Forward reachability analysis of max-plus linear systems using max-plus polyhedra*

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1 Forward reachability analysis of max-plus linear systems using max-plus polyhedra* Guilherme Espindola Winck, Mehdi Lhommeau Laurent Hardouin Laboratoire Angevin de Recherche en Ingénierie des Systèmes, Université d Angers, France ( s: guilherme.espindolawinck@etud.univ-angers.fr, mehdi.lhommeau@univ-angers.fr, laurent.hardouin@univ-angers.fr) Department of Automation and Systems Engineering - CTC-DAS, School of Control and Automation Engineering, Federal University of Santa Catarina - UFSC Florianópolis, SC, Brazil ( gwinck@grad.ufsc.br) Abstract: Lorem ipsum dolor sit amet, qui utinam aliquid epicuri et, hinc epicurei ei duo, invidunt probatus periculis pro ad. Doctus mentitum at mea, malis detraxit an duo. Essent facilis nominati ex vim, justo dolorem gubergren nec ea, harum corrumpit quo ea. Vix no ullum dolor graecis, at eum agam iudico mandamus, cu hinc noster sed. Nihil iisque vim at. Usu vidit movet rationibus an, quo alia homero epicuri id. Saperet commune te qui, no maiorum euripidis sit. Sit habeo invidunt legendos et, nec error noster urbanitas ex. Vel ea nostro delicatissimi, nam solet epicuri efficiantur an. Nam aeque nusquam ex. Keywords: (max,+) algebra, Discrete Event Dynamic Systems, Forward Reachability Analysis, Max Plus Linear Systems, Max Plus Polyhedra 1. INTRODUCTION Discrete Event Dynamic System (DEDS) subject is a subclass of dynamic systems where the state evolution depends entirely on the occurrence of asynchronous discrete events over time, which the decision-free can be modeled by a subclass of timed Petri nets, namely Timed Event Graphs (TEGs) and described in terms of Max Plus Linear (MPL) equations by a linear way. These kind of systems are used to model manufacturing systems, telecommunications networks, railway networks, and parallel computing (Baccelli et al. (1992), Sec.1.2) by other words systems that cannot be described by physics equations. The MPL equations are based in the MPL algebra, it is an idempotent semiring, an algebraic structure also called dioid (Baccelli et al. (1992), Chap. 3), that has two basics operators, the maximization and addition, and describes two phenomenas, respectively the Synchronization Phenomena, it means the start of a task waits for the previous set of tasks, and the Delay Phenomena, the time to complete a task is equal to the sum between the starting time and the task duration. Maximization, is the sum ( ). Addition, is the product ( ). By using MPL algebra we transform non-linearities into a linear form, by example max(a, b) = a b, and the linearity has performance analysis (Heidergott et al. (2006)) and control theory as the classical linear system theory. So it is possible to design an optimal open loop control (Cohen et al. (1999), Lhommeau et al. (2005), Menguy et al. This work was supported by the Brazilian Capes Agency through the Brafitec program. (2000)) as well optimal state-feedback control in order to solve the model matching problem (Cottenceau et al. (2001), Lhommeau et al. (2003), Maia et al. (2003,2005)). Reachability analysis is a fundamental problem when we talk about DEDS, because when we design a control system a key problem is to ensure the running of the system. For that purpose, it is useful to construct a TEGs of the system and extract certain properties like deadlock directly on the model. This is known as model checking. Exist a number of ways to determine by example if a state is reachable from a set of initial conditions, known as forward reachability problem. In Gazarik et al. (1999) residuation is used to determine if a state is reachable form a single initial condition. In Gaubert and Katz (2003), it is shown that if the initial set is a rational semi module the reachable set is also a rational semi module. These authors mention that this set has a simple shape and suggest that an efficient numerical method remains to be designed. In Lu et al. (2012) reachability problem is tackled by considering max-plus polyhedra, a more general kind of set than semimodule. In Adzkiya et al. (2014) the forward reachability problem addressed by considering an initial set depicted as the union of difference bound matrices (DBM). The authors state that a max-plus polyhedra can be depicted as an union of DBM and claim that their approach is more general than the one using max-plus polyhedra, the price to pay being the number of DBM to handle.

2 2. MATHEMATICAL BACKGROUND In this section we recall some of the notions we shall use in the following (Cohen et al. (2001)). 2.1 Max-Plus Algebra Let IR max denote the set IR { }, with two internal operations: sum( ) and product(, but this symbol is often omitted). The max-plus algebra is the semiring (IR max,, ), also called dioid (Baccelli et al. (1992), Def.4.1). { a b. = max(a, b) a b. = a + b where ε = and e = 0, are respectively null element (O) and the identity element (1). a IR max the following properties are respected: a ε = a e a = a (2) a ε = ε a a = a ( : Idempotent) IR max is linearly ordered with respect to, the partial order relation is given by a b a = a b, and in IR max coincides with in IR. 2.2 Complete dioid This natural order endows an idempotent semiring with a sup-semilattice structure (for which x y = x y is the least upper bound of x and y), and, in the case of IR max, it suffices to add + (denoted T) 1 to the set to obtain a complete sup-semilattice (in which arbitrary subsets have a least upper bound). The corresponding semiring will be denoted IR max. It is a standard result that complete sup-semilattices are also (complete) lattices, for which the greatest lower bound of x and y is denoted x y, i.e, the least upper bound of the (nonempty) subset of all elements which are less than x and y. We say that an idempotent semiring is complete 2 when it is complete as an ordered set, and when the product distributes over arbitrary sups. For instance, the semiring IR max is complete. (Notice that, in IR max, since O is absorbing, ε T = + = ε = O = ). It is straightforward to extend addition and multiplication to rectangular matrices. In particular, making the semiring of scalars IR max act on the additive monoid (IR n max, ) of n-dimensional columns vectors by multiplication, we equip IR n max with a structure of (free, finitely generated) semimodule, in which addition is idempotent. The max-plus algebra can be extend to matrices as follows. Given A, B IR n p max and C IR p q max then: [A B] ij = a ij b ij and [A C] ij = p k=1 a ik c kj. 1 A semiring T has the greatest element denoted by T (T is called the top element of T ) i.e. T = x T x = +. 2 For complete dioids, the partial order relation is: a b = a b a b = b a. (1) Example 1. Given A = C = ( 1 e 2 ). Then: A B = A C = ( ) 2 3 e ε e 4, B = 4 1 ε ( ) ε e e ε 3 e = ε e ( ) 1 ε e e ( ) 2 3 e 3 4 4, 4 1 e ( ) ( ) e e 2 3 ε 1 e e 4 2 = e ε Residuation Theory - Cohen et al. (2001), Sec and Residuated maps A mapping f : U X between two ordered sets is residuated if it is isotone (that is, order preserving), and if, for all x X the subset {u U f(u) x} admits a maximal element, denoted f (x). The isotone mapping f : X U is called the residual of f. The residual f is the only isotone mapping satisfying the following properties: 3 f f M1 n, f f M1 n (3) A simple characterization holds in the case of complete lattices. Before considering it, let us introduce some terminology. When U and X are lattices, we say that f : U X is a ( or sup)-morphism if f(u v) = f(u) f(v) u, v U (same terminology with ). When the lattices U and X are complete, we say that f is( or sup)-continuous if f preserves least upper bounds of arbitrary sets (specializing this property to the empty set we get f(ε) = f(sup ) = sup = ε, where, ε denotes the bottom element of an ordered set). The dual property for is called ( or inf)-continuity. These properties are called lower( ) and upper( ) semicontinuity, respectively (Baccelli et al. (1992), Def.4.43). Semimodules A finitely generated semimodule V IR n max is the set of linear combinations of a finite family {u 1,..., u p } of vectors of IR n max: V = { p i=1 λ iu i λ 1,..., λ p IR max }. In matrix terms, V can be identified to the column space { or image of the } n p matrix A = [u 1,..., u p ], V = I(A) =. Ax x IR p max. Finally, if U and X are semimodules, we say that f is linear if it is an additive morphism and, in addition, f(α u) = α f(u) with α a scalar and u U. Now, returning to our residuation summary, f is residuated if f is ( or sup)-continuous. In particular, linear mappings between free finitely generated semimodules are residuated. The following identities can be easily derived from (3): 3 Where M1 n is the identity map, without reference to the underlying set, which should be clear from the context.

3 f f f = f, f f f = f, (h f) = f h (4) where f, h are residuated mappings with f : U X, h : X Y. The notion of dually residuated mapping is defined naturally by reversing the order in the above definitions. See Baccelli et al. (1992) Sec.4.4 for details. We use the notation f for the dual residual of f. An immediate consequence of characterization (3) and its dual is that a residuated map f is itself dually residuated and (f ) = f. Scalar Residuation The idempotent law induces that a semiring has an order structure, i.e:, a b = a b a b = b a, and considering the mapping L a : x a x 4 for some given a is such that L a (c b) = L a (c) L a (b), (a (c b)) = a c a b, as shown in (4) this theory is suitable to consider inversion problem over idempotent semiring. Its residual L a 5 is denoted y a \y and is actually the classical definition of subtraction of a from y with the additional rule (which results from the very definition): ε \ε = T (i.e, ( ) = + = + to be contrasted with ε T = ε which may also, ambiguously be written as + (+ ) = + = ). Inequality a x b Over a complete idempotent semiring T, inequality a x b admits a greatest solution, denoted, x = a \b. Example 2. In IR max, if a and b are finite, we have a \b. = b a and a b. = min(a, b). Then the inequality 5 x 3 admits a greatest solution x = 5 \3 = 3 5 = 2. It achieves equality in the scalar case. 2.4 Matrix Residuation Similar considerations apply to L a by a rectangular matrix M IR rect max are obtained as follows: Given A IR m n max, B IR m p max and C IR n p max, the entries of C = A \B are given by: (Matrix Left Residuation) m C 1 i n,1 j p = (A ki \B kj ) (5) k=1 Where A \B is the greatest matrix solution X such AX B. Therefore, calculating (5) amounts to performing a kind of the (left) matrix product (L a ) of B by the transpose A where the scalar multiplication ( ) is replaced by (left) scalar division (L a) and scalar addition ( ) is replaced by lower bound ( ) 6 (Baccelli et al. (1992), Lemma.4.83). Example 3. Given A = A \B = ( 2 5 ε ) ( 2 \6 ε \4 1 \9 5 \6 3 \4 8 \9 ) ( 6 and B = 4 9 ) ( 4 =. 1). Then: 4 L a is the Left multiplication by a. 5 L a is the Left division by a. 6 i.e. A \B = A T B. Where is the min product between two matrices. In IR max, if a and b are finite, then: a b. = b a. 2.5 Max-Plus Linear System The autonomous model of an MPL system is given by: x(k) = A x(k 1) (6) where the elements a ij of matrix A IR n n max represents the minimal delay between two events. The event-number is represented by k N and the state vector x IR n max is a dater. It means x = [x i (0) x i (1) x i (2)... x i (k th)]. The non-autonomous model of an MPL system is defined by considering an external input u in (6) : x(k) = A x(k 1) B u(k) (7) where u is often used to control the system, and B IR n m max. Any non-autonomous MPL system can be transformed into an augmented MPL model by considering F = (A B) 7 IR n (n m) max and y(k 1) = (x(k 1) T u(k) T ) T (Baccelli et al. (1992), Sec.2.5.4). Then, as follows: x(k) = F y(k 1) (8) 3. TROPICAL POLYHEDRA CONES AND TROPICAL POLYHEDRON The element of R d max can be thought as points of an affine space, or as vectors. The addition and multiplication can be naturally extended to R d max. Given two elements x, y R d max, x y is the element with entries x i y i. Similarly, the multiplication of a vector x R d max by a scalar λ R max is denoted by λx, and is the element with entries λx i. Finally, the tropical addition is extended to the Minkowski sum of two sets S, S R d max, denoted by S S, and defined as {x x (x, x ) S S }. A set C R d max is said to be a tropical convex set if for all u, v C and λ, µ R max such that λ µ = 1, λu µv C. Tropical convex sets are defined as the tropical analogues of convex sets. In tropical linear combinations, the requirement that λ, µ be nonnegative is omitted. Indeed, 0 = λ holds for all scalar λ R max. The set of combinations λu µv for λ µ = 1 represents the tropical segment joining the two elements u and v. Example 4. Figure 1 depicts three kinds of tropical segments in dimension R 2 max. Fig. 1. The three kinds of generic max-plus segments in R 2 max. 7 M m: augmented matrix M with m. Fig. 2. A tropical cone in R 2 max.

4 The tropical convex hull co(s) of a subset S R d max is the set of the tropically convex combinations α 1 x 1... α p x p, where p 1, x 1,..., x p S, α 1,..., α p R max and α 1... α p = 1. Similarly, given a subset S R d max, the tropical convex cone cone(s) of S is defined as the set of the tropically convex combinations α 1 x 1... α p x p where p 1, x 1,..., x p S and α 1,..., α p R max. Example 5. An example of tropical convex cone in R 2 max is given by Figure 2. A tropical convex set (resp. tropical convex cone) is said to be finitely generated if it is the the form co(s) (resp. cone(s)) for some finite subset S R d max. Given a tropical convex set C R d max, an element x C is said to be an extreme point (or vertex) of C if for all u, v C and λ, µ R max such that λ µ = 1, x = λu µv x = u or x = v. Analogously, when C is a tropical convex cone, a non-null element x C is said to be an extreme generator of C if for all u, v C, x = u v x = u or x = v. In this case, the set of the form {λx λ R max } is said to be an extreme ray of C, and the vector x is a representative of this ray. Tropical polyhedra cone are the analogues of convex polyhedra in tropical algebra. Tropical polyhedra are defined as the intersection of finitely many tropical halfspaces. A tropical halfspace is a set of the vectors x = (x i ) R d max verifying an inequality constraint of the form : a i x i b i x i, 1 i d 1 i d where a = (a i ), b = (b i ) R 1 d max. It can be equivalently written as the set of the solutions of a system of inequality constraints A x B x, where A = (a ij ) and B = (b ij ) are n d matrices with entries in R max. It is important to note that in R d max, systems of equality and inequality constraints are equivalent. Indeed, any inequality can be written as an equality since y z y = y z. As a consequence, systems of inequality and equality constraints have the same expressiveness. For shake of readability, tropical polyhedral cones will be often referred to as polyhedral cones or cones. Polyhedral cones are known to be generated by their extreme rays Gaubert and Katz (2007b); Butkovič et al. (2007). A finite set G = (g i ) i I of vectors is said to generate a polyhedral cone C if each g i belongs to C, and if every vector x of C can be written as a tropical linear combination i λ ig i of the vectors of G (with λ i R max ). A tropical polyhedron of R d max is the affine analogue of a tropical polyhedral cones. It is defined by a system of inequalities of the form A x c B x d, where A, B R n d max and c, d R n max. Each row of the system is an affine inequality, and corresponds to an tropical affine halfspace, that is, a set consisting of the element x = (x i ) R d max verifying an inequality constraint of the form : a i x i c b i x i d, 1 i d 1 i d where a = (a i ), b = (b i ) R 1 d max and c, d R max. The description of tropical polyhedra and polyhedral cones as intersections of half-spaces is said to be external. Moreover, tropical polyhedra and polyhedral cones admit an internal representation by means of finitely many points and rays, as established by Gaubert and Katz (2011). This corresponds to the tropical analogue of the Minkowski- Weyl theorem. Then, the tropical polyhedral of R d max are precisely the sets of the form co(p ) cone(r), where P and R are finite subsets of R d max. A fundamental problem in the computational aspects of tropical polyhedra is given a tropical polyhedron, how to compute an internal representation from an external representation. Indeed, the tropical analogue of the Minkowski-Weyl (see Gaubert and Katz (2007a)) states that tropical polyhedra and tropical polyhedral cones are exactly finitely generated tropical convex sets and tropical convex cones respectively : Theorem 1. (Gaubert and Katz (2007a)). The tropical polyhedra of R d max are precisely the sets of the form co(p ) cone(r) where P and R are finite subset of R d max. Theorem 2. (Gaubert and Katz (2007a)). The tropical polyhedral cones of R d max are precisely the sets of the form cone(g) where G is a finite subset of R d max. More generally, every tropical polyhedron of R d max can be represented by a tropical polyhedra cone of R d+1 max thanks to an analogue of the homogenization method used in classical case. Indeed, a non-homogeneous system A x b = C x d can be associated to the homogeneous system (A b) z = (C d) z of R d+1 max. That is why, in the sequel, we will only state the main results for cones. We next introduce the tropical analogue of the double description method proposed by Allamigeon et al. (2010). The tropical analogue double description method computes a minimal generating set of a polyhedral cone, starting from a system of tropically linear inequalities defining it. The tropical double description method is conceptually similar to the one proposed in Butkovič et al. (2007), being capable of generating the entire set of solutions by solving the system row-by-row. It uses, however, a more elaborated approach for solving each equation, using the concept of extreme rays. This leads to a more compact representation of the intermediate solutions set and thus the method has a substantially better average complexity than the Elimination Method. The algorithm given in Allamigeon et al. (2010) is based on an incremental technique based on successive elimination of inequalities. Given a polyhedral cone defined by a system of p constraints, it computes by induction on k = 1,..., p a generating set G k of the intermediate cone defined by the first k constraints. Passing from the set G k to the set G k+1 relies on a result which, given a polyhedral cone C and a tropical halfspace H = {x a x b x}, allows to build a generating set G of C H from a generating set G of C. Let us denote by ε i 1 i d the element of Rd max whose i-th coordinate is equal to 1, and the other coordinates to 0. Intuitively, the set ε i 1 i d is a tropical analogue of the canonical basis. Then, the following theorem describes the whole tropical double description method:

5 Theorem 3. ( Allamigeon et al. (2011)). Let C be a tropical polyhedral cone generated defined as the set {x R d max A x B x}, where A, B R p d max (with p 0). Let G 0,..., G p be the sequence of finite subsets of R d max defined as follows : G 0 = { ε i} 1 i d, G i = {g G i 1 A i g B i g} = {(A i h) g (B i g) h g, h G i 1, A i g B i g, and A i h > B i h}, for all k = 1,..., p, where A i and B i are the i-th rows of A and B. Then, C is generated by the finite set G p. Example 6. We will illustrate the results on the tropical polyhedron P defined by the system (9), or equivalently on its homogenized cone C defined by the system (10). { 0 x1 + 3 x max(x 2, 0) 0 max(x 1, x 2 2) (9) { x3 x x 1 max(x 2, x 3 ) x 3 max(x 1, x 2 2) (10) It is important to note that this inductive approach produces redundant generators. To eliminate this redundant generators, the principle is the following : an element h is extreme in the cone generated by a given set H if and only if h can not be expressed as the tropical linear combination of the elements of H which are not proportional to it. This property can be checked using residuation (see Butkovič et al. (2007) for algorithmic details). 4. REACHABILITY ANALYSIS Zones and Polyhedra A set { of initial conditions } is defined by a zone as R = X 0 x IR n max : S where S = {Ax b}. Where A IR m n max, b IR m 1 max and = {<,, >, }. A single point of initial condition x 0 is located in a zone of constraints R, which means x 0 X 0. A zone R can be defined as a max-plus polyhedron using Algorithm (??). Dynamical System Let us consider an autonomous system x(k + 1) = A x(k) where A IR n n max. 4.1 Direct Image I(X k 1 ) - Forward Reachability With focus on autonomous MPL systems, given a set of initial conditions X 0, the reach set X k is recursively defined as the image of X k 1 : X k = I(X k 1 ) = {A x : x X k 1 } = A X k 1 (11) Lemma 1. Let be f k : IR n max IR n max a linear map, P = co(v ) cone(w ) a max-plus polyhedron. Then: f k (P) = co(f k (V )) cone(f k (W )) (12) k {1, 2,..., k th} event We can prove it as follows: Application Function: f k (H) = A H (13) Applying f k (P) then, P(V, W ) = co(v ) cone(w ) A P(V, W ) = A (co(v ) cone(w )) As we know a function on classical algebra f(x) = a g(x) is equal to f(x) = g(a x) if x f(x) is a linear map. Then, A P(V, W ) = co(a V ) cone(a W ) and finally: f k (P) = co(f k (V )) cone(f k (W )) Finally we obtain a new Polyhedron P k where the set of states X k P k k {1, 2,..., k th} event. 4.2 One-shot computation of the reach set In this section we design a procedure for computing the reach set for a specific event step N using a tailored (oneshot) procedure. Given a set of initial conditions X 0, we compute the reach set at event step N using: X N = (I I)(A) = I N (A) = { A N x : x X 0 } Where A N is the N-th power of A. 8 Applying (13): (14) f N (P 0 ) = A N P 0 (15) Then, A N P 0 (V 0, W 0 ) = co(a N V 0 ) cone(a N W 0 ) and finally: f N (P 0 ) = co(f N (V 0 )) cone(f N (W 0 )) (16) We obtain a new Polyhedron P N where the set of states X N P N defined by the initial Polyhedron P 0 with X 0 P NUMERICAL REPRESENTATION OF A MAX-PLUS POLYHEDRON - DRAWING A POLYHEDRON Using (??): V = { v 1,..., v p} A = {α 1,..., α p } Where p i=1 α i = 1 α i 0 i {1,..., p} W = { w 1,..., w q} B = {β 1,..., β q } Where β j IR max j {1,..., q} 8 Given m IN, the m-th max-algebraic power of A IR n n max is denoted by A m and corresponds to A A (m times). Notice that 1... O A 0 is n-dimensional max-plus identity matrix: O... 1 n n

6 = α 1 v 1... α p v p β 1 w 1... β q w q p = max α i v i, i=1 q β j w j (17) j=1 By repeating it N times we obtain a set of points such that by approximation represents the polyhedron P. 5.1 Drawing 2D Example Example 7. Given the Polyhedron P hexagon: representing a (( ) ( ) ( )) (( )) P hex O = co,, cone O (18) By using the procedure presented in Sec. 5 we obtain the following drawing of the hexagon: 1 O O O O 1 O O ) O 1 O 2 1 O x 2 O = O 1 6 O 1 ( x1 ) x 2 ( x1 6.2 Polyhedron P X0 = co(v ) cone(w ) Using Algorithm (??): P X (21) (( ) ( ) ( )) (( )) O = co,, cone O 6.3 Computing I(X 0 ) (22) Then, the Direct Image of X 0 is the polyhedron P X1 that is defined and represented by Sec. 8: X 1 = I(X 0 ) = {A x : x X 0 } = A X 0 (23) P X1 = ( [( ) 2 = co A 4 ( ( )) O cone A O ( ) 1, 4 ( )]) 1, 6 (24) Then: P X1 (( ) ( ) ( )) (( )) O = co,, cone O (25) Fig. 3. Example of numerical representation of a max-plus hexagon 6. RUNNING A NUMERICAL EXAMPLE WITH A DETERMINISTIC SET OF INITIAL STATES Example 8. Given the MPL system below: x(k + 1) = And the set of initial conditions: [ ] 2 5 x(k) (19) 3 3 X 0 = { } x IR 2 max : 0 x 1 2, 4 x 2 6; (20) The goal is compute the Forward Set, then for this purpose we create a matrix model of the Box X 0 and finally we are able to use Algorithm (??) and calculate the Direct Image for k times (events). 6.1 Model Ax b = Cx d Fig. 4. Polyhedra X 0 and X 1 7. CONCLUSION Lorem ipsum dolor sit amet, quo ne mutat putant quaestio, an summo noluisse eum. Sit ei iudicabit reprehendunt, ad solum nemore cotidieque eos. In reque minimum eum,

7 decore possit minimum no usu. Ex facete oportere pro, te qui sale tempor definitiones. Pro no tantas primis constituto, mei an nibh tota. Pro ad idque iusto, legere viderer te cum. Eu legere euripidis nec. Veniam sensibus appellantur ad eam, te cum minimum indoctum tractatos. No dolor sententiae nam, dicit graeci vix no. Eum legimus antiopam rationibus ei, ex ullum minim euismod est, ut alia sensibus duo. Ut cum saepe admodum accusam, vim purto rebum malorum id, at dolorum habemus philosophia nam. At nusquam electram assentior nam, malorum postulant his in. Esse simul inimicus ea cum, fierent signiferumque et quo, ea ludus ceteros interesset sit. Sumo velit an sea, quas possim eos ex, est modo eruditi delenit no. In mutat modus cum, ridens vituperatoribus ei qui. Cu sea erat reque eligendi, an est docendi tibique voluptatum. Appareat legendos vim an. Te ubique impetus voluptatum nec, ei mel meis nobis. Pro nostrud debitis in. Illud recteque mel ad, vix lorem argumentum sadipscing et. Ne illum consulatu moderatius sea. Eu nostrum menandri lobortis eam, aliquip admodum id duo, pri ea alia platonem pertinacia. Duo ad. REFERENCES Adzkiya, D., De Schutter, B., Abate, A., Apr Forward reachability computation for autonomous maxplus-linear systems. In: Proceedings of the 20th International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS 2014). Grenoble, France, pp Adzkiya, D., De Schutter, B., Abate, A., Computational techniques for reachability analysis of max-pluslinear systems. Automatica 53 (3), Allamigeon, X., Gaubert, S., Goubault, E., Inferring min and max invariants using max-plus polyhedra. In: Proceedings of the 15th International Symposium on Static Analysis. SAS 08. Springer-Verlag, Berlin, Heidelberg, pp Allamigeon, X., Gaubert, S., Goubault, É., The Tropical Double Description Method. In: Marion, J.-Y., Schwentick, T. (Eds.), 27th International Symposium on Theoretical Aspects of Computer Science. Vol. 5 of Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, pp URL /2443 Allamigeon, X., Gaubert, S., Katz, R., The number of extreme points of tropical polyhedra. Journal of Combinatorial Theory, Series A 118 (1), URL article/pii/s Baccelli, F., Cohen, G., Olsder, G., Quadrat, J., Synchronization and Linearity : An Algebra for Discrete Event Systems. Wiley and Sons. Butkovič, P., Hegedus, G., : An elimination method for finding all solutions of the system of linear equations over an extremal algebra., pp Butkovič, P., Schneider, H., others, Generators, extremals and bases of max cones. Linear algebra and its applications 421 (2-3), URL article/pii/s Cohen, G., Gaubert, S., Quadrat, J.-P., Max-plus algebra and system theory: Where we are and where to go now. Annual Reviews in Control 23, Cohen, G., Gaubert, S., Quadrat, J.-P., Hahn- Banach separation theorem for max-plus semimodules. J.L. Menaldi, E. Rofman, A. Sulem (Eds.), Optimal Control and Partial Differential Equations, IOS Press, pp Cottenceau, B., Hardouin, L., Boimond, J.-L., Ferrier, J.- L., August Model Reference Control for Timed Event Graphs in Dioid. Automatica 37, Dyhrberg, J., Lu, Q., Madsen, M., Ravn, S., Fahrenberg, U., Computations on Zones using Max- Plus Algebra. In: Proc. 22nd Nordic Workshop in Programming Theory (NWPT10). URL pdf Gaubert, S., Methods and Applications of (max,+) Linear Algebra. Research Report RR-3088, INRIA, projet META2. URL Gaubert, S., Katz, R., Reachability and invariance problems in max-plus algebra. In: Positive Systems, Proceedings of the First Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2003), Rome, Italy. pp Gaubert, S., Katz, R., 2007a. The minkowski theorem for max-plus convex sets. Linear Algebra and its Applications 421 (2), , special Issue in honor of Miroslav Fiedler. Gaubert, S., Katz, R., Minimal half-spaces and external representation of tropical polyhedra. Journal of Algebraic Combinatorics 33 (3), Gaubert, S., Katz, R. D., 2007b. The minkowski theorem for max-plus convex sets. Linear Algebra and its Applications 421 (2-3), , special Issue in honor of Miroslav Fiedler. Gazarik, Michael, J., Kamen, Edward, W., Reachability and observability of linear systems over max-plus. Kybernetika 35 (1), [2] 12. Heidergott, B., Olsder, G., van der Woude, J., Max Plus at Work: Modeling and Analysis of Synchronized Systems : a Course on Max-Plus Algebra and Its Applications. No. v. 13 in Max Plus at work: modeling and analysis of synchronized systems : a course on Max-Plus algebra and its applications. Princeton University Press. Lhommeau, M., Hardouin, L., Cottenceau, B., Optimal control for (max, +)-linear systems in the presence of disturbances. In: Positive Systems, Proceedings of the First Multidisciplinary International Symposium on Positive Systems: Theory and Applications (POSTA 2003), Rome, Italy, August 28-30, pp Lhommeau, M., Hardouin, L., Ferrier, J.-L., Ouerghi, I., Interval analysis in dioid: Application to robust open-loop control for timed event graphs. In: Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC th IEEE Conference on. pp Lu, Q., Madsen, M., Milata, M., Ravn, S., Fahrenberg, U., Larsen, K. G., Reachability analysis for timed

8 automata using max-plus algebra. The Journal of Logic and Algebraic Programming 81 (3), Maia, C., Hardouin, L., Mendes, R., Cottenceau, B., December Optimal Closed-loop Control of Timed Event Graphs in Dioids. IEEE T.A.C. 48 (12), Maia, C., Hardouin, L., Santos-Mendes, R., Cottenceau, B., On the model reference control for maxplus linear systems. In: Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC th IEEE Conference on. pp Menguy, E., Boimond, J. L., Hardouin, L., Ferrier, J. L., nov Just in time control of linear systems in dioid: cases of an update of reference input and uncontrollable input. IEEE TAC 45 (11), Paffenholz, A., Polyhedral geometry and linear optimization. Preliminary version of July. URL bc125c152423aaaeed5e63c3cd396faa7742.pdf

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