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Multi - Model intergration and hierarchical control of DEDS
MULTI-MODEL INTEGRATION AND HIERARCHICAL CONTROL OF Of DEDS HULTI-I«)DEL
Zheng Yingping Institute of Automation. Automation, Academia Sinica. Sinica, Beijing, Beijing. China theory. problems arise about the relationship between Abstract. After a rapid development period of DEDS theory, theories. and the possibility of building bUilding an 'unified' theory. Based on the many existing models and theories, recent works towards this goal, goal. some consideration are discussed in this paper. The main points are: The complexity of DEDS and the necessity of multi-model description and hierarchical control framework, A discussion on the existing theoretical models and control problems to reveal the possible connections between them when taken as 'knowledge blocks', Integration methodology and the importance of human interference and man-machine interaction, interaction. AI techniques and expert system framework would be appropriate for such an integrated theory. Keywords. tion.
System), System complexity, control, System integraDEDS(Discrete Event Dynamic System). complexity. Hierarchical control. Int~ra
control problems should also be solved in a hierarchical way, and beside these. these, to be consistent with the real way. complexity of DEDS, it should also be able to use human intelligence experience, ••• ) (knowledge, experience •••• and skills efficiently in a systematic way. Some points about this will be further discussed in In the follOWing sections.
I NTRODUCTI ON Great progresses have been achieved in DEDS research during the past decade. Many problems have been solved by different models. tools and points of research groups with diverse models, view. It is well known that a great deal of theoretical models have been proposed (e.g. see Ho. Ho, 1989; Ho, 1990). 1990), 1989: Ho. each has itself's advantages and shortcomings, could not be substituted each other. There were quite some efforts made to seek for an unified model or theory. But this is too difficult to be done. done, as indicated by Y.C.Ho (1989), diffioult (1989). due to the lack of any topological structure for its inherently discrete state space, space. thus leads to serious problem of computational complexity, complexity. due to the continuity of most performance measures causing difficulties in mathematical analysis, and due to the existence of the stochastic analysis. disturbance, the complicated hierarchical structures, structures. and the involved dynamic behaviors.
SYSTEM COMPLEXITIES AND THE RELEVENT RESEARCH APPROACHES In system scienceL science. a 'system' 'system" is defined as an integrated entity organized via interconnected components to have some collective functions. The duty of system science is to study behavior in systems. the variety of dynamic behavlor systems, the relationship function. thus between system structure and its behavior and function, to design. deSign, construct and govern systems according to some desired specifications. This Is is a newly developed field, and has close relationship interdisciplinary research field. with control science. It could provide views on the variety of systems. systems, more profound understanding of the general movement, and research methodology for principles on system movement. control. as well as DEDS theorists. control,
As a compromise. compromise, it is suggested to classify the existing DEDS models in to three levels: the logical level. level, the algebraic level and the stochastic performance level. Models in In
same level serve similar purposes,
and the interconnec-
tions and the overlaps between them are attracting many sti II much to be done researchers' interests. But there are still for having a consistent, consistent. integrated model system.
One of the most important points in system science is the understanding of the hierarchy of the different levels of complexities in systems, systems. and the necessity of utilizing different approaches to solve problems with different level complexities.
To imagine the form or the structure of such a universal model or theory. theory, let us consider a practical process of analysis, design and operation of a real reai DEDS (e.g. a compuanalysis. ter integrated Manufacturing system). An exspert (or a group of experts) should know the system specification and technical constraints firstly. Then he has some knowledges in his brain such as a conceptual framework about the problem, problem. or process movement. natural man-made laws of the movement, experiences and available technical measures. He will face a sequence of problems. problems, arising from stages of preliminary (qualitative, design, quantitative (detailed) (qualitative. structural) design. design, ••• , till ti I I implementation and verification. He should design ••••• choose some appropriate model and theoretical tools for each eacb stage, combining with his experience. experience, to solve specific problem. He might need to return to and revise the solution for a former stage if he meets essential difficulty for going on. Such revises are always based on the results at next'stage. and he will never go in to fine the 'direct next'stage, details before getting some ideas on the system's global words, he should work stage by structure and behavior. In words. stage hierarchically. This is a systematical approach, approach. only experience, the theoretical methods combined with knowledge, knowledge. experience. could be utilized efficiently.
A 'simple system' consists usually a small number of components. their connections and the system movement are components, governed by some well known laws and can be described pretty well by some easily manipulable mathematical relations. Examples of such systems are many electric-mechanical systems and some industrial units. They can be described by differential equations or transfer functions. functions, analysed via well-developed frequency-domaln frequency-domain method, and controlled simply by PlO PID regulators successfully in most cases. For for multivariable multi variable systems or large-scale systems, things are more complicated. Many highly sophisticated mathematical methods would be needed. needed, more uncertainties should be taken ruethods simplification account of, Simplification and approximation will be necessary for trade-off between computation cost and system But there are some underlying beliefs performance. underlYing supporting these approaches. That is. is, Jhat What we need is only a good mathematical model, model. by using advanced mathematical techniques and very powerful computational machine, machine. all problems could be solved, If some thing has been simplified. simplified, the error can be estimated or keeped in a tolerable limit, used, than its If some iterative algorithm has been used. convergence should be proved. Unfortunately the situations
An DEDS integrated model system shouid should not only include all the theoretical models in a hierarchical framework. framework, its
-7-
are usually not like this in the real world. big gap between theory and practice.
it thus
I. At logical level.
makes
(I) Ramadge & Wonham's Formal language/automata theory the most fundamental framework of the logical level model and the supervisory control theory: Its extensions to Rabin automata. Buchi automata. etc •• (2) Petri nets. a more concise model. phenomena such as concurrence could be studied. visual display of dynamic processes, Its extensions tc colored or other modified Petri nets.
Great progresses have been made in the study of complex systems, or more precisely complex behaviors in systems recently. Theories on phenomena such as catastrophe and bifurcation, dissipative structure, synergetics and other self-organizing or evolution processes, chaos and spin Illass. etc.. are developed very quickly. They unfold many fashionable pictures from the worlds of physics. chemistry. biology and even human being activi ty. and attract much Interests from both theo 'sts and practitioners.
When time variable is considered, (3) Temporal logic. (4) Clock automata. (5) Timed-Petri net.
Yet most of these theories deal with only systems consisting of only subsystems in a two-level hierarchy. though the number of the subsystems might be gigantic. Such systems are called 'simplex giant systems' by T.S. Tsien (1989). or described as 'large system with flat structure' by II.Simon (I9g1l. And the notion of the really 'complex' system is suggested to be left for those with hierarchical structure of more levels. the number of lower-level subsystems governed by each high-level subsystem may not be very large. but they are well organized to form a very effective functional structure. Such systems may have plenty of complicated behaviors and the ability to evolve to higher form of organization. Examples of such systems are life-and eco-
2. At the algebraic level. to generate the dynamic trajectories by a small set of algebraic rules. (1) Finitely recursive process. (2) Communication sequential process. And when the time variable is considered. (3) Min-Max algebra. Dioid.
-systems, earth/atmosphere system, nervo and brain system, socio-economic systems. etc •. Beside these, many man-made
Also some models described by special languages.
technological systems and human decision-making organizations should also belong to this category. and these could just be the prototype of our nED:; research.
(4) Condition/event systems (R.S.Sreenivas & B.Krogh. 1990) (5) Hybrid dynamical systems (Benveniste 1990).
In compar i son wi th other comp I ex sys terns as Inen t i oned above.
3. At the stochastic performance level. the models are more suitable to practical situations.
DEDS problems seem more treatable. the feature of 'man-made' provides more measures for systems' design and control,
yet
they have many i mpor tan t app I i ca t ions to moder n techno I 010 ical problems.
(I) Queueing networks. (2) ~arkov chain or GSHP. (3) Simulation methods. including models and algorithms. random seed generation. post data proces~ing. etc •. (4) Perturbation analysis. an important method in statistical experiments.
The complexities of DEDS could be summar ized as follows, (I)
The multi level hierarchical structure;
(2) The totally discrete state space, causing various combina tor j a I compu ta t i on comp lex it i es, the mode I reduc t i on techniques and heuristic solution will be needed: (3) Many qualitatively different models are required by different problems arising from analysis, design and control at different levels. and these models are inter-
Here we would not to give the detai led description of these models, instead. just to emphasize some common points or them and the related control problems. There are some ingredients common in all model/theories.
connected in many intricate ways;
(4) The common feature of the man-made systems, the 'well-organized complexity', the implication of the human factor and the necessity of man-mdchine interactions.
(I) The concepts of Event and State. Here by ·state". the physical state of the system is refered, such as the server's states of ·work·, "idle" or "down', and the buffer's state is described by the number or customers. The ·state variables' in the M'ax-algebra linear system formulation are in fact the time instants of occurrence of the new events, which might belter be called 'time state' of DEDS. we suggest. The event is assumed to occur instantaneously and asynchronously according to some deterministic or stochastic mechanisms. The "event series· and the ·state series· constitute the system 'trajectory' completely. The interaction between events and states (as wel I as phyical time in cases except in the untimed logical level) determine a systemJs behavior and performance.
And the approaches for solving DEDS problems should include, (1) An
integrated
system consisting
of
both
qualitative
mathematical models and quantitative rules of
experien-
ces and expert know ledges: (2) Man-machine interaction to take full advantages of the human ability in judgement and decision-making,as well as
the
computer's abi I i ty in mass information
proces-
sing; (3) Experiments via computer simulation (possibly via practical condition. too) for verification of theory and revealling new phenomena. Before get into deta i Is abou t these. let us fir s t I y rev i se the already existing models and theories in the next section.
(2) The state transition mechanism. This is a mapping from old state and the triggering event to the new state. It could be represented in terms or tables, some concise mathematical operations, or even programming languages. It may be a single-value or a general set-value mapping' may be deterministic of stochastlc in property. Some part of the mechanism can not be changed due to natural laws or other constraints. and some others could be designed by people. Some qualitative, structural properties of the dynamic mapping are
EXISTING MODELS AND THEORIES There are many relevent models and the associated mathematical methods and theories for the DEDS study, they could be classified (may be not completely and not exactly) as follows.
-8-
here only that a profound understanding of 'general system theory' (see e.g. Mesarovic 1975) will be very helpful in this respect. We "ill return to this topic la ter.
analyzed and designed via the logical level models, but some detailed design or parameter optimization should be done via algebraic level or stochastic performance level models combined with some quantitative mathematical methods.
All these models/theories could be viewed as 'kno"ledge blocks'. and organized via AI or knowledge engineering techniques to constitute an integrated DEDS theory.
(3) Information acquisition for feedback control. The event/state observation may be complete or not. the dynamic trajectory of event and for state could be reconstructed from the observed information or not. The problem is simi lar to that in traditional control problem. Time delay in the observation and estimation process should some time be taken account of. Statistical inferences are required to be done on-line very quickly. These are research problems for different levels of models.
DYNAHIC AND CONTROL ISSUES There are many different definitions proposed in literatures for concepts on dynamic properties such as stability. con trollabi I i ty. observabl lit). etc.. To clear the vagueness and inconsistence in these definitions. we need a IDOre general understanding of th"se concepts. A set - theoretical framework proposed by M.D.Mesarovic (1975) seems appropriate for this purpose.
(4) Control mechanisms for DEDS. For the models at the logical level, a supervisory control can be realized by disabling some controlable events based on event/state observation information. In the Petr i net set, the higher level's schedul ing command may enter the system through the 'external input places' in the form of a sequence of input token.
In his framework. a system is defined as a relation between its conponent sets. i.e. a subset in the Cartesian product of the component sets. These sets could be. for example. input object set X, output object set Y and some structural parameter set P, when an input-output or cause-consequence system is considered. The input object may be divided further into sets of the control input U and the disturbance D. Thus a system S can be represented formally as
Routing. planning and scheduling are very important for a high quality system at the design stage. When a system is working in an environment with uncertainly, the on -IIIIe redesign of routing and work-schedule will keep, even improve the system's performance. The implementa tion of such strategies can usually be reduced to selection of customer class from the waiting queues and dispatch of them to some determined server at specific time.
Se::: X x Y x p; U x D x Y x P.
(I)
A system's dynamic property or performance can be characteri zed by evaluatio~ function G, XxYxP -+ V. Its value thus can be determined in turn by U. D and P. if the output Y is also determined completely by them. that is (2)
(5) Research purpose. mulat ion.
system specification and problem for The desired property or performance could be represented v E V'C: V. wnere V' is a subset of V.
All models and theories are developed for design. analy sis and control of systems to satisfy some required specifications, which are given in terms of sets of requirmenls on system functions, structure, constraints and performance indecies. Some specification/for one model/theory may be given by another model/theory as design result, and thus they are inte rconnected each other. The
problem
formulation should not
c;ystem specification and requir ement,
only
reflect
A specific goal v E V' is defined as controllable. if there exist u E U such that (2) is val id for the given p and any allowable d. If for any v E V' such u exists. the system is called completely controllable. This definition of the controllability encompasses not only the clas.ical definitions of the 'point-ta - point' controllability and the 'functional' controllability. but also the definition of 'controllable sublanguage' for the automata model of DEDS (see e.g. Ramadge. 1987>. of the definition for the 'time-state' in the Hax-algebra model. etc ..
the
but also be trac -
table in order to obtain some useful results to the research purpose.
serve
In Hesarovic's framework, the concept stability is always related to the deviation of evaluation v caused by errors in perturbation d (Liapunov stability or input-output stabili ty) of in parameter p (structural stabi I i ty). Then the mapping ~v ; F( ~ d. ~p) has to be studied, and the measure fS' r _ d~v _i ations _ _~h,,-uJd iJ,,_def.ined _i'Lter1!ls _O( _ne il'.~QcorJ!~.9 subset systems N(v), (d),N(p) in sets V,D and P respectively. A system, evaluated by v. is stable relative to d (or p) if and only if. for any set V'EN(v). there exists set D'EN(d) (or P' E:N(p» such that if ~dED'(or ~pEP').then ~vEV'.
(6) Mathematical tools, theoretical method and resul ts. Hany mathematical branches ar e found useful in DEDS research. They could be roughly classified into t .. o olasses. One class is for system's discrete evendriven features.
which
has same root as that for
as
theoretical
computer science. Among them are formal lan guage/automata theory. petri - net. communication sequential process. temporal logic, as well as combinatorial optimization algorithms and computation complexity. Another class is for the continuous performance analy sis. which includes stochastic processes and statistic analysis, probabilistic convergence and gradient evaloa tion. eet.. Hax - algebra is some how. special, which is similar in form to the 'normal' linear algebra on real number field, but is closely related to graph theory and discrete mathematics. A bridge connecting these two classes is necessary for DEDS research, and would be one of the sources of difficulties in this area.
Similarly, the observability of a system can be defined as the possibility of distinguishing its evaluation v from observation on its output y. and so on. By different definitions or'the evaluation function v; G(' .. ). these definitions of system dynamic properties can be well appl ied to different DEDS models at different levels. Of course. they can be studied only by utilizing various different mathematical methods. The control problems for DEDS are also formulated in different ways for different models at each level.
There are plenty of theoretical methods and results in DEDS research, they are interconnected each other as described in the previous last point. It is emphasized
many
At the I('gical level, in the formal language/automata formu-
--9-
lati on. a control pa tt e rn y , L;( - {D. tl de ter min es the enabl ing or disabl ing of B. contr o llabl e e vent i n L;,. It combines with an automaton t o form a contr o lled DES G,. A supervisory controller then is de fin ed as ano ther aut omat on S wi th the same input event set L;. S's state x E X is fedback via mapping (j> ,X~r to control the trajectory of Gc. To specif y the desired s ystem behavi or. a sublanguage L. represe nting the 'legal behavi or' a nd a s ublanguage L. representing the 'minimal acceptable behavi or' are given. The supervisory control problem is de fined to construct a proper supe rvisor S for give n Gc such that their generated language L(S/ Ge) is in between L. and L., • (3)
Many modified and extended versions ar e proposed literature. see e.g. Ramadge (1987). Cieslak Tsitsiklis (1989). Li(1990 ) . Fa(1989). etc ..
in the (1988).
CONCLUSION . AN INTEGRATED DEDS THEORY From the discussions above. we believe that an integrated DEDS theory will be formed in the near further. It is necessary since any individual model/theory is not enough solving any real. practical problems. Human for experts' knowledge is crucial for a correct problem formulation . right ways of problem solving. and judgement of the applicability of the obtained results. And the whole process usually goes through different levels iteratively. by using various methods. It is also possible due to the inherent connections existing between these models/theories. which are recently faSCinating topics studied in the l i terature. AI technique and expert system framework would be suitable for this purpose. A knowledge base should contain both qualitative and quantitative models/theories. their connecti ons and applications. A friendly man - machine interface will be needed. The traditional idea about a 'theoretical system' should be changed. Cooperation between people from AI . OR and control fields is essential.
Also at this level. for retri net mode l. the s upervis ory control problem has als o been formulated (Chen. 1990). Here the languages consist of strings of tran s iti ons. design problems for different contr o ller s . the higher - level' s scheduling command input in the f orm o f token sequence through the 'external input places' . etc . .
REFERENCES
After the l ogical structur e has been de termine d. detailed design problems o f contr o lling syste m be havior and optimi z ing system pe rf ormance unde r uncertaint y s hould be consi dered. In the determi n i s t ic ca se there are ,
Benveniste . A. & P.Le Guernic (1988). Hybrid dynamical systems theory and nonlinear dynamical systems over finite fields. Proc. 27th CDC. Austin. Texas. Chen. Ha o- xun (1990). PhD Thesis. Xi-an Jiao-tong University. Xi - an. China. Cieslak.R • • C.Desclaux. A. Fawaz & P.Varaiya (1988). Supervisory control of discrete processes with partial observations. Trans.IEEE AC- 33. 249-260 . a.Jing- huai (1990). PhD Thesis. Inst. of Automation. Academia Sinica. Beijing. China . Gershwin.S.B. (1989) . Hierarchical flow control, a framework for scheduling and planning discrete event in manufactu r ing systems. Proc.lEEE.77- I. 195- 209. Ho . V. C. (1989). Scanning the issue. Dynamics of discrete event systems. Proc. IEEE.77 - 1.3-6. Ho.V . C. (1990). Analyzing complexity and performance in a manmade world . An introduction to discrete event dyna mic systems. Preprints 11th IFAC World Congress. Vol.l. 66 - 70. Li. Vong - hua (1990). PhD Thesis. Beijing Univ . of Aero- and Astronautics. Beijing. China. Luh.P.B. et.at. (1988). Parallel machine scheduling using Lagrangian r e laxati on. Proc. lnt.Conf. on CIM. Tr oy. Me sar ovic.M . D. (1975) . General syste ms theory , Mathematical foundations. Academic Press. Perkins.J . & P.R.Kumar (1989) . Stable. distributed. real - time scheduling of flexible manufacturing / assembly / disassembly systems. Trans.lEEE AC- 34 - 1. Ramadge.P.J. & W.M.Wonham (1987). Supervisory contr o l of a class of discrete-event processes. SIAM J.Control opt i m. 25. 206-230. SimonY. (1981l. The science of artificial. MIT Press. Sreen i vas.R.S. & B.H.Krogh (1990). Condition / event systems. submitted to J.DEDS . Theory & Appllications. Sun.X.a .• a.H.Yu & Y.P.Zheng (1990) . Single- machine scheduling method based on neural - net. Proc.2nd National Conf. on Neural Net. Beij i ng. Is i en . H. S. (1989). Pr i va te communica t ion. Isitsikl isd.N. <1989>' On the control of discrete event dynamic systems. J.KBth.Control Signals Systems. Vol.2 95-107. Wang.X . S. & a.S . Chen (1990). Performance evaluation for priority rules in job shop scheduling problems. Proc. Symp. CIMS-China 90. Beijing. Xiang . D (1990) . M.Sc. Thesis. Inst. of Automation. Academla Sinica. Beijing. China. Vu.J . X.
(I) Real - time re sc hedul ing strateg ies. Since most of the scheduling algorithms ar e of NP complexity, special tec hnique s ar e needed for the real - time so lution. Some o f them are , I) Methods of size reduc ti on f or lar ge-s cale int eger pr ogramming pr obl e ms . For example . a me thod based on 'Lagra nge relaxati on- dual - decompos iti on' work s well in practice. see Luh (1988). 2) Heur i s tic rules based algorithms . Among them the most popular one may be the 'branch and bound' method. which is in fact a famil y o f algorithms . adopted t o vari ous pr oblems with the help o f des igne r's e xperiece and skill • . Many expe ri e nt a ! scheduling rul es ha ve b"e n ap plied effectively in practice for quite a l ong time . they could be implemented in in computer and become 'control s trategies ' for DEDS. There are some interes ting comparative studies on thi s issue. see e . g. Wang (1990) • 3) Fast par a ll e l algorithms a nd t ec hniques. Ne w mechani s m such as hype rcube or ne ur a l ne t could be used for schedui ing pr oblem. see e.g. Sun (1990). (2) Distributed real-time feedba ck scheduling strategies . This kind of s trategies was propos ed by J.R.Perkin s and P.R. Kumar (1989) and succeeding researchers. Strategies such as CAF. CLB. CLW (Perking. 1989). LLSD. CLWL. (Xiang. 1990). HWB. CPW (Vu, 1991). et al have been proposed to stabili ze the producti on pr ocess and the in process invent ory.
(3) Control problems in Max-algebra formulation. usually with respect to j ob compl e ti on times or the stationary working processes of DEDS. Stoc hastic control might be a more important issue at the detailed deSign stage when more practical effects should be considered. PA is a very effective method fo r para meter optimizati on. and can be used further for on - line sensitivity' estimation and self-optimizing control. Multi-time-scale fl ow contr o l should be put in a hierarchical framework and stochastic environment (Gershwin,1989) . Topics such as stochastic ordering and optimal routing control. optimal task distributi on or service as s ignment. optimai dy nami~ scheduling •• • •• have been extensively discussed. People are more concerned about contr o l problems rather than modeling and analysis.
-10-
Zheng Yingping Institute of Automation. Automation, Academia Sinica. Sinica, Beijing, Beijing. China theory. problems arise about the relationship between Abstract. After a rapid development period of DEDS theory, theories. and the possibility of building bUilding an 'unified' theory. Based on the many existing models and theories, recent works towards this goal, goal. some consideration are discussed in this paper. The main points are: The complexity of DEDS and the necessity of multi-model description and hierarchical control framework, A discussion on the existing theoretical models and control problems to reveal the possible connections between them when taken as 'knowledge blocks', Integration methodology and the importance of human interference and man-machine interaction, interaction. AI techniques and expert system framework would be appropriate for such an integrated theory. Keywords. tion.
System), System complexity, control, System integraDEDS(Discrete Event Dynamic System). complexity. Hierarchical control. Int~ra
control problems should also be solved in a hierarchical way, and beside these. these, to be consistent with the real way. complexity of DEDS, it should also be able to use human intelligence experience, ••• ) (knowledge, experience •••• and skills efficiently in a systematic way. Some points about this will be further discussed in In the follOWing sections.
I NTRODUCTI ON Great progresses have been achieved in DEDS research during the past decade. Many problems have been solved by different models. tools and points of research groups with diverse models, view. It is well known that a great deal of theoretical models have been proposed (e.g. see Ho. Ho, 1989; Ho, 1990). 1990), 1989: Ho. each has itself's advantages and shortcomings, could not be substituted each other. There were quite some efforts made to seek for an unified model or theory. But this is too difficult to be done. done, as indicated by Y.C.Ho (1989), diffioult (1989). due to the lack of any topological structure for its inherently discrete state space, space. thus leads to serious problem of computational complexity, complexity. due to the continuity of most performance measures causing difficulties in mathematical analysis, and due to the existence of the stochastic analysis. disturbance, the complicated hierarchical structures, structures. and the involved dynamic behaviors.
SYSTEM COMPLEXITIES AND THE RELEVENT RESEARCH APPROACHES In system scienceL science. a 'system' 'system" is defined as an integrated entity organized via interconnected components to have some collective functions. The duty of system science is to study behavior in systems. the variety of dynamic behavlor systems, the relationship function. thus between system structure and its behavior and function, to design. deSign, construct and govern systems according to some desired specifications. This Is is a newly developed field, and has close relationship interdisciplinary research field. with control science. It could provide views on the variety of systems. systems, more profound understanding of the general movement, and research methodology for principles on system movement. control. as well as DEDS theorists. control,
As a compromise. compromise, it is suggested to classify the existing DEDS models in to three levels: the logical level. level, the algebraic level and the stochastic performance level. Models in In
same level serve similar purposes,
and the interconnec-
tions and the overlaps between them are attracting many sti II much to be done researchers' interests. But there are still for having a consistent, consistent. integrated model system.
One of the most important points in system science is the understanding of the hierarchy of the different levels of complexities in systems, systems. and the necessity of utilizing different approaches to solve problems with different level complexities.
To imagine the form or the structure of such a universal model or theory. theory, let us consider a practical process of analysis, design and operation of a real reai DEDS (e.g. a compuanalysis. ter integrated Manufacturing system). An exspert (or a group of experts) should know the system specification and technical constraints firstly. Then he has some knowledges in his brain such as a conceptual framework about the problem, problem. or process movement. natural man-made laws of the movement, experiences and available technical measures. He will face a sequence of problems. problems, arising from stages of preliminary (qualitative, design, quantitative (detailed) (qualitative. structural) design. design, ••• , till ti I I implementation and verification. He should design ••••• choose some appropriate model and theoretical tools for each eacb stage, combining with his experience. experience, to solve specific problem. He might need to return to and revise the solution for a former stage if he meets essential difficulty for going on. Such revises are always based on the results at next'stage. and he will never go in to fine the 'direct next'stage, details before getting some ideas on the system's global words, he should work stage by structure and behavior. In words. stage hierarchically. This is a systematical approach, approach. only experience, the theoretical methods combined with knowledge, knowledge. experience. could be utilized efficiently.
A 'simple system' consists usually a small number of components. their connections and the system movement are components, governed by some well known laws and can be described pretty well by some easily manipulable mathematical relations. Examples of such systems are many electric-mechanical systems and some industrial units. They can be described by differential equations or transfer functions. functions, analysed via well-developed frequency-domaln frequency-domain method, and controlled simply by PlO PID regulators successfully in most cases. For for multivariable multi variable systems or large-scale systems, things are more complicated. Many highly sophisticated mathematical methods would be needed. needed, more uncertainties should be taken ruethods simplification account of, Simplification and approximation will be necessary for trade-off between computation cost and system But there are some underlying beliefs performance. underlYing supporting these approaches. That is. is, Jhat What we need is only a good mathematical model, model. by using advanced mathematical techniques and very powerful computational machine, machine. all problems could be solved, If some thing has been simplified. simplified, the error can be estimated or keeped in a tolerable limit, used, than its If some iterative algorithm has been used. convergence should be proved. Unfortunately the situations
An DEDS integrated model system shouid should not only include all the theoretical models in a hierarchical framework. framework, its
-7-
are usually not like this in the real world. big gap between theory and practice.
it thus
I. At logical level.
makes
(I) Ramadge & Wonham's Formal language/automata theory the most fundamental framework of the logical level model and the supervisory control theory: Its extensions to Rabin automata. Buchi automata. etc •• (2) Petri nets. a more concise model. phenomena such as concurrence could be studied. visual display of dynamic processes, Its extensions tc colored or other modified Petri nets.
Great progresses have been made in the study of complex systems, or more precisely complex behaviors in systems recently. Theories on phenomena such as catastrophe and bifurcation, dissipative structure, synergetics and other self-organizing or evolution processes, chaos and spin Illass. etc.. are developed very quickly. They unfold many fashionable pictures from the worlds of physics. chemistry. biology and even human being activi ty. and attract much Interests from both theo 'sts and practitioners.
When time variable is considered, (3) Temporal logic. (4) Clock automata. (5) Timed-Petri net.
Yet most of these theories deal with only systems consisting of only subsystems in a two-level hierarchy. though the number of the subsystems might be gigantic. Such systems are called 'simplex giant systems' by T.S. Tsien (1989). or described as 'large system with flat structure' by II.Simon (I9g1l. And the notion of the really 'complex' system is suggested to be left for those with hierarchical structure of more levels. the number of lower-level subsystems governed by each high-level subsystem may not be very large. but they are well organized to form a very effective functional structure. Such systems may have plenty of complicated behaviors and the ability to evolve to higher form of organization. Examples of such systems are life-and eco-
2. At the algebraic level. to generate the dynamic trajectories by a small set of algebraic rules. (1) Finitely recursive process. (2) Communication sequential process. And when the time variable is considered. (3) Min-Max algebra. Dioid.
-systems, earth/atmosphere system, nervo and brain system, socio-economic systems. etc •. Beside these, many man-made
Also some models described by special languages.
technological systems and human decision-making organizations should also belong to this category. and these could just be the prototype of our nED:; research.
(4) Condition/event systems (R.S.Sreenivas & B.Krogh. 1990) (5) Hybrid dynamical systems (Benveniste 1990).
In compar i son wi th other comp I ex sys terns as Inen t i oned above.
3. At the stochastic performance level. the models are more suitable to practical situations.
DEDS problems seem more treatable. the feature of 'man-made' provides more measures for systems' design and control,
yet
they have many i mpor tan t app I i ca t ions to moder n techno I 010 ical problems.
(I) Queueing networks. (2) ~arkov chain or GSHP. (3) Simulation methods. including models and algorithms. random seed generation. post data proces~ing. etc •. (4) Perturbation analysis. an important method in statistical experiments.
The complexities of DEDS could be summar ized as follows, (I)
The multi level hierarchical structure;
(2) The totally discrete state space, causing various combina tor j a I compu ta t i on comp lex it i es, the mode I reduc t i on techniques and heuristic solution will be needed: (3) Many qualitatively different models are required by different problems arising from analysis, design and control at different levels. and these models are inter-
Here we would not to give the detai led description of these models, instead. just to emphasize some common points or them and the related control problems. There are some ingredients common in all model/theories.
connected in many intricate ways;
(4) The common feature of the man-made systems, the 'well-organized complexity', the implication of the human factor and the necessity of man-mdchine interactions.
(I) The concepts of Event and State. Here by ·state". the physical state of the system is refered, such as the server's states of ·work·, "idle" or "down', and the buffer's state is described by the number or customers. The ·state variables' in the M'ax-algebra linear system formulation are in fact the time instants of occurrence of the new events, which might belter be called 'time state' of DEDS. we suggest. The event is assumed to occur instantaneously and asynchronously according to some deterministic or stochastic mechanisms. The "event series· and the ·state series· constitute the system 'trajectory' completely. The interaction between events and states (as wel I as phyical time in cases except in the untimed logical level) determine a systemJs behavior and performance.
And the approaches for solving DEDS problems should include, (1) An
integrated
system consisting
of
both
qualitative
mathematical models and quantitative rules of
experien-
ces and expert know ledges: (2) Man-machine interaction to take full advantages of the human ability in judgement and decision-making,as well as
the
computer's abi I i ty in mass information
proces-
sing; (3) Experiments via computer simulation (possibly via practical condition. too) for verification of theory and revealling new phenomena. Before get into deta i Is abou t these. let us fir s t I y rev i se the already existing models and theories in the next section.
(2) The state transition mechanism. This is a mapping from old state and the triggering event to the new state. It could be represented in terms or tables, some concise mathematical operations, or even programming languages. It may be a single-value or a general set-value mapping' may be deterministic of stochastlc in property. Some part of the mechanism can not be changed due to natural laws or other constraints. and some others could be designed by people. Some qualitative, structural properties of the dynamic mapping are
EXISTING MODELS AND THEORIES There are many relevent models and the associated mathematical methods and theories for the DEDS study, they could be classified (may be not completely and not exactly) as follows.
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here only that a profound understanding of 'general system theory' (see e.g. Mesarovic 1975) will be very helpful in this respect. We "ill return to this topic la ter.
analyzed and designed via the logical level models, but some detailed design or parameter optimization should be done via algebraic level or stochastic performance level models combined with some quantitative mathematical methods.
All these models/theories could be viewed as 'kno"ledge blocks'. and organized via AI or knowledge engineering techniques to constitute an integrated DEDS theory.
(3) Information acquisition for feedback control. The event/state observation may be complete or not. the dynamic trajectory of event and for state could be reconstructed from the observed information or not. The problem is simi lar to that in traditional control problem. Time delay in the observation and estimation process should some time be taken account of. Statistical inferences are required to be done on-line very quickly. These are research problems for different levels of models.
DYNAHIC AND CONTROL ISSUES There are many different definitions proposed in literatures for concepts on dynamic properties such as stability. con trollabi I i ty. observabl lit). etc.. To clear the vagueness and inconsistence in these definitions. we need a IDOre general understanding of th"se concepts. A set - theoretical framework proposed by M.D.Mesarovic (1975) seems appropriate for this purpose.
(4) Control mechanisms for DEDS. For the models at the logical level, a supervisory control can be realized by disabling some controlable events based on event/state observation information. In the Petr i net set, the higher level's schedul ing command may enter the system through the 'external input places' in the form of a sequence of input token.
In his framework. a system is defined as a relation between its conponent sets. i.e. a subset in the Cartesian product of the component sets. These sets could be. for example. input object set X, output object set Y and some structural parameter set P, when an input-output or cause-consequence system is considered. The input object may be divided further into sets of the control input U and the disturbance D. Thus a system S can be represented formally as
Routing. planning and scheduling are very important for a high quality system at the design stage. When a system is working in an environment with uncertainly, the on -IIIIe redesign of routing and work-schedule will keep, even improve the system's performance. The implementa tion of such strategies can usually be reduced to selection of customer class from the waiting queues and dispatch of them to some determined server at specific time.
Se::: X x Y x p; U x D x Y x P.
(I)
A system's dynamic property or performance can be characteri zed by evaluatio~ function G, XxYxP -+ V. Its value thus can be determined in turn by U. D and P. if the output Y is also determined completely by them. that is (2)
(5) Research purpose. mulat ion.
system specification and problem for The desired property or performance could be represented v E V'C: V. wnere V' is a subset of V.
All models and theories are developed for design. analy sis and control of systems to satisfy some required specifications, which are given in terms of sets of requirmenls on system functions, structure, constraints and performance indecies. Some specification/for one model/theory may be given by another model/theory as design result, and thus they are inte rconnected each other. The
problem
formulation should not
c;ystem specification and requir ement,
only
reflect
A specific goal v E V' is defined as controllable. if there exist u E U such that (2) is val id for the given p and any allowable d. If for any v E V' such u exists. the system is called completely controllable. This definition of the controllability encompasses not only the clas.ical definitions of the 'point-ta - point' controllability and the 'functional' controllability. but also the definition of 'controllable sublanguage' for the automata model of DEDS (see e.g. Ramadge. 1987>. of the definition for the 'time-state' in the Hax-algebra model. etc ..
the
but also be trac -
table in order to obtain some useful results to the research purpose.
serve
In Hesarovic's framework, the concept stability is always related to the deviation of evaluation v caused by errors in perturbation d (Liapunov stability or input-output stabili ty) of in parameter p (structural stabi I i ty). Then the mapping ~v ; F( ~ d. ~p) has to be studied, and the measure fS' r _ d~v _i ations _ _~h,,-uJd iJ,,_def.ined _i'Lter1!ls _O( _ne il'.~QcorJ!~.9 subset systems N(v), (d),N(p) in sets V,D and P respectively. A system, evaluated by v. is stable relative to d (or p) if and only if. for any set V'EN(v). there exists set D'EN(d) (or P' E:N(p» such that if ~dED'(or ~pEP').then ~vEV'.
(6) Mathematical tools, theoretical method and resul ts. Hany mathematical branches ar e found useful in DEDS research. They could be roughly classified into t .. o olasses. One class is for system's discrete evendriven features.
which
has same root as that for
as
theoretical
computer science. Among them are formal lan guage/automata theory. petri - net. communication sequential process. temporal logic, as well as combinatorial optimization algorithms and computation complexity. Another class is for the continuous performance analy sis. which includes stochastic processes and statistic analysis, probabilistic convergence and gradient evaloa tion. eet.. Hax - algebra is some how. special, which is similar in form to the 'normal' linear algebra on real number field, but is closely related to graph theory and discrete mathematics. A bridge connecting these two classes is necessary for DEDS research, and would be one of the sources of difficulties in this area.
Similarly, the observability of a system can be defined as the possibility of distinguishing its evaluation v from observation on its output y. and so on. By different definitions or'the evaluation function v; G(' .. ). these definitions of system dynamic properties can be well appl ied to different DEDS models at different levels. Of course. they can be studied only by utilizing various different mathematical methods. The control problems for DEDS are also formulated in different ways for different models at each level.
There are plenty of theoretical methods and results in DEDS research, they are interconnected each other as described in the previous last point. It is emphasized
many
At the I('gical level, in the formal language/automata formu-
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lati on. a control pa tt e rn y , L;( - {D. tl de ter min es the enabl ing or disabl ing of B. contr o llabl e e vent i n L;,. It combines with an automaton t o form a contr o lled DES G,. A supervisory controller then is de fin ed as ano ther aut omat on S wi th the same input event set L;. S's state x E X is fedback via mapping (j> ,X~r to control the trajectory of Gc. To specif y the desired s ystem behavi or. a sublanguage L. represe nting the 'legal behavi or' a nd a s ublanguage L. representing the 'minimal acceptable behavi or' are given. The supervisory control problem is de fined to construct a proper supe rvisor S for give n Gc such that their generated language L(S/ Ge) is in between L. and L., • (3)
Many modified and extended versions ar e proposed literature. see e.g. Ramadge (1987). Cieslak Tsitsiklis (1989). Li(1990 ) . Fa(1989). etc ..
in the (1988).
CONCLUSION . AN INTEGRATED DEDS THEORY From the discussions above. we believe that an integrated DEDS theory will be formed in the near further. It is necessary since any individual model/theory is not enough solving any real. practical problems. Human for experts' knowledge is crucial for a correct problem formulation . right ways of problem solving. and judgement of the applicability of the obtained results. And the whole process usually goes through different levels iteratively. by using various methods. It is also possible due to the inherent connections existing between these models/theories. which are recently faSCinating topics studied in the l i terature. AI technique and expert system framework would be suitable for this purpose. A knowledge base should contain both qualitative and quantitative models/theories. their connecti ons and applications. A friendly man - machine interface will be needed. The traditional idea about a 'theoretical system' should be changed. Cooperation between people from AI . OR and control fields is essential.
Also at this level. for retri net mode l. the s upervis ory control problem has als o been formulated (Chen. 1990). Here the languages consist of strings of tran s iti ons. design problems for different contr o ller s . the higher - level' s scheduling command input in the f orm o f token sequence through the 'external input places' . etc . .
REFERENCES
After the l ogical structur e has been de termine d. detailed design problems o f contr o lling syste m be havior and optimi z ing system pe rf ormance unde r uncertaint y s hould be consi dered. In the determi n i s t ic ca se there are ,
Benveniste . A. & P.Le Guernic (1988). Hybrid dynamical systems theory and nonlinear dynamical systems over finite fields. Proc. 27th CDC. Austin. Texas. Chen. Ha o- xun (1990). PhD Thesis. Xi-an Jiao-tong University. Xi - an. China. Cieslak.R • • C.Desclaux. A. Fawaz & P.Varaiya (1988). Supervisory control of discrete processes with partial observations. Trans.IEEE AC- 33. 249-260 . a.Jing- huai (1990). PhD Thesis. Inst. of Automation. Academia Sinica. Beijing. China . Gershwin.S.B. (1989) . Hierarchical flow control, a framework for scheduling and planning discrete event in manufactu r ing systems. Proc.lEEE.77- I. 195- 209. Ho . V. C. (1989). Scanning the issue. Dynamics of discrete event systems. Proc. IEEE.77 - 1.3-6. Ho.V . C. (1990). Analyzing complexity and performance in a manmade world . An introduction to discrete event dyna mic systems. Preprints 11th IFAC World Congress. Vol.l. 66 - 70. Li. Vong - hua (1990). PhD Thesis. Beijing Univ . of Aero- and Astronautics. Beijing. China. Luh.P.B. et.at. (1988). Parallel machine scheduling using Lagrangian r e laxati on. Proc. lnt.Conf. on CIM. Tr oy. Me sar ovic.M . D. (1975) . General syste ms theory , Mathematical foundations. Academic Press. Perkins.J . & P.R.Kumar (1989) . Stable. distributed. real - time scheduling of flexible manufacturing / assembly / disassembly systems. Trans.lEEE AC- 34 - 1. Ramadge.P.J. & W.M.Wonham (1987). Supervisory contr o l of a class of discrete-event processes. SIAM J.Control opt i m. 25. 206-230. SimonY. (1981l. The science of artificial. MIT Press. Sreen i vas.R.S. & B.H.Krogh (1990). Condition / event systems. submitted to J.DEDS . Theory & Appllications. Sun.X.a .• a.H.Yu & Y.P.Zheng (1990) . Single- machine scheduling method based on neural - net. Proc.2nd National Conf. on Neural Net. Beij i ng. Is i en . H. S. (1989). Pr i va te communica t ion. Isitsikl isd.N. <1989>' On the control of discrete event dynamic systems. J.KBth.Control Signals Systems. Vol.2 95-107. Wang.X . S. & a.S . Chen (1990). Performance evaluation for priority rules in job shop scheduling problems. Proc. Symp. CIMS-China 90. Beijing. Xiang . D (1990) . M.Sc. Thesis. Inst. of Automation. Academla Sinica. Beijing. China. Vu.J . X.
(I) Real - time re sc hedul ing strateg ies. Since most of the scheduling algorithms ar e of NP complexity, special tec hnique s ar e needed for the real - time so lution. Some o f them are , I) Methods of size reduc ti on f or lar ge-s cale int eger pr ogramming pr obl e ms . For example . a me thod based on 'Lagra nge relaxati on- dual - decompos iti on' work s well in practice. see Luh (1988). 2) Heur i s tic rules based algorithms . Among them the most popular one may be the 'branch and bound' method. which is in fact a famil y o f algorithms . adopted t o vari ous pr oblems with the help o f des igne r's e xperiece and skill • . Many expe ri e nt a ! scheduling rul es ha ve b"e n ap plied effectively in practice for quite a l ong time . they could be implemented in in computer and become 'control s trategies ' for DEDS. There are some interes ting comparative studies on thi s issue. see e . g. Wang (1990) • 3) Fast par a ll e l algorithms a nd t ec hniques. Ne w mechani s m such as hype rcube or ne ur a l ne t could be used for schedui ing pr oblem. see e.g. Sun (1990). (2) Distributed real-time feedba ck scheduling strategies . This kind of s trategies was propos ed by J.R.Perkin s and P.R. Kumar (1989) and succeeding researchers. Strategies such as CAF. CLB. CLW (Perking. 1989). LLSD. CLWL. (Xiang. 1990). HWB. CPW (Vu, 1991). et al have been proposed to stabili ze the producti on pr ocess and the in process invent ory.
(3) Control problems in Max-algebra formulation. usually with respect to j ob compl e ti on times or the stationary working processes of DEDS. Stoc hastic control might be a more important issue at the detailed deSign stage when more practical effects should be considered. PA is a very effective method fo r para meter optimizati on. and can be used further for on - line sensitivity' estimation and self-optimizing control. Multi-time-scale fl ow contr o l should be put in a hierarchical framework and stochastic environment (Gershwin,1989) . Topics such as stochastic ordering and optimal routing control. optimal task distributi on or service as s ignment. optimai dy nami~ scheduling •• • •• have been extensively discussed. People are more concerned about contr o l problems rather than modeling and analysis.
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