Proceedings of the 2001 IEEE International Symposium on Intelligent Control September 5─7, 2001 • México City, México
Optimal Sensor Choice for Observability in Free-choice Petri Nets L. Aguirre-Salas, O. Begovich and A. Ramirez-Trevifio
CINVESTAV Guadalajara Av. Lopez Mateos Sur 590, Guadalajaxa, Jal., 45090, Mexico E-mail:
[email protected] Abstract--This paper s t u d i e s t h e o p t i m a l sensor choice for observability p r o b l e m in D i s c r e t e E v e n t S y s t e m s (DES) m o d e l e d by Interpreted Free-choice Petri N e t s (IFCN). Taking advantage o f t h e fact that, eventd e t e c t a b i l i t y is a necessary and sufficient c o n d i t i o n for observability in live, cyclic and b o u n d e d IFCN, a structural characterization of e v e n t - d e t e c t a b i l i t y and a s i m ple p o l y n o m i a l a l g o r i t h m to choose a m i n i m a l cost sensor configuration for observability in this class o f nets are provided. T h i s a l g o r i t h m is successfully applied to an illustrative e x a m p l e . Keywords--Discrete E v e n t S y s t e m s , P e t r i N e t s , Interpreted Petri N e t s , ObservabUity, E v e n t - D e t e c t a b i l i t y , Sensor Choice. I. INTRODUCTION An interesting problem in Discrete Event Systems ( D E S ) is concerned with the selection of the lowest cost sensor configuration satisfying that all internal signals of a given system can be determined. This problem has been addressed from several points of view. In [1] and [2], authors use Finite State Machines ( F S M ) to model a D E S . T h e y present algorithms to choose a minimal set of measurable events such that the resulting a u t o m a t a is observable in the sense of the definition presented in [3], where the complete determination of the occurrences of non-measurable events or the exact reconstruction of the system state are not required. Using the same formalism, in [4] Bavishi and Chong make a partition of the state set according with a priori known possible failure scenarios. Based on this knowledge, they present an algorithm to determine the minimal number of event sensors needed to distinguish between the events belonging to different failure scenarios. In the other hand, modeling with timed F S M , in [5], Park and Chong provide an algorithm to compute a minimal cost event and state sensor configuration such that the occurrence of timed events can be reconstructed. Although the F S M is suitable for describing D E S , its application is limited to small size systems, since the models result quite large when the size of the system grows. In order to cope with the state explosion problem, Petri nets ( P N ) are being used as modeling forrealism for D E S . The P N formalism provide a clearer
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graphical descriptions and simple and sound mathematical support, allowing to represent DES properties such as process synchronization, concurrence, etc. [6]. T h e approach herein presented uses Interpreted Petri nets ( I P N ) [7] to represent D E S . T h e I P N are an extension of P N allowing to represent the input signals associated to transitions and the output signals generated when a marking is reached. Based on this modeling tool, this work presents a structural characterization of event-detectability and then an algorithm determining a minimal cost configuration of sensors such that the observability of a live, bounded and cyclic Interpreted Free-choice Petri Net ( I F C N ) is preserved. This algorithm has a polynomial complexity and exploits the structural characterization of event-detectability herein presented. T h e o u t p u t of the algorithm is a set of places and transitions t h a t must be measurable to maintain the observability property of the I F C N and, at the same time, obtaining the minimal cost sensor set. This paper is organized as follows. Section II presents an overview of several P N and I P N basic concepts and introduces the necessary notation. In Section III, the concepts of event-detectability, observabflity and a sufficient condition for observability are reviewed. In Section IV, a structural characterization of event-detectability and an algorithm to choose a minimal cost sensor configuration for observabflity are derived. Finally, Section VI presents the conclusions of this work. II. BASIC DEFINITIONS This section briefly introduces the main concepts related with P N and I P N . Definition 1: A Petri net structure is a 4-tuple N ---( P, T, I, O ), where P = {Pl , P2 , ... , P~ } and T ----{tl,t2,...,t,~} are finite sets of elements called places and transitions, respectively, such that P N T ---- 0; I :P x T , {0, 1} is a function representing the arcs going from places to transitions aad O : P x T ~ {0,1} is a function representing the arcs going from transitions to places. A P N structure can be also represented by its incidence matr/x C = [C~j]nxm, where c~j = O ( p ~ , t j ) I(pi,tj). Let x, y e P U T , t h e n " (x) = {y [ there is an
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arc from y to x} and (x) • = {y t there is an arc from x represented as a matrix ~o = [~0~j]~x~, where if the marto y} are the sets of predecessors and successors of the king of the j-th place is available at the output then the node x, respectively. A subnet of a net N = (P, T, I, O) i-th row of ~o, denoted as ~o~,is equal to the transpose of is a 4-tuple Q = ( P ' , T ' , I ' , O ' ) , where P ' C P, T' C T t h e j - t h elementary vector ej (ej [3"]= 1 and ej [i ¢ j] = 0, and Vp 6 P ' , Vt e T' it holds that I'(p,t) = I(p,t) i = 1, 2 , . . . , n). It means that, the markings of certain and O'(p,t) = O(p,t). The net Q is also called the places are available at the output of the net through the subnet generated by the set of nodes X = P ' U T'. function ~o, then the following definition arise. Definition 5: A place pi E P is said to be measurable, A free-choice net (FC) is a P N such that Vp~ E P, if 3j 6 1 , 2 , . . . , r such t h a t ~oj = eT; otherwise, pl is {(p~)" { > 1 --~ Vt~ e (p~)', {• (t~){ = 1. The marking of a P N is a function M : P --* Z + called non-measurable. Thus, T = Tc U Tu, where Tc and T~ are the sets of that assigns to each place of N a non-negative number of tokens, represented by black dots. The marking at controllable and uncontrollable transitions, respectively. a moment k is often represented by the vector Me = Similarly, T = T m U Tnm and P = I'm U Pnm, where [Mk(Pl)...Me(P,~)] T. A P N system (Y, Mo) is a net N Tm and Pm are the sets of measurable transitions and places, respectively; while T~m and Pnm are the sets of with an initial marking M0. Definition 2: An Interpreted Petri Net ( I P N ) is a 7- non-measurable transitions and places, respectively. In this paper, the measurable nodes are depicted as white tuple O = (N, E, T, @, A, D, ~) where nodes, while the non-measurable nodes are depicted as • N = (N', Mo) is a P N system; • ~. = {al,a~, ...,au} is the transition input alphabet, dark nodes. The marking of an I P N evolves according to the folwhere a~ is a transition input symbol; lowing rules: a transition tj 6 T of an I P N is ena• T = {T1, T 2 , . . . , Tv} is the transition output alphabet, bled at a marking M if Vp~ 6 P, M(p~) ~ I ~ , t j ) . where T~ is a transition output symbol; If tj is an enabled controllable transition and the sym• @ = {¢i, ¢2, ...,¢s} is the marking output alphabet, bol A(tj) = a~ ¢ e is present then tj must fire; otherwhere ¢~ is a marking output symbol; wise, if tj is an enabled uncontrollable transition then • A : T --* E U {e} is a function that assigns an input it can be fired. In both cases, if tj fires at a marking symbol to each transition of the net, where e represents Mk, then a new marking Mk+l is reached, which can be an internal system event. This hmction has the following computed using the dynamical part of the state equarestriction: Vt~,t~ E T, j ¢ k if I~i,t~) = I(p~,te) ~ 0 tion: M~+i = M~ + Cv~, where re(i) = 1, for i = j and and both A(t3), A(te) ¢ ~, then A(t~) y£ A(te); re(i) = 0, otherwise. The notation M tj ~ M ' repre• D : T --* T U {~} is a function t h a t assigns an output symbol to each transition of the net. If a transition t~ sents the fact that tj is enabled at M and, when it is fires then ~j = D(t~) or the Symbol e are presented at fired, the marking M ' is reached. the output of the system; The incidence matrix C can be split into two matri• ~o: R ( N ' , Mo) -~ {@ U {e}} ~ is a function t h a t assigns ces: C = [C~iCD], where C e and C ° are formed by the an output symbol to each reachable marking of the net, columns of the non-measurable and measurable transiwhere R ( N ' , Mo) is the teachability set of (N', M0) and tious, respectively. Hence, the state equation of an I P N the symbol e represents a null measurement. D D can be written as :Me+l = M e + C v e + C e ev e and Remark 3: To enhance the fact that there exists an initial marking in an I P N , it will be written (Q, M0) Ye+l ----~ M e + i . A firing sequence of an I P N (Q, M0) is a sequence instead of Q : (N, E, T, @, A, D, ~). The transition input alphabet E of an I P N can be a = t~tj...te such t h a t M0 t~ Mi tj ) ... tk ) M e . thought as a set of actuator signals attached to the tran- The set of all firing sequences is called firing language t~,M1 t j , . . , t~, sitious of the net. Similarly, the output alphabets 7 and £(Q, M o ) = { a l a = t i t j . . , t ~ A M o of the net can be thought as sets of event and state Mw}. The input language of (Q, Mo) is £~n(Q, Mo) = sensor signals for transitions and places, respectively. In {A(t~)A(tj)...A(te)lt~½...tk 6 £(Q, Mo)}. this context, it is possible to distinguish between controlI I I . O B S E R V A B I L I T Y ISSUES lable and uncontrollable transitions and between measurable and non measurable nodes of the net as establiThis section reviews the main concepts and results shed in the following definitions. related with observability property. Observabflity is a Definition 4: A transition t~ 6 T is said to be con- property of dynamic systems t h a t establishes the possitrollable, if A(ti) ¢ e, and uncontrollable, otherwise. A bility of determining the system states t h a t cannot be transition tj 6 T is said to be measurable, if D(t~) y£ ~, measured. and non-measurable, otherwise. Definition 6: An I P N given by (Q, M0) is observable This paper focuses in the case where @O{e} = Z + and at k steps if and only if Vw 6 £~n(Q, Mo) 3z, such that wz e £ ~ ( Q , Mo), Izl
