Path-complete positivity of switching systems
PPath-complete positivity of switchingsystems
F. Forni ∗ , R. M. Jungers ∗∗ , R. Sepulchre ∗∗∗∗
University of Cambridge, United Kingdom([email protected]). ∗∗ Universit´e catholique de Louvain, Belgium([email protected]) ∗∗∗
University of Cambridge, United Kingdom([email protected])
Abstract:
The notion of path-complete positivity is introduced as a way to generalize theproperty of positivity from one LTI system to a family of switched LTI systems whose switchingrule is constrained by a finite automaton. The generalization builds upon the analogy betweenstability and positivity, the former referring to the contraction of a norm, the latter referringto the contraction of a cone (or, equivalently, a projective norm). We motivate and investigatethe potential of path-positivity and we propose an algorithm for the automatic verification ofpositivity.
Keywords:
Positivity, Path-complete Lyapunov functions, Switching systems, Monotonicity,Perron-Frobenius theory.1. INTRODUCTIONPositivity is a classical concept of linear system theory.It originates in the many examples of system dynam-ics whose state variables remain positive along trajec-tories, and finds its theoretical foundations in Perron-Frobenius theory. In a nutshell, under mild assumptions,the solutions of a positive system converge to a dominanteigendirection in the positive orthant Luenberger (1979).Positivity has known a renewed interest in the recentyears for its advantageous computational scalability overgeneral linear systems Rantzer (2015). As a geometricconcept, positivity is primarily about the contraction ofa cone under the action of a linear map. The positiveorthant is a cone of special interest, but Perron-Frobeniustheory owes fundamentally to the geometric contractionof a cone more than to an algebraic property of matriceswith positive elements.It is the same contraction property that makes positiv-ity the infinitesimal (or differential) characterization ofmonotonicity : the order preserving property of a mono-tone map is equivalent to a positivity property for thelinearized map. This geometric viewpoint on positivity isat the root of the differential positivity theory recentlyintroduced in Forni and Sepulchre (2016) to characterizeand study the asymptotic properties of nonlinear systemswhose trajectories infinitesimally contract a smooth conefield. It has proven quite insightful to think of differentialpositivity as an analog of differential stability, or contrac-tion analysis. In one case, one studies the contraction ofa smooth norm field, e.g. a Riemannian metric, while in (cid:63)
R.J. is currently on sabbatical at UCLA, Department of ElectricalEngineering, Los Angeles, USA. His work is supported by the FrenchCommunity of Belgium and by the IAP network DYSCO. He is aFulbright Fellow and a FNRS Fellow. the latter case, one studies the infinitesimal contractionof a cone field. This insight points to a basic but profoundsimilitude between stability and positivity : two contrac-tion properties, that only differ by the geometric natureof the object that is contracted.The present paper draws upon this analogy to generalizethe concept of positivity from a single matrix (or linear op-erator) to a family of matrices. Such a generalization hasreceived considerable attention in the context of stability,but much less in the context of positivity. In particular,we focus in the present paper on the recent frameworkof path-complete Lyapunov analysis, which is a unifyingapproach to study the stability of a switched systemwhose switching rule is constrained by a finite automaton.Our goal here is to mimick this framework when the norm contraction underlying stability is replaced by a cone contraction underlying positivity.The paper is organized as follows. In Section 2, we brieflyrecall the notion of positivity, and its links with stability.Then, in Section 3, we naturally draw on this parallel tointroduce our main concept: path-complete positivity. InSection 4 we explain what this concept implies in termsof dynamical systems and control, and finally Section 5touches upon the algorithmic problem of recognizing thisproperty for a given set of matrices.2. POSITIVITY VERSUS STABILITYBoth stability and positivity are classical notions in linearsystems analysis. We review basic notations and terminol-ogy and stress the analogy between these two propertiesin the elementary context of a linear time-invariant (LTI)system x + = Ax . a r X i v : . [ c s . S Y ] N ov tability refers to the invariance of a norm, i.e. a ball in the state-space. The restriction to quadratic norms | x | P := √ x T P x (where P is a positive definite matrix)is no loss of generality for LTI systems, in which casethe invariance condition corresponds to the (Lyapunov)inequality A T P A − γP (cid:22) , ≤ γ ≤ . The case γ = 1 only ensures invariance (i.e. Lyapunovstability) whereas the case γ < V ( x ) := x T P x is also called a (quadratic) Lyapunovfunction.Fundamentally, positivity is the analog property when theball is replaced by a cone. In this paper, a cone
K ⊆ R n always means a convex pointed solid cone. Recall that aset K is a convex cone if αx + βy ∈ K for all x, y ∈ K andall α, β >
0. Pointed means −K ∩ K = { } . K is solid if itcontains n linearly independent vectors.A linear system is positive with respect to K (in short, K -positive) if A K ⊆ K . Positivity only ensures the invariance condition, while strict positivity also enforces contraction , by requiringthat the boundary of the cone is mapped into the interiorof the cone A K ⊆ int K . Positivity has a natural metric characterization based onthe Hilbert metric d K associated to the cone K . Definition 1.
Bushell (1973a) Given a cone
K ∈ R n , thecorresponding Hilbert metric is given by d K ( x, y ) := log (cid:18) M K ( x | y ) m K ( x | y ) (cid:19) ∀ x, y ∈ K where M K ( x | y ) = inf { λ | λy − x ∈ K} = inf { λ | λy ∈ x + bdr K} ; m K ( x | y ) = sup { µ | x − µy ∈ K} = sup { µ | µy ∈ x − bdr K} . We take M K ( x | y ) = ∞ if ∀ λ > , λy / ∈ x + K .The Hilbert metric is in fact a distance among rays of thecone , satisfying the property d K ( αx, βy ) = d K ( x, y ) forany positive scaling α and β . It is theferore a distancein the projective space. In short, contraction of a ball ismeasured by a norm distance, whereas contraction of acone is measured by a projective distance. The Hilbertmetric characterizes the contraction of a cone in the sameway as a Lyapunov function characterizes the contractionof a ball, as shown in the following theorem. Theorem 1.
Bushell (1973a) Consider a matrix A ∈ R n × n . If A is K -positive, then there exists γ < x, y ∈ K d K ( Ax, Ay ) ≤ γd K ( x, y ) . (1)Moreover, the smallest γ satisfying the equation abovesatisfies γ = tanh 14 D A K where D A K := sup x,y ∈ int K d K ( Ax, Ay ) . Clearly γ < D A K < ∞ , that is, whenever A K ⊆ int K . In what follows we will say that K is a γ - contracting cone for the linear map A whenever (1) holds. Proving the contraction of a map is a fundamental way ofcharacterizing the existence of a fixed point. Contractionof a ball implies that the iterated map eventually shrinksto a point. This is the essence of Lyapunov theory.Likewise, contraction of a cone implies that the iteratedmap eventually shrinks to a ray (a point in the projectivespace). This is the essence of Perron-Frobenius theory.For a LTI system, both stability and positivity have aspectral characterization. Exponential stability (or con-traction) means that all the eigenvalues have a strictlynegative real part, while strict positivity (or projectivecontraction) means that the matrix A has a dominanteigenvector in the interior of the cone.3. CONTRACTION AND PATH-CONTRACTIONThere exists an extensive literature devoted to generaliz-ing the stability of a single matrix (in the sense recalledin the previous section) to a finite (or even compact) setof matrices A σ ∈ R n × n , σ ∈ Σ := { , . . . , N } ⊂ N . See forinstance Liberzon (2003); Jungers (2009) and referencestherein. One obvious application is the stability analysisof switched systems x + = A σ x where the update rule isallowed to switch among the considered set of matrices.Drawing upon the analogy stressed above, this sectiongeneralizes positivity to a set of matrices. A straightforward extension with respect to the previoussection is to study uniform positivity of a family ofmatrices with respect to a common cone.Not surprisingly, strict positivity of each matrix (possiblywith respect to different cones) is necessary but notsufficient for uniform strict positivity. And proving theexistence of a common invariant cone for a set of lineardynamics is hard. Actually, the existence question isalgorithmically undecidable Protasov (2010); Protasovand Voynov (2012); Rodman et al. (2010), very much forthe same reasons as its companion question of uniformnorm contraction (see Blondel and Tsitsiklis (2000), or(Jungers, 2009, Section 2.2.3)).It would certainly be of interest to revisit the largebody of literature on uniform stability in the light of theanalog question of uniform positivity. Even the questionof defining a joint projective radius for a family of positivesystems in analogy to the ‘joint spectral radius’ defined fora family of stable systems seems valuable and not entirelystraightforward. We do not pursue this question in thepresent paper and leave it for future research.
Uniform positivity or uniform stability is too conservativeof a property for the many applications where the switch-ing rule is not arbitrary. This has long been acknowledgedin the literature of switched systems, see for instanceEssick et al. (2015); Bliman and Ferrari-Trecate (2003);Lin and Antsaklis (2009), where the permissible sequenceof switches is typically modeled by a finite automaton.onsider a class of switching linear systems representedby x ( k + 1) = A σ ( k ) x ( k ) (2)where σ ∈ Σ := { , . . . , N } ⊂ N and each A σ is a n × n matrix. For a switching signal σ ( · ) : N → Σand any initial condition x ∈ R n , the unique solution x ( · ) : N → R n of (2) is called a trajectory of the system.We say that the system is a constrained switching system if the sequences σ (0) σ (1) . . . generated by the switchingsignal σ ( · ) belongs to a regular language L r .Thus, σ ( · ) is generated by any finite-state automaton( Q, Σ , δ ) that accepts the same regular language L r , where Q is the set of states, Σ is the alphabet and δ ⊆ Q × Σ × Q isthe transition relation. We say that such an automaton is path-complete to emphasize the fact that its paths capturea complete description of the allowed behaviors of theswitching signal. We will denote any labeled transitionby the compact notation i σ → j ∈ δ . A finite sequence oftransitions from i to j will be represented by i σ ...σ r −→ j .The complexity of the switching behavior is modulatedby the automaton. An example is in Figure 1. Arbitraryswitches between two matrices A and A are easilycaptured by the automaton on the left. In contrast, theautomaton on the right enforces a switching behavior witha strict alternation between 0 and 1. A mixed situation isprovided by the automaton in the middle, whose switchessequences allow for any repetition of 1 separated byisolated zeros. q { , } q q q q Fig. 1. Automata with different path restrictions.The case of unconstrained switches is typical of ro-bust analysis where parametric uncertainties are mod-eled via nondeterministic switches among a family of lin-ear systems Liberzon (2003). Constrained switches arisefrom literature on hybrid/cyber-physical systems Essicket al. (2015); Bliman and Ferrari-Trecate (2003); Linand Antsaklis (2009). In constrained switching systems,specific sequences of operations are captured by suitablebranches of the automaton. Restrictions on paths couldbe used to model forms of ergodicity in the sequence ofmatrix operations, or to model the alternation betweenperiods of local/isolated operations and periods of collec-tive computations.
Since the nineties, several methods have been proposed forthe stability analysis of switched systems with or withoutrestrictions on the switching rules Bliman and Ferrari-Trecate (2003); Daafouz et al. (2002); Essick et al. (2014);Branicky (1998). We briefly summarize the recently pro-posed framework of path-complete Lyapunov functions,Ahmadi et al. (2014), that provides a unifying approach,and generalizes these techniques.
Definition 2.
Consider a constrained switching systemand let ( Q, Σ , δ ) be any path-complete automaton. A path-complete Lyapunov function is a multiple Lyapunov function given by a finite set of homogeneous positivedefinite functions ( V i ) i ∈ Q , V i : R n (cid:55)→ R + , such that V j ( A σ x ) ≤ γV i ( x ) . for each transition i σ → j ∈ δ and each x ∈ R n .The reason of this definition lies in the following theorem. Theorem 2. (Ahmadi et al. (2014)).Consider a constrained switching system and let ( Q, Σ , δ )be any path-complete automaton. The existence of a path-complete Lyapunov function for γ = 1 is a valid criterionfor the stability of the switching system. Asymptoticstability requires 0 ≤ γ < We follow the approach of path-complete Lyapunov func-tions to define the corresponding notion for positive sys-tems. Once again, the key step is to substitute cones tonorms.
Definition 3.
Consider a constrained switching systemand let ( Q, Σ , δ ) be any path-complete automaton forthis system. The constrained switching system is path-complete positive with respect to the set of cones K := {K q | q ∈ Q } if A σ K i ⊆ K j for each transition i σ → j ∈ δ . Strict path-completepositivity further requires that A σ K i ⊆ int K j for each transition i σ → j ∈ δ .The definition above reduces to positivity when eachcone in the set K is identical. Path-complete positivityis a proper generalization of positivity: Example 1 belowdiscusses the case of a path-complete positive switchingsystem that cannot be positive with respect to a commoncone. Example 1.
Consider the constrained switching system x + = A σ x with A = (cid:20) (cid:21) A = (cid:20) (cid:21) ;and suppose that the automaton in Figure 2 is pathcomplete.The system cannot be strictly positive with respect to acommon cone since the dominant eigenvector e of thematrix A is a non-dominant eigenvector of the othermatrix A and viceversa. It turns out that the system is Each K i is a pointed, convex, solid cone. q Fig. 2. One of the automata generating σ in Example 1.strictly path-complete positive with respect to the familyof cones ¯ K := {K , K } where K := { x ≥ , | x | ≤ x }K := { x ≥ , | x | ≤ x / } . One can check that the path-complete inclusions aresatisfied with such values of K , K : Indeed, followingthe automaton paths, any ( x , x ) ∈ K is mapped into( x +1 , x +2 ) ∈ int K by A : A K = (cid:8) x +1 = 5 x ≥ , | x +2 | = | x | ≤ x = x +1 / < x +1 / (cid:9) ⊆ int K . In a similar way,any ( x , x ) ∈ K is mapped into ( x +1 , x +2 ) ∈ int K by A : A K = { x +1 = 5 x ≥ , | x +2 | = | x | ≤ x / x +1 / Theorem 3. Consider a constrained switching system, let( Q, Σ , δ ) be any path-complete automaton, and supposethat the constrained switching system is path positive withrespect to the set of cones K := {K q | q ∈ Q } . Then, thereexists 0 ≤ γ ≤ i σ → j of theautomaton, d K j ( A σ x, A σ y ) ≤ γd K i ( x, y ) ∀ x, y ∈ K i . (3)Furthermore, strict path positivity guarantees 0 ≤ γ < Proof: Following the proof argument for Theorem 3.1in Bushell (1973a), one shows that path positivity guar-antees m K i ( x | y ) ≤ m K j ( A σ x | A σ y ) ≤ M K j ( A σ x | A σ y ) ≤ M K i ( x | y ) for each transition i σ → j ∈ δ , which directlyimplies (3) for γ = 1.For strict path positivity (3) with 0 ≤ γ < q ∈ Q , define the oscillation osc K q ( x | y ) := M K q ( x | y ) − m K q ( x | y ). Theorems 4 and 5 in Bauer (1965)show that osc K j ( A σ x | A σ y ) ≤ N ij ( A σ )osc K i ( x | y ) for each i σ → j ∈ δ , where the oscillation ratio 0 ≤ N ij ( A σ ) < A σ K i ⊆ int K j . This result is well known for positiveoperators from a cone into itself. The proof argument inBauer (1965) extends to the case of positive operatorsbetween two different cones. Finally, using the proof ar-gument of Lemma 3 in Bushell (1973b) one shows that d K j ( A σ x | A σ y ) ≤ N ij ( A σ ) d K i ( x | y ) for each i σ → j ∈ δ .Thus, γ := max i σ → j ∈ δ N ij ( A σ ) < (cid:50) At each transition i σ → j strict positivity guarantees thatthe linear map A σ is a contraction on the rays of thecones, in the sense of the adapted Hilbert metrics d K i . Itis easy to prove, by induction, that any pair ( x ( · ), y ( · )) oftrajectories of the system associated to the same switchingsignal σ ( · ) and such that x (0) , y (0) ∈ K q (0) \ { } satisfylim k →∞ (cid:12)(cid:12)(cid:12)(cid:12) x ( k ) | x ( k ) | − y ( k ) | y ( k ) | (cid:12)(cid:12)(cid:12)(cid:12) = 0 . (4)Equation (4) makes clear that a strictly positive systemasymptotically ‘forgets’ its initial condition, as it con-verges to a unique steady state solution in the projectivespace, for every switching signal.Note that the projective contraction property does notenforce convergence to a fixed point. For example, astraightforward consequence of the theorem is that eachcyclic path q σ ...σ r −→ q defines a corresponding path-dependent Perron-Frobenius eigenvalue and eigenvector, λ σ ...σ k and v σ ...σ r , such that A σ r . . . A σ v σ ...σ r = λ σ ...σ k v σ ...σ r (since ¯ A := A σ , . . . , A σ r is necessarily a strictly posi-tive matrix). Denoting rays by [ x ] := { λx | λ > } , asimple permutation of indices shows that [ v σ ...σ r σ ] =[ A σ v σ ...σ r ], [ v σ ...σ r σ σ ] = [ A σ v σ ...σ r σ ], and so on.Indeed, all the path-dependent Perron-Frobenius eigen-vectors on a cyclic path define an invariant sequence ofrays. Such sequence is also an attractor of the system.Thus, trajectories along these cycles either converge tozero or to a limit cycle of r rays. In that sense, path-positivity retains the fundamental contraction propertyof a positive system.5. ALGORITHMS FOR DECIDING POSITIVITYTesting the existence of a common invariant or contractivecone is hard Protasov (2010). In fact Protasov provedthat the question of whether a set of matrices has aninvariant cone is Turing-undecidable. His constructionsuggests that the question is hard when the matricesshare a common invariant linear subspace. For matricesthat do not share a common invariant subspace, wealgorithmically test whether a given set of matrices hasa common γ -contracting cone, for a given contractionratio 0 < γ < 1. We only discuss the algorithm in thecase of uniform positivity and leave for future work ageneralization to path-complete positivity. A single matrix admits a contracting cone if and only if ithas a leading eigenvector. An obvious necessary conditionfor uniform strict positivity w.r.t. a common cone K istherefore that each system A σ has a leading eigenvector.We introduce a corresponding splitting of the state-space,which relies on the eigenstructure of A σ . Definition 4. For any positive matrix A σ , we define the invariant splitting of R n ( V σ , N σ ) as the pair of two σ − invariant subspaces of dimension 1 and n − V σ is defined as the span of the Perron Frobeniuseigenvector of A σ . N σ is the unique n − A σ such that V σ ∩ N σ = { } (for example N σ could be defined by the columns of the coordinatetransformation that brings A σ into its real Jordan form).An elementary necessary condition is as follows. Proposition 4. If a cone K is invariant for the matrix A σ ,then necessarily K (cid:92) N σ = { } . Corollary 5. (Basic test). If a set of matrices M = { A σ } share a common contracting cone, then they all havea strictly dominant eigenvalue, and the correspondingeigenvector v σ does not belong to any N σ (cid:48) for any σ (cid:48) (cid:54) = σ. The basic idea of our algorithm below is to start from aninner bound, and proceed by forward iteration (i.e. applyour matrices to this inner bound) in order to enlarge it.For the initial inner bound, one can start with the convexhull of the leading eigenvectors of the matrices, whichmust be in any invariant cone. Given the set of leadingeigenvectors { v i } , it is not clear however whether to use v i or − v i in the initial inner bound. We resolve this choiceas follows: pick any matrix A σ and define w as the normalvector to the invariant subspace N σ . Then for each leadingeigenvector of the matrices A i , pick the orientation v i suchthat v Ti w > . We formalize the argument in the followingproposition. Proposition 6. (The orientation trick). Suppose that A , A ∈ R n × n have a common contracting cone K , andnote v , v the leading eigenvectors of A , A . Supposewithout loss of generality that v ∈ K . Then, with thenotations of Definition 4, v is also in K if and only if( w T v ) · ( w T v ) > , where w is the normal vector to N . Proof: If: one has either v ∈ K , or − v ∈ K . Now, if( w T v ) · ( w T v ) > , it means that ( w T v ) · ( w T ( − v )) < , and then there exist α, β > αv + β ( − v ) ∈ N , and thus ( − v ) cannot belong to K . Only if: Suppose by contradiction that ( w T v ) · ( w T v ) < . Then there exist α, β > αv + βv ∈ N , andthis contradicts v , v being in K , because K (cid:84) N = { } . (cid:50) By construction, the convex hull of leading eigenvectors(selected with the proper orientation) is an invariantcone. It thus provides an inner bound for the contractingcone K . However, the following proposition shows thatthis cone cannot be contracting, even if there exists acontracting cone. Proposition 7. Consider a set of matrices M ∈ R n × n , andthe set of leading eigenvectors v i ∈ R n of the matrices in M . Suppose that K = conic-hull (cid:91) A ∈M ∗ ,v i Av i is a closed convex pointed cone . Then, K is an invariantcone, but not a γ − contracting cone for any γ < . M ∗ is the set of all the products of matrices of M . Proof: It is obvious that K is invariant by definition of K . Now, let us suppose by contradiction that MK ⊂ int K . Since every v i is an eigenvector of some matrix A ∈ M , conic-hull { v i } is not contracting. Thus, thereexists some x ∗ ∈ K \ int K , x ∗ / ∈ conic-hull { v i } . Bydefinition of K , for any (cid:15) > , there is a x (cid:48) ∈ K , A ∈ M such that | Ax (cid:48) − x ∗ | < (cid:15). This is in contradiction withthe fact that MK ⊂ int K , MK being a finite unionof closed sets strictly contained in K . (cid:50) Example 2. Consider the set of matrices M = { A , A } .A = (cid:20) . 65 0 . (cid:21) A = (cid:20) . . (cid:21) . The leading eigenvectors are v = [1 . , T and v =[1 , . T . M possesses an invariant cone, which isconic-hull { v i } . However, the cone K (cid:15) = conic-hull { [ (cid:15), T , [1 , T } , for (cid:15) > (cid:15) = 0 . { v i } , it remains stuck in the conedelimited by these two vectors: K = conic-hull (cid:91) A ∈M ∗ ,v i Av i = conic-hull { v i } . That is, K is an invariant cone, but not contracting. Itis however included in the contracting cone K (cid:15) . (cid:121) In any contracting cone, Theorem 1 implies a uniformupper bound D A K on the distance between two points in A K . This bound on the distance is useful to build a largerinner bound on the contracting cone K : indeed, an upperbound on the distance between two points translatesgeometrically into a lower bound on the distance betweenany of these points and the boundary of the consideredcone (see Definition 1 of the Hilbert metric, and theproof of Lemma 8 below). Thus, we can leverage thisinformation in order to inflate the cone, by ‘pushing theboundaries’ of our inner bound. We formalize this in thenext lemma: Lemma 8. Let K be a γ -contracting cone for a set of linearmaps M , and take a matrix A ∈ M . For any x, y ∈ K ,consider x (cid:48) = Ax, y (cid:48) = Ay. Suppose that x (cid:48) − y (cid:48) / ∈ K ;then, for any ρ ≥ exp( D A K ) we have that y (cid:48) + 1 ρ − y (cid:48) − x (cid:48) ) ∈ K . Proof: From Definition 1 we have 0 ≤ m ( Ax | Ay ) < Ax − Ay / ∈ K . Thus, M ( Ax | Ay ) = M ( Ax | Ay ) m ( Ax | Ay ) m ( Ax | Ay ) ≤ exp( D A K ) m ( Ax | Ay ) ≤ exp( D A K ) . Furthermore, M ( Ax | Ay ) Ay − Ax ∈ K , thus ρAy − Ax ∈ K since ρ ≥ M ( Ax | Ay ) . Finally, writing ρAy − Ax + Ay − Ay ∈ K , we obtain Ay + ρ − ( Ay − Ax ) ∈ K . (cid:50) Lemma 8 provides a way to widen any inner bound of K in such a way that the widened cone is still a subset of K .ndeed, if an inner bound is not contractive, we can usethe lemma to widen its boundary slightly outwards beforepursuing the forward iteration algorithm. Lemma 8 andLemma 9 below are at the core of Algorithm 1, whichdecides in finite time whether a given set of matrices hasa common γ -contracting cone, as clarified in Theorem 10. Lemma 9. Let K be a cone in R n , x, y ∈ int K , and sup-pose that d K ( x, y ) > . Then, for any ( n − − dimensionalhyperplane H such that H (cid:84) K = { } , there exists a λ > y − λx ∈ H. Proof: For λ very small, we have y − λx ∈ K ; for λ verylarge, we have y − λx ∈ −K . Thus, by continuity, theremust be a λ such that y − λx ∈ H. (cid:50) In the next theorem, we suppose that the matrices do nothave zero eigenvalues, nor common invariant subspace.These are technical assumptions that hold for genericmatrices. Theorem 10. Consider a set of positive matrices M withnonzero eigenvalues and no common invariant subspace.Given a contraction ratio γ , Algorithm 1 decides infinite time whether the set of matrices has a common γ -contracting cone. • It returns a γ − contracting cone provided that sucha cone exists. • If there is no γ − contracting cone, it returns ‘NO’, ora δ − invariant cone, for γ < δ < , if it has foundone. Data : A set of matrices M , a number γ ∈ (0 , Result : Outputs YES if the set of matrices has a γ − contracting cone (and returns a descriptionof the invariant cone). If the set of matrices hasa strictly invariant cone, but no γ − contractingcone, it may return NO, or a δ -invariant conefor some δ > γ . begin K = conic-hull { v | v : leading eigenvector of A ∈M} % vectors v are picked according to Proposition 6 ρ ≥ exp(4tanh − ( γ )) t = 0 while true doif K t T N σ = { } for some N σ then Output NO; Exit. endif K t is strictly invariant then Output YES; Exit. end K t +1 = conic-hull { S A ∈M A K t ∪ K t } For all A ∈ M , for all vertices y, x ∈ K t such that ( Ay − Ax ) ∈ N σ for some σ ∈ M , K t +1 = conic-hull {K t +1 ∪ { Ay + ( Ay − Ax ) / ( ρ − } t = t + 1; endend Algorithm 1. An algorithm for deciding joint positivity Proof: The algorithm iteratively computes inner bounds K t for K . We first prove that indeed K t are valid inner bounds (provided that there indeed exists a contractingcone K ). We then prove that one of these inner bounds K t must be contracting for some t (and not only invariant).Thus, the algorithm will terminate with an effectivecontracting cone.The algorithm starts with K as a first inner bound (if K is not a convex pointed cone, then one directly concludesthat the set of matrices does not have a common invariantcone). Thus, suppose that K is a valid inner bound byProposition 7. Now, at every step, with an initial innerbound K t , the algorithm performs two operations. First,it takes the union of K t with all the images of this set K t +1 = conic-hull { (cid:91) A ∈M A K t ∪ K t } , which is clearly still an inner bound, by definition of acontracting cone. Then, for any two points x, y ∈ K t , itadds the point { Ay + ( Ay − Ax ) / ( ρ − } to K t +1 . For practical efficiency, the true algorithm can only do itfor points Ay, Ax that are vertices of the new inner bound.Also, one has to scale x in order to ensure the condition( Ay − Ax ) ∈ N σ , but, provided that Ax and Ay are notparallel, this is always possible by Lemma 9 above. Finallynote that there must always be an Ay ∈ K t +1 \ int K t +1 (if not, K t +1 would be contracting) and Ax non-alignedwith Ay (because the matrix A has nonzero eigenvalues).In turn, the condition ( Ay − Ax ) ∈ N σ implies that Ay − Ax / ∈ K (Proposition 4), and we can apply Lemma8. These new added points are guaranteed to be in theinvariant cone K , by Lemma 8, and this proves that K t +1 is still an inner bound.We now prove that this procedure generates a contract-ing cone after a finite number of steps (if there existsone). Suppose the contrary. Then, the inner bounds K t converge towards a cone K ∞ which is invariant, but notcontracting. Consider a vertex z of K ∞ , which is suchthat Az ∈ K ∞ \ int K ∞ . That is, Az is in the boundary of K ∞ . This implies that the inflating step in the algorithmis such that for all x ∈ K ∞ , Ax is in the same face of K ∞ as Az (because in the opposite case, the inflationstep would ‘push’ Az out of K ∞ , and Az would not be avertex anymore). K ∞ being of nonempty interior (becausethe matrices have no nontrivial invariant subspace), thisimplies that A has zero eigenvalues, a contradiction.In conclusion, the algorithm cannot converge to a noncontracting invariant cone. Thus, if there exists a con-tracting K , since K t are valid inner bounds (i.e. containedin K ), they will either converge to K , or the algorithm willstop before (having found another contracting cone). If,on the other hand, there is no invariant cone, the ‘innerbounds’ will keep growing until they intersect some N σ , and the algorithm will stop, concluding that there is no γ − contracting cone. (cid:50) 6. CONCLUSIONS AND FURTHER DIRECTIONS.In this work, we have introduced the concept of path-complete positivity, which generalizes the notion of pos-itivity. We showed that this notion can be useful, forinstance for (constrained) switching systems, for whichwe provide an example of system which is not positive,but yet, is path-complete positive. We showed that path-complete positive systems inherit much of the nice proper-ies of positive systems, and we sketched an algorithm todecide whether a switching system has an invariant cone.Our algorithm is inspired from the similar, and muchmore studied, problem of proving stability for switchingsystems. It proceeds by forward propagation, which is awell-known technique for proving stability of a switchingsystem. However, the positivity problem is more tricky,for several reasons: first, contrary to the stability prob-lem, one cannot take an arbitrary norm for initializing aforward propagation procedure. 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