Controllers for Robotics and. Manufacturing Devices with Temporal. Logic and the ControlD System. Marco Antoniotti.


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1 Synthesis and Veriæcation of Discrete Controllers for Robotics and Manufacturing Devices with Temporal Logic and the ControlD System. by Marco Antoniotti September 1995 A dissertation in the Department of Computer Science submitted to the faculty ofthe Graduate School of Arts and Science in partial fulællmentof the requirements for the degree of Doctor of Philosophy at New York University Approved: Professor Bhubaneswar Mishra Research Advisor
2 cæ Marco Antoniotti All Rights Reserved 1995
3
4 To my sister Franca, who made this a better world To my nephew Tommaso and my niece Agnese, who will make this a better world To Mary, who makes this a better world iv
5 Acknowledgments I am really indebted to my advisor Bud Mishra for his continuous support and friendship during myyears as a graduate student. He has been a been a guide through æelds that were unknown to meandhas been a guide during troubled times. There are still many æelds I will need to wander through and I am sure I will ænd his traces there. Iwant tothank Professor Mohsen Jafari of Rutgers University for his help with the CRAMTD project and for kindness and enthusiasm since our ærst meetingat a conference. A person who deserves much credit for my achievements isstefania Bandini of the Universitaça degli Studi di Milano, who has always been able to keep me onmytoes and to come tomy aid whenever necessary. She has always been the source of precious advice. Dr. R. Kurshan of AT&T Bell Laboratories was very kind andhelpful duringtheænal phaseofmythesis by providing valuable information on many technical details and developments inthe theory of related topics. Iwant tothank Professor J. T. Schwartz for his valuable commentson my work. Professor R. Wallace is the person who introduced me tothe Walking Machine problem. He designed and worked on the hardware and was very helpful in giving me feedback inorder to keep my research ærmly set on a pragmatic foundation. v
6 Friends like Giovanni Gallo and Alberto Policriti are a blessing for anybody who is doing research andwhoneeds a companion to sneak out tothe movies every once in a while. Iwant tothank Professor K. Perlin, Professor S. Mallat, Professor D. Shasha and Professor R. Boppana for their kindness and for their suggestions over the past years. Special thanks go to many people in Milan who haveinvarious ways contributed to mysuccess as a graduate student: Professor G. Mauriofthe Dipartimento di Scienze dell'informazioneofthe Universitaça degli Studi di Milano and all the friends at Quinary S.p.A., where the machines are named after Lisp functions. Thanks are also due to Anina KarmenMeade ofthe Department of Computer Science of NYU for her patience with myoverly complicated administrative problems andtofred Hansen for all his work inthe Robotics Research Laboratory. A group of people who deserves thanks and other things is constituted by my roommate over the years: Sunder Sethuraman, a great mathematician and abetter soccer player than I'll ever be; David Bacon, Ron Even and Marek Teichmann, mythree oæce mates who endured meandtoto Paxia, who I had to endure. Finally, Ithank Mary, whomilove, who loves meand, luckily, thinks the factorial of any numberis7. vi
7 Contents 1 Introduction 1 2 Manufacturing, Robotics, Control Theory and Veriæcation Models and Control : : : : : : : : : : : : : : : : : : : : : : : : Classical Control Theory : : : : : : : : : : : : : : : : : : : : : Control Theory Applied to DES : : : : : : : : : : : : : : : : : Building Supervisory Controllers for CDES : : : : : : : Veriæcation and Temporal Logic : : : : : : : : : : : : : : : : : Temporal Properties of Systems : : : : : : : : : : : : : Temporal Logics: Linear and Branching Time : : : : : Theorem Proving andmodel Checking : : : : : : : : : Veriæcation with Automata Theory : : : : : : : : : : : Reductions and Compositional Veriæcation : : : : : : : Real TimeVeriæcation : : : : : : : : : : : : : : : : : : Role of Veriæcation Tools : : : : : : : : : : : : : : : : : Hybrid Systems Veriæcation and Control : : : : : : : : : : : : 35 vii
8 3 Temporal Logic Supervisor Synthesis with Veriæcation Notational Preliminaries : : : : : : : : : : : : : : : : : : : : : CDES Notions and Notations : : : : : : : : : : : : : : CTL Direct Synthesis of Supervisors : : : : : : : : : : : : : : : Controlled Semantics as Model Restriction : : : : : : : Main Objectiveand Notable Problems : : : : : : : : : Inductive Construction of the Controlled Semantics : : Circumventingthe Problems : : : : : : : : : : : : : : : CTL Soundness of Controlled Semantics : : : : : : : : : Simple CTLSynthesis Algorithm : : : : : : : : : : : : : : : : : Existence of a Supervisor Synthesis Algorithm : : : : : 88 4 The ControlD System Revised Algorithm and The Implementation : : : : : : : : : : State Space Traversing Algorithm : : : : : : : : : : : : Tradeoæs in the Treatment of Disjunctive Forms : : : : Comparison with Standard Supervisor Synthesis Algorithm : : : : : : : : : : : : : : : : : : : : : : : : : : : : Open Problem: Symbolic Representation : : : : : : : : Veriæcation and Synthesis : : : : : : : : : : : : : : : : : : : : The Environment : : : : : : : : : : : : : : : : : : : : : : : : : ControlD Components anduserinterface : : : : : : :109 viii
9 5 Building a DiscreteèHybrid Controller for a Walking Machine A Brief History of Walking Machines : : : : : : : : : : : : : : Problems for Walking Machines : : : : : : : : : : : : : : : : : Leg Behavior Models and Gaits : : : : : : : : : : : : : StabilityofWalking Machines : : : : : : : : : : : : : : Synchronization and Realtime Control : : : : : : : : : Building Walking Machines Controllers with ControlD : : : : Discrete Controller for the Walking Machine : : : : : : Continuous Control Constraints : : : : : : : : : : : : :129 6 Ensuring Failure Behavior for the CRAMTD Manufacturing Line CRAMTD Project Tray Packing Model : : : : : : : : : : : : : Failure Behavior Control : : : : : : : : : : : : : : : : : : : : : Modeling thetray Packing Line : : : : : : : : : : : : : CTL Speciæcation of Behavior : : : : : : : : : : : : : : Concluding Remarks : : : : : : : : : : : : : : : : : : : : : : :143 7 Conclusion and Future Work 144 ix
10 List of Figures 1.1 The walking machine example : : : : : : : : : : : : : : : : : : Schematic drawing ofthe tray pack lineofthe CRAMTD project The standard arrangement of plant G and supervisor hr;'i. : Schematic æowchart of the Supervisor building steps. : : : : : A case illustrating the requirements for AX supervisors : : : : A counterexample for the intuitive supervisor map construction for disjunctions : : : : : : : : : : : : : : : : : : : : : : : : Restrictions on the language satisfying Aëb U aë. : : : : : : : : Controlled Semantics for :f without uncontrollable events : : Controlled Semantics for :f with uncontrollable events : : : : An illustration of the problems with the controlled semantics for negation : : : : : : : : : : : : : : : : : : : : : : : : : : : : A problematic case involving the supervisor map assignments needed to satisfy AXèpè and f ç AëAXèpè U që. : : : : : : : : A problematic case for the semantic of the EU and AU operators. 83 x
11 4.1 Schematic of the Model Restriction Supervisor Synthesis èmrssè Algorithm : : : : : : : : : : : : : : : : : : : : : : : : : : : : : The label state graph procedure : : : : : : : : : : : : : : : : The function label AU. : : : : : : : : : : : : : : : : : : : : : The procedure label AU next. : : : : : : : : : : : : : : : : : The procedure model synth : : : : : : : : : : : : : : : : : : Asnapshot of the ControlD graphical environment : : : : : : : ControlD edit window : : : : : : : : : : : : : : : : : : : : : : State diagram representing the movementofasingle leg. : : : Hexapod tripod gait : : : : : : : : : : : : : : : : : : : : : : : Hybrid Controller Architecture for Walking Machine : : : : : : Mini Actuator Leg Prototype 1 : : : : : : : : : : : : : : : : : The FSM model of a Leg with uncontrollable event slip : : : : Schematic representation of the walking machine : : : : : : : : Simpliæed Geometric Model of the Walking Machine : : : : : : Schematic of the tray packing line ofthe CRAMTD project : : FSM model for the CRAMTD CheckWeight Station : : : : : FSM model for the CRAMTD Rejectdiverter : : : : : : : : : Model of the measurement communication : : : : : : : : : : : ControlD window with CRAMTD constraints : : : : : : : : :142 xi
12 List of Tables 2.1 Syntax and informal Semantics for PLTL : : : : : : : : : : : : Syntax and informal Semantics for CTL : : : : : : : : : : : : : 27 xii
13 Chapter 1 Introduction A person's ëproductivity" can be measured in many ways. However, independent ofhow this productivityismeasured, it appears that the advent of modern computing devices and software architectures has improved this ability byseveral factors of magnitude 1. One ofthe aims of computer science has been to provide better software tools that would increase our productivity èand hopefully, leave us more time to engage in leisurely activities such as philosophizing or æshingè. Two desirable characteristicsofsuch tools are their ëusefulness" in solving a given problem and the ësoundness" of the theory on which they are founded. 1 In spite ofthe fact that computers have been around for decades, there seems to be a folk consensus that this result has been actually achieved only recently outside research institutions and very hightech industries ècfr. Business Week, July 17th, 1995: cover story on Wages in Americaè. 1
14 This dissertation will describe how problems in robotics and manufacturing prompted the developmentofatheory of Discrete Event Systems èdesè, how this theory has been applied and how achange in perspective leads to the construction of algorithms and tools which improveonthe original formulation. Two motivating examples form robotics and manufacturing will be used to substantiate the claims to bemade. The ærst one isanapplication to the problem of synthesizing a controller for the synchronization of leg movements ofawalking machine. The second isanapplication to the construction of control software for a food processing manufacturing line. èthe latter work has been carried out incollaboration with the DepartmentofIndustrial EngineeringofRutgers Universityè. In the walking machine example the challenge is to build a discrete controller capable of producing reasonable gaits which follow simple principles of coordination. Moreover, the system presents many problems whichhave been recently addressed as ëhybrid" between the continuous and the discrete viewpoints. Figure 1.1 shows a sample graphic output ofthe system. Chapter 5 contains a more thorough description of the example. Manufacturing systems pose very interesting problems from the viewpoint of coordination and failure detection. Figure 1.2 showsaschematicofatray packing lineofthe Combat Ration Automated Manufacturing Technology Demonstration ècramtdè of Rutgers University. Chapter 6 contains a more 2
15 Figure 1.1: The walking machine example. Spacing Conveyor B Reject Conveyor F A Seamer and Main Motor Phasing Conveyor Filling Equipment E Filling Conveyor A C D Reject Diverter I Lids Conveyor H Discharge Conveyor Checkweigher Mound Detector Figure 1.2: Schematic drawing of the tray pack line of the CRAMTD project. 3
16 detailed description of the application. The CRAMTD system is composed of manysubsystems, each of which can be represented by a ænite state model. The result ofthe controller synthesis process for such aplant is packaged in intermediate format which can be translated into avariety of ëprogrammable logic controllers" èplcè and other software tools. In this case, the problems arise from the combinatorial explosion due to the nature of the model. The dissertation is organized as follows. Chapter 2 contains an introduction that provides contexts for the developmentsdescribed later in the dissertation. It contains a brief literature review and adescription of standard tools: e.g. control theory, Ramadge and Wonham's theory of controlled discrete event systems, temporal logic, and veriæcation. Chapter 3 contains a novel interpretation of propositional temporal logic as a tool for the synthesis of controllers for discreteevent systems. Chapter 4 describes an algorithm for controller synthesis and its implementation using the ControlD toolset, and then compares it with the ëstandard" algorithm for the same problem. Chapters5and 6 contain the descriptions of the walking machineand food manufacturing examples, respectively. 4
17 Chapter 2 Manufacturing, Robotics, Control Theory and Veriæcation This thesis studies the semiautomated synthesis and veriæcation of control systems for robotics and manufacturing devices using formal methods in a discrete framework, and bears some resemblance to the theory of controlled discrete event systems ècdesè of Ramadge and Wonham's ë67ë. The aim is to study the integration of CDES theory with the techniques developed for the speciæcation and veriæcation of discrete event systems èdesè. Many of these techniques rely on the application of someæavor of temporal logic ë35ë. This chapter begins with adescription of the main concepts underlying the dissertation, followed by a review of the literature on various related subjects. It is worthwhile to note in advance that such review will necessarily be partial and incomplete, becauseofitsthe scope and the sheer amount of 5
18 material available in the literature. 2.1 Models and Control The notions of robotics and manufacturing devices or systems encompass many artifacts which are currently employed in the production and servicing activities of the economyofmanynations. Giving a precise deænition of these terms is therefore a self defeating task. Nevertheless, science and engineering have provided many abstractions and tools to model such a wide variety of systems. For the purpose of this dissertation, the robotics and manufacturing systems that will be considered, can be formally described in terms of ænite state machines èfsmè ë41ë. Such discretedescription will represent a certain ëview" of the comprehensivebehavior of the robotics or manufacturing system. I.e., it represents amodel of certain behavioral aspects ofthe device. This model is discrete because byitsnature theunderlying formal tool èfsmè is. Continuous characteristics will be best described by diæerent formal tools èe.g. ordinary diæerential equations, linear and nonlinear control theoryè. Models of robotics and manufacturing systems serve diæerent purposes. They may simply be a description of the relationships among the components of a system, to beusedasadocumentation. More often, these models serve the purpose of analyzing the system in order to either explain, predict, or control its behavior. The nature of these models depends on which ofthese 6
19 tasks èor their combinationè is to be performed. Nowadays, explanation, prediction and control are supported by combinations of hardware and software embodied by computer systems. There is therefore a focus on those modelingtools that presentdesirable computational properties such aslow algorithmic complexity. This dissertation will focus on an approach tothe control task based on a computational model whose characteristics make it relatively practical to implement while maintaining expressiveness. The basis of this computational model is the theory of CDES developed by Ramadge and Wonham in the mid 80's. This theory is deeply rooted in the classical control theory from whichit borrows terminology and key concepts. It will be shown that by dropping the traditional control theory heritage, the resulting computational model gains in expressiveness and eæciency. To substantiate these ændings and demonstrate the feasibilityofthe approach, an implementation of the algorithms has been incorporated in the ControlD tool. The tool has been used to construct the controllers of two systems: a walking machine ë12ë and amanufacturing line ë10ë. Before discussingthese results, an introduction to thekey concepts aswell as a èincompleteè review of the related èand vastè literature touching these subjects isinorder. 7
20 2.2 Classical Control Theory The main ideas used in controlling physical devices can be traced back to Maxwell's governor example which also introduced the concept of feedback control for dynamical systems. This traditional control theory approach has been extensively formalized èoriginally by Shannon, Kalman and others í see ë44, 77, 71ëè and has resulted in rich linear and nonlinear theories treating continuous laws. These theories introduce manykey concepts that will be brieæy reviewed in this section, in order to clarify their use in other parts ofthis dissertation. The following brief introduction is closely patterned after ë32, 57ë. Avery abstract view of control theory considers a system which evolves over time èt è while producing outputs èy èinresponseto inputs èuè. The outputs are also dependent onthe system state. Usually, inputs, outputs and system state are denoted with uètè, yètè, and xètè, i.e. as functions of time. The exact nature of these functions and the underlying representation of time are the basis for taxonomy of control models. The evolution of the values produced by these functions are called trajectories, orhistories, ortraces. The focus is usually on statespace trajectories H X 4 = fhx : T! Xg; and ofoutput histories H Y 4 = fhy : T! Y g: 8
21 The dependency of state xètè on the inputs is usually denoted by a state transition equation xètè =çèt; t 0 ;xèt 0 è;uètèè; or by a state evaluation equation xèt 0 è = x 0 ; x 0 ètè = fèt; xètè;uètèè: These functions are coupled with a output equation yètè =gèt; xètè; uètèè: ç, f, and g are application dependent functions. With these deænitions it is now possible to formulatethe control problem for a dynamical system by distinguishing agoal subset of the trajectories in H X. Controlling a dynamical system is equivalent to constraining it within a trajectory in the goal subset. According tothe nature of the goal subset, several æavors of the control problem can be formulated. Two classical ones are the servo problem and the stateavoidance problem. Servo Problem. Given a reference trajectory g 2 H x, deæne the goal subset as G = fh X such that jh X, gjçæ; æ é 0g: I.e. the trajectory g must be ëtracked" as closely as possible. 9
22 Stateavoidance Problem. Given a set of states Q ç X, deæne the goal subset as G = fh X such that 8t:h X ètè 62 Qg: I.e. the trajectory never reaches a state inq èprovided that x 0 62 Qè. In order to constrain the system within the bounds speciæed by the trajectory goal set G, actions must be taken. These actions are represented as inputs tothe system and are referred to ascontrol laws or control policies. Control laws are chosen given the ëcurrent" state ofthe system, or the the ëcurrent" output orboth. I.e. feedback is used in order to decide the ënext" input tothe system. Often, goal trajectories may be unattainable. Control theory deænes notions and methods to describe this situation. The key notion is that ofcon trollability of a system. A point inthe timespace hç;xi2t æ X is said to be controllable with respect to asetc ç X of target states if and only if there exists a control action æ such that ataëtime" téç the resulting point ht; x 0 i in the timespace is within C. hç;xi æ ; ht; x 0 i; and ht; x 0 i2c: A dynamical system is completely controllable if and only if it is possible to transfer any state xèt 0 è 2 X to anyother state inx in a ænite amount of ëtime". 10
23 The deænition of controllability relies on precise information about the state ofthe system. Such information may not be available. The problem of recovering such information is the observability problem of a system. Controllabilityand observability may be related by the socalled duality principle. A wide range of physical systems can be described and controlled in terms of linear, timeinvariant models. For this class of models, which are characterized by a linear matrix form of the deæning state andoutput equations, a very rich set of sophisticated mathematical tools has been developed over the years. Nonlinear dynamical systems present many more diæculties. Classical Control Theory Deæning Traits Control theory has been historically interested in continuous systems èover time T è, i.e. systems whose models are represented with the tools of diæerential equations and linear algebra. Another characteristic of control theory is the uniformity of representation. This principle applies both to basic research incontrol theory and in the practice of designing controllers for physical systems èrobotics and manufacturingè. It is common practice to represent uètè;gètè;fètè, and the control task using the same underlying formalism. This practice has been carried over to the modelingand control of systems which cannot be promptly represented withthetools of continuous mathematics. Section 2.3, contains some introductory remarks about the reformulation 11
24 of control theory for discrete event systems. 2.3 Control Theory Applied to DES While continuous control theory has been very successful, manymodern complex physical devices have proven to be not amenable to itstechniques. There are essentially two sources of problems. Many ofthese devices can be properly described only in a discrete orhybrid èi.e., mixture of discrete and continuous dynamicsè setting. The plant has to bemodeled in terms of a discrete set of states and transitions 1, which poses many problems. Secondly, the behavior desiredofthe ultimate system tends to be fairly complex. In their original work, Ramadge and Wonham ë66ë describe a reinterpretation of the key concepts ofcontrol theory for systems whose underlying dynamics is represented in terms of formal languages. The system to be controlled is considered a generator ègè of a language L èsee ë41ëè. This choice is well suited to represent the discretenature of a wide variety of systems. Ramadge and Wonham introduced the term CDES to indicate this class of systems. As a standard example, consider a machineonashop æoor. A high level 1 Theword discrete assumes two diæerent meanings in control theory_it is used toindicate a discretization of time T,orasadeæning characteristic of the underlying state space. In this dissertation, unless explicitly noted, the term ëdiscrete" always refers to system modeled with anunderlying discrete state space. 12
25 model of its operation may be given in terms of three states idle, running, faulty. The normal operation of the machineisanalternation of the ærst two states, controlled by start and stop signals. Every once in while, the machine will break down, causing the model to movetothe faulty state. In its simplest form, a CDES is deæned in terms of its generator G =èæ;s;æ;s 0 è: æisthe alphabet of the generator èsimilar to the outputs Y of a control theory speciæcationè, whose elements are called events. S is a ænite set of states èthe analogous of Xè. æ : ææs! S is the unregulated transition function, deæned in the standard way. Thestate s 0 is the initial state. The generator is also called, with anobvious analogy, the plant to be controlled. The main assumption on CDES is that the alphabet set is partitioned into two subsets. æ c ç æisasubset of controllable events, æ u =æ, æ c is a subset of uncontrollable events. This assumption is the basis of the whole CDES theory. The control law to beapplied to the plant can disable events inæ c preventing them from being generated. This disabling action constitutes the input ofthe system èin analogy to Uè. An admissible control for a CDES is a set of disabling actions and can simply be represented as a subset of æ c or as a function æ :æ c!f0; 1g: 13
26 A ç 2 æ is enabled if æèçè = 1,disabled otherwise. The set of all æ's is denoted by,. The analogue of the state equation f is the controlled state transition function æ c, which uses the deænition of æ with the extension that æ :æ u! f1g. i.e. æ c :,æ S æ æ! S: 8 é p if æèq; çè = p and æèçè = 1; æ c èæ;q;çè= é:? otherwise: By substituting æ with æ c in the deænition of G, the controlled generator G c is obtained. With these deænition it is now possible to formulate the control problem for CDES's. I.e. design a supervisor device that selects the control inputs in such a way that the given CDES behaves in obedience to various constraints Building Supervisory Controllers for CDES Theanalogy with continuous control theory is carried over in the speciæcation of a supervisor for a CDES plant G c. Since the plant isagenerator of a language èæëgëè, it is natural to assume adevice observing, orrecognizing, the events generated and producing the control inputs asneeded. Such adevice is represented as a recognizer R =èæ;r;ç;r 0 è: 14
27 hr;'i ç admissible control æ G c Figure 2.1: The standard arrangement of plant G and supervisor hr;'i. R is the recognizer set of states, ç is its transition function and r 0 is its initial state. The recognizer has an associated map ' : R!,; which represents the control law for the CDES. The pair hr;' is the supervisor for G c. ' is called the supervisor map and represents the state feedback law for the plant. The supervisor is coupled with the plant inthe standard arrangementshown in ægure 2.1. With these deænitions, Ramadge and Wonham developatheory which gives guarantees about the existence and the ëconstructibility" of supervisors. 15
28 CDES Theory: Main Results The main result ofthe theory of CDES is the existence theorem for supervisors ècfr. ë66ëè. The result is based on a notion of controllability of languages. Assumethe standard language union, intersection, concatenation and preæx closure operations described in ë41ë. If K and L are languages over an alphabet æ partitioned between controllable and uncontrollable events, then the language K is said to becontrollable if çkæ u ë L ç ç K: This condition ensures that given any sequence of events inthe preæx of K èkè, ç any subsequent uncontrollable event will not produce a behavior that a recognizer for K would fail to detect. The other main result ofthe theory concerns the structure of the family of controllable sublanguages of a given language K. This family is closed under union and intersection and contains a supremal element. The supremal controllable sublanguage of K, denoted by K ",istherefore an approximation to K. The existence of this supremal element isthe key to the construction of the actual supervisor for a wide variety of systems. A Pragmatic Methodology The existence of the supremal controllable sublanguage of a given language K, implies a procedure for the construction of a supervisor for a given system. 16
29 Start Build Model of System Language L Build Model of "Desired Behavior" Language K yes no Is K Controllable? no Is K acceptable? yes Build approximation K yes Is K acceptable? no Done Figure 2.2: Schematic æowchart of the Supervisor building steps. The supervisor hr;'i realizes the control law for a given set of constraints. These constraints are expressed in terms of a language K. Ifthis language is not controllable, then its approximation K " can be built. This leads to a pragmatic procedure for the construction of supervisors èsee Figure 2.2è. Such a procedure is not an algorithm directly implementable. It is a guideline for a practitioner employing these notions to actually build a controller. In particular, the step checking for the ëgoodness" of the controlled behavior is completely up to the human. Ramadge and Wonham take this fact into account intheir theory by introducing the notion of ëacceptable" language ècfr. ë66ëè. The procedure is the basis for the design of a software environment for 17
30 the synthesis of discrete controllers for CDES's. In ë13ë a system is described that performs this ëaiding" task. The process of building a supervisor proceeds in two steps: ærst a supremal sublanguage for the speciæed K is built, then the actual supervisor map ' is synthesized. There are restrictions to the kind oflanguages that can be eæectively used with this procedure. The theory works well for regular languages. For larger classes of languages èe.g. contextfreeè the uniqueness of K " cannot be guaranteed. These results are contained in ë64ë. In the same paper, one can ænd an algorithm for the construction of the supremal controllable sublanguage of a K " with respect to a preæx closed language æëgë. This algorithm assumes an FSM representation of G and R and produces the desired automata in OèjSj; jrjè, where jsj is the cardinality of the set of states of the generator G and jrj is the cardinality ofthe set of states of the recognizer R. The synthesis algorithm proposed in this dissertation improves slightly on this result. Though the overall asymptotic complexityofthe proposed synthesis algorithm does not change, it will be argued that the diæerent representation used can signiæcantly improvethe practicalityofthe resulting tool. Other Developments in Ramadge and Wonham's Theory The part of the theory of CDES's described so far is only the basis which constitutes the background for the development of this dissertation. An overview 18
31 of other developmentsinthe theory can be found in ë67ë. The other topics which have been investigated in the CDES framework regard observability and modular synthesis of supervisors ë63ë. The issue of decentralized supervision has been also investigated ë68ë. In this case the question asked is whether the action of several supervisors acting locally can achieve the same eæect as a centralized one. Another research direction concerns what kind of extensions can be introduced in the CDES framework preserving its basic characteristics. A crucial question concerns what classes of languages besides the regular ones admit the unique supremal sublanguage construction or what are the characteristics of other extensions to the basic model. For instance,!languages èand Bíuchi automataè as a basis for CDES are considered in ë65ë, and Vector Addition Systems are considered in ë47, 48ë. The basic CDES model does not include time as a component. The only notion of the passage of timeisderived from the generation of events bythe plant. In this direction, a standard extension to the basic CDES model with discrete time ticks is presented in ë18ë. On the foundation of CDES, other extensions and reinterpretation of the theory have been proposed, mainly with the aim to reuse the wealth of knowledge developed within the temporal logic community. This dissertation falls in this last category. Additional references will be given in Section
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