PhD Course on Discrete event and hybrid systems

Transcription

1 PhD Course on Discrete event and hybrid systems Discrete event systems Discrete event systems (DES) theory is the theoretical foundation to the study of dynamical man-made systems, whose evolution is related to the occurrence of asyncronous events, usually not numerical. The most relevant application domains are: manufacturing, process control, supervisory systems, software engineering, transportation and so on. As a result of the combinatoric nature of the analysis, optimization and control problems of DES, the theory of DES has developed slowly with respect to the theory of timedriven systems. During the last fifteen years, to address the requirements from industry of formal methods for the study of DES, the international scientific community has devoted a great effort in this respect, and several important results have been obtained. In particular, solid theoretical foundations have been given within the framework of supervisory control, state observation and fault diagnosis. Hybrid Systems Recent technological innovations have caused an ever increasing interest in the study of hybrid systems (HS). The peculiarity of hybrid systems is the interaction between continuous-time dynamics (governed by differential or difference equations), and discrete dynamics and logic rules (described by temporal logic, finite state machines, if-then-else conditions, discrete events, autonomous or forced switching, etc.). The research domain of HS is attracting an ever increasing number of researches. However, several important problems are still open in this framework, such as optimal control and analysis of many relevant properties, such as stability and Zeno phenomena. In particular, the latter is strictly related to the sliding mode theory, and many of its relevant results can be used or extended. Objectives of the course The main objective of this course is that of collecting together researches from several universities, working within the framework of discrete and hybrid systems, so as to provide an outline of the main theoretical and application results in these areas. All lectures will be taken at a quite didactic level, so that also PhD students not working in these topics, will be able to understand the proposed results. Dates and location All lectures will take place at the University of Cagliari, Faculty of Engineering, Piazza D Armi, Cagliari, Italy in the following dates: March 21, March 27, May 22, June 7, June 20, July 17, Organizers: Carla Seatzu, Elio Usai Dep. of Electrical and Electronic Engineering University of Cagliari, Italy {seatzu, eusai}@diee.unica.it

2 Lecturers Giorgio Bartolini Dip. Ingegneria Elettrica ed Elettronica, Università di Cagliari, Italy; URL: Francesco Basile Dip. Ingegneria dell Informazione e Ingegneria Elettrica, Università di Salerno, Italy; fbasile@unisa.it URL: Maria Paola Cabasino Dip. Ingegneria Elettrica ed Elettronica, Università di Cagliari, Italy; cabasino@diee.unica.it Daniele Corona Delft Center System and Control, Technische Universiteit, Delft, The Netherlands; d.corona@dcsc.tudelft.nl URL: Alessandro Giua Dip. Ingegneria Elettrica ed Elettronica, Università di Cagliari, Italy; giua@diee.unica.it URL: Christoforos N. Hadjicostis Dep. of Electrical and Computer Engineering, University of Illinois, Urbana-Champaign, USA; chadjic@control.csl.uiuc.edu URL: Stéphane Lafortune Dep. of Electrical Engineering and Computer Science, University of Michigan, Ann-Arbor, USA; stephane@eecs.umich.edu URL: Arie Levant Applied Mathematics Dept., School of Mathematical Sciences, Tel-Aviv University, Israel; levant@post.tau.ac.il URL: Cristian Mahulea Dep.to Informática e Ingeneiría de Sistemas, Centro Politécnico Superior, Zaragoza, Spain; cmahulea@unizar.es URL: Andrea Paoli Dip. Ingegneria Elettronica, Informatica e Sistemistica, Università di Bologna, Italy; apaoli@deis.unibo.it URL: Giovanni Michele Pinna Dip. Informatica e Matematica, Università di Cagliari, Italy; gmpinna@unica.it Alessandro Pisano Dip. Ingegneria Elettrica ed Elettronica, Università di Cagliari, Italy; pisano@diee.unica.it URL: Carla Seatzu Dip. Ingegneria Elettrica ed Elettronica, Università di Cagliari, Italy; seatzu@diee.unica.it URL:

3 Elio Usai Dep. of Electrical and Electronic Engineering, University of Cagliari, Italy; URL: Arjan van der Shaft Department of Mathematics and Computing Science, University of Groningen, The Netherlands; URL: Zhenyu Yang Dep. of Software and Media Technology, Esbjerg Technical Inst., Aalborg University, Denmark; URL:

4 Detailed Program March 21, A. Giua Observers for Petri nets G.M. Pinna The theory of regions and the synthesis of nets from computations F. Basile Supervisory control of Petri nets based on monitor places March 27, M.P. Cabasino Identification of place/transition nets C. Hadjicostis Coding approaches to reliable descrete event systems design C. Mahulea Optimal control of continuous Petri nets May 22, A. Van Der Shaft Analysis and control of complementarity hybrid systems D. Corona Adaptive cruise controller for a Smart car: A comparison benchmark for MPC-PWA control methods A. Van Der Shaft Composition and bisimulation of hybrid systems June 7, C. Seatzu Optimal control of switched systems Z. Yang On the controllability and fault tolerance of hybrid dynamical systems E. Usai Zeno phenomena in hybrid systems and sliding mode behaviours June 20, A. Paoli Supervisory control of discrete event systems S. Lafortune Diagnosis of discrete event systems S. Lafortune Decentralized control of discrete event systems July 17, G. Bartolini Simplex sliding mode control method for nonlinear multiinput uncertain systems A. Levant Homogeneous discontinuous control A. Pisano Second-order sliding modes in mechanical and electromechanical systems Basic principles and implementation results

5 Abstracts of talks March 21, 2007 Alessandro Giua, Observers for Petri nets State estimation is a fundamental issue in systems theory. Reconstructing the state of a system from available measurements may be considered as a self-standing problem, or it can be seen as a prerequisite for solving a problem of different nature, such as stabilization, state-feedback control, diagnosis, filtering, and others. Despite the fact that the notions of state estimation, observability and observer are well understood in time-driven systems, in the area of discrete event systems there are relatively few works addressing these topics and several problems are still open. In this talk I will present two approaches to observer design for Petri net models. In the first approach, inspired by systems theory, the initial marking (i.e., the initial state) is assumed unknown, while the firing of all transitions can be completely observed. The purpose of the observer is that or reconstructing the initial marking, from which the current one can be uniquely determined given the observed sequence of transition firings. In the second approach, inspired by the notion of nondeterministic automata typical of computer science, the initial marking is known. However, as the net evolves, the current marking is usually unknown due partial observation: some transitions firings may generate no observable event, or two different transition firings may generate the same observable event. The purpose of the observer is that or keeping track of all possible markings that may have been reached from the initial one with a sequence of firings consistent with the observed behavior. G. Michele Pinna, The theory of regions and the synthesis of nets from computations In this talk I will review various approaches to the net synthesis based on the notion of regions, introduced by Ehrenfeucht and Rozenberg. The net synthesis problem can be described as follows: from a suitable representation of nets computations is it possible to figure out which system, represented as a Petri net, has actually produced such computations? The first attempt can be traced back to the notion of non sequential process of Petri nets, where a suitable labeling was defined on a representation of computation based on partial orders enriched with informations on the resources used. The notion of resource is quite crucial in Petri nets: elementary nets or condition/events nets identify resources with the holding of certain conditions, whereas other Petri nets models such as place/transition nets allow for multisets of resources. Petri nets computations are in fact driven by the available resources, but these resources are usually abstracted from the computations. In general computations are represented in a much more abstract way, where the notion of resource is hidden, e. g. in the case of Petri Nets transition systems, causal trees, marking graphs or suitable automata (e.g. concurrent, step, higher dimensional or event automata). In these more abstract representations of computations resources can be fruitfully identified with the notion of region. After introducing the notion of regions, I will review various approaches to the synthesis and various categorizations of computations in the light of this notion. I will show also possible generalizations and applications to other fields like Business Process Management. Some ideas about an incremental calculus of regions will be also presented. Francesco Basile, Supervisory control of Petri nets based on monitor places This talk deals with the problem of enforcing generalized mutual exclusion constraints (GMEC) on place/transition nets with uncontrollable transitions. First some literature methods that address this problem or are related to it are briefly recalled. Then, an efficient control synthesis technique which enforces GMEC constraints by introducing monitor places to create suitable place invariants is presented in detail. The method has been shown to be maximally permissive and to give a unique control structure in the case that the set of legal markings is controllable. This is not true for uncontrollable specifications, but the class of monitor places enforcing an uncontrollable specification can be parameterized with respect to the solution of a linear system of equations. This can help to solve the problem to choose the best monitor based solution for a given GMEC according to a suboptimal criterion. If the classical partition of the event set into controllable and uncontrollable events from supervisory control theory is replaced by

6 associating a control and observation cost to each event the supervisory control problem can be formulated as an optimal control problem. Monitor places which enforce the constraint are devised as a solution of an integer linear programming problem whose objective function is expressed in terms of the introduced costs. March 27, 2007 Maria Paola Cabasino, Identification of place/transition nets In this talk we examine the problem of identifying a Petri net system, given a finite language that it generates. Firstly we consider the problem of identifying a free labeled Petri net system, namely all transition labels are distinct. The set of transitions and the number of places is assumed to be known, while the net structure and the initial marking are computed solving an integer programming problem. Then we show how this approach can be extended in several ways introducing additional information about the model (structural constraints, conservative components, stationary sequences) or about its initial marking. Furthermore, we show how the approach can also be generalized to the case of labeled Petri nets, where two or more transitions may share the same label. In particular, in this case we impose that the resulting net system is deterministic. In both cases the identification problem can still be solved via an integer programming problem. Finally, we show how given an automaton that represents the coverability graph of a net we are able to solve the problem of determining a net system whose coverability graph is isomorph to the automaton. Our approach requires solving an integer programming problem whose set of unknowns contains the elements of the pre and post incidence matrices and the initial marking of the net. Christoforos Hadjicostis, Coding approaches to reliable descrete event systems design Fault tolerance has been a long standing necessity in system design and operation. In systems with memory (i.e., state), however, modular redundancy and other traditional approaches to fault tolerance are undesirable not only because they are expensive but also because they rely heavily on the assumption that the error-correcting (e.g. voting) mechanism is fault-free. This talk presents a general framework that systematically addresses these issues in fault-tolerant discrete-event dynamic systems. By replacing the original system with a coded, redundant implementation that retains the original functionality and state, we are able to exploit violations on the state encoding of this redundant implementation and develop error detection, correction and/or reconfiguration techniques. Unlike traditional methodologies that rely on concurrent checking at the end of each event epoch, our approach allows the construction of redundant systems in which detection and identification of errors is based on non-concurrent checks. Thus, the checker of the resulting design can operate at a slower speed than the rest of the system, which relaxes the stringent requirements on its reliability. We demonstrate this approach in the context of linear dynamic systems and finite automata. Cristian Mahulea, Optimal control of continuous Petri nets Continuous Petri nets (contpn) were introduced as an approximation to deal with the state explosion problem which can appear in discrete event models. When time is introduced, the flow through a fluidified transition can be defined in many ways, the most used in literature are constant and variable speed, which can be seen as some kind of finite and infinite server interpretations of the transitions behaviour. The first point of the talk is to introduce these two semantics, explain some important properties and show that piecewise behaviours are obtained for both semantics. Then, it is proved that for a broad class of nets, timed PN under infinite server semantics is a more accurate approximation of the discrete nets. Starting with the crucial question of how to control a timed contpn system, an approach based on the idea of slowing down the firing flow of transitions is considered. Firstly, we abstract the constraints on the states and inputs and we try to understand the behavior of contpn and interpret classical results in the contpn case. In particular, the poles of the underlying linear systems are studied and an interesting finding is that the non-controllable ones are zero-valued. Considering then the constraints, the steadystate control is study trying to characterize the equilibrium states for a constant control action. For some

7 particular net subclasses, unique solutions are algebraically obtained. A LPP is purposed to computes the optimal steady-state for a given control action. May 22, 2007 Arjan van der Shaft, Analysis and control of complementarity hybrid systems In recognition of the fact that many systems contain both continuous and discrete aspects, considerable study has been devoted to "hybrid systems". The formulation of equations of motion for hybrid systems in explicit form, including the event rules and the description of the continuous dynamics for every possible mode, is in many cases a formidable task, and there is a clear need for devices that enable the modeler to work in what might be called a "high-level language". A formalism that can be used for this purpose is the so-called complementarity formalism. The formalism is applicable to a broad class of physical hybrid systems, as well as to hybrid systems described by an underlying dynamics subject to piecewise-linear constraints. This talk surveys some of the key notions in the complementarity modeling and analysis of hybrid systems. Furthermore, we describe the relation between complementarity systems and the framework of port-hamiltonian systems, which turns out to be a happy marriage. The theory of port-hamiltonian systems offers tools for control, in particular for stabilization, based on the identified Hamiltonian structure of the system. We discuss the potential of these techniques for control of complementarity port- Hamiltonian systems. Daniele Corona, Adaptive cruise controller for a Smart car: A comparison benchmark for MPC- PWA control methods The design of an adaptive cruise controller (ACC) for a Smart car, which is a type of small car, is proposed as a benchmark set-up for several existing control methods based on model predictive control (MPC), developed for nonlinear and piecewise affine (PWA) systems and on a tuned proportional-integral (PI) action. Each of these methods has been already applied to specific case studies, different from method to method. We propose therefore the study of implementing and comparing them over a common benchmark, allowing to assess their main properties, characteristics and strong/weak points. The ACC aims to achieve the tracking of a leading vehicle and energy saving, within specific constraints, that model physical limitations, safety/comfort issues, environmental protection and mechanical stress of the vehicle. In simulation, a realistic model of the Smart, that involves gear box and engine nonlinearities, is considered. A description of the methods to be compared is presented, and the comparison keys, obtained under a given simulation scenario, are collected in a table. In particular, trades-off between complexity and accuracy of the solution, as well as computational aspects are highlighted. Arjan van der Shaft, Composition and bisimulation of hybrid systems A common theme in the theory of concurrent processes and in systems and control theory is to characterize systems which are 'externally equivalent'. The idea is that we only want to distinguish between two systems if the distinction can be detected by an external system interacting with these systems. This is a fundamental notion in design, enabling us to take a 'divide and rule' strategy, and in analysis, allowing us to switch between externally equivalent representations of the same system and to reduce sub-systems to externally equivalent but simpler ones. In concurrent processes the crucial notion in this endeavor is the concept of bisimulation which expresses when a sub-process can be considered to be externally equivalent to another (hopefully simpler) process. On the other hand, classical notions in systems and control theory are state space equivalence of dynamical systems, and reduction of a dynamical system to an equivalent system with minimal state space dimension. These notions have been instrumental in e.g. linking input-output models to state space models, and in studying the properties of interconnected systems. In this talk we will show how the notion of bisimulation for concurrent processes can be extended to continuous dynamical systems, and how the developed notion unifies the concepts of state space equivalence and reduction. Furthermore we show how by merging this notion with the standard notion of bisimulation for concurrent one obtains a structural bisimulation notion for hybrid systems.

8 Finally we address the 'control by composition' problem in this context: Given a hybrid system with part of the external variables accessible for controller interaction, and a hybrid controller system to-bedesigned. What are the achievable closed-loop systems up to bisimulation? June 7, 2007 Carla Seatzu, Optimal control of switched systems In this talk we focus on a particular class of hybrid systems, namely switched piecewise affine autonomous systems, and present an optimal control procedure whose objective is that of minimizing a quadratic performance index over an infinite time horizon. We first show how an optimal solution can be computed under the assumption that the switching sequence has a finite length, and the decision variables are the switching instants and the sequence of operating modes. Then, we generalize this approach to the case of an infinite number of admissible switches. The main advange of such a procedure, that is based on dynamic programming arguments, is that it identifies the regions of the state space where an optimal mode switch should occur, therefore providing a state feedback control law. An application to the design of semiactive suspension system for vehicles is also presented. Zhenyu Yang, On the controllability and fault tolerance of hybrid dynamical systems Hybrid Control Systems (HCS) are usually referred to as control systems that involve both continuous and discrete dynamics as well as continuous and discrete controls. The controller in HCS not only supervises the dicrete transitions, but also controls the continuous behaviors, and in most cases these two functionalites need to be cooperated. Therefore, the controllability concept of HCS should reflect the influence of input signals on the evolution of continuous and disrecte states as well. This talk will focus on the following isses. (1) What's the meaning of controllability in HCS? (2) How to analyze this kind of system property? First of all, a unified approach for controllability analysis for a general class of HCS will be presented. Then, an algebraic methods/criteria will be introduced for a class of linear switching systems. (3) Fundemental analysis of Fault Tolerant HCS (FT-HCS). The analysis of the reconfigurability of FT-HCS will be introduced based on the proposed controllability concept and methods. Elio Usai, Zeno phenomena in hybrid systems and sliding mode behaviours It is well known that in some cases hybrid systems can lead to the so called Zeno phenomenon, i.e., theoretically infinite frequency switching appears in the system behaviour. Such a behaviour can be considered as a mathematical modelling of a solution of a differential inclusion, which is common to appear in constrained mechanical systems. Since this behaviour has quite interesting properties of invariance and robustness, it can be useful to force, even artificially, infinite frequency switching in the control system; this is the usual working state of variable structure systems with sliding modes. In this lecture the origin of natural and artificial sliding modes are described, ant their main characteristics and features are discussed by means of simple examples. June 20, 2007 Andrea Paoli, Supervisory control of discrete event systems to be defined

9 Stephane Lafortune, Diagnosis of discrete event systems We are interested in the detection of ``significant'' events, such as faults, in technological systems whose dynamics are modeled in the framework of discrete event systems. In the first part of the talk, we will review the salient features of a methodology for fault diagnosis of discrete event sytems termed the ``Diagnoser Approach.'' This approach has been successfully used in several domains, incuding document processing systems and intelligent transportation systems. In the second part of the talk, we will present some recent extensions of this methodology regarding the ability to distribute the diagnosis function among a set of diagnoser modules in the case of large systems composed of several interconnected components. Stephane Lafortune, Decentralized control of discrete event systems This talk will present a critical overview of key results on the control of partially-observed distributed discrete event systems. Both decentralized and distributed control architectures will be considered. In decentralized architectures, a set of local supervisors work jointly to enforce a global specification on the controlled behavior; these supervisors have different information structures and do not communicate in real-time. The state-of-the art in decentralized control will be reviewed with focus on the role of inference and on the curse of undecidability. In ``distributed" architectures, the supervisors are allowed to exchange information in real-time, leading to what are called networked systems. In several classes of networked systems, energy, bandwidth, and/or security often require to minimize communications among supervisors. The intricacies of the synthesis of minimum communication policies will be illustrated. Recent results on this topic will be presented. July 17, 2007 Giorgio Bartolini, Simplex sliding mode control method for nonlinear multiinput uncertain systems The methodology known as Sliding Modes Control consists in the implementation of motions artificially constrained to surfaces in the state space. The constraints are chosen such that the reduced order free motion(zero dynamics) is characterized by good properties in particular stability, precise tracking and insesitivity to disturbances. The attained zerodynamics can be suitable to be furtherly dealt with by other synthesis methods. The attainment of the constrained motion must be achieved in finite time despite of uncertainties, disturbances and exogenous disturbances. This problem requires the solution of differential inequalities which unavoidably leads to discontinuous control laws. Considering multiinput nonlinear uncertain systems with vector-constraint equations a particular discontinuous control strategy,based on the properties of simplices of vectors, appears to be very promising. As the first step the constraint vector space is partioned in non overlapping regions with disjoint interiors constituted by the cones positively spanned by the vectors of the simplex except one at a time. At any cone it is associated a fixed vector of the simplex. The resulting variable stucture control system can be viewed also as a "switched control " or a "quantized control" system discontinuous on the boundaries of the cones and at the origin of the considered space. It has been proved, by Liapunov-like methods adapted to the theory of differential equation with discontinuous r.h.s., that any system belonging to a rather wide class can be forced to satisfy the chosen constraints in a finite time interval. This method will be further endowed with an antichattering procedure and used to deal with the control of systems in regular form non affine in the control law. Interesting application to the control of systems with monodirectional actuators will be also provided. Arie Levant, Homogeneous discontinuous control Homogeneity features of dynamic systems are found to provide for a number of general practically important features. In particular, asymptotic stability implies finite time convergence, if the system

10 homogeneity degree is negative; the asymptotic accuracy is calculated in a very general way in the presence of input noises, delays, discrete sampling and switching. A general uncertain single-input-single-output regulation problem is shown to be only solvable by means of discontinuous control via the so-called high-order sliding mode. The homogeneity approach allows to facilitate design and investigation of new high-order sliding-mode controllers, featuring such attractive properties as practical continuity of the control in the presence of noises. Robust output-feedback controllers are produced, provided robust exact homogeneous differentiators are used. The asymptotic accuracy of the controllers is shown to be the best possible under given circumstances. Simulation results are presented demonstrating the feasibility of the approach. Alessandro Pisano, Second-order sliding modes in mechanical and electromechanical systems Basic principles and implementation results This lecture is devoted to address some practical control problems involving mechanical and eletromechanical systems operating under uncertainty conditions. In particular, the effective application of second-order sliding mode control (2-SMC) techniques is demonstrated. The main theoretical tools involved will be introduced first, namely: i. The "Dynamical 2-SMC", an approach providing superior robustness performance while avoiding discontinuities in the generalized control forces. ii. The "Observer-based" and "Parallel-filter" 2-SMC, two approaches to deal with systems with high relative degree undergoing partial state-feedback. iii. The "Frequency-shaped 2-SMC", a design tool which can effectively account for the presence of parasitic actuator dynamics. Each topic will be discussed making reference to practical examples involving: hydrojet-actuated marine vessels, electrohydraulic "double stage" valves, robotic master/slave haptic interfaces, active pantographs for high-speed train transportation systems. Experimental results are given, and the most relevant implementation issues emerged in the experimental activities will be addressed in some detail.