UNIVERSITY OF MICHIGAN DEPARTMENT OF ELECTRICAL ENGINEERING AND COMPUTER SCIENCE LECTURE NOTES FOR EECS 661 CHAPTER 1: INTRODUCTION TO DISCRETE EVENT SYSTEMS Stéphane Lafortune August 2006
References for Chapter 1: Textbook, Chapter 1: Section 1.3 A Multidisciplinary Area: Discrete Event Systems Operations Research Computer Science DES Systems & Control What: Discrete State Space (logical, symbolic variables) Event-driven Dynamics Why: Technological Systems, Computer Control Large, Complex Systems: they need to be analyzed, diagnosed, controlled, and optimized S. Lafortune - Last revision: August 2006 1
Where: Inherently Discrete Systems: computer systems, communication networks, automated manufacturing systems (cell and factory levels), software systems. Systems with Continuous and Discrete Variables (hybrid systems), modeled as DES at a certain level of abstraction, e.g., for the higher level control logic: automated manufacturing systems (machine and cell levels), process control, transportation systems. Embedded systems ; networked systems. How: Mathematical Modeling, Analysis, Verification, Diagnosis, Controller Design, Optimization, Simulation S. Lafortune - Last revision: August 2006 2
SUPERVISORY CONTROLLER COORDINATION REAL-TIME CONTROL DIAGNOSTICS FAILURE RECOVERY Conceptual Control System Architecture: Commands INTERFACE Observable events EQUIPMENT CONTROLLERS CONTROLLER SYSTEM S. Lafortune - Last revision: August 2006 3
Some Examples The Heating System of a Heating, Ventilation, and Air Conditioning (HVAC) Unit FAN HTG. COIL VALVE PUMP BOILER CONTROLLER The operation of the unit is monitored by a set of sensors. The issue of interest: Fault Diagnosis. Specifically: diagnose occurrence of sharp faults during the on-line operation of the unit. Examples of faults: stuck failures of valves, on-off failures of pumps, controllers, sensors, etc. Implementation: diagnostics module in the control logic. S. Lafortune - Last revision: August 2006 4
PUMP PON POFF P1 P2 PON VALVE V3 CV, OV POFF SO1 SO2 OV LOAD CV V1 V2 OV FOFF L0 FOFF CV SC1 SC2 S P I S P D S P I V4 L1 L2 CV, OV S P D Models of the Components of the HVAC System: FOFF F1 FON FOFF FAN CFOFF F2 FON FON C21 FOFF C22 FOFF BON B1 BOFF BOFF BOILER SPD C23 SPI SPD SPI C24 B2 BON FON S P I OV PON BON C1 C2 C3 C4 C5 C6 FOFF S P D S P I S P D CFON C10 BOFF C9 POFF C8 CV C7 C11 FON C12 SPI SPD C13 OV C14 PON C15 BON C16 SPD SPI C20 OV C19 PON C18 BON C17 CONTROLLER S. Lafortune - Last revision: August 2006 5
1 N < FON, NF > Part of the Diagnoser for the Heating System (HVAC Unit) F1: SO F2: SC F3: CFON F4: CFOFF 7 N 8 F1 9 F2 37 F3 38 F1 F3 39 F2 F3 85 F4 86 F1 F4 87 F2 F4 10 N 11 F1 12 F2 40 F3 41 F1 F3 42 F2 F3 A < SPI, NF > 13 N 14 F1 43 F3 44 F1 F3 16 N 17 F1 46 F3 47 F1 F3 19 N 20 F1 58 F3 59 F1 F3 22 N 25 N 27 F2 < OV, NF > < PON, F > < BON, F > < SPD, F > < CV, N F > < POFF, NF > < BOFF, NF > 28 N 29 F1 30 F2 < SPI, NF > < FON, NF > B < FOFF, NF > < BON, F > 4 N 5 F1 6 F2 34 F3 35 F1 F3 36 F2 F3 82 F4 83 F1 F4 84 F2 F4 < SPD, NF > 28 N 29 F1 30 F2 49 F3 50 F1 F3 51 F2 F3 88 F4 89 F1 F4 90 F2 F4 < OV, NF > 52 F3 53 F1 F3 54 F2 F3 < PON, F > < BON, F > < SPI, F > < OV, F > < PON, F > < BON, F > C < SPD, F > 55 F3 56 F1 F3 67 F3 68 F1 F3 70 F3 71 F1 F3 73 F3 74 F1 F3 76 F3 77 F1 F3 46 F3 47 F1 F3 E < FON, NF > 1 N 2 F1 3 F2 79 F4 80 F1 F4 81 F2 F4 < FOFF, NF > < PON, NF > 57 F2 F3 69 F2 F3 72 F2 F3 75 F2 F3 < BON, NF > < SPI, NF > < OV, NF > < PON, NF > < BON, NF > 78 F2 F3 48 F2 F3 < BON, NF > < SPD, NF > D 7 N 8 F1 9 F2 < OV, NF > 1 N 2 F1 3 F2 58 F3 59 F1 F3 < OV, F > 60 F2 F3 < OV, NF > 10 N 11 F1 12 F2 < PON, F > 61 F3 62 F1 F3 < PON, F > 63 F2 F3 < PON, NF > 13 N 14 F1 < BON, F > 64 F3 65 F1 F3 66 F2 F3 16 N 17 F1 < SPD, F > 19 N 20 F1 < CV, N F > S. Lafortune - Last revision: August 2006 6
A Small Telephone System 1 2 1 2 0 0 The network has screening, forwarding, and multi-way calling capabilities. The issue of interest: Feature Interactions. Specifically: detection and resolution of logical conflicts (interactions) between options (features). Implementation: correct design of the (modular) software programs that run at the switches. S. Lafortune - Last revision: August 2006 7
Model of User 0 in a Telephone System: Model of User 1 at Switch 0 in Telephone System: req00 FWD_TO_0 FWD_TO_1 req01 FWD_TO_2 req02 REQ fwd001 fwd002 fwd020 fwd021 nocon00 req00 fwd010 con01 nocon01 onh0 req01 fwd012 REQ_0 REQ_1 REQ_2 nocon0 nocon0 CON INIT offh0 con02 nocon02 fh0 dfh0 req02 nocon0 con10 nocon10 fwd101 fwd102 NOT_REQ onh1 INIT req10 offh1 fh1 dfh1 S. Lafortune - Last revision: August 2006 8
G C 0 G 0 A Control Architecture for Approaching this Problem:... S TCS... S OCS...... S POTS-4 S. Lafortune - Last revision: August 2006 9
Other examples: Railway Connections and Time Tables 1 The network of railway connections is closed and each line has a fixed number of trains. The inter-station travel times are known and deterministic. The objective is to design satisfactory time tables for the trains. Specifications include: certain trains have to wait for one another to allow change overs. Constraints: want system to operate fast, but also want perturbations to completely disappear in finite time. Issues of interest: how do perturbations to the time table propagate, what limits the minimum operation time, where would it be helpful to add trains, etc. Approach: write equations for the departure times of the trains, using maximum and addition. 1 Example due to G. J. Olsder S. Lafortune - Last revision: August 2006 10
Dispatching Control in an Elevator System 2 Events: hall call, car call, car arrives at floor i, etc. States: position of car k, number of passengers waiting at floor i, etc. (very large state space!) Control problem: which car to send where so as to achieve satisfactory performance? Performance measures: average waiting time (until car comes), average service time (until car delivers to desired floor), fraction of passengers waiting more (on average) than one minute, etc. Probabilistic formulation: passenger arrival rates at floors, probability distribution for destination floors, load times and travel times, etc. Common solution: threshold-based control, i.e., hold a car until a threshold is reached. The issue is then to determine this threshold and automatically adjust it in real-time, based on observed passenger arrival rates. 2 Example due to C. Cassandras S. Lafortune - Last revision: August 2006 11
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The Three Levels of Abstraction in Modeling DES Sample Paths of Discrete Event Systems x(t) x6 x5 x4 x3 x2 x1 t1 t2 t3 t4 t5 t6 t7 t e1 e2 e3 e4 e5 e6 e7 e Describe this sample path by the timed sequence of events that it contains: s t e = (e 1, t 1 )(e 2, t 2 )(e 3, t 3 )(e 4, t 4 )(e 5, t 5 )(e 6, t 6 )(e 7, t 7 ) S. Lafortune - Last revision: August 2006 14
The behavior of a given DES is described as follows: Timed Language: set of all timed sequences of events that the DES can generate/execute Stochastic Timed Language: a timed language with a probability distribution function defined over it Language: a timed language where the timing information has been deleted, i.e., it is a set of sequences, or traces, of events. Formal language theory: Finite set of events E : {e 1, e 2,..., e n } s e = e 1 e 2 e 3 e 4 e 5 e 6 e 7 Set of all finite strings of event in E: E - Kleene-closure A language L is a subset of E : L E S. Lafortune - Last revision: August 2006 15
This leads to the three complemetary levels of abstraction at which DES are studied. Logical level: the language model is used to study properties that concern event ordering only; e.g., consider the telephone system example, as well as the HVAC unit example (diagnosis). Priorities, mutual exclusion, deadlock, livelock, occurrence of unobservable events, etc. Temporal level: the timed language model is used to study properties that concern the timing of the events; e.g., consider the railway network example. Deadlines, cycle times, effect of perturbations, etc. Stochastic level: the stochastic timed language model is used to study properties that concern the expected behavior of the system under the given statistical information; e.g., consider the elevator example. Average delay, throughput, and other relevant performance measures. N.B.: Discrete Event Simulation usually refers to the stochastic level. Question: How to represent [(stochastic) timed] languages? S. Lafortune - Last revision: August 2006 16
Discrete Event Modeling Formalisms Formal classes of models that represent [(stochastic) timed] languages State-based formalisms: define a state space and specify the state transition structure (i.e., (out state, event, in state) triples) that represents the language. Automata (or State Machines) and Petri Nets are widely used. Trace-based formalisms: use (recursive) algebraic equations on the events to represent the traces in the language (i.e., no explicit state ). Often referred to as Process Algebras. Communicating Sequential Processes (CSP) is a well-know formalism in this category. We will study: (untimed and timed) automata [modeling, analysis, diagnosis, supervisory control] (untimed and timed) Petri nets [modeling, analysis, some control] timed event graphs, a special case of timed Petri nets [analysis using max-plus algebra] We illustrate the above modeling formalisms for the (familiar) example of the dining philosophers. S. Lafortune - Last revision: August 2006 17
Automata models of two philosophers (P1, P2) and two forks (F1, F2) P1 1T 1f1 1I1 1f 1f2 1E P2 2T 2f1 2I1 2f 2f2 2E 1f2 1I2 1f1 2f2 2I2 2f1 F1 1f1,2f1 F2 1f2,2f2 1A 1U 2A 2U 1f,2f 1f,2f S. Lafortune - Last revision: August 2006 18
Composition of the four automata: P 1 P 2 F 1 F 2 1f (1T,2T,1A,2A) 1f1 1f2 1f1 2f1 1f2 1f2 2f1 (1E,2T,1U,2U) (1I2,2I1,1U,2U) 2f 2f2 2f2 1f1 2f2 (1I1,2I2,1U,2U) 2f1 (1T,2E,1U,2U) S. Lafortune - Last revision: August 2006 19
Petri net model of one philosopher and two forks if2 if1 fork 2 available holding fork 2 if eating thinking if1 if2 holding fork 1 fork 1 available S. Lafortune - Last revision: August 2006 20
Petri net model of two philosophers and two forks fork 2 available 1f2 1f1 2f1 2f2 1f 2f 1f1 1f2 2f2 2f1 fork 1 available philosopher 1 philosopher 2 S. Lafortune - Last revision: August 2006 21
Recursive equation model of two philosophers and two forks P1 = (1f1 1f2 E1 1f2 1f1 E1) E1 = (1f P1) P2 = (2f1 2f2 E2 2f2 2f1 E2) E2 = (2f P2) F1 = (1f1 1f F1 2f1 2f F1) F2 = (1f2 1f F2 2f2 2f F2) SY STEM = P1 P2 F1 F2 In general, we get a set of equations of the form: X = f(x) Y = g(x) where X is a vector of processes and f must contain. S. Lafortune - Last revision: August 2006 22
How to Compare Modeling Formalisms? Descriptive Power: Language complexity or class of languages that a (finite) model can represent. Finite-state automata: Regular Languages R Labeled Petri Nets: PN L R. Algebraic Structure: Formal operations that permit to build complex systems by interconnecting simple systems and that allow to manipulate a model for analysis and synthesis purposes. R has nice properties: closed under union, concatenation, intersection, parallel composition, complementation w.r.t. E. These operations can be implemented using finite-state automata. PN L does not enjoy such nice properties. However, Petri nets have intrinsically modular structure: e.g., system decomposition by means of place-bordered Petri nets. S. Lafortune - Last revision: August 2006 23