Index A abstraction, 14 alarm pattern, 309 algorithm controllability language, 53 alphabet, 24 approximation l-complete, 14 arc of a Petri net, 192 assembly, 198 atomic firing of a transition, 323 automaton ε-automaton, 26 canonical, 30 composition of automata, 97 deterministic, 25 language, 85 marked, 101 minimal, 30 nondeterministic, 26 path, 85 probabilistic, 89 projected, 67 pushdown, 34 real-time, 34 state-partition, 70 trajectory, 85 trimmed, 98 weighted, 89 B basis marking, 283 285 basis reachability graph, 289 behaviour, 8 generated by infinite state machine, 13 generated by Mealy automaton, 8 boundedness, 206 structural, 222, 223 branching process, 306 C characteristic system, 352 choice, 198 Church-Turing-Rosser Thesis, 37 clock, 169 closed-loop system behavior, 51 co-set, 306 cut, 307 communication event-based synchronous protocol, 142 state-based synchronous protocol, 132 articulate, 129 feasible protocol, 135 concatenation, 24 concurrency, 197, 303, 305 co-set, 306 concurrent, 306 concurrent composition of Petri nets, 241 242 concurrent enabling, 324 condition, 305
472 Index conditional controllability, 153 conditional decomposability, 149 conditional normality, 161 conditional observability, 161 cone, 304 configuration, 307 of event structure, 307 prime, 304, 307 conflict, 196, 198, 306 behavioral, 196 self-conflict, 305 structural, 196 conflict relation, 305 conservativeness structural, 223 conservativity strict, 205 consistency, 224 consistent firing sequence, 281 marking, 281, 282 reachable marking, 281 continuous Petri net lim-live, 370 lim-reachability, 369 attribution, 391, 397 bounded, 370, 371 choice free, 369 consistent marking, 402 continuization, 365 deadlock, 370 diagnoser, 403 diagnosis state, 403, 404 equal conflict, 369 fluidization, 365 implicit place, 372 initial marking, 365 join free, 369 marked graph, 412 marking, 368 proportional equal conflict, 369 reachability, 369 reachability set, 371 rendez-vous, 367 siphon, 371 spurious discrete solution, 371 spurious marking, 367 spurious solution, 371 state equation, 371 state estimation, 402 strongly connected p-component, 395 structurally bounded, 370, 371 structurally live, 371 structurally persistent, 376 synchronic relation, 367 synchronic theory, 367 trap, 372 control architectures coordination control, 111 distributed control, 110 distributed control with communication, 110 hierarchical control, 111 control objectives fairness, 112 non starvation, 112 control of RAS, 272 D-process-place, 274, 275 D-resource-place, 274, 275 integer linear programming problem, 272 iterative control policy, 272, 273 synthesis of live S 4 PR net systems, 275 thieves of a siphon, 273 virtual resource, 257, 273, 274 control synthesis global, 121 modular, 121 controllability of a Petri net, 244, 249 251 coobservability, 114 C & P, 114 conditional, 129 unconditional, 129 coordinator, 149 cut, 307 D deadlock, 210 decidability, 37 derivation tree, 33 deterministic Petri net, 321 deterministic timed Petri net, 326 determinization, 87 of probabilistic automaton, 91 diagnosability, 94 A-diagnosability, 96 AA-diagnosability, 96
Index 473 for discrete event systems, 313 probabilistic, 95 weak F-, 314 N-, 314 diagnosabiliy weak, 313 diagnosabilty of Petri nets, 294 diagnoser, 86 probabilistic, 93 diagnosis, 86, 348 using fluid Petri nets, 293 using Petri nets, 286 diagnosis set, 310 diagnosis state, 286 diagnosis using Petri nets decentralized, 295 dioid, 437, 453 complete dioid, 437 max-plus algebra, 437 min-plus algebra, 437 minmaxgd, 443 disassembly, 198 discrete event, 3 discrete event system decentralized, 110 distributed, 110 discrete-event system, 109 distributed system, 97 E e-vector, 282 minimal, 282 enabled, 304 epsilon-reduction, 87 probabilistic, 90 event, 305, 306 event graph, 433 event set, 24 event sets agree on controllability, 121 agree on observability, 121 event structure, 306 configuration, 307 prefix, 307 example alternating bit protocol, 108 underwater vehicles, 108 explanation, 282, 309, 310 minimal, 282, 284 extension, 306 F factor, 24 fault class, 286 fault transition, 286 finite causes, 306 firing sequence, 195, 196, 224 empty, 195 repetitive, 206 increasing, 207 stationary, 207, 217, 224 firing time interval, 345 firing time of a transition, 320 firing vector, 214 flow relation, 304 fluid Petri net, see continuous Petri net function computable, 37 constructible, 39 G generator controlled deterministic finite, 49 deterministic, 11 deterministic finite, 47 nondeterministic, 12 generators conditionally independent, 149 grammar, 31 context-free, 32 context-sensitive, 32 left-linear, 35 linear, 32 right-linear, 35 type 0, 31 type 1, 32 type 2, 32 graph basis reachability, 289 coverability, 199 marked, 214, 230 reachability, 199
474 Index H high-throughput screening, 445 home marking, 249 home space, 249 homomorphism, 24 I idempotent semiring, 437 implicit place, 225 infimum, 58 infix, 24 interleaving, 307 J j-vector, 283 285 justification, 284 K K-diagnosability, 294 Kleene star, 452, 455, 456 L L-observer, 155 labeling function, 280 labeling function of Petri nets, 236 language, 8, 24, 85 L-closed, 49 L-marked, 49 accepted, 26 controllable, 53 decidable, 36, 37 decomposable, 116 generated by Mealy automaton, 8, 9 marked by generator, 11 marked by Mealy automaton, 8, 9 observable, 85 of a marked net, 196 of Petri nets, see Petri net languages product of languages, 101 recognizable, 26 recursive, 36 recursively enumerable, 36 regular, 25 stopped, 90 strongly decomposable, 116 visible, 85 weighted, 89 lattice, 58 complete, 58 complete lattice, 437 sub-semilattice, 437 letters, 24 linear set, 249 liveness, 209, 210 structural, 224 liveness in RAS, 265, 267 characterization of liveness in S 3 PR, 265 in S 4 PR, 266 in S 5 PR, 266 empty siphon, 265 insufficiently marked siphon, 266 m-process-enabled transition, 266 m-resource-disabled transition, 266 M marked graph strongly connected timed, 329 timed, 329 marked net, 194 marking, 194 X invariant, 221 blocking, 244 controllable/uncontrollable, 245 dead, 210 enabling a transition, 194 final, 236 initial, 194 potentially reachable, 215 reachable, 196, 304 spurious, 215 matrix of incidence, 214 216 post, 192 pre, 192 max-plus algebra, 437 γ-transform, 442 eigenproblem, 439 eigenvalue, 439 eigenvector, 439 max-plus linear system, 438
Index 475 Mealy automaton deterministic, 6 nondeterministic, 8 memory policy, 325 min-plus algebra, 437 δ-transform, 443 MinMaxGD, 443, 461 mirror image, 24 modular DES modularly controllable, 121 modular system, 97 monoid, 437 monolithic supervisory design, 243 Moore automaton deterministic, 10 nondeterministic, 10 morphism, 24 mutual exclusion, 198 N net, 304 homomorphism, 304 marking, 304 occurrence, 305 occurrence net, 305 ordinary, 304 Petri net, 304 verifier, 313 non-anticipation, 6 strict, 10 nonblockingness of a Petri net, 244, 249 normality, 77 O observability, 72 observer, 69, 86 distributed, 98 probabilistic, 90 observer property, 155 occurrence net, 305 co-set, 306 output control consistency, 156 P P-decreasing, 216, 222, 224 P-increasing, 216, 220 P-invariant, 216, 218, 220, 221, 223 parallelism, 197 partial observation, 67 periodic series, 457 Petri net, 191, 304 acyclic, 229 binary, 205 bounded, 205 conservative, 206 dead, 209 deadlocking, 210 deterministic, 238 free-choice, 231 generator, 236 labeled, 236 languages, 236 241 live, 209 not quasi-live, 209 ordinary, 228 place/transition, 192 pure, 194, 228 quasi-live, 209 repetitive, 206 restricted, 228 reversible, 208 safe, 205, 303, 304 strictly conservative, 205 structurally bounded, 222 structurally live, 224 structure, 192 194 system, 194 time/timed, see time/timed Petri net token, 304 unfolding, 306 philosophers problem, 262 place, 304 implicit, 225 of a place/transition net, 192 structurally implicit, 225 poset, 58 postset, 304 powers, 24 prefix, 24 of event structure, 307 preset, 304 prime configuration, 307 prime event structure, 306
476 Index problem complete, 40 existence supervisory control, 52 halting problem, 37 Post s correspondence, 38 supremal supervisory control complete observations, 61 procedure design supervisor, 56 process branching, 306 product labeled, 308 synchronous, 308 projection, 24 properties of RAS, 265, 269 deadlock-freeness, 267 directedness, 268 livelocks, 267 liveness monotonicity, 267 reversibility, 268, 269 t-semiflow realizability, 268, 269 pruning, 309 pullback, 308 pumping lemma, 29, 33 for Petri nets, 252 Q quantisation, 3 quotient, 24 R RAP, see resource allocation problem RAS, see resource allocation system RAS models, 258 abstraction methodology, 258, 259, 261 acceptable initial marking, 260 idle place, 259, 262 iterative state machine, 261 L-S 3 PR, 264 modularity, 259, 261 PC 2 R, 261, 262 potentially acceptable initial marking, 263 process place, 259, 262 resource lending, 262 resource place, 259, 262 S 3 PR, 258, 264 S 4 PR, 259 S 5 PR, 264 reachability of a marking, 214, 221, 229 reachability set, 196 reachable, 304 reduction, 40 refinement, 14 region, 180 regular expressions, 25 relation observational indistinguishability, 69 partial order, 58 tolerance, 69 repetitiveness, 206, 207 structural, 224 reserved marking, 323 residuation theory, 452 resource allocation problem, 257 resource allocation system, 257, 258 classes, see RAS models control, see control of RAS liveness analysis, see liveness in RAS properties, see properties of RAS reversal, 24 reversibility, 207, 209 run of event structure, 307 S self-loop, 193, 228 sequentiality, 197 server semantics, 324 set X invariant, 221 controllable sublanguages, 59 partially-ordered sublanguages, 58 potentially reachable, 215, 221 prefix-closed controllable sublanguages, 59 reachability, 221 signal, 3 continuous-valued, 3 discrete-valued, 3 quantised, 3 signal aggregation, 4
Index 477 siphon, 226 minimal, 226 state class graph, 351 state estimation, 85 distributed, 98, 101 state machine, 214, 229 state space explosion, 301 stationarity, 207 stochastic Petri net, 321 stochastic timed Petri net, 332, 334 string, 24 sub-word, 24 subautomaton, 81 substitution, 24 suffix, 24 supervisory control complete observations, 50 nonblocking, 51 of Petri nets, 235, 242 254 supremum, 58 symbols, 24 synchronization, 197 synchronization graph, 433 synchronous product, 69, 97 system l-complete, 15 coordinated, 150 hybrid, 13 time invariant, 13 T T-decreasing, 216, 217 T-increasing, 216, 217 T-invariant, 216, 217 TCPN, see timed continuous Petri net time configuration, 355 time Petri net, 321, 338 340 time-trace, 346 timed automaton, 172 deterministic, 184 timed continuous Petri net, 368, 373, 376 adding noise, 382 approximation under ISS, 376 bounded input controllable, 410 chaotic model, 373 configuration, 374, 375, 387, 393, 400, 409, 413 configuration matrix, 374 control action, 409 control methods, 414, 419 controllability, 409, 410 controllability with uncontrollable transitions, 412 distinguishable configuration, 393 distributed control, 422 finite firing semantics, 372 firing semantics, 372 fluidization, 372 generic observability, 398 homothetic properties, 374 incidence matrix, 383 infinite firing semantics, 373 model predictive control, 415 mono-t-semiflow, 373, 375, 422 monotonicity, 375 observability, 391, 394 396, 404 observability matrix, 391, 392 observers, 399 on-off control, 418 product semantics, 373 reachability set, 374 redundant configuration, 387, 388 redundant region, 389 region, 374 steady state, 373, 374 stochastic T-timed Petri net, 366 structural observability, 397 structurally timed-live, 376 terminal strongly connected p-component, 397 throughput, 374, 408 timed continuous Petri net, 373 timed reachability, 404 timed-deadlock-free, 376 timed event graph, 433 timed marked graph, 329 timed Petri net, 321, 323 337 timed transition system, 171 timing structure of a Petri net, 320 token, 194, 304 trace, 303 Mazurkiewicz, 307 transition, 304 dead, 209 deterministic, 322 enabled, 194
478 Index fault, 286 firing, 195 immediate, 322 live, 209 observable, 280 of a place/transition net, 192 quasi-live, 209 silent, 280 source, 195 stochastic, 322 unobservable, 280 transition of a Petri net controllable/uncontrollable, 243 trap, 226 minimal, 226 trimming of a Petri net, 244, 246 248, 251 253 tuple of supervisors maximal solution, 118 Nash equilibrium, 118 strong Nash equilibrium, 118 Turing machine, 35 deterministic, 35 universal, 37 twin machine, 95 twin plant, 95 U unfolding, 306 V verifier net, 295 W word, 24 empty, 24 length, 24 Z zone, 177