Petri Nets-Based Computer Aided Synthesis of Control Systems for Discrete Events Dynamic Systems

Copyright © IFAC Computer Aided Design in Control Systems. Swansea. UK. 1991

PETRI NETS-BASED COMPUTER AIDED SYNTHESIS OF CONTROL SYSTEMS FOR DISCRETE EVENTS DYNAMIC SYSTEMS F. Capkovic Institute of Technical Cybernetics. Slovak. Academy of Sciences. Brarislava. Czechoslovakia

Absj:r~ct -,An idea of utilizing the Petri nets-based analytical system approach to modelling the discrete event dynamic systems and its application to the automation of their control systems synthes i s is presented and elaborated in this paper. Not to be dependent on partial features of a problem to be dealt with a general abstract discrete dynamic system is built by means of the so called bounded ordinary Petri nets to model the mentioned types of systems to be controlled. In attempting to treat problems of the computer aided synthesis of their contro l systems by means of a uniform analytical technique, we find that the automatic step-by-step analysis of the model gives us the expected procedure. This procedure can serve as a kernel of a decision support algorithm. It is developed to improve and generalize the previous author's ( 1988 ) procedure by means of utilizing the main principle of the automatic reasoning presented and elaborated in recent author's works ( 1990 ) . Its applicability is illustrated on a simple example of the automatic robot assembly.

Keywords . Computer-aided design; control systems; control theory; discrete events dynamic systems; mode lling; Petri nets; process control; simulation; system analysis. INTRODUCTION

ported by the author's works ( 1988, 1989, 1990, and 1991). Although the monography by Peterson (1981) offers many details about the different types of the PN we prefer the vector notation of the abstract dynamic discrete system describing the PN in analytical terms. It seems to be more convenient from the system theory and control theory points of view. Thus, we obtain an analogy to the s tate equation of the discrete-time system which is widel y used in those theories. Since the systems discrete in nature exhibit their changes over time asynchronously, as the results of the operation of internal and external influences, the time cannot be expressed explicitly by any monotonically flowing independent variable of the system. It is only an implicit quantity. Therefore, the independent variable of such a type of the discrete systems can only represent a step of the system dynamics development.

Continuous and discrete-time systems have been dominant s ubject matters of control theory. Consequently, there are many methods both for analysis of such types of systems and for their control systems synthesis. In case of systems whi ch are discrete by nature, i.e. the so called discrete events dynamic systems (DEDS ) the situation is different. Until recently they had been situated out of the control theory centre of interest. At present, when the complexity of discrete production processes (DPP), E.p. flexible manufacturing systems (FMS), is rapidly increasing, it is unthinkable to perform the synthesis of their control systems intuitively. It is necessary to treat this question more seriously - see the challenge (1987) . The first fundamental question that arises in this way is how to construct general descriptive models of the DE OS behaviour . Many mathematical methods in this area are perfected every year - e .g. Cohen and co-workers ( 1985), Inan and Var iya ( 1988) etc. The Petri nets (PN ) can also be, and are being, effectively used owing to their ability to express the DE OS asynchronous behaviour and concurrent operation of their subprocesses - see e.g. Va lette and colleagues ( 1985), Ichikawa and Hiraishi (1988), and man y other works. Such emphasis on the PN should not confuse their proper role. The PN are not a substitute for mas tery of subject matter, experience or judgment, neither are they a substitute for proper research in the area of the DE OS themselves. They are only aids to research and seem to be an integral part of va lid research in the DEDS control theory. Use of the PN based methods belongs to the sphere of the discrete mathematics applications.

However, the mathematical model of the discrete plant itself is not the aim of our effort. The model is only a means of finding the way how to reach the our actua l aim - the CAD system for the control systems synthesis. It is natural to expect that the knowledge of the model of the controlled discrete system will allow the automatic analysis of the system properties by means of simulation. The automatic step-by-step analysis of the model gives us an efficient support for decisionmaking in the process of the control algorithm synthesis. It represents a kernel of the CAD algorithm helping us to find the correct sequence of control interferences with operating the DEDS. PETRI NETS-8ASED MODELS OF THE DE OS

This paper is an attempt to find a su itable general exact methodology, from the system theory point of view, in order to have a possibility of the analytical formulation of the problems concerning the control systems synthesis for the DEDS. On the basis of such an approach a computer-aided design (CAD ) algorithm can be realized to automatize the DEDS control systems synthesis. This paper is sup-

The DEDS consists of a finite number (e .g. n) of the elementary subprocesses ( operations, activities) which may vary over time concurrently and their starti ng and ending may be asynchronous - i.e. such systems change their states owing to the asynchronous discrete events expressed by means of a finite number (e.g. m) of discrete variables. Knowing the nature of the general analogy between the DE OS ele313

The formalism presented here is a general one. It covers considerably wide class of the PN (ordinary ones - especially bounded and safety, logical ones - bivalued or fuzzy, etc.) according to actual specifying the values of the variables and parameters. However, such a form of the formalism is not suitable for practical using. It must be concretized by means of the mentioned system approach. Hence, we can distinguish two basic forms of the mathematical description of the PN as follows 1) the form suitable for the ordinary PN

mentary subprocesses and the PN positions and the analogy between the DE OS discrete events and the PN transitions we shall describe the structure and the dynamic behaviour of the DE OS in analytic terms by means of an abstract discrete dynamic system. As to their structure the PN are directed bipartite graphs with two types of nodes (positions and transitions) and two types of edges (oriented arcs from the positions to the transitions and the oriented arcs from the transitions to the positions ) . The PN dynamics is characterized by moving tokens among positions by firing some transitions. Many details about different types of the PN and about their properties can be found in the monography by Peterson ( 1981 ) . To prepare the way for the applications of the PN theory to the study of the DEDS structure and dynamics we particularly wish to emphasize our view on the fundamental conception of the PN, since it is this which yields the above mentioned analytical expression of the DE OS model.

(3)

x

where

t j' j

where X = {xo' xl'

xN } is a set of the state vecx tors of the positions in different situations uN J is a set of the state vecu tors of the transitions in different situations is a transition function of the S:XXU-X PN x0 is an initial state vector of the PN positions Nx' Nu are some integers

l}

=

1, m.

2) the form suitable for the logical PN xk+ 1 = xk or B and uk x

klk=D

with

,k = 0, N

(4)

= x 0

B = GT or F F and uk < = xk

s )T s = 0, Nx Pn 1 r )T ur r = 0, Nu t t1' t~ m T symbolizes the vector or matrix transposition 6' s EfD,l, ... ,c },i=l,n;s=D,N x Pi Pi are the states of the PN elementary positions. They express the numbers of tokens placed into the rings representing the positions. c is a capacity vector of the p PN positions. Its elements express the maximal numbers of tokens which are allowed to be placed into positions J

B = GT - F

G is the Cm x n)-dimensional arcs incidence matrix. Its elements g .. E.[O, I}, i = 1, m; j = 1, n, IJ express the presence or the absence of the connections emerging from the transitions t., i = 1, m, and entering the positions p . , j = 1,ln. J B is the (n x m)-dimensional structural matrix given by means of the matrices F, G

( 2)

i~E{D,

0

with K being the discrete step of the system dynamics development. It is different from the discrete time, because the system is strictly asynchronous N is an integer - the number of the steps xkEO X is the n-dimensional state vector of the system in the step k uk ~ U is the m-dimensional control vector of the system in the step k F is the (n x m)-dimensional arcs incidence matrix. Its elements fijE fo, 1), i = 1, n ; j = 1, m, express the presence ( f .. = 1) or the absence ( f .. = 0) of the connec 1J tions emerging the posiIJ tions Pi' i = 1, n, and entering the transitions

SPN: (p, T, F, G:> , P n T = 0 , F n G = 0 (l) where is a finite set of the PN posiP (PI' tions T {t , t m } is a finite set of the PN tranl sitions n, m are some integers; 0 is an empty set F '=: P ><: T is a set of arbitrary interconnections (arcs) oriented from the positions to the transitions G ~ T ~ P is a set of arbitrary arcs oriented from the transitions to the positions ~ symbolizes the Cartesian product of the sets. The PN dynamics DP.N can be formally represented by the following quadruplet

6~ ,

x

<: F•u = x k k

Proceeding in purely formal fashion, the PN structure SpN can be expressed formally as a quadruplet

xs

=

klk=D

6

where or, and are, respectively, the operator of the logi--- cal additioning and logical multiplying in the corresponding type of logic - i.e. in the bivalued logic or in the fuzzy one. It must be said that the components of the vectors and the elements of the matrices in (4) are considered to be logical too. In the bivalued logic they acquire their values from the set {O, l}. In the fuzzy logic they acquire their values from the interval of real numbers 0, 1

<

>.

In this paper we shall use the so called bounded PN - i.e. the ordinary PN with c ~ 1, i = 1, n. The Pi positions of such type of the PN will represent the elementary subprocesses of the DE OS ( e.g. the elementary operations of the FMS ) and the transitions will represent the discrete events which can happen during the DE OS operation (e.g. the starting or ending the elementary operations of the FMS). Regarding its generality the presented analytical model ( 3) can be utilized for describing a sufficiently wide class of the DPP.

, j = 1, m; r = 0, Nu

are the states of the PN elementary transitions . They show us whether the corresponding transitions are closed ( D) or opened ( l) .

314

ons, i.e. the positions containing at least one token . It is clear that the n-dimensional vector -k -kT (7) (Y l' ... , Yn) Yk

COMPUTER AIDED SYNTHESIS OF THE DE OS CONTROL SYSTEMS Before indicating how to alter the behaviour of the DE OS in some fashion, let us analyse some as pects of the OEOS peculiarity in connection with their control. We have to be aware of the OEOS asynchronous behaviour. As we can see, neither a "trajectory" in order to have the s tate vector behave in a prescribed fashion nor a precise terminal state in classical understanding are prescribed. On the other hand, the actual state of the OEOS dynamics development points out, in a step-by-step fashion, the possibilities how the control vector should be synthetized, i.e. which discrete events should be performed in order to transfer the system in question from the existing state to another one, if it is still actually possible. Although the above presented possibility of the analytical expression of the OEOS structure and dynamics itself does not yet any recipe for the automatic control of the system dynamics development in the steps k = 0, 1, ... , N, by means of the step-by-step analysis of the model we can obtain an analytical procedure of the control strategy synthesis . The only way to realize the step-by-step development of the process dynamics is to generate the control vector u in each s tep . The k question as to how such an automatic procedure can be built must now be faced . In what follows, we shall show a very simple technique in order to answer this question. It is performed on the basis of the DE OS simulation. It is simplier than the technique by Capkovic ( 19B8 ) , because we use the results in the area of the so called automatic reasoning presented by this author ( 1989, 1990, 1991 ).

k

Its nonzero components point out the trans itions which are not able to be opened in the step k the transitions having passive at least one of their input positions. Using the auxiliary vector zk z.k J

{~

wk = zk =

k)T Yn

, j

l, m

J

~

zk = 11. m - zk

(10 )

(11)

i=l where

These vectors include opening all possible combinations of the transitions belonging to the mentioned basis Tb and they create the set of potentially possible control strategies in the step k: U~. Only when these control vectors are tested with respect to the inequality conditions associated to the rlationship ( 3) we can obtain (after elimination of the unsuitable vectors) the actual set of the realizable control strategies U~ ( in the step

(6)

, i

(9)

This auxiliary vector is not yet the control vector u . Its nonzero components point out only a set Tb k of transitions which represents a basis for the control vector construction. The number of potentially possible control vectors in the step k is given by the number

Consider the auxiliary vector with dimensionali ty n as follows

0

k (z l' ... , / m) T k if v j > 0 k if v. = 0

we can obtain the vector

( 5)

if ok > 0 Pi ok if 0 Pi

k Yi

(8)

Its nonzero components point out the transitions having active at least one of their input positions. However, instead we are interested in the transitions having active all of their input positions - the transitions which are able to be opened in the step k ( i.e. the so called enabled transitions ) . Naturally, one way to obtain such transitions is to find the transitions which are not able to be opened in this step and eliminate them. Here we have no difficulty in understanding the following procedure including the above mentioned process of the automatic reasoning.

1

k Yi = 1

~(:t )

wk = FT. xk

{

~

being the operator of the logical negation being n-dimensional constant vector with all components equal to 1 will point out the passive positions, i.e. the positions without the tokens (empty positions ) . Regarding the structural matrix F and the vector (7) we can obtain another m-dimensional auxiliary vector

can be obtained in the following way

with components k Yi

-k Yi

n - Yk

~n

k T

k (Y l'

Yk

~

( wl ' · · · ' wm)

Yk

~

with

Consider that in the step k = 0 the system (3) is in an initial state represented by means of the state vector x = xo. Its components express the k actual states of the DE OS elementary subprocesses by means of the tokens placed into the corresponding positions. We wish to proceed, if possible, from the state xk to another s tate xk+l ' xk . It was shown in the author's work ( 1988 ) that regarding the structural matrix F and the actual state vector xk the m-dimensional auxiliary vector wk

11.

Yk

k) . This set is constructed from such control vec tors which satisfy the inequality conditions. Henk+l ce, we can create the set X of the reall y reaR chable states in the next step of the system dynamics development in such a way that for an y vector

1, n

u E Uk we enumerate the corresponding vector xk+l k R k by means of the equation ( 3) . When the set UR con-

Its nonzero components point out the active positi-

315

tains only one control vector uk ' this vector can be used for control of the DE OS in the step k. When the set U~ contains several realizable control vectors a metarule must be defined to choose one of them. As to the metarule formulation we have to distinguish two main cases as follows 1) when the metarule is given in advance by means of the only possible prescribed sequence of technological operations or when it is self-evident under given circumstances in an actual situation occuring durIng the process dynamics development (e.g. in order to avoid technological nonsenses ) 2) when the metarule is not s o clearly evident in the actual situation or when several apparently eqUIvalent metarules (making sense ) are possible under given circumstances. While the first case is very simple from the metarule choice point of view, the second one is more complicated from this angle. In such a case we can utilize the information embraced in the mentioned set x~+l , because this set affords the absolutly entire spectrum of the further one-step development possibilities of the sys tem in question ( from the step k to the step k+l ) . When we are not sure of the choice, inspite of the analys is of this set we can (i n the debatable cases) use two- or more- ' -steps "excursion" to the future system dynamics development - i.e. to the steps k+2, k+3, . . . . After the suitable metarule choice we have to return, of course, to the actual step k because it is the step for which the metarule is' chosen. We must say that because of the breadth of the various OEOS (different types of the DPP or FMS ) mutually different as to the type of the eleme~tary subprocesses or operations, any general analytical crIterIon cannot be formulated in order to chooce the metarule. Even a formulation of such an general crIterIon IS practically impossible also for the single steps of a more complicated discrete system alone, because of both the possible wide spectrum of its various elementary operations (tooling different shapes~ assembly of different parts, di s assembly of dIfferent parts, etc. ) and the difficulties to determine a sequence of their performing in the strictly asynchronous process. However, on the other hand, the presented model-based step-by-step justifying of the complex DE OS operation can also help us to discover a new interest ing poss ibilIty as to the sequence of the technological operatIons or, at least, a better one in comparison with that intended in the original design process of the OEDS structure .

trol vectors

- If (only one realizable control vector uk E U~ ) THEN GO TO Label 2 enumerating the responses of the system ( 3) on the realizable control vectors from the set U~ and creating the set x~+l of the really reachable states in the s tep k+l determination of the metarule (either on the basis of experience in the OEOS operation or on the bas i s of the se t x~+l analysis ) the choIce of the corresponding control vector uk Label 2: realization of the control vector uk computing the new state vector xk+l

- If ( xk+l ' xk ) or ( xk+l ' xt ) THEN BEGIN

~k + 1

GO TO Label 1

ENO-

output of the pairs (uk ' xk ) , k

0, N

END The foregoing procedure can serve as a kernel of a CAD system determined for the OEOS control systems synthesis. From this point of view it is not so much any fixed analytic technique as a s tate of mind. Consequently, it is natural to expect that the theory of this type would be illustrated on an examle . In what follows we shall study a s imple case of the FMS to familiatize the reader with both the proposed PN-based modelling and the model-based examination of the OEOS dynamics development in order to synthetize the sequence of the control interferences with their operation. A CASE STUDY Consider. the automatic robot assembly, more precisely the dIsassembly and subsequent assembly, concernIng the problem of replacing the old bearing B on the axle A - see Fig. 1 - by the new bearing E. ThIS problem was studied from another point of view by Zhang (1989 ) . As we can see, the bearing B cannot be grasped by the robot and handled directly, but another bearing 0 and the sleeve C must be removed from the axle A before it. This is made in the disassembly process. Only when the axle A is free of these parts the assembly process can be performed - i.e. the bearing E can be put on the axle A and subsequently, the sleeve C and the bearing 0 can be put on the axle A too, in order to completize the device. The situation before disassembly of the bearing B to be replace is on Fig.la ) and the s ituation after assembly of the correct bearing E on Fig.lb).

A rough flow chart of the algorithm of the OEOS control systems synthesis can be verbally described as follows. Algorithm of the control synthesis START

k =0 x k = Xo

defining a desirable terminal state vector x

t

Label 1: creating the auxiliar y vector Yk on the baof ( 6) Yk = ~ Yk T vk = F . Yk creating the auxiliary vec t or zk on the basis 01 ( 9) wk = zk = ~ zk generating the set U~ of the potentially possible control vectors by means of ( 11 ) testing the realizability inequalities associated to 0) creating the set U~ of the realizable con-

~is

a)

Fig. 1.

316

b)

The scheme of the handled object.

see in Fig. 2., the matrix has the following nonzero elements: f 13 , f24' f 31 , f32' f33' f34' f44' fS2' fS7' f 5S ' f65' f66' f 7S ' fS2' f93' f96' f97' f lo ,7' flo,S' f 11 ,1. The nonzero elements of the matrix G are the following: g15' g25' g29' g34'

The graphical representation of the corresponding PN-based model is given in Fig. 2.

g39' g44' g47' gS2' gS3' g56' g63' g69' g6,10' g73' g75' g7S' g7,10' gS3' gS,lO' gS,ll·

•

•

)0-.----.

The active positions in Fig . 2 symbolizes the initial state of the object presented in Fig.la). It can be written mathematically by the vector xo = ( 1 1 4 0 0 1 0 1 0 0 l )T The Fig.lb) shows the terminal state of the object. It can be mathematically expressed as follows x = (0 I 4 lOO 0 1 0 1 l ) T t The positions capacity vector has the form c

( 1 141 1 1 1 1 1 1 l ) T

p

Using the above introduced algorithm we have K=0

o0

0 0 UT

yo

Cl 1 1

yo

(0 001 1 0

v

(0

1 3 2)T

z

(0

UT

0 0

o0

0 0

O)T

0 0 0 0 O) T

w0

Cl

Tb

[ t l } , u0 = w0

The inequality F.u k ~ xk ' checking the existence of conflict situations is satisfied (because only one transition is enabled), as well as the inequality

T

Fig. 2.

c .

.

G .u k = c p - xk ' checklng the overflow of the POS1tions capacity. Consequently, U is the regular o control vector. We can imbed it into ( 3). Hence,

The PN-based model of the FMS operation

K= 1 The states of the elementary subprocesses of this DPP can be represented by means of n = 11 PN positions as follows : PI - B is on A; P2 - A is present in the working (disassembly or assembly) place; P3 - an actual arrangement of the group of parts in working place consists of at least one part; P4 - B is not on A; Ps - D is not on A; P6 - E is not on A; P7 - A is entirely free of the parts and it

Xl

Cl

o1

0 1 0

VI

Cl 0 1 1 1

2 UT

wI

(0 1 000

3

o0

o O)T

O)T

{ t21 ' u I = w1 The conflicts testing inequality is satisfied. The overflow testing one is not satisfied. However, inspite of this fact, the possible overflow is treated by means of clossing the loop between t2 and Tb

is out of the working place; Ps - C is on A; Ps -

Ps ( t 2 , PS ) - i.e. a token placed in P5 making enabled the transition t2 leaves P5 in order to flow

- C is on A; P9 - C is not on A; PlO - E is on A; Pll - D is on A.

over t2 and then, after passing t 2 , it immediately returns to the empty P5 again. It means that no overflow occures. In such a way the unsatisfied condition points out a need of a correction of the model structure. It would be performed automatically. In the other words, the model would be able of a "self-correction" of its own structure. Thus ,

The discrete events which should occure during the process operation can be expressed by means of the forllowing m = S transitions: t1 - start of taking off 0 from A; t2 - start of taking off C from A; t3 - start of taking off B from A; t4 - start of removing A from the working place ( from the disassembly one); ts - start of putting A into the wor-

K= 2

king place (in to the assembly one); t6 - start of putting E on A; t7 - start of putting C on A; ts - start of putting 0 on A. The PN structure can be represented by means of the oriented arcs inciriency matrices F, G. As we can

x2

Cl

2 0 1 lOO I

v2

Cl

oI

w2

(0 0 1 0 0 1 0 O)T

\ 317

o O)T

I 0 1 UT

{ t 3 , t 6 } , N = 2 , Np t

3

U~ ,{ u~, u~, u~}, u~ , (0 0 1 0 0 0 0 0) T u~

u

( 0 0 0 0 0 1 0 O)T, u~ , ( 0 0 1 0 0 1 0 O)T

2

Cl

3 0

CONCLUSIONS The PN-based model of the DEDS was established to describe the DEDS structure and dynamics development in analytical terms. The analytical approach offers ~ formalism to 1) describe the DEDS structure and dynamics in a unific form; 2) perform the DEDS automatic analysis and their dynamic evaluation; 3) provide a uniform way for the control strategy synthesis. A simple analytical procedure for the control strategy synthesis is presented. It can serve as the kernel of the CAD system for the DEDS control systems. Its application is illustrated on the simple FMS case study. This procedure improves that one presented by author (1988 ) , namely by means of decreasing the number of possibilities which have to be tested. The improvement is achieved by using the so called automatic reasoning process based on the inference mechanism of the PN dynamic behaviour.

000

we can see that the last state represents the technological nonsense (B and E are simultaneously on A without D and C). Hence, we can formulate the metarule: "the disassembly has to be finished before starting assembly". Consequently we have u u~ ,(0 0 1 0 0 0 0 o)T 2 K 3 (0 1 1 1 1 1 0 0 1 0 O)T {t4' t J ,

U~

{u~, u~,

REFERENCES

uU

By means of the above indicated procedure we have the demand to close the loop (t 4 , P4)' After application of the rule: "the assembly cannot be performed in disassembly place" we obtain the control

K

4

ColI. of authors. (1987). Challenge to control: A collective view. IEEE Trans. Aut. Contr., AC-32, 27S-28S Capkovic, F. (1988). A decision support algorithm for flexible manufacturing systems control. Computers in Industry, IQ, l6S-l7o Capkovic, F. (1989). An analytical model of rule-based types of reasoning and decisionmaking. In Plander, I. (Ed. ) , Artificial Intelligence and Information-Control Systems of Robots - 89. Proc. S-th Int. Conf. AIICSR-89, Strbske Pleso, Czecho-Slovakia. November 1989. North-Holland, Amsterdam. 369-372 Capkovic, F. (1990a ) . A system approach to knowledge representation and its application in modelling the automatic reasoning and decisionmaking. In Proc. 6-th Int. Symp. System-Modelling-Control, zakotane, Poland, October 1990. 64 69 Capkovic, F.1990b). A study of decductive types of automatic reasoning by means of simulation. Proc. IMACS Europ. Simul. Meet. Problem Solving by Simulation, August 1990, Esztergom, Hungary. 171-177 Capkovic, F. (1991 ) . A representation of rule-based types of reasoning by means of an abstract dynamic system development. Systems Science, 12, No 4, in print Cohen, G., D.Dubois, J.P.Quadrat, and M.Viot. (198S ) A linear system theoretic view of discrete event processes and its use for performance evaluation. IEEE Trans. Aut. Contr., AC-30, 210-220 Ichikawa, A.,and K.Hiraischi.~8 ) . Analysis and control of discrete events systems by Petri nets. In Discrete Events Systems: Models and Applications. Springer Verlag, New York. Inan~nd P.Variya. (1988 ) . Finitely recursive process models for discrete events systems. IEEE Trans. Aut. Contr., AC-33, 626-639 Valette, M., M.Courvoisier, H.Demnon,andJ.M.8igou. (198S ) . Putting Petri nets to work for controlling FMS. In Proc. Int. Symp. Circuits Syst. Kyoto, Japan, vol. 2. . 929-932 Peterson, J.L. (1981). Petri Net Theory and th~ Modeling of Systems. Prentice Hall, New York. Zhang, W. (1989). Representation of assembly and automatic robot planning by Petri nets. IEEE Trans. Syst., Man,and Cybern. SMC-19, No 2, 418-422

000 O)T

u3

(0 0 0

x4

(0 0 0 1 1 1 1 0 1 0 D)'

Tb

{ t s ' t6 },

T

U~

=

f u14 '

u24 , un

Having closed the loop (t s ' P6) we have, after the elimination of both the simultaneous firing t ' t6 s and a technological nonsense,the control (0 0 0 0 1 0 0 o)T K

s

(0 1 1 1 1 1 0 0

0 O)T

After formulation the rule: " the disassembly process is finished, start the assembly process" we have the following control vector

K

6

o O)T

Us

(0 0 000

x6

(0 1 2 1 100 0 1

Tb

{ t 4' t 7 , t 8 } , Nt

o)T 3, Np , 7

6 - {I UP u6 , u62 , u63 , u4 , uS' u6 ' u~ } 6 6 6 We find the demands of closing the loops (t , PS)' 7 ( t 7 , PlO)' (t 8 , PlO)' After the elimination of the nonsenses and applying the rule: "the assembly process is running, the disassembly operations cannot be performed" we obtain the control vector ( 0 0 0 0 0 0 1 o)T K

7

x

7

( 0 1 3 1 1 001 0

' ( 0 0 0 0 0 0 0 l )T

we obtain the terminal state x . Therefore, the prot cess of the control strategy synthesis can be finished after N ' 7 steps of the process dynamics development.

The conflicts testing inequality is not satisfied for the last vector - namely because of P9 being common input position for t , th ( the token cannot 3 oetween these two be divided). We have an optlon transitions as to which of them will be fired. Although the second inequality is not satisfied for any of these two possibilities, the situation is treated by closing the loops (t 3 , P9)' (t 6 , P9)' After utilizing the set of really reachable states consisting of the following state vectors x~ ( 0 1 1 1 0 0 0 o)T x3

7

o)T

Tb {t2,t4,t8}'U~ {u~, u~} After the elimination of the simultaneous firing t 2 , t8 we find, on the basis of the set of really reachable states, that aplying the control vector

318