On Timed Event Graphs Stabilization by Output Feedback in Dioid

Transcription

1 n Timed Event Graphs Stabilization by utput Feedback Dioid Bertrand Cottenceau, aurent Hardou, Jean-ouis Boimond To cite this version Bertrand Cottenceau, aurent Hardou, Jean-ouis Boimond. n Timed Event Graphs Stabilization by utput Feedback Dioid. 1st IFAC Symposium on System Structure and Control, Workshop on (max,+) algebras, Aug 2001, Prague, Czech Republic. pp.x-x, <hal > HA Id hal https//hal.archives-ouvertes.fr/hal Submitted on 17 Jul 2013 HA is a multi-disciplary open access archive for the deposit and dissemation of scientific research documents, whether they are published or not. The documents may come from teachg and research stitutions France or abroad, or from public or private research centers. archive ouverte pluridisciplaire HA, est destée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 N TIMED EVENT GRAPH STABIIZATIN BY UTPUT FEEDBACK IN DIID Bertrand Cottenceau, aurent Hardou, Jean-ouis Boimond 1 1 aboratoire d'ingenierie des Systemes Automatises, 62, avenue Notre-Dame du lac, ANGERS, FRANCE, Tel (33) , Fax (33) [bertrand.cottenceau, laurent.hardou, jean-louis.boimond]@istia.univ-angers.fr Abstract This paper deals with output feedback synthesis for Timed Event Graphs (TEG) dioid algebra. The feedback synthesis is done order to stabilize a TEG without decreasg its origal production rate, optimize the itial markg of the feedback, delay as much as possible the tokens put. Keywords Timed Event Graphs, (max,+) algebra, Residuation, Stability, Feedback Synthesis. 1 Introduction We are terested here the problem of Timed Event Graphs (TEG) stabilization. We rst recall that a TEG is a Petri net whose each place has one upstream transition and one downstream transition. This class of Petri nets admits a lear representation on (max, +) or (m, +) algebra [1] [4]. Property of stability is closely related to TEG structure. A TEG is said to be structurally stable if its markg (i.e., its number of tokens) remas limited for all rg sequence of put transitions (this denition is troduced [1, chap. 6]). The problem of TEG stabilization has been considered by Cohen et al. [3] and more recently by Commault [5]. Commault obtas a sucient condition of stability for TEG. Such a condition is satised if TEG is made strongly connected by addg paths (i.e., successions of places and transitions) between the output and the put of the TEG. Consequently, each place of the resultg TEG necessarily belongs to a circuit and its markg is then bounded. In addition, it is shown [1] that a controllable and observable TEG can be made stable, by addg an output feedback, without alterg its own production rate. Gaubert has shown [9] that the number of tokens that must be placed the feedback, order to stabilize a TEG, is a resource optimization problem which can be formulated as an teger lear program. The approach presented here is based, on the one hand, on Gaubert's work [9] and, on the other hand, on the work itiated [7]. The objective is here to synthesize a dynamic feedback which mimizes the number of tokens required, under the constrat that feedback keeps the origal throughput. In section 2, we will recall the algebraic tools necessary to feedback synthesis. We will briey recall, section 3, TEG modelization over dioid J; K and some periodic properties of TEG. In section 4, we will present how an existg feedback a TEG can be improved and the way which this can be applied to the problem of TEG stabilization. 2 Algebraic tools The reader is vited to consult [1] or [4] for a complete presentation of the followg theoretical recalls. 2.1 Dioid Theory Denition 1 (Dioid, Complete Dioid) A dioid D is a set endowed with two ternal operations denoted (addition) and (multiplication), both associative and both havg a neutral element denoted and e respectively such that is commutative and idempotent (8a 2 D; a a = a), is distributive with respect to and is absorbg for the product (8a 2 D; a = a = ). A dioid (D; ; ) is said to be complete if it is closed for nite sums and if multiplication distributes over nite sums too. The sum of all its elements is generally denoted T. Denition 2 (rder relation) A dioid is endowed with a partial order denoted and dened by the followg equivalence a b () a = a b. Denition 3 (Subdioid) et (D; ; ) a dioid and C D. (C; ; ) is said subdioid of D if ; e 2 C, and C is closed for and. Theorem 1 (Kleene star theorem) The implicit equation x = ax b dened over a complete dioid admits x = a b as least solution with a = i0 ai.the star operator is usually called Kleene star.

3 2.2 Residuation Theory In ordered set, equations f(x) = b may have either no solution, one solution, or multiple solutions. In order to give always a unique answer to this problem of mappg version, residuation theory [2] provides, under some assumptions, either the greatest solution ( accordance with the partial order) to the equation f(x) b or the least solution to f(x) b. Denition 4 (Isotone mappg) A mappg f de- ned over ordered sets is said to be isotone if a b ) f(a) f(b). Denition 5 (Residuation) et f E F, with (E; ) and (F; ) ordered sets. Mappg f is said residuated if for all y 2 F, the least upper bound of the subset fx 2 Ejf(x) yg exists and lies this subset. It is then denoted f ] (y). Mappg f ] is called the residual of f. When f is residuated, f ] is the unique isotone mappg such that f f ] Id and f ] f Id Theorem 2 ([1]) et f (D; ; ) (C; ; ) a mappg dened over complete dioids. Mappg f is residuated if, and only if, f() = and, 8A D, f( x2a x) = x2a f(x). Corollary 1 et a x 7 a x and R a x 7 x a dened on a complete dioid. Mappgs a and R a are both residuated. Their residuals will be denoted respectively ] a(x) = a nx and R a(x) ] = x=a Proof by denition, is absorbg for and product distributes over sums complete dioids. 2.3 Mappg restriction Denition 6 et f E F a mappg and A E a subset. We will denote f ja A F the mappg dened by equality f ja = f IdjA where IdjA A E is the canonical jection. Identically, let B F with Imf B. Mappg Bj f will be dened by equality f = IdjB Bj f where IdjB B F is the canonical jection. Proposition 1 et D a complete dioid and D sub a complete subdioid of D. Then, the canonical jection Idjsub D sub D; x 7 x is residuated. Its residual will be denoted Pr sub. Proof sce D sub is a subdioid of D and is complete, the result is immediate accordg to theorem 2. 3 TEG description dioid Max J; K 3.1 Dioid J; K. The put-output behavior of a TEG may be represented by a transfer relation some particular dioids. Hereafter, we will essentially represent TEG behavior on dioid J; K. et us recall that dioid Max J; K is formally the quotient dioid of B J; K, set of formal power series two variables (; ) with Boolean coef- cients and with exponents Z, by the equivalence relation xry () ( 1 ) x = ( 1 ) y (see [1],[4] for an exhaustive presentation). Dioid J; K is complete with a bottom element = +1 1 and a top element T = et us consider a representative s = i2n f(n i; t i ) n i t i B J; K of an element belongg to J; K. The support of s is then dened as f(n i ; t i )jf(n i ; t i ) 6= g and the valuation (resp. degree) of this element, denoted val (s) (resp. deg (s)) as the lower bound (resp. upper bound) of its support. A series of J; K is said polynomial if its support is nite. When an element of J; K is used to code a set of formations concerng a transition of a TEG, then a monomial k t may be terpreted as the k th event occurs at least at date t. 3.2 Realizability, Periodicity and Rationality Denition 7 (Causality) et h 2 J; K. h is causal either if (h = ) or (val (h) 0 and h val (h) ). The set of causal elements of J; K has a complete dioid structure. This dioid will be denoted + J; K. A matrix is said causal if each of its entries is causal. Denition 8 (Periodicity) et h 2 J; K. h is periodic if it exists two polynomials p and q, and a monomial r = such that h = p qr. The ratio = = is called the production rate of the series. The set of periodic series of J; K has a dioid structure denoted per J; K. A matrix H 2 J; Kpm is said periodic if all its entries are periodic. The production rate of this periodic matrix is then dened as = m ij. 1ip;1jm Denition 9 (Realizability) H 2 J; Kpm is said realizable if it exists four matrices A1, A2, B and C with entries f; eg such that H = C(A1 A2) B. Remark 1 In other words, there is a TEG whose transfer is H. Denition 10 (Rational) et h 2 J; K. h is rational if it may be written as a nite composition of sums, products and Kleene stars of element belongg to the set f; e; ; g. A matrix is said rational if all its entries are rational.

4 The followg theorem recalls that the put-output transfer of a TEG is characterized by periodic properties. Theorem 3 ([4]) et H 2 equivalent H is periodic and causal H is rational H is realizable. J; Kpm. Are Proposition 2 The canonical jection Id j+ + J; K J; K; x 7 x is residuated. Its residual will be denoted Pr + (x). Proof accordg to theorem 2, it suces to remark that canonical jection veries 8A + J; K; Id j+ ( x2a x) = x2a x. Practically, for all x 2 Pr + (x) is obtaed by J; K, the computation of Pr + ( f(n i ; t i ) ni ti ) = g(n i ; t i ) ni ti where i2n i2n g(n i ; t i ) = f(ni ; t i ) if (n i ; t i ) (0; 0) otherwise Theorem 4 ([8],[10]) et s1; s2 2 per J; K. Then, s1 ns2 2 per J; K. Proposition 3 et s 2 Pr + (s) 2 per J; K a periodic series. rat J; K is the greatest rational element less than or equal to s. Proof (sketch of proof) see [6] for further details. The proof consists remarkg that 8s 2 per J; K, Pr + (s) belongs to per J; K too. Moreover, Pr + (s) 2 + J; K. Accordg to theorem 3, such an element is then rational. 2 rat J; K. The ele- Proposition 4 et a; b ment Pr + (a nb) is the greatest rational solution of a x b. In that sense, we can consider that rat a rat J; K rat J; K; x 7 ax is residuated. Proof sce a and b are rational, they are periodic too (cf. theorem 3). Therefore, accordg to theorem 4, a nb is a periodic element but not necessarily causal 1. Furthermore, accordg to proposition 3, Pr + (a nb) is then the greatest rational solution of a x b. 1 for stance, and 2 2 are periodic and causal series, nevertheless 2 2 = = 1 1 is not causal.. 4 Feedback Synthesis for TEG 4.1 Greatest feedback In previous section, we have recalled that a TEG can be represented by its put-output transfer. For stance, considerg a TEG with m puts and p outputs, its put-output behavior may be simply written Y = HU, with H 2 rat J; K pm a rational matrix. Figure 1 represents the block diagram of a system U V Y H Figure 1 System H with an output feedback F denoted H on which has been added an output feedback F. By applyg theorem 1, closed-loop transfer of g. 1 is Y = H(F H) U rat J; K pm is the open-loop trans- rat J; K mp is the output feedback where H 2 fer and F 2 transfer. F ater on, we will denote M H the followg mappg M H J; Kpm J; Kmp X 7 H(XH) The mappg M H clearly represents the way which a feedback F modies the closed-loop transfer of a system H. In particular, M H is isotone sce it is a composition of isotone mappgs. Remark 2 M H (X) may also be written (HX) H sce H(XH) = H HXH HXHXH = (HX) H. Thanks to theorem 2, one can check that M H, de- ned over complete dioids, is not residuated. Indeed, M H (a b) 6= M H (a) M H (b). Nevertheless, the followg result shows that there exists a restriction of M H that is residuated. Proposition 5 et us consider mappg ImM H jm H J; Kmp X M H ( 7 H (XH) ImM H jm H is residuated and its residual is ] ( ImMH jm H ) M H ( J; Kmp ) X J; Kmp ) 7 H nx=h J; Kmp Proof this result rests on a and R a residuation (cf. corollary 1). It suces to show that equality H(XH) H(aH) (1)

5 admits a greatest solution 8a 2 J; Kmp. By considerg the Kleene star operator, (1) amounts to satisfyg the nite sequence of equalities HXH H(aH) ; H(XH) 2 H(aH) ; etc. Indeed, once the rst one is satised, the second one follows sce H(XH) 2 = (HXH)(XH) H(aH) (XH) = (Ha) HXH sce (Ha) H = H(aH) (Ha) H(aH) = H(aH) (ah) = H(aH) sce (ah) (ah) = (ah) The same holds true recursively for the next equalities. Hence we can concentrate on the rst one only, and clearly H n(h(ah) )=H provides the answer. Proposition 6 et us consider a TEG whose transfer is H 2 rat J; K pm endowed with an output feedback whose transfer is F 2 Then, rat J; K mp. ^F + = Pr + (H nm H (F )=H) is the greatest realizable feedback such that M H (F ) = M H ( ^F+ ). Proof clearly, M H (F ) 2 ImM H. So, accordg to proposition 5, sce ImMH jm H is residuated, equation M H (X) M H (F ) (2) admits ^F = H nm H (F )=H as greatest solution. In particular, sce for X = F the equality of (2) is veried, ^F is then the greatest solution to equation M H (X) = M H (F ). In other hand, M H (F ) is realizable, then periodic (cf. theorem 3), sce it represents the closed-loop transfer. Therefore, accordg to theorem 4, H nm H (F )=H is a periodic matrix but not necessarily causal matrix. Accordg to proposition 3, ^F+ = Pr + (H nm H (F )=H) is the greatest rational solution of M H (X) = M H (F ). Remark 3 Another terpretation consists sayg that for any realizable system H closed by a realizable feedback F, there is an optimal realizable feedback preservg the transfer of closed-loop system. Sce ^F + F, the system ^F+ delays the put of tokens system H, compared to the feedback F, while ensurg the same output. So, compared to the system F, the feedback ^F+ decreases the number of tokens, or their sejourn times, the system H. 4.2 Stabilization of TEG For TEG, stability property essentially means that tokens do not accumulate denitely side the graph or dierently that, for all puts, markg remas bounded. This property is obtaed when all transitions re with the same average frequency. A TEG is said structurally controllable (resp. observable) if every ternal transition can be reached by a direct path from at least one put transition (resp. is the orig of at least one direct path to some output transition)(see [1]). It has been showed that a structurally controllable and observable TEG can be made stable by addg an output feedback [3] [10]. Indeed, as soon as all transitions belongs to a sgle strongly connected component, the TEG is stable. Therefore, it suces that output feedback makes the TEG strongly connected to enforce stability. Moreover, stability may be obtaed order to preserve itial TEG production rate. The followg theorem, comg from [1], formalizes this result. Theorem 5 Any structurally controllable and observable event graph can be made ternally stable by output feedback without alterg its origal throughput Resource optimization feedback Accordg to theorem 5, a TEG can be made stable while preservg its trsic throughput. bviously, this feedback stabilization requires some amount of itial tokens feedback arcs. In manufacturg context, for stance when a TEG describes a production system, the itial feedback markg can represent some resources like transport means (used to convey parts) or recyclable maches. Consequently, it is particularly signicant to limit as much as possible their number. Here, we consider the problem of feedback markg mimization under both constrats of TEG stabilization and production rate preservg. This resource optimization problem, described more precisely thereafter, is tackled 2, and solved, by Gaubert [9]. et us consider a TEG made up of m puts and p outputs. Arcs provided with a place are added between outputs and puts so as the TEG becomes strongly connected 3. When strongly connectedness is reached, the problem consists calculatg the mimal number of tokens to be placed each of these arcs order to preserve the throughput of the open-loop system. The transfer of feedback system can be represented by a matrix F = F ij 2 M axrat J; K pm where F ij = qij if q ij tokens are itially allocated to place located between output j and put i, and F ij = if there is no arc. The problem lies the computation and mimization of q = fq ij g order that closed-loop system keeps the same production rate as the open-loop 2 other authors have solved such a problem but not necessarily with (max,+) approaches. 3 Practically, it is not always necessary to connect all outputs to all puts to obta strongly connectedness.

6 one. Gaubert [9] has shown that such a problem may be solved as an teger lear programmg problem where the lear cost function is J(q) = X i=m;j=p i=1;j=1 ij q ij ; with ij a price associated to each resource, and the constrat is (q) ; where is the production rate of the open-loop system and (q) is the production rate with feedback. If we denote w Nc (q) (resp. w Tc ) the (classical) sum of tokens (resp. holdg times) a circuit c, then w Nc (q) (q) = m ; c w Tc i.e., for each circuit the followg constrat will be satised w Nc (q) w Tc The solution of this teger lear program yields to q ij tokens that must be placed each feedback arc. We denote F R this feedback. Then, F R ensures closedloop stability, preserves the same production rate and mimizes the cost function Synthesis of a greater stabilizg feedback We propose here to improve the feedback obtaed above by computg the greatest dynamic feedback which preserves M H (F R ). Proposition 7 et us denote F R a feedback loop obtaed by solvg a resource optimization problem. The feedback loop ^FR+ = Pr + (H nm H (F R )=H) is the greatest realizable feedback such that M H (F R ) = M H ( ^FR+ ). Proof direct from proposition 6. This feedback can be seen as a renement to the solution brought by Gaubert [9]. Indeed, as we have explaed remark 3, feedback ^FR+ veries ^F R+ F R. Therefore, feedback ^FR+ releases put rgs latter than with feedback F R while ensurg the same output and the same resource number each feedback. Indeed, sce the itial markg (i.e., the resource number) of a path described by a periodic series s is equal to val (s), we obta ^FR + FR () 8i; j ^FR + ij FR ij ) 8i; j val ( ^FR + ij ) val (FR ij ) The last statement means that the resource number of each path of feedback ^FR+ is less than or equal to the ones of F R. In the other hand, M H ( ^FR+ ) = M H (F R ), and val (F Rij ) is the mimal number of tokens which allows to mimize J(q) while preservg the production rate. This latest statement leads to equality val ( ^FR+ ij ) = val (F Rij ) Illustrative example We present here how the precedg results can be implemented. et us consider the structurally controllable and observable TEG drawn solid les g.2. Its transfer matrix J; K22 is 9 () H = 5 () 15 ( 2 5 ) From this transfer matrix, we deduce that the TEG production rate is = 2=5 (see denition 8). This TEG represents a production unit with 4 maches denoted M1 to M4. Because of the dierence of production rates of maches constitutg this workshop, one notices that TEG model is not stable. Indeed, by rg all puts an nity times at a given date we can observe an accumulation of tokens upstream mache M 4. Therefore, stability of that system can be obtaed by addg an output feedback. It is suf- cient to make the TEG strongly connected to ensure its stability. In that particular case, the TEG becomes strongly connected by addg a feedback of the form F = q 11 q21 q22 We consider here the resource optimization problem order to mimize the followg cost function J(q) = q 11 + q 21 + q 22 (i.e., ij = 1). This problem can be solved by considerg the sum of tokens and temporization of each elementary circuit 4 which yields to the TEG production rate denoted (q) (q) = m 2 5 ; q 11 9 ; q 21 5 ; q Therefore, for q = (4; 2; 6), cost J(q) is mimum, i.e., F R = This stabilizg feedback that keeps origal throughput and mimizes resources number (tokens) is drawn dotted les g. 2. n the basis of this solution F R (obtaed by lear programmg approach) and accordg to proposition 7, we can rene this result by computg ^FR+ = Pr + (H nf R =H). We do not detail calculus here. The result obtaed is ^F R+ = (e )(2 5 ) A realization of that system is drawn g.3. Remark 4 We can notice that feedback ^FR+ has an arc y1 u2 that does not exist feedback F R. 4 the naive enumeration of elementary circuits is simpler than writg the lear program. But, for large graphs, such an enumeration becomes practically impossible (for a complete graph with n vertices, the enumeration complexity is ((n 1)))). Gaubert's approach [9] allows to consider only n 2 equalities.

7 K K & % % K 0 20 dates K Figure 2 System H with feedback F R Figure 3 System H with feedback ^FR+ Figure 4 represents the v1 rg sequence with ^FR+ (dotted le) and the v1 rg sequence with F R (solid le) for the same put u1 (dashed le). Clearly, feedback ^FR+ delays tokens entrance system H. For lack of place, we have not described rg sequences v2, y 1 nor y 2 for that simulation. We can only assert that outputs are identical both cases and that sequence v2 is not improved by the feedback ^F R+. References [1] F. Baccelli, G. Cohen, G.J. lsder, and J.P. Quadrat. Synchronization and earity An Algebra for Discrete Event Systems. John Wiley and Sons, New York, [2] T.S. Blyth and M.F. Janowitz. Residuation Theory. Pergamon Press, xford, [3] G. Cohen, P. Moller, J.P. Quadrat, and M. Viot. ear system theory for discrete-event systems. In 23rd IEEE Conf. on Decision and Control, as Vegas, Nevada, events Figure 4 v1 with ^FR+ (dotted les), v1 with F R (solid les) and u1 for both systems (dashed les). [4] G. Cohen, P. Moller, J.P. Quadrat, and M. Viot. Algebraic Tools for the Performance Evaluation of Discrete Event Systems. IEEE Proceedgs Special issue on Discrete Event Systems, 77(1)39{58, January [5] C. Commault. Feedback stabilization of some event graph models. IEEE Trans. on Automatic Control, 43(10)1419{1423, ctober [6] B. Cottenceau. Contribution a la commande de systemes a evenements discrets synthese de correcteurs pour les graphes d'evenements temporises dans les diodes. Phd thesis ( french), ISTIA Universite d'angers, [7] B. Cottenceau,. Hardou, J.. Boimond, and J.. Ferrier. Synthesis of greatest lear feedback for timed event graphs dioid. IEEE Trans. on Automatic Control, 44(6)1258{1262, June [8] S. Gaubert. Theorie des systemes leaires dans les diodes. Phd thesis ( french), Ecole des Mes de Paris, Paris, [9] S. Gaubert. Resource optimization and (m,+) spectral theory. IEEE TAC, 40(11)1931{1934, November [10] Max Plus. Second rder Theory of M-lear Systems and its Application to Discrete Event Systems. In Proceedgs of the 30th CDC, Brighton, England, December 1991.