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 description of hybrid systems. Amode graphically describing the causality relationship be-tween the variables is the hybrid Petri net (HPN). A modein which the concept of time is introduced into the HPN isthe timed HPN (THPN). In this paper, with regard to theTHPN with an external input place (THPNIP), state feed-back control is discussed in the case where the controlspecifications are given by a description on the attainableensemble. In THPNIP, in general there is no guarantee thatthe maximum allowable feedback, which is the maximumelement in the ensemble of the state feedback satisfying thecontrol specification, will exist. Hence, the necessary andsufficient conditions are presented for the existence of themaximum allowable feedback. © 2003 Wiley Periodicals,Inc. Electron Comm Jpn Pt 3, 86(10): 1–7, 2003; Publishedonline in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.10011

Key words: hybrid Petri net; state feedback con-trol; maximum allowable feedback; hybrid system; discretephenomenon system.

1. Introduction

Systems in which discrete variables and continuousvariables are mixed are called hybrid systems [1]. In the

field 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 theory, a modelingtool to graphically express the cause and result relationshipsbetween variables and easily express even the asynchro-nous nature and parallel nature is the Petri net (PN) [2].Although many models to describe 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 description Q as acontrol specification, Q is satisfied at all firings reachablefrom the firing satisfying Q in a closed loop system with astate feedback f, such an f is called the permissible feed-back (PF) [5]. In general, there is more than one PF for acontrol-invariable description, and the largest elements arecalled the largest PFs. A largest PF is not necessarily unique.If a largest PF is unique, then it is called the maximum PF(MPF) [6]. The MPF is the feedback that satisfies thecontrol specification, maximizes the reachable ensemblesin the closed loop system, and allows the firing of mosttransitions at each firing. Hence, MPF is the most desirablefeedback. In a PN with an external input place (PNIP), theMPF does not necessarily exist in general. Ushio indicatedthat the existence of the MPF is expressed by an extremelysimple relationship of the PFs on the ensemble called weakinterference [6]. However, in order to study weak interfer-ence it is necessary to construct all PFs. Hence, this method

© 2003 Wiley Periodicals, Inc.

Electronics and Communications in Japan, Part 3, Vol. 86, No. 10, 2003Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J84-A, No. 10, October 2001, pp. 1243–1250

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is not practical. In contrast, Takai and colleagues havepresented the necessary and sufficient conditions for studyof the existence of the MPF without the configuration ofthe PF [7]. However, in principle, the PNIP is a logicalmodel that does not contain a concept of time, but informa-tion on time is important in the real-time control of adiscrete event system. Takae and colleagues studied thereal-time control of a timed Petri net with an external inputplace [8]. A model of the HPN with the concept of timeintroduced is the timed HPN (THPN) [3]. Hence, the modelwith the concept of time introduced into an HPN with anexternal input place (HPNIP) can be expressed by a timedHPNIP (THPNIP). In the present paper, state feedbackcontrol in the THPNIP is studied. The necessary and suffi-cient conditions for the existence of the MPF are presented.

2. The Hybrid Petri Net

In the PN, the state of the variables and its change areexpressed by the connection relationship of the nodes andshifts of the tokens called place and transition [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 discrete(D) place and discrete (D) transition and those for thecontinuous variables are called the continuous (C) placeand continuous (C) transition [3]. In this paper, they aregraphically expressed as shown in Fig. 1. If Z

+ is theensemble of positive integers and ℜ+ is the ensemble ofpositive real numbers, then

[Definition 1] HPNIP is expressed as

where P is the finite ensemble of the places, T is the finiteensemble of the transitions, and P ∩ T = ∅ and P ∪ T ≠∅. Also, h: P ∪ T → {D, C} is a hybrid function thatindicates whether each node is associated with a discretevariable or a continuous variable. I: P × T → ℜ+ ∪ {0}denotes the weight of the input branch of the transition. Ifh(Pi) = D, the constraint I(Pi, Tj) ∈ Z

+ ∪ {0} is added.Further, O: P × T → ℜ+ ∪ {0} denotes the weight of the

output branch of the transition. If h(Pi) = D, the constraintO(Pi, Tj) ∈ Z

+ ∪ {0} is added. When h(Pi) = D ∧ h(Tj) = C,the branch connecting Pi and Tj is the permitted branchI(Pi, Tj) = O(Pi, Tj). Pcp is the ensemble of the external inputD places. Icp: Pcp × T → {0, 1} is the function assigning apermitted branch with a weight Icp(Pcp, Tj) = 1 to the tran-sition. The permitted branch allows firing of the transitionTj if 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 in 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 thefiring for the external input D place by the state feedback(described later) for each external input D place Pcpi

∈ Pcp.M0 is the initial firing of the internal place. The initial firingof the internal D place is a nonnegative integer, while theinitial firing of the internal C place is a nonnegative realnumber.

In contrast to the model without time in Definition 1,Alla and co-workers have introduced the concept of time[3]. In order to express continuous dynamics on a directedclosed path consisting of transitions enabling weak firing,it is necessary to introduce a delay in the firing of a Ctransition [9]. When state feedback control is considered,the transition is fired even if, in the method of introducinga delay to the firing time, such an occurrence is preventedduring the delay between the firing decision and actualfiring. Therefore, in this paper, a method for forbidding theuse of a token for a certain time after firing is introduced.Hence, the firing of the THPN is expressed as M = Ma + Mu,where Ma and Mu denote available firing and unavailablefiring at the time of firing of the transition. Also, •Tj andTj

• are the ensembles of the input and output places of thetransition Tj.

[Definition 2] THPNIP is expressed as

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 all internal input placesPi ∈ •Tj, then the D transition Tj is called marking-enabledat time t.

(2) If Mcp(Pcpi) ≥ Icp(Pcpi

, Tj) for all external input Dplaces Pcp

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

(1)

Fig. 1. Nodes of HPNs.

(2)

2

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

(4) Tj that can be fired at time t is fired at the sametime t. When Tj is fired, I(Pi, Tj) usable tokens are takenaway from each internal input place Pi ∈ •Tj and O(Pi, Tj)unusable tokens are added to each internal output placePi ∈ Tj

•.(5) The unusable tokens added to the internal output

place Pi by firing remain unusable during the delay time ofdi. After di, these unusable tokens become usable.

The timed C transition becomes continuously firedaccording to the firing speed. For this transition, there aretwo types of firing, strong firing and weak firing, accordingto the marking of the internal input place [3]. If firing isenabled at the maximum speed for this marking, this caseis called strong firing-enabled. If firing is possible at a speedless than the maximum value, this case is called weakfiring-enabled [3]. In the following, the definition of thefiring rule is presented.

[Definition 4] (Firing rule of timed C transition)

(1) A C transition is called strong firing-enabled attime t, if it is a source transition or Ma(Pi) ≥ I(Pi, Tj) for allof the internal input D places Pi ∈ •Tj and Ma(Pi) > 0 for allof the internal input C places Pi ∈ •Tj. Then, the firing speedvj(t) of Tj is vj(t) = Vj.

(2) Tj is called weak firing-enabled at time t, if thereexists a token changing from unusable to usable at eachinternal input C place Pi ∈ •Tj such that Ma(Pi) ≥ I(Pi, Tj)and Ma(Pi) = 0 for all internal input D places Pi ∈ •Tj. Then,the firing speed vj(t) of Tj is

where minPi

is taken for all internal input C places Pi ∈ •Tj

such that M(Pi)a = 0. On the other hand, pvi(t) denotes the

speed at which the unusable tokens become usable in Pi attime t.

(3) Tj is called marking firing-enabled if it is strongfiring-enabled and weak firing-enabled.

(4) If Mcp(Pcpi) ≥ Icp(Pcpi

, Tj) for all external input Dplaces Pcpi

∈ •Tj, then Tj is called firing-enabled at time t.(5) Tj that is marking-enabled and control-enabled is

firing-enabled.(6) Tj that is firing-enabled at time t is fired at the same

time t. If Tj continues to fire from the time t for δ, the usabletokens of I(Pi, Tj) × ∫t

t+δ vj(τ)dτ within each internal input Cplace Pi ∈ ⋅Tj are removed and the unusable tokens ofO(Pi, Tj) × ∫t

t+δ vj(τ)dτ are added to each internal output Cplace Pi ∈ T j

• .

(7) The unusable tokens added to each internal outputC place Pi ∈ T j

• remain unusable during the time di. Theseunusable tokens become usable after di is passed.

For the continuous firing of a C transition Tj causingvariations of the real valued tokens, the internal input Dplace Pi ∈ •Tj is coupled to Tj with an allowed branch.Therefore, it should be noted that the tokens within Picannot be removed.

Let the closed packet of the attainable ensemble ofHtim be R(Htim). If the transition Tj ∈ T is marking-enabledat the marking M ∈ R(Htim), we write M[Tj >. Further, if themarking is transitioned to M′ by firing of Tj, let M[Tj > M′.In this paper, simultaneous firing of several transitions isconsidered. Let the power ensemble, that is, the ensembleconsisting of all partial ensembles of T, be 2T. If the suben-sembles T

sub ∈ 2T of a certain transition are simultaneouslyfiring-enabled at M ∈ R(H

tim), then we write M[T sub>. Fur-

ther, if the marking is transitioned to M′ by simultaneousfiring of T

sub, we write M [T sub > M′. The subensemble

Tb ⊆ 2T is defined as follows: Tb = {Tbj ∈ 2T|M ∈ R(H

tim)such that M[Tbj

> exists}. Further, each Tbj ∈ Tb is divided

into Tbj = DTbj

∪ CTbj. Here, DTb

j denotes the ensemble of D

transitions and CTbj denotes the ensemble of C transitions.

3. State Feedback Control

3.1. Descriptor

Let the ensemble of descriptors on R(Htim) be B ={0, 1}R(Htim). The fundamental two-term operations denial,crossing, and coupling on B are expressed by –, ∧, and ∨[5–8]. Only descriptors such that the ensemble of truemarkings is a closed ensemble are considered. In the fol-lowing, unless there is a danger of confusion, the ensembleof markings satisfying the descriptor Q is expressed by Q.For arbitrary Q1, Q2 ∈ B, the semiordering relationship ≤on B is defined as Q1 ≤ Q2 when Q1 ∧ Q2 [5–8]. For eachTbj

∈ Tb, the descriptor DTbj is defined as follows [5–8]:

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

Tbj are marking-enabled at the same time as the marking M

and is 0 otherwise.Let us define the transform wpTbj

on B as follows. IfM[Tbj

> M′ and Q(M′) = 1 on M ∈ R(Htim) for Tbj = 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 at marking M′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 at M

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∈ R(Htim) at time t for Tbj = DTbj ∪ CTbj

and further if CTbj

continues to fire from M′ and Q(M′′) = 1 at marking M″ attime t + dt, then wpTbj

(Q)(M) = 1. Otherwise, wpTbj (Q)(M)

= 0. Hence, wpTbj (Q)(M) is a transformation that is 1 for the

transition to a state satisfying Q by simultaneous firing ofseveral transitions Tbj

at marking M, and 0 otherwise.wlpTbj

on B is defined as follows [5–8]:

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

transition to a state satisfying Q by the concurrent firinginability of multiple transitions Tbj

at the marking M, or bysimultaneous firing of Tbj

, and is 0 otherwise.

3.2. State feedback

The ensemble Tc of the controllable transitions andTu of the uncontrollable transitions are defined as follows[6–8]:

Further, for each Tbj ∈ Tb, the subensemble of the external

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

Hence, cpTbj is the ensemble of the external input D places

for which branches exist to the transitions contained in Tbj.

Clearly, cpTbj = ∅ for Tbj

∈ 2Tu ∩ Tb.

Note that Γ is the ensemble of the control patterns andis given by a power ensemble of Pcp. Hence, the controlpattern γ ∈ Γ indicates an ensemble of the external input Dplaces into which tokens are inserted. The state feedback isdefined as a mapping from each attainable marking ofH

tim to the control pattern. Hence, the ensemble of statefeedbacks is the ensemble of mappings from R(Htim) to Γand is written as ΓR(Htim). The closed loop system of the statefeedback f ∈ ΓR(Htim) applied to H

tim is written as H tim|f.

The semiordered relationship ≤ in ΓR(Htim) is definedas follows. In an arbitrary M ∈ R(Htim), f1 ≤ f2 when f1(M)⊆ f2(M) for f1, f2 ∈ ΓR(H

tim). The sum f1 + f2 for eachΓR(H

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

The marking Mcp of the external input D place at timet is given as follows by the state feedback f ∈ ΓR(H

tim):

3.3. Control invariance and maximumallowable feedback

For each Tbj ∈ Tb and f ∈ ΓR(H

tim), the descriptor fTbj

∈B is defined as follows [5–8]:

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

transitions Tbj are control-firing-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 ensemble of PFs for Q be F(Q). Note that f ∈F(Q) such that f (≠ f) ∈ F(Q) for f ≥ f does not existis called a maximum PF of Q [6–8].

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

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

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

∗ = Rf}. Since the numberof tokens that can be inserted into the external input Dplace is at most 1, a maximum element f∗ always existsin f (f*). If there exists f satisfying f ≥ f∗, then Rf = Rf∗

and f = f∗ from the definition of f∗. Hence, there alwaysexists a maximum PF in F(Q). "

If a maximum PF exists uniquely, it is called an MPF[6–8]. In general, an MPF does not always exist.

[Lemma 2] Let the descriptor 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) An arbitrary f, g ∈ F(Q) satisfies the followingexpressions for arbitrary Tbj

∈ Tb:

(3)

(4)

(5)

4

(Proof) First, sufficiency is proven. From Lemma1, a maximum PF always exists. It is now assumed thattwo maxima PFf1 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. Therefore, by virtue of Eq. (5), ¬ (f1 + f2)Tbj (M)

= 0. Similarly for f2 ∈ F(Q), f2Tbj (M) = 0. Hence, 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 interfering [6]. From the proof for Lemma 2, thefollowing system can easily be proven.

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

Let cpTj be the ensemble of external input D placesconnected to each controllable transition Tj ∈ Tc. Then, theensemble T(Pcp

i) ⊆ Tc of the transitions is defined as fol-

lows for each Pcpi ∈ Pcp [7, 8]:

Therefore, T(Pcpi) is an ensemble of the transitions control-

led only by the external input D place Pcpi.

Further, for each Pcpi ∈ Pcp, the transformation cwlpPcp

ion B is defined as follows [7, 8]:

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

transitions Tbj controlled only by the external input D place

Pcpi at the marking M are marking-enabled at the same time

or are transitioned to the state satisfying Q by simultaneousfiring of Tbj

and 0 otherwise.Let M

cwlpP

cpi be the ensemble 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 descriptor 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 attainable marking. WhenTb

j ∩ Tc ⊆/ T(Pcp

i) or M ∉ M

cwlpPcp

i, ¬ fTbj

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

∩Tc ⊆ (Pcpi

) and M ∈ M cwlp

Pcp

i. It is self-evident from Eq. (3)

that Eq. (4) holds if DTbj (M) = 0. Let us consider the case

of DTbj (M) = 1. Let us introduce the decomposition

Tbj = Tbjc ∪ Tbju

. Here, Tbjc ⊆ Tc and Tbju ⊆ Tu. SincecwlpPcpi

(Q)(M) = 1 from M ∈ M cwlp

Pcpi, it is found that

wlpTbjc (Q)(M) = 1. Therefore, Q(M′) = 1 at each marking

attainable by the firing of Tbjc 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 descriptor Q be control-invari-

ant. Then, the following three conditions are equivalent.

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

j ∈ Tb, arbitrary f, g ∈ F(Q) satisfy

the following:

(3) For arbitrary Tbj ∈ 2T

c ∩ 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), Tb

j ∈ Tb such that

(6)

(7)

(8)

(9)

5

Since Q is control-invariant, it is possible to assumewithout loss of generality that Tbj

⊆ Tc. Let us considerPcpi

∈ cpTbj. If T(Pcp

i) = ∅, cwlpPcpi

(Q)(M) = 1 from Eq. (6).Let us consider 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 ∈ 2T

c ∩ 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 System 1, h ∈ F(Q). Therefore, wlpTbj

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

3.4. Example

Let us consider the existence of the maximum allow-able feedback in THPNIP shown in Fig. 2. The initialmarking 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 maximumallowable 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 at 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 include control for the case in which only part of themarkings are observable, and control for the case in whichcontrol specifications including time as well as markingsare 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 modu-lar control synthesis in Petri nets with external inputplaces. 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.Maximum allowable feedback in timed Petri net withexternal 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 THPNIP.

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AUTHORS (from left to right)

Atsushi Tanaka (member) completed the M.S. program at Kinki University in 1997 and the doctoral program at OsakaUniversity in 2001, receiving a D.Eng. degree. He is a lecturer in the CG Research Department, Kinki College of Computersand Electronics. He has been engaged in research on discrete phenomenon systems, net theory, and hybrid systems. He is amember of the Systems, Control and Information Society, the Society of Instrument and Control Engineers, and the InformationProcessing Society.

Toshimitsu Ushio (member) graduated from the Department of System Engineering of Kobe University in 1980 andcompleted the doctoral program in 1985. He then became a research associate at the University of California, Berkeley. Afterserving as a research associate at Kobe University, a lecturer and associate professor at Kobe Women’s University, and anassociate professor at Osaka University, he has been a professor on the Faculty of Engineering Science, Osaka University, since1997. His research interests are discrete phenomenon systems, nonlinear phenomena, and hybrid systems. He is a member ofthe Society of Instrument and Control Engineers, the Information Processing Society, the Systems, Control and InformationSociety, IEEE, and AMC.

Shinzo Kodama (member) graduated from the Faculty of Science and Engineering of Waseda University in 1955 andobtained a Ph.D. degree from the University of California. After serving as a professor at Osaka University, he has been aprofessor on the Faculty of Science and Engineering of Kinki University since 1995. He was a visiting associate professor atthe University of California in 1969–70. He has been engaged in research on control theory and net theory. He is a Life Fellowof IEEE, a fellow of the Society of Instrument and Control Engineers, and an honorary member of the Systems, Control andInformation Society.

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