ERRATUM: “State Feedback Control of Timed Hybrid Petri Nets”Volume 86, Number 10, pages 1–7

Atsushi Tanaka,1 Toshimitsu Ushio,2 and Shinzo Kodama3

1Kinki College of Computers and Electronics, Sakai, 593-8326 Japan2Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531 Japan

3Graduate School of Engineering, Kinki University, Higashi-Osaka, 577-8502 Japan

The corrections to the above article were inadvertently omitted from the published version. We reproduce the entirepaper here with the corrections.

We apologize for any inconvenience caused to the authors and readers.

© 2003 Wiley Periodicals, Inc.DOI 10.1002/ecjc.10158

Electronics and Communications in Japan, Part 3, Vol. 87, No. 1, 2004

82

State Feedback Control of Timed Hybrid Petri Nets

Atsushi Tanaka,1,* Toshimitsu Ushio,2 and Shinzo Kodama3

1Kinki College of Computers and Electronics, Sakai, 593-8326 Japan

2Graduate School of Engineering Science, Osaka University, Toyonaka, 560-8531 Japan

3Graduate School of Engineering, Kinki University, Higashi-Osaka, 577-8502 Japan

SUMMARY

A system in which discrete variables and continuousvariables are mixed is called a hybrid system. Many modelshave been proposed for the description of hybrid systems.A model graphically describing the causality relationshipbetween the variables is the hybrid Petri net (HPN). Amodel in which the concept of time is introduced into theHPN is the timed HPN (THPN). In this paper, with regardto the THPN with an external input place (THPNIP), statefeedback control is discussed in the case where the controlspecifications are given by a predicate on the reachable set.In THPNIP, in general there is no guarantee that the maxi-mum permissive feedback, which is the maximum elementin the set of the state feedback satisfying the control speci-fication, will exist. Hence, the necessary and sufficientconditions are presented for the existence of the maximumpermissive feedback. © 2003 Wiley Periodicals, Inc.Electron Comm Jpn Pt 3, 87(1): 83–89, 2004; Publishedonline in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10011

Key words: hybrid Petri net; state feedback con-trol; maximum permissive feedback; hybrid system; dis-crete event system.

1. Introduction

Systems in which discrete variables and continuousvariables are mixed are called hybrid systems [1]. In thefield of system control theory, there are two types of ap-proaches to hybrid systems. One is the approach fromnonlinear system theory and the other is that from discreteevent system theory. In discrete event system theory, amodeling tool to graphically express the cause-and-resultrelationships between variables and easily express even theasynchronous nature and concurrency is the Petri net (PN)[2]. Although many models describing hybrid systems havebeen proposed [1], this paper deals with systems modeledby the hybrid PN (HPN) [3] as an extension of the PN andstudies its state feedback control. Ramadge and Wonhamhave proposed two control methods, supervisor control [4]and state feedback control [5], in discrete event systemsmodeled by automata. If, for a given predicate Q as a controlspecification, Q is satisfied at all markings reachable fromthe marking satisfying Q in a closed-loop system with astate feedback f, such an f is called the permissive feed-back (PF) [5]. When there exists some PF of Q, such a Qis called control-invariant. In general, there is more thanone PF for a control-invariant predicate, and the maximalelements are called the maximally PFs. A maximally PFis not necessarily unique. If a maximally PF is unique,then it is called the maximum PF (MPF) [6]. The MPF isthe feedback that satisfies the control specification,maximizes the reachable set in the closed-loop system,and allows the firing of most transitions at each marking.Hence, MPF is the most desirable feedback. In a PN with

© 2003 Wiley Periodicals, Inc.

Electronics and Communications in Japan, Part 3, Vol. 87, No. 1, 2004Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J84-A, No. 10, October 2001, pp. 1243–1250

*Currently with Ohara College of Information Systems, Osaka, 556-0011Japan

83

an external input place (PNIP), the MPF does not necessar-ily exist in general. Ushio indicated that the existence of theMPF is expressed by an extremely simple relationship onthe set of PFs called weak interaction [6]. However, in orderto study weak interaction, it is necessary to construct allPFs. Hence, this method is not practical. In contrast,Takai’s group has presented the necessary and sufficientconditions for the study of the existence of the MPFwithout the configuration of the PF [7]. However, inprinciple, the PNIP is a logical model that does notcontain a concept of time, but information on time isimportant in the real-time control of a discrete eventsystem. Takae and colleagues studied the real-time controlof a time Petri net with an external input place [8]. A modelof the HPN with the concept of time introduced is the timedHPN (THPN) [3]. Hence, the model with the concept oftime introduced into an HPN with an external input place(HPNIP) can be expressed by a timed HPNIP (THPNIP). Inthis paper, state feedback control in the THPNIP is studied.The necessary and sufficient conditions for the existence ofthe MPF are also presented.

2. The Hybrid Petri Net

In the PN, the state of the variables and its change areexpressed by the connection relationship of the place andtransition nodes and by the shift of the tokens [2]. The PNconsisting of the portion expressing the discrete variablesand that for the continuous variables is called the HPN [3].The nodes for the discrete variables are called the discreteplace (D-place) and discrete transition (D-transition) nodesand those for the continuous variables are called the con-tinuous place (C-place) and continuous transition (C-tran-sition) nodes [3]. In this paper, they are graphicallyexpressed as shown in Fig. 1. If Z

+ is the set of positiveintegers and ℜ+ is the set of positive real numbers, then

[Definition 1] HPNIP is described by

where P is the finite set of the places, T is the finite set ofthe transitions, and P ∩ T = ∅ and P ∪ T ≠ ∅. Also, h: P∪ T → {D, C} is a hybrid function that indicates whethereach node is associated with a discrete variable or a con-

tinuous variable. I: P × T → ℜ+ ∪ {0} denotes the weightof the input arc of the transition. If h(Pi) = D, the constraintI(Pi, Tj) ∈ Z

+ ∪ {0} is added. Further, O: P × T → ℜ+ ∪{0} denotes the weight of the output arc of the transition.If h(Pi) = D, the constraint O(Pi, Tj) ∈ Z

+ ∪ {0} is added.When h(Pi) = D ∧ h(Tj) = C, the arc connecting Pi and Tj isthe permission arc I(Pi, Tj) = O(Pi, Tj). Pcp is the set of theexternal input D-places. Icp: Pcp × T → {0, 1} is the func-tion assigning a permission arc with the weightIcp(Pcp, Tj) = 1 from the external input D-place to the tran-sition. The permission arc allows firing of the transition Tjif a token exists at the external input D-place Pcp

i, and

forbids firing of Tj otherwise, so that the transition iscontrolled. However, there is no change of the token insidePcp

i due to firing of Tj. Since the on/off control structure of

the event, Mcp: Pcp → {0, 1} is a function determining themarking by the state feedback (described later) for eachexternal input D-place Pcpi

∈ Pcp. M0 is the initial mark-ing of the internal places. The initial marking of the internalD-places is a nonnegative integer, while the initial markingof the internal C-places is a nonnegative real number.

In contrast to the model without time in Definition 1,Alla’s group has introduced the concept of time [3]. In orderto express continuous dynamics on a directed circuit con-sisting of weakly enabled transitions, it is necessary tointroduce a delay in the firing of a C-transition [9]. Whenstate feedback control is considered, the transition fireseven if, in the method of introducing a delay to the firingtime, such an occurrence is prevented during the delaybetween the firing decision and actual firing. Therefore, inthis paper, a method for forbidding the use of a token for acertain time after firing is introduced. Hence, the markingof the THPN is expressed as M = Ma + Mu, where Ma andMu denote available marking and unavailable marking atthe time of firing of the transition. Also, •Tj and Tj

• are theset of the input and output places of the transition Tj.

[Definition 2] THPNIP is described by

where Tempo: T → ℜ+ is a function assigning a positive realnumber to each C-transition Tj, while Tempo(Tj) = Vj indi-cates the maximum firing speed of Tj. Delay: P → ℜ+ ∪{0} is the function assigning a nonnegative real number toeach internal place Pi, and Delay(Pi) = di is the delay timeof the token supplied to Pi by firing of the input transitionuntil it can contribute to firing of the output transition.

[Definition 3] (firing rule for timed D-transition)

(1) If Ma(Pi) ≥ I(Pi, Tj) for each internal inputplace Pi ∈ •Tj, then the D-transition Tj is calledmarking-enabled at time t.

(1)

Fig. 1. Nodes of Hybrid Petri Nets.

(2)

84

(2) If Mcp(Pcpi) ≥ Icp(Pcpi

, Tj) for all external input D-place Pcp

i ∈ •Tj, then Tj is called control-enabled at time t.

(3) Tj that is marking-enabled and control-enabled isfiring-enabled.

(4) Tj that is firing-enabled at time t fires. When Tj

fires, I(Pi, Tj) available tokens are taken away from eachinternal input place Pi ∈ •Tj and O(Pi, Tj) unavailable to-kens are added to each internal output place Pi ∈ Tj

•.(5) The unavailable tokens added to the internal out-

put place Pi by firing remain unavailable during the delaytime of di. After di, these unavailable tokens become avail-able.

The timed C-transition fires according to the firingspeed continuously. For this transition, there are two typesof enabling, strong enabling and weak enabling, accordingto the marking of the internal input place [3]. If firing isenabled at the maximum speed for this marking, this caseis called strong enabling. If firing is possible at a speed lessthan the maximum value, this case is called weak enabling[3]. In the following, the definition of the firing rule ispresented.

[Definition 4] (Firing rule for timed C-transition)

(1) A C-transition is called strongly enabled at timet, if it is a source transition or Ma(Pi) ≥ I(Pi, Tj) for eachinternal input D-place Pi ∈ •Tj and Ma(Pi) > 0 for eachinternal input C-place Pi ∈ •Tj. Then, the firing speed vj(t)of Tj is vj(t) = Vj.

(2) Tj is called weakly enabled at time t, if there existsa token changing from unavailable to available at eachinternal input C-place Pi ∈ •Tj such that Ma(Pi) = 0 andMa(Pi) ≥ I(Pi, Tj) for each internal input D-place Pi ∈ •Tj.Then, the firing speed vj(t) of Tj is

where minPi

is taken for every internal input C-placePi ∈ •Tj such that Ma(Pi) = 0. On the other hand, pvi(t)denotes the speed at which the unavailable tokens becomeavailable in Pi at time t.

(3) Tj is called marking-enabled if it is strongly en-abled and weakly enabled.

(4) If Mcp(Pcpi) ≥ Icp(Pcpi

, Tj) for each external inputD-place Pcpi

∈ •Tj, then Tj is called control-enabled at timet.

(5) Tj that is marking-enabled and control-enabled isfiring-enabled.

(6) Tj that is firing-enabled at time t fires. If Tj

continues to fire from time t for δ, the available tokens ofI(Pi, Tj) × ∫t

t+δ vj(τ)dτ in each internal input C-place

Pi ∈ •Tj are removed and the unavailable tokens ofO(Pi, Tj) × ∫t

t+δ vj(τ)dτ are added to each internal outputC-place Pi ∈ T j

•.(7) The unavailable tokens added to each internal

output C-place Pi ∈ T j• remain unavailable during time di.

These unavailable tokens become available after di ispassed.

For the continuous firing of a C-transition Tj causingvariations of the real-valued tokens, the internal input D-place Pi ∈ •Tj is coupled to Tj with a permission arc. There-fore, it should be noted that the tokens in Pi cannot beremoved.

Let the closure of the reachable set of Htim beR(Htim). If the transition Tj ∈ T is marking-enabled at themarking M ∈ R(Htim), let M[Tj >. Further, if the marking istransitioned to M′ by firing of Tj, let M[Tj > M′. In this paper,simultaneous firing of several transitions is considered. Letthe power set, that is, the set consisting of all subsets of T,be 2T. If the subset T

sub ∈ 2T is marking-enabled atM ∈ R(H

tim), then let M[T sub > M ′. Further, if the marking

is transitioned to M′ by firing of T sub, let M [T

sub > M′. Thesubset Tb ⊆ 2T is defined as follows:

Tb = {Tbj ∈ 2T|M[Tbj

> for some M ∈ R(H tim)}

Further, each Tbj ∈ Tb is divided into Tbj

= DTbj ∪ CTbj

. Here,DTbj

denotes the set of D-transitions and CTbj denotes the set

of C-transitions.

3. State Feedback Control

3.1. Predicate

Let a set of predicates on R(Htim) be B = {0, 1}R(Htim).The fundamental Boolean two-term operators negation,conjunction, and disjunction on B are expressed by ¬, ∧,and ∨ [5–8]. Only predicates such that the set of truemarkings is a closed set are considered. In the following,unless there is a danger of confusion, the set of markingssatisfying the predicate Q is expressed by Q. For arbitraryQ1, Q2 ∈ B, the partial order relation ≤ on B is defined asQ1 ≤ Q2 when Q1 ∧ Q2 = Q1 [5–8]. For each Tbj

∈ Tb, thepredicate DTbj

is defined as follows [5–8]:

Hence, DTbj (M) is a predicate that is 1 if the transitions Tbj

are marking-enabled at the same time for the marking Mand is 0 otherwise.

Let us define the transformer wpTbj on B as follows.

If M[Tbj > M′ and Q(M′) = 1 on M ∈ R(Htim) for Tb

j =

85

DTbj, then wpTbj

(Q)(M) = 1. If Tbj continues to fire from M

∈ R(Htim) at time t for Tbj = CTbj

and Q(M′) = 1 for markingM′ at time t + dt for 0 < dt ≤ δ with a sufficiently small δ >0, then wpTbj

(Q)(M) = 1. If M[DTbj > M′ and Q(M′) = 1 for

M ∈ R(H tim) at time t for Tbj

= DTbj ∪ CTbj and further if

CTbj continues to fire from M′ and Q(M′′) = 1 for marking

M″ at time t + dt, then wpTbj (Q)(M) = 1. Otherwise, wpTbj

(Q)(M) = 0. Hence, wpTbj (Q)(M) is a transformer that is 1

for the transition to a state satisfying Q by simultaneousfiring of several transitions Tbj

for marking M, and 0 other-wise.

The transformer wlpTbj on B is defined as follows

[5–8]:

Hence, wlpTbj (Q)(M) is a transformer that is 1 for the

transition to a state satisfying Q by simultaneous firing ofTbj

or the concurrently firing-disabled transitions Tbj for the

marking M, and is 0 otherwise.

3.2. State feedback

The set Tc of the controllable transitions and Tu of theuncontrollable transitions are defined as follows [6–8]:

Further, for each Tbj ∈ Tb, the subset of the external input

D-place cpTbj ⊆ Pcp is defined as follows [7, 8]:

Hence, cpTbj is the set of the external input D-places for

which arcs exist to the transitions contained in Tbj. Clearly,

cpTbj = ∅ for Tbj

∈ 2Tu ∩ Tb.

Note that Γ is the set of the control patterns and isgiven by a power set of Pcp. Hence, the control pattern γ ∈Γ indicates a set of the external input D-places into whichtokens are inserted. The state feedback is defined as amapping from each reachable marking of H

tim to the controlpattern. Hence, the set of state feedbacks is the set ofmappings from R(Htim) to Γ and is written as ΓR(Htim). Theclosed-loop system of the state feedback f ∈ ΓR(Htim) appliedto H

tim is written as H tim|f.

The partial order relation ≤ in ΓR(Htim) is defined asfollows. In an arbitrary M ∈ R(Htim), f1 ≤ f2 when f1(M) ⊆f2(M) for f1, f2 ∈ ΓR(H

tim). The sum f1 + f2 for each f1, f2 ∈ΓR(H

tim) is defined as follows. At each M ∈ R(Htim),

The marking Mcp of the external input D-places at time t isgiven as follows by the state feedback f ∈ ΓR(H

tim):

3.3. Control invariance and maximumpermissive feedback

For each Tbj ∈ Tb and f ∈ ΓR(H

tim), the predicate fTbj

∈B is defined as follows [5–8]:

Hence, fTbj (M) is a predicate with a value of 1 if several

transitions Tbj are control-enabled at the marking M and of

0 otherwise.If the state feedback f ∈ ΓR(H

tim) satisfies the follow-ing expression for an arbitrary Tbj

∈ Tb, f is called PF ofQ [5–8]:

Equation (4) implies that all markings reachable from anarbitrary marking satisfying Q at H

tim|f satisfy Q. If PF ofQ exists, Q is said to be control-invariant [5–8]. If Q iscontrol-invariant, more than one PF exists in general.Let the set of PFs for Q be F(Q). Note that f ∈ F(Q)such that f ∈ F(Q) such that f (≠ f) ∈ F(Q) for f ≥ fdoes not exist is called a maximally PF of Q [6–8].

[Lemma 1] Let the predicate Q be control-invariant.Then, a maximally PF always exists in F(Q).

(Proof) For arbitrary f ∈ F(Q), let Rf = R(Htim|f). It isclear that Rf is a closed subset of Q. If R(Q) ={Rf |f ∈ F(Q)}, then there exists a maximal elementRf

∗ ( f ∗ ∈ F(Q)) in R(Q) because Q is a closed set. Further,let F̂ (f*) = {f ∈ F(Q)|Rf

∗ = Rf}. Since the number of tokensthat can be inserted into the external input D-place is atmost 1, a maximal element f∗ always exists in F (f*). Ifthere exists f satisfying f ≥ f∗, then Rf = Rf∗ and f = f∗ fromthe definition of f∗. Hence, there always exists a maximalPF in F(Q). "

If a maximally PF exists uniquely, it is called a MPF(maximum permissive feedback) [6–8]. In general, a MPFdoes not always exist.

[Lemma 2] Let the predicate Q be control-invariant.Then, the necessary and sufficient condition for the exist-ence of the MPF is that Q satisfies the following condition(C).

(C) Arbitrary f, g ∈ F(Q) satisfy the following expres-sion for an arbitrary Tbj

∈ Tb:

(3)

(4)

86

(Proof) First, sufficiency is proven. From Lemma1, a maximal PF always exists. It is now assumed thattwo maximal PF f1 and f2 exist such that f1 ≠ f2. If fm =f1 + f2, then fm is clearly not a PF. Hence, there existTbj

∈ Tb and M ∈ R(Htim) such that Q(M) = 1 and wlpTbj

(Q)(M) ∨ ¬ fmTbj

(M) = 0. Then, wlpTbj (Q)(M) = 0. Also, since

Q ≤ wlpTbj (Q) ∨ ¬ f1Tbj

holds from the fact that f1 ∈ F(Q),f1Tbj

(M) = 0. Similarly for f2 ∈ F(Q), f2Tbj (M) = 0. Therefore,

by virtue of Eq. (5), ¬ (f1 + f2)Tbj = 1. This results in fmTbj

(M)= (f1 + f2)Tbj

(M) = 0. Finally, wlpTbj (Q)(M) ∨ ¬ fmTbj

(M) =1, which is a contradiction. Next, necessity is proven. It isclear that Eq. (5) holds for M ∈ R(Htim) satisfying Q(M) =0 or wlpTbj

(Q)(M) = 1. Hence, in what follows, let usconsider M ∈ R(Htim) satisfying Q(M) = 1 and wlpTbj

(Q)(M)= 0. Let the MPF of Q be fm. Then, from the definition ofthe PF, wlpTbj

(Q)(M) ∨ ¬ fmTbj

(M) = 1. Since wlpTbj (Q)(M)

= 0, it is found that fmTbj

(M) = 0. On the other hand, fromthe definition of MPF and that of the sum of the statefeedbacks, fm ≥ f + g for arbitrary f, g ∈ F(Q). Hence, sincefmTbj

(M) = 0, (f + g)Tbj

(M) = 0 is obtained. From this, Eq. (5)holds. "

If Q satisfies condition (C) in Lemma 2, Q is said tobe weakly interactive [6]. From the proof for Lemma 2, thefollowing corollary can easily be proven.

[Corollary 1] If condition (C) in Lemma 2 holds, f +g ∈ F(Q) for arbitrary f, g ∈ F(Q).

Let cpTj be the set of external input D-places con-nected to each controllable transition Tj ∈ Tc. Then, the setT(Pcp

i) ⊆ Tc of the transitions is defined as follows for each

Pcpi ∈ Pcp [7, 8]:

Therefore, T(Pcpi) is a set of the transitions controlled only

by the external input D-place Pcpi. Further, for each Pcpi

∈Pcp, the transformer cwlpPcp

i on B is defined as follows [7, 8]:

Hence, cwlpPcpi (Q)(M) is a transformer that is 1 if several

transitions Tbj controlled only by the external input D-place

Pcpi at the marking M are concurrently marking-disabled or

are transitioned to the state satisfying Q by simultaneousfiring of Tbj

and 0 otherwise.

Let M cwlpPcpi be the set of the markings M ∈ R(Htim)

such that cwlpPcpi (Q)(M) = 1 for each Pcp

i ∈ Pcp. The basis

feedback bPcpi: R(H

tim) → Γ is defined as follows:

Hence, bPcpi(M) is the feedback that determines the insertion

of only one token into the external input D-place Pcpi for the

marking M such that cwlpPcpi (Q)(M) = 1.

[Lemma 3] Let us assume that the predicate Q iscontrol-invariant. Then, for an arbitrary Pcpi

∈ Pcp, bPcpi

∈ F(Q).(Proof) Let the basis feedback for Pcpi

∈ Pcp be f.Hence, f = bPcpi

. It is sufficient to prove that Eq. (4) issatisfied for an arbitrary Tb

j ∈ Tb. When Tbj

∩ Tc = ∅,namely, Tb

j ⊆ Tu, the control-invariance of Q allows

Let M ∈ R(Htim) be an arbitrary reachable marking. WhenTb

j ∩ Tc ⊆/ T(Pcp

i) or M ∉ M

cwlpPcpi, ¬ fTbj

(M) = 1 from Eq.(7). Therefore, it is sufficient to consider the case of ∅ ≠Tbj

∩ Tc ⊆ (Pcpi) and M ∈ M

cwlpPcpi. It is self-evident fromEq. (3) that Eq. (4) holds if DTbj

(M) = 0. Let us consider thecase of DTbj

(M) = 1. Let us introduce the decompositionTbj

= Tbjc ∪ Tbju. Here, Tbjc ⊆ Tc and Tbju ⊆ Tu. Since

cwlpPcpi (Q)(M) = 1 from M ∈ M

cwlpPcpi, it is found thatwlpTbjc

(Q)(M) = 1. Therefore, Q(M′) = 1 at each markingM′ reachable by the firing of Tb

jc within a sufficiently short

time. Further, due to the control-invariance of Q, wlpTbju

(Q)(M′) = 1. Hence, Eq. (4) is satisfied. "[Theorem 1] Let the predicate Q be control-invariant.

Then, the following three conditions are equivalent.

(1) The MPF exists.(2) For an arbitrary Tb

j ∈ Tb, arbitrary f, g ∈ F(Q)

satisfy the following:

(3) For an arbitrary Tbj ∈ 2Tc ∩ Tb, the following

holds:

(Proof) The equivalence of conditions (1) and (2) isobvious from Lemma 2. Hence, let us show that condition(2) holds when condition (3) is valid. Suppose that condi-tion (2) does not hold. Then there exist f, g ∈ F(Q), M ∈R(Htim), and Tb

j ∈ Tb such that

(6)

(7)

(8)

(9)

(5)

87

Since Q is control-invariant, it is possible to assume withoutloss of generality that Tbj

⊆ Tc. Let us consider Pcpi ∈ cpTbj

.If T(Pcp

i) = ∅, cwlpPcpi

(Q)(M) = 1 from Eq. (6). Let usconsider the case of T(Pcp

i) = ∅. Since (f + g)Tbj

(M) = 1,either f(M)(Pcpi

) = 1 or g(M)(Pcpi) = 1. Since f, g ∈ F(Q),

wlpTbj

g (Q)(M) = 1 for an arbitrary Tbj

g ∈ 2T(Pcpi). Therefore,

cwlpPcpi (Q)(M) = 1. Therefore, from Eq. (8) wlpTbj

(Q)(M)= 1. But this contradicts Eq. (9). Next, let us show thatcondition (3) holds if condition (2) is satisfied. Let usassume that Eq. (8) does not hold. Then there exist M ∈R(Htim) and Tb

j ∈ 2Tc ∩ Tb such that

From Lemma 3, bPcpi ∈ F(Q) for each Pcp

i ∈ cpTb

j. From Eqs.

(7) and (11),

Therefore, if

then h(M) = cpTbj, so that hTbj

(M) = 1. On the other hand,from Corollary 1, h ∈ F(Q). Therefore, wlpTbj

(Q)(M) = 1,which contradicts Eq. (12). "

3.4. Example

Let us consider the existence of the maximum per-missive feedback in THPNIP shown in Fig. 2. The initial

marking is M0 = (0, 0). Let the control specification be 0 ≤M(P2) ≤ 4. At the marking M = (4, 3) reachable from M0,Q(M) = 1. The marking-enabled transitions at M are T1, T2,and T3. Hence, for T1, cwlpPcp

1 (Q)(M) = 1(Pcp

1 ∈ cpT1) and

for T2, cwlpPcp2 (Q)(M) = 1(Pcp

2 ∈ cpT2). Since wlp{T

1,T

2,T

3}

(Q)(M) = 0, Eq. (8) does not hold. Hence, the maximumpermissive feedback does not exist.

4. Conclusions

In this paper, state feedback control is studied in thecase where the control specification is given only on themarking in a THPNIP. The necessary and sufficient condi-tions are derived for the existence of an MPF. In general,there are infinitely many markings that satisfy the controlspecifications. The development of an algorithm to checkthis condition is a future topic of study. Also, since the statefeedback presented in this paper is time-invariant, Zenobehavior applies. The introduction of time-varying statefeedback to avoid this situation is a future topic. Furtherstudies for the future include control for the case in whichonly part of the markings are observable, and control forthe case in which control specifications including time aswell as markings are given.

REFERENCES

1. Grossman RL, Nerode A, Ravn AP, Rischel H (edi-tors). Hybrid systems. LNCS, Vol. 736. Springer;1993.

2. Murata T. Petri nets: Properties, analysis and appli-cations. Proc IEEE 1989;77:541–580.

3. Bail JL, Alla H, David R. Hybrid Petri nets. Proc ECC91, p 1472–1477, Grenoble.

4. Ramadge PJ, Wonham WM. Supervisory control ofa class of discrete event processes. SIAM J ControlOptimiz 1987;25:206–230.

5. Ramadge PJ, Wonham WM. Modular feedback logicfor discrete event systems. SIAM J Control Optimiz1987;25:1202–1218.

6. Ushio T. Maximally permissive feedback and modularcontrol synthesis in Petri nets with external input places.IEEE Trans Autom Control 1990;35:844–848.

7. Takai S, Ushio T, Kodama S. Concurrency and maxi-mally permissive feedback in Petri nets with externalinput places. Int J Control 1994;60:617–629.

8. Takae A, Takai S, Ushio T, Kumagai S, Kodama, S.Maximally permissive feedback in timed Petri netwith external input places. Trans IEICE 1995;J78-A:1593–1600.

9. Tanaka A, Ushio T, Kodama S. Conflict resolution incontinuous Petri nets. Proc 4th Int Symp ArtificialLife and Robotics (AROB’99), p 114–117, Oita.

(10)

(11)

(12)

Fig. 2. A timed hybrid Petri net with external inputD-places.

88

AUTHORS (from left to right)

Atsushi Tanaka (nonmember) completed the doctoral program in the Faculty of Engineering Science at Osaka Universityin 2001, receiving a Ph.D. degree. He is a lecturer at Ohara College of Information Systems. He has been engaged in researchon discrete event systems, net theory, and hybrid dynamical systems. He is a member of IPSJ.

Toshimitsu Ushio (member) received his B.S., M.S., and Ph.D. degrees from Kobe University in 1980, 1982, and 1985.He was a research assistant at the University of California, Berkeley in 1985. From 1986 to 1990, he was a research associateat Kobe University, and became a lecturer at Kobe College in 1990. He joined Osaka University as an associate professor in1994, and is currently a professor. His research interests include nonlinear oscillation and control of discrete event systems. Heis a member of SICE, ISCIE, IEEE, and ACM.

Shinzo Kodama (nonmember) graduated from the Faculty of Science and Engineering at Waseda University in 1955 andcompleted the doctoral program at the University of California, Berkeley in 1963, receiving a Ph.D. degree. In 1962, he becamea lecturer on the Faculty of Engineering at Osaka University, and a professor in 1974. In 1995, he became a professor on theFaculty of Science and Engineering at Kinki University. From 1969 to 1970 he was a visiting associate professor at the Universityof California. He has been engaged in research on control theory and net theory. He is currently a professor emeritus of OsakaUniversity and a Life Fellow of IEEE.

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