆ . Fora finite set ω ⊆ ∆ , we consider the following sampled quasi-linear optimization problem: P jsr ( ω ) : min P = P (cid:62) ∈ R n × n ,γ ≥ ( γ, (cid:107) P (cid:107) F ) s.t. P ∈ X := { P : P (cid:23) I ∧ (cid:107) P (cid:107) F ≤ C } , ( Ax ) (cid:62) P ( Ax ) ≤ γ x (cid:62) P x, ∀ ( A, x ) ∈ ω, (4)for some fixed parameter C ≥ n . Note that P jsr ( ω ) is a sampled, data-driven version of the clas-sical quadratic Lyapunov framework for the approximation of the Joint Spectral Radius (JSR) ofthe switched linear system defined by A ; see, e.g., Jungers (2009, Theorem 2.11). The JSR is aubiquituous measure of stability of switched linear systems (Blondel and Nesterov, 2005; Parriloand Jadbabaie, 2008; Jungers et al., 2017); it also appears in other areas of hybrid system control,like wireless networked control (Berger and Jungers, 2020).In order to apply the results from Section 3 on P jsr ( ω ) , we make the following assumption onthe matrices in A . First, let us introduce the notion of Barabanov matrix.
Definition 7
A matrix A ∈ R n × n is said to be Barabanov if there exists a symmetric matrix P (cid:31) and γ ≥ such that A (cid:62) P A = γ P . Assumption 8
There is no Barabanov matrix in A . We claim that Assumption 8 is not restrictive in most of the practical situations. To motivate thisclaim, we provide an equivalent characterization of Barabanov matrices in the proposition below,whose proof can be found in Appendix A. For further work, we plan to investigate the possibility torelax or remove this technical assumption.
Proposition 9
A matrix A ∈ R n × n is Barabanov if and only if it is diagonalizable and all itseigenvalues have the same modulus. We now show that Assumption 8 ensures that Assumption 5 holds for (4).
Proposition 10
Consider the sampled problem (4) . Let Assumption 8 hold. Then, for every N ∈ N ∗ , the vector ¯ ω N ∈ ∆ N is non-degenerate with probability one. We will need the following lemma.
Lemma 11
Let P ( x , . . . , x n ) be a nonzero polynomial on R n . The zero set of P , i.e., the set ofpoints x ∈ R n such that P ( x ) = 0 , has zero Lebesgue measure. We skip the proof of this well-known fact (see, e.g., Teschl, Problem 2.15).
Proof of Proposition 10
Let ≤ i ≤ N − . Let us look at the probability that β := { δ , . . . , δ i } is an essential set for P jsr ( ω ) and that δ N is in another essential set. This probability is smaller than
10. I.e., P = P ⊗ P where P and P are the uniform distributions on A and S respectively. HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION or equal to the probability that β is an essential set for P jsr ( ω ) and that ( Ax ) (cid:62) P ( Ax ) = γ x (cid:62) P x ,where ( γ, P ) = Opt jsr ( β ) and δ N = ( A, x ) .Assume that the above probability is nonzero. Then, since A is finite, that there is A ∈ A suchthat ( Ax ) (cid:62) P ( Ax ) = γ x (cid:62) P x for all x in a set S ⊆ S with nonzero measure. Thus, by Lemma 11,it holds that ( Ax ) (cid:62) P ( Ax ) = γ x (cid:62) P x for all x ∈ S . This contradicts the assumption that thereis no Barabanov matrix in A . Hence, the probability that β is a basis for P jsr ( ω ) and that δ N is inanother basis is zero. Since β and δ N were arbitrary, this concludes the proof.Theorem 6 can thus be applied to P jsr ( ω ) . Corollary 12
Consider the sampled problem (4) . Let Assumption 8 hold. Let N ∈ N , N ≥ d := n ( n +1)2 , and let ε ∈ (0 , . Then, P N ( { ¯ ω N ∈ ∆ N : V jsr (¯ ω N ) > ε } ) ≤ Φ( ε, d − , N ) , (5) where V jsr (¯ ω N ) = P ( { δ ∈ ∆ : Cost jsr ( ω N ∪ { δ } ) > Cost jsr ( ω N ) } ) . Remark 13
We note the improvement of the right-hand side term of (5) , compared to Kenanianet al. (2019, Theorem 10); this term becomes Φ( ε, d − , N ) instead of Φ( ε, d, N ) in Kenanianet al. (2019). This is due to the improvement of the bound on the cardinality of essential sets ofquasi-linear problems; see Theorem 2. From Corollary 12, we deduce the following probabilistic guarantee on the upper bound on theJSR of the switched linear system given by A , that we can get from the solution of the sampledproblem P jsr ( ω ) . The derivation of this result follows the same lines as in Kenanian et al. (2019,Theorems 14 and 15), so that the details are omitted here. Corollary 14
Consider the sampled problem (4) . Let Assumption 8 hold. Let N ∈ N , N ≥ d := n ( n +1)2 , and let ε ∈ (0 , . Then, for all ¯ ω N ∈ ∆ N , except possibly those ¯ ω N in a subset Ω ⊆ ∆ N with measure P N (Ω) ≤ Φ( ε, d − , N ) , it holds that ρ ( A ) ≤ γ ∗ (cid:46)(cid:114) − I − (cid:16) εκ ( P ∗ ) m ; d −
12 ; 12 (cid:17) , where ( γ ∗ , P ∗ ) = Opt jsr ( ω N ) , κ ( P ) = (cid:113) det( P ) λ min ( P ) n , I − is the inversed regularized incompletebeta function and ρ ( A ) is the JSR of the switched linear system defined by A .
5. Numerical experiments: consensus of hidden network
We consider the problem of consensus in a switching and hidden network. The interaction betweenthe nodes in the network over time can be modeled as a switched linear dynamical system: x ( t + 1) = A σ ( t ) x ( t ) , x ( t ) ∈ R n , A σ ( t ) ∈ A := { A , . . . , A m } ⊆ R n × n , where x ( t ) is the state vector ( n is the number of nodes) at time t and A σ ( t ) is the interactionmatrix at time t , with A i being unknown row-stochastic matrices, i.e., A i = , i = 1 , . . . , m ,
11. See, e.g., Kenanian et al. (2019, Definition 2). HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION where is the all-one vector in R n . The goal is to verify that x ( t ) = A σ ( t − · · · A σ (1) A σ (0) x (0) converges to c for some c as t → ∞ . As shown by Jadbabaie et al. (2003), this question boilsdown to the computation of the JSR of A (cid:48) := { A (cid:48) , . . . , A (cid:48) m } ⊆ R n − × n − where A (cid:48) i = BA i B (cid:62) ,for i = 1 , . . . , m , and B ∈ R n − × n is a fixed orthogonal matrix ( BB (cid:62) = I n − ) with kernelspanned by . In our experiment, we consider a network of nodes, switching among modes, asshown in Figure 2. The possible networks are not known, and only the state of the different agentsis available. Hence, we use the data-driven framework in Section 4 to estimate the JSR of A (cid:48) .Figure 2: Example of switching network with modes.First, we sample a data set of N pairs: ( x i , y i ) , with i = 1 , . . . , N , where x i is sampled uni-formly at random on S and y i = A σ i x i , with σ i sampled uniformly at random in { , . . . , m } . Thisdata set is projected onto R n − as follows: ( x i , y i ) (cid:55)→ ( x (cid:48) i , y (cid:48) i ) where x (cid:48) i = Bx i and y (cid:48) i = By i and B is as above. We then solve the problem in Section 4 with the projected data set. We fix theconfidence level at β = 0 . . The probabilistic upper bound on the JSR obtained from Corollary 14is shown in Figure 3 for different sizes of the sample set. For a comparison, the bound of Kenanianet al. (2019) is also given. While both bounds converge when the number of samples increases, thebound in this paper requires fewer samples to deduce convergence of the system to consensus, withthe same confidence level.Figure 3: Data-driven upper bounds on the JSR for different sizes of the sample set. Bound 1 refersto the bound of this paper, Bound 2 refers to the bound of Kenanian et al. (2019), and the dash-dottedline is the bound computed from the white-box model, using the JSR toolbox (Jungers, 2009).
12. The orthogonality of B is important to ensure that x (cid:48) i / (cid:107) x (cid:48) i (cid:107) is distributed uniformly on S . HANCE - CONSTRAINED QUASI - CONVEX OPTIMIZATION
6. Conclusions
In this work, we generalized the theory of chance-constrained optimization to quasi-convex prob-lems, and pushed further the effort initiated in Kenanian et al. (2019), demonstrating its use fordata-driven stability analysis of complex systems. More precisely, we introduced the class of quasi-linear optimization problems, which is a subclass of quasi-convex problems. We particularizedand improved some classical results of quasi-convex programming to this class. This allowed usto extend the results of chance-constrained convex optimization to quasi-linear optimization prob-lems. Thriving on this, we provided a proof of concept that quasi-linear problems are useful fordata-driven control applications. In particular, we applied our framework to the problem of JSRapproximation of black-box switched linear systems, introduced in Kenanian et al. (2019).For future work, we plan to investigate other applications of chance-constrained quasi-linearoptimization for data-driven control. For instance, we believe that by replacing the conic con-straints with their sampled counterpart, one could transform many optimization problems in controltheory into quasi-linear programs, and then use chance-constrained optimization to bridge the gapbetween the original and the sampled formulations. We also plan to investigate the possibility ofrelaxing or removing the assumption that there are no Barabanov matrices involved in the switchedlinear system. Finally, we plan to provide other approaches for the data-driven stability analysisof switched linear systems based on chance-constrained quasi-linear optimization (e.g., thriving onsum-of-square optimization or path-complete Lyapunov frameworks).
Appendix A. Proof of Proposition 9
First, we prove the if direction: Assume that A is diagonalizable and all its eigenvalues have thesame modulus. Then, there is T ∈ R n × n invertible such that A = T − DT , where D ∈ R n × n is block-diagonal with diagonal blocks of size or , corresponding to eigenvalues with the samemodulus. Denote this common modulus by γ . Now, let P = T (cid:62) T , which is positive definite. Weverify that A (cid:62) P A = T (cid:62) D (cid:62) DT = γ T (cid:62) T = γ P . Hence, A is Barabanov.Now, we show the only if direction: Assume that A (cid:62) P A = γ P for some P (cid:31) and γ ≥ .Let P = L (cid:62) L be a Cholesky factorization of P . It follows that B (cid:62) B = γ I , where B = LAL − .If γ = 0 , this implies that B = 0 and thus A = 0 , proving the only if direction when γ = 0 .If γ > , this implies that B/γ is a unitary matrix. It follows that B is diagonalizable and all itseigenvalues have modulus γ . Now, since A is similar to B , the same holds for A , proving the onlyif direction when γ > . References
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