Control Synthesis of Discrete Manufacturing Systems using Timed Finite utomata JROSLV FOGEL Institute of Informatics Slovak cademy of Sciences ratislav Dúbravská 9, SLOVK REPULIC bstract: - n application of timed game automaton to the control synthesis of discrete manufacturing systems is presented in the paper. This approach adopts the algorithm for optimal control synthesis of timed automata proposed in [] as a game-theoretic approach between controller the model of the plant. Key- Words: - Timed automat control synthesis, model verification, production systems.. Introduction t this time, there exist two main approaches to the control synthesis of timed discrete event dynamic systems. The first language - based approach is the extension of the basic Ramadge- Wonham supervisory control theory with time constraints can be found in [5]. In this approach the supervisory control is based on the theory of formal languages. The second approach is state - based. The authors of [,] use a game - theoretic formulation of the control synthesis problem. In this approach, the interaction between a controller the plant can be seen as some variant of the two- person antagonistic game. strategy for a given game is a rule that tells the controller how to choose between possible actions in any game position for finite state games is done using the max- min principle of [6]. The manufacturing systems are composed of the number of distributed production devices, which can be modeled by timed automata. The composite model of the production process can be obtained as the synchronous composition of the individual models. The behaviors of the composite model can be specified for example, by timed computational tree logic (TCTL) verified using some of the known model checking techniques. t this time, there exist several modeling verification tools also known as formal methods, which are available on the net. For timed automata these are for example, Kronos [3], Uppaal [4]. The availability of such powerful verification techniques enables to design the model with correct quantitative timing information before its using for the optimal controller strategy extraction. The rest of the paper is organized as follows: In section, we introduce the Timed game automaton describe the synthesis algorithm given in []. In section 3, the application of the proposed algorithm in the control of discrete manufacturing systems is described. The illustrative example is given in section 4.. Time optimal control strategy for timed game automaton The theory of timed automata was first described in [7]. Timed automata are finite automata equipped with time variables also called clocks. Clocks grow uniformly when the automaton is in some state. Clocks constraints may be state invariants or the transition guards that specify when the transition is enabled. Some clocks may be reset when the transition is taken. Time games between timed automata its environment are modeled by timed game automaton [,]. Timed game automaton (TG) is a tuple TG=(Z,,, T, T, δ, ρ) Z Q X is a zone, Q X are the state clock spaces,, are distinct action alphabets, are controller action are uncontrolled action of the environment, T Q X ε T Q X ε are timing constraints, ε, ε are {ε} {ε}, ε is an action, δ: Q ε ε Q ρ: Q ε ε J(X) are transition reset functions, J(X) determines the functions which resets the clocks. Further requirements given in [] are the following: the automaton is strongly non-zeno, the set T (a)={x: (a) T } is a k- zone similar T (. The winning game problem of a player (rachystochronic problem) is formulated as follows: Given a TG a set F Q X find a strategy T * T for player which allows him to reach the target set F as fast as possible whatever player does.
The algorithm computes the value function f * : Q X R + that allows to reach F from ( in no more than f * ( time. The algorithm assumes that in the configuration ( the value function f is known in each time or discrete step (defined in []). For those configurations from which it is not possible to reach F the function f has the value infinity. The algorithm is given in [] as follows: Initialization step when ( F f ( otherwise α act( < πidle( { ε} > πidle( α( α act( { ε} = πidle( < = πidle( = αact( = { a T max f ( δ' ( = πact( f ( )}. ε b The authors show that the algorithm converges if function f used in the algorithm belong to the class of k- simple functions, which is closed under operator π. k - simple functions are defined as Iteration steps repeat n:= n+ ; f n := π ( f n- ); until f n = f n- ; Strategy extraction is f * := f n ; T * = α (f * ) The operators are defined as follows: π ( f ) = min{ f, πact( f ), πidle( f ) } πact( = min max f ( δ '( ) π idle a ε b ( = inf v( t R+ v( = max(sup g( τ ), t + f ( x + ) τ < t g( max( τ + f ( δ '( x + τ, ε, )) = b The strategy extraction operator is given as () ( if a = b = ε, ( ε ) T δ '( ( δ (, ρ( ( ) otherwise ci when x Di f ( dj xlj when x Ej D i, i=,...,m E j, j=,...,n are k- zones, E j {x x lj k} c i, d j N { }. subset of X is called k-zone if it can be obtained as a oolean combination of inequalities of the form x i c, x i > c, x i - x j c, c {,,..., k}. 3. pplication of the algorithm in the production process control The discrete production process consists of the group of distributed production devices co-operating mutually. Every device can be modeled as TG. The complex model of the production process can be defined as the synchronous parallel composition of the individual TG i as follows. Let TG = (Z,,, T, T, σ, ρ ) TG = (Z,,, T, T, σ, ρ ) then the parallel composition is defined as TG TG =(Z,,, T, T, σ, ρ), Q = Q Q, X =X X, Z=Q X, =, =, T T T T T T T T when a when a when a when b when b when b
{ q', s' q' Q, s' Q} if a or b δ ( = q', δ ( s, = s' δ ( s, { q', s q' Q, s Q} if a or b δ ( = q' { s' q Q, s' Q} if a or b δ ( s, = s' The parallel operator can be easily spread to n devices. In this application, the action from can be interpreted as the control actions that start the technological operations, meantime the actions from are reaction of the environment as e.g., information about the operation finishing, some sensor or break-down information, etc., obtained by the monitoring subsystem. During the model development process, it is necessary to create the correct model expressing all technological requirements as e.g., timing constraints, sequences of technological operations, model safety, etc. The model, which satisfies these requirements, can be obtained in the verification process also known as model checking for finite state models. s we mentioned before, there exist several programming tools using the description language based on the timed automata model. The specification language of the model behaviors usually used is based on the real-time temporal logic as e.g., TCTL (time computational tree logic). Such modeling verification tools are already mentioned Kronos Uppaal. Using these tools those state transitions can be found by reachability analysis from which the model cannot reach the target set. For these states, the values of the function f have to be set equal infinity for every transition. 4. n illustrative example The working cell contains two numerically controlled machines M M. Each machine may process two types of parts P P. The parts enter the incoming buffer with regard to part's type in alternating way. The capacity of the buffer is one. The buffer is getting free when a part begins to be processing on some machine that is simultaneously the signal for the entrance of a new part into the buffer. The machines can process both types of parts with the different periods of processing d ij, i=,; j=,, i is the type of the machine j is the type of the part. fter the operations on the machines are finished, the parts enter the outgoing buffer if this is free. The capacity of the outgoing buffer is also one. We suppose that the buffer will be made automatically until time T. The goal is to find the control strategy starting the operations of the parts processing on the machines such, that the parts' frequency entrance into the incoming buffer is maximal. The following constraints must be fulfilled in each production cycle. t the same time, each machine can process only one part. Fig.. Timed automata model of M
Fig.. Timed automata model of M The signal about the operation finishing on a given machine arrives after the period d ij has elapsed. The models of machines M, M incoming outgoing buffers UF in, UF out are shown in Figures - 3 in the form of timed finite automata. Here start ij is the control action machine Mi starts work on a Pj- part while finish ij is the uncontrolled action machine Mi finishes working on a Pj-part. is the uncontrolled event signalling that the outgoing buffer is free, P j is the event signalling which type of part is in the incoming buffer. The meaning of the automaton states is evident from their names. The initial states are marked by small arrow. In the wait state, the machine is waiting for the signal maximally T time units. The time constants have the following meaning: T, T are upper - bound time constants during which it is advantageous for the machine (from the point of view of the production cycle minimisation) to wait for the part of the other type then to perform the operation on the part actually being situated in the buffer. These constant can be determined from the technological parameters d ij T. Where T is upper- bound time constant during which the outgoing buffer is getting free. T is the waiting time of the part in the buffer for processing on the machine. The objective of the control strategy is to minimize the total sum of T for a given time horizon. ccording the operation processing on machine Mi, the strategy for machine Mj, j i is determined as follows: Mj is waiting for the part of other type in the idle state or Mj is performing the operation on the part actually placed in the incoming buffer. The level of the part processing on machine Mi is given by the value d ij - y y is the actual value of the time variable during the model's visit in the busy state. The optimal waiting time t of the machine is time that gives the infimum of the function v( in the relation (). The complex model of the production cell is given as the parallel composition of the individual devices M M UF in UF out. The value functions f for the state transitions of the machines M M are shown in Tables. For buffers' transitions their values are equal. The described algorithm can provide the following decisions. For example, let the model is in the configuration (busy, idle, full, ) in time given by the local time variables (y [d -λ, d ], z=, u [,T], x=), λ [d, ] is the level of the part P processing on machine M. Let the part P is in the incoming buffer d <d d <d. In the case, when λ is sufficiently small it is advantageous for M to wait in the state idle for part of the type than to process part P actually placed in the buffer. The results of the algorithm follow. The results of the algorithm for the control strategy extraction (case ) were obtained in the simulation process. For the given technological values d ij T, the objective function Q = T was computed for time horizon of parts then compared with fixed strategy (case, each part is processed on that machine which processing time is shorter). The signal was generated romly with the uniform distribution function from the interval [,T ] T was equal 5 time units. The values of the processing intervals were d =, d = 4, d = 5, d = 5 time units. The function Q computed in case has the value Q = 847 which is smaller than the value Q = 9 computed in case.
5. Conclusion The described method of the production modelling control has the following advantages: it enables to use the existing modelling tools for the model development its verification from the point of view of the control synthesis it renders time-optimal trajectory for reaching the target configuration of the model. Table. states y actions idle start j (, T ] ε busy (,d ) ε [d, ) busy (,d ) ε [d, ) wait d (d, d +T ] ε (d, d +T ] wait d (d, d +T ] ε (d, d +T ] Table. states z actions idle start j (, T ] ε busy (,d ) ε [d, ) busy (,d ) ε [d, ) wait d (d, d +T ] ε (d, d +T ] wait d (d, d +T ] ε (d, d +T ] actions finish finish action s finish finish f T - y d - y d d - y d d + T -y T d + T -y T f T - z d - z d d - z d d + T -z T d + T -z T References: [] E. sarin, O. Maler, s Soon as Possible: Time Optimal Control for Timed utomata. In: Hybrid Systems: Computation Control, LNCS 569, Springer 999, pp. 9-3. [] E. sarin, O. Maler,. Pnueli, J. Sifakis, Controller Synthesis for Timed utomata. In: Proc. IFC Symposium on Structure Control, Elsevier 998, pp. 469-474. [3] S. Yovine, Kronos: Verification Tool for Real- Time Systems. In: Int. Journal on Software Tools for Technology Transfer, Vol.. No. /, October 997. [4] K. Larsen, P. Pettersson, Wang Yi, Uppaal in a Nutshell. In: Int. Journal on Software Tools for Technology Transfer, Vol.. No. /, October 997, pp. 34-5. [5].. rin W.M. Wonham, The Supervisory Control of Timed DES, IEEE Transactions on utomatic Control, Vol.39, No., 994, pp. 39-34. [6] J. von Neumann O. Morgenstern, Theory of Games Economic ehavior, Princ. University Press, 944. [7] R. lur D. L. Dill, Theory of Timed utomat Theoretical Computer Science 6, 994,pp.83-35. Fig. 3. Timed automata model of UF out UF in