626:. One also notices from basic convex analysis that the complementarity relation can equivalently be rewritten as the inclusion into a normal cone, so that the bouncing ball dynamics is a differential inclusion into a normal cone to a convex set. See Chapters 1, 2 and 3 in Acary-Brogliato's book cited below (Springer LNACM 35, 2008). See also the other references on non-smooth mechanics.
141:, a physical system with impact. Here, the ball (thought of as a point-mass) is dropped from an initial height and bounces off the ground, dissipating its energy with each bounce. The ball exhibits continuous dynamics between each bounce; however, as the ball impacts the ground, its velocity undergoes a discrete change modeled after an
605:
Such a contact model does not incorporate magnetic forces, nor gluing effects. When the complementarity relations are in, one can continue to integrate the system after the impacts have accumulated and vanished: the equilibrium of the system is well-defined as the static equilibrium of the ball on
653:
impossible. Instead, the tools are analyzed for their capabilities on benchmark problems. A possible theoretical characterization of this is algorithms that succeed with hybrid systems verification in all robust cases implying that many problems for hybrid systems, while undecidable, are at least
559:
It is noteworthy that the dynamical model is complete if and only if one adds the contact force between the ground and the ball. Indeed, without forces, one cannot properly define the bouncing ball and the model is, from a mechanical point of view, meaningless. The simplest contact model that
693:
models. These methods generate traces of system behaviors in discrete event system manner which are different from discrete time systems. Detailed of this approach can be found in references and the software tool
637:
There are approaches to automatically proving properties of hybrid systems (e.g., some of the tools mentioned below). Common techniques for proving safety of hybrid systems are computation of reachable sets,
70:, or of electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena.
487:
331:
603:
560:
represents the interactions between the ball and the ground, is the complementarity relation between the force and the distance (the gap) between the ball and the ground. This is written as
251:
405:
519:
889:
940:
939:
Alur, Rajeev; Courcoubetis, Costas; Halbwachs, Nicolas; Henzinger, Thomas A.; Ho, Pei-Hsin; Nicollin, Xavier; Olivero, Alfredo; Sifakis, Joseph; Yovine, Sergio (1995),
624:
556:
amount of time. In this example, each time the ball bounces it loses energy, making the subsequent jumps (impacts with the ground) closer and closer together in time.
539:
521:
is a dissipation factor. This is saying that when the height of the ball is zero (it has impacted the ground), its velocity is reversed and decreased by a factor of
197:
170:
351:
925:
840:
62:). Often, the term "hybrid dynamical system" is used instead of "hybrid system", to distinguish from other usages of "hybrid system", such as the combination
1199:
Thomas A. Henzinger, Peter W. Kopke, Anuj Puri, and Pravin
Varaiya: What's Decidable about Hybrid Automata, Journal of Computer and System Sciences, 1998
1217:
Stefan
Ratschan: Safety verification of non-linear hybrid systems is quasi-decidable, Formal Methods in System Design, volume 44, pp. 71-90, 2014,
662:
Two basic hybrid system modeling approaches can be classified, an implicit and an explicit one. The explicit approach is often represented by a
353:
is the acceleration due to gravity. These equations state that when the ball is above ground, it is being drawn to the ground by gravity.
970:
904:
1179:
1079:
1023:
410:
256:
1099:"Dynamical systems coupled with monotone set-valued operators: Formalisms, Applications, well-posedness, and stability"
548:
behavior. Zeno behavior has a strict mathematical definition, but can be described informally as the system making an
712:
563:
931:
1208:
Martin Fränzle: Analysis of Hybrid
Systems: An ounce of realism can save an infinity of states, Springer LNCS 1683
1245:
675:
17:
205:
122:
359:
1250:
874:
639:
492:
869:
811:
1054:
1255:
1240:
826:: MATLAB toolbox for verification of hybrid systems with respect to temporal logic specifications
755:: A MATLAB Toolbox for reachability analysis of cyber-physical systems, including hybrid systems
1049:
609:
884:
879:
524:
43:
732:(Discrete Event System) modeling and simulation oriented to the simulation of hybrid systems
1098:
1041:
907:(PDMP), an example of a (stochastic) hybrid system and a generalization of the jump process
864:
175:
148:
51:
978:
Goebel, Rafal; Sanfelice, Ricardo G.; Teel, Andrew R. (2009), "Hybrid dynamical systems",
674:. The implicit approach is often represented by guarded equations to result in systems of
38:
that exhibits both continuous and discrete dynamic behavior – a system that can both
8:
643:
631:
142:
59:
1045:
764:
685:
As a unified simulation approach for hybrid system analysis, there is a method based on
1121:
995:
823:
336:
86:
1158:
Branicky, Michael S. (2005), Hristu-Varsakelis, Dimitrios; Levine, William S. (eds.),
795:
770:
1175:
1125:
1075:
1019:
956:
679:
999:
773:: C++ library for state set representations for hybrid systems reachability analysis
749:
library for (numerically rigorous) reachability analysis of nonlinear hybrid systems
689:
formalism in which integrators for differential equations are quantized into atomic
667:
1218:
1167:
1113:
1059:
1011:
987:
960:
952:
859:
663:
35:
817:
199:
be the velocity of the ball. A hybrid system describing the ball is as follows:
114:
113:
Hybrid systems have been used to model several cyber-physical systems, including
55:
910:
1222:
1063:
1015:
1234:
1090:
Building
Software for Simulation: Theory, Algorithms, and Applications in C++
913:, an example of a (stochastic) hybrid system and a generalization of the PDMP
544:
The bouncing ball is an especially interesting hybrid system, as it exhibits
138:
1171:
991:
894:
807:
545:
1159:
836:
789:
898:
606:
the ground, under the action of gravity compensated by the contact force
67:
1138:
678:(DAEs) where the active equations may change, for example by means of a
1117:
63:
27:
Dynamical system that exhibits continuous and discrete dynamic behavior
1069:
1010:, Lecture Notes in Applied and Computational Mechanics, vol. 35,
901:), an example of a (stochastic) hybrid system with zero flow component
541:. Effectively, this describes the nature of the inelastic collision.
965:
725:
695:
671:
650:
649:
Most verification tasks are undecidable, making general verification
720:
126:
938:
801:
1032:
Kofman, E (2004), "Discrete Event
Simulation of Hybrid Systems",
752:
145:. A mathematical description of the bouncing ball follows. Let
776:
930:, IEEE Computer Society Press, pp. 278–292, archived from
716:
924:
Henzinger, Thomas A. (1996), "The Theory of Hybrid
Automata",
890:
Behavior trees (artificial intelligence, robotics and control)
767:: A tool for overapproximating reachability of hybrid automata
761:: A tool for reachability analysis of nonlinear hybrid systems
746:
742:
97:
hold, while discrete transitions can occur as soon as given
729:
690:
686:
101:
are satisfied. Discrete transitions may be associated with
846:
927:
11th Annual
Symposium on Logic in Computer Science (LICS)
843:
of correct-by-construction controllers for hybrid systems
783:
85:. The state changes either continuously, according to a
482:{\displaystyle x_{1}^{+}=x_{1},x_{2}^{+}=-\gamma x_{2}}
758:
326:{\displaystyle {\dot {x}}_{1}=x_{2},{\dot {x}}_{2}=-g}
612:
566:
527:
495:
413:
362:
339:
259:
208:
178:
151:
93:. Continuous flow is permitted as long as so-called
977:
77:of a hybrid system is defined by the values of the
1164:Handbook of Networked and Embedded Control Systems
618:
597:
533:
513:
481:
399:
345:
325:
245:
191:
164:
1008:Numerical Methods for Nonsmooth Dynamical Systems
1232:
1096:
598:{\displaystyle 0\leq \lambda \perp x_{1}\geq 0.}
1070:Francois E. Cellier and Ernesto Kofman (2006),
1005:
629:
137:A canonical example of a hybrid system is the
1087:
941:"The algorithmic analysis of hybrid systems"
657:
394:
375:
240:
221:
1166:, Boston, MA: Birkhäuser, pp. 91–116,
1097:Brogliato, Bernard; Tanwani, Aneel (2020),
1006:Acary, Vincent; Brogliato, Bernard (2008),
1053:
964:
923:
1157:
89:condition, or discretely according to a
14:
1233:
1031:
905:Piecewise-deterministic Markov process
820:: Polyhedral hybrid automaton verifier
804:: Verification tool for hybrid systems
779:: A toolbox for set-based reachability
784:Temporal Logic and Other Verification
246:{\displaystyle x\in C=\{x_{1}>0\}}
1139:IEEE CSS Committee on Hybrid Systems
1034:SIAM Journal on Scientific Computing
24:
917:
798:: Model checker for hybrid systems
792:: Nonlinear hybrid system verifier
400:{\displaystyle x\in D=\{x_{1}=0\}}
25:
1267:
1132:
701:
514:{\displaystyle 0<\gamma <1}
1160:"Introduction to Hybrid Systems"
676:differential algebraic equations
132:
736:
1211:
1202:
1193:
1151:
172:be the height of the ball and
13:
1:
1144:
980:IEEE Control Systems Magazine
706:
1074:(first ed.), Springer,
1072:Continuous System Simulation
957:10.1016/0304-3975(94)00202-T
945:Theoretical Computer Science
7:
853:
728:: General-purpose tool for
715:: Hybrid system solver for
108:
10:
1272:
875:Variable structure control
1223:10.1007/s10703-013-0196-2
1064:10.1137/S1064827502418379
1016:10.1007/978-3-540-75392-6
870:Variable structure system
658:Other modeling approaches
830:
619:{\displaystyle \lambda }
407:, jumps are governed by
1172:10.1007/0-8176-4404-0_5
1092:(first ed.), Wiley
992:10.1109/MCS.2008.931718
534:{\displaystyle \gamma }
1246:Differential equations
849:: State-space explorer
640:abstraction refinement
620:
599:
535:
515:
483:
401:
347:
327:
253:, flow is governed by
247:
193:
166:
1088:James Nutaro (2010),
885:Cyber-physical system
880:Joint spectral radius
621:
600:
552:number of jumps in a
536:
516:
484:
402:
348:
328:
248:
194:
192:{\displaystyle x_{2}}
167:
165:{\displaystyle x_{1}}
44:differential equation
865:Sliding mode control
644:barrier certificates
610:
564:
525:
493:
411:
360:
337:
257:
206:
176:
149:
79:continuous variables
1046:2004SJSC...25.1771K
897:(in the context of
459:
428:
143:inelastic collision
60:difference equation
1118:10.1137/18M1234795
814:for hybrid systems
616:
595:
531:
511:
479:
445:
414:
397:
343:
323:
243:
189:
162:
1251:Dynamical systems
1181:978-0-8176-4404-8
1081:978-0-387-26102-7
1025:978-3-540-75391-9
680:hybrid bond graph
654:quasi-decidable.
346:{\displaystyle g}
305:
270:
16:(Redirected from
1263:
1225:
1215:
1209:
1206:
1200:
1197:
1191:
1190:
1189:
1188:
1155:
1128:
1103:
1093:
1084:
1066:
1057:
1040:(5): 1771–1797,
1028:
1002:
974:
969:, archived from
968:
935:
860:Hybrid automaton
664:hybrid automaton
625:
623:
622:
617:
604:
602:
601:
596:
588:
587:
540:
538:
537:
532:
520:
518:
517:
512:
488:
486:
485:
480:
478:
477:
458:
453:
441:
440:
427:
422:
406:
404:
403:
398:
387:
386:
352:
350:
349:
344:
332:
330:
329:
324:
313:
312:
307:
306:
298:
291:
290:
278:
277:
272:
271:
263:
252:
250:
249:
244:
233:
232:
198:
196:
195:
190:
188:
187:
171:
169:
168:
163:
161:
160:
121:, logic-dynamic
115:physical systems
73:In general, the
50:(described by a
42:(described by a
36:dynamical system
21:
1271:
1270:
1266:
1265:
1264:
1262:
1261:
1260:
1231:
1230:
1229:
1228:
1216:
1212:
1207:
1203:
1198:
1194:
1186:
1184:
1182:
1156:
1152:
1147:
1135:
1101:
1082:
1026:
920:
918:Further reading
856:
839:: Tool for the
833:
786:
739:
709:
704:
660:
635:
630:Hybrid systems
611:
608:
607:
583:
579:
565:
562:
561:
526:
523:
522:
494:
491:
490:
473:
469:
454:
449:
436:
432:
423:
418:
412:
409:
408:
382:
378:
361:
358:
357:
338:
335:
334:
308:
297:
296:
295:
286:
282:
273:
262:
261:
260:
258:
255:
254:
228:
224:
207:
204:
203:
183:
179:
177:
174:
173:
156:
152:
150:
147:
146:
135:
111:
99:jump conditions
81:and a discrete
28:
23:
22:
15:
12:
11:
5:
1269:
1259:
1258:
1256:Control theory
1253:
1248:
1243:
1241:Systems theory
1227:
1226:
1210:
1201:
1192:
1180:
1149:
1148:
1146:
1143:
1142:
1141:
1134:
1133:External links
1131:
1130:
1129:
1094:
1085:
1080:
1067:
1055:10.1.1.72.2475
1029:
1024:
1003:
975:
936:
919:
916:
915:
914:
911:Jump diffusion
908:
902:
892:
887:
882:
877:
872:
867:
862:
855:
852:
851:
850:
844:
832:
829:
828:
827:
821:
815:
812:Theorem prover
805:
799:
793:
785:
782:
781:
780:
774:
768:
762:
756:
750:
738:
735:
734:
733:
723:
708:
705:
703:
702:Software Tools
700:
668:hybrid program
659:
656:
634:
628:
615:
594:
591:
586:
582:
578:
575:
572:
569:
530:
510:
507:
504:
501:
498:
476:
472:
468:
465:
462:
457:
452:
448:
444:
439:
435:
431:
426:
421:
417:
396:
393:
390:
385:
381:
377:
374:
371:
368:
365:
342:
322:
319:
316:
311:
304:
301:
294:
289:
285:
281:
276:
269:
266:
242:
239:
236:
231:
227:
223:
220:
217:
214:
211:
186:
182:
159:
155:
134:
131:
110:
107:
26:
18:Hybrid systems
9:
6:
4:
3:
2:
1268:
1257:
1254:
1252:
1249:
1247:
1244:
1242:
1239:
1238:
1236:
1224:
1220:
1214:
1205:
1196:
1183:
1177:
1173:
1169:
1165:
1161:
1154:
1150:
1140:
1137:
1136:
1127:
1123:
1119:
1115:
1111:
1107:
1100:
1095:
1091:
1086:
1083:
1077:
1073:
1068:
1065:
1061:
1056:
1051:
1047:
1043:
1039:
1035:
1030:
1027:
1021:
1017:
1013:
1009:
1004:
1001:
997:
993:
989:
985:
981:
976:
973:on 2010-01-27
972:
967:
962:
958:
954:
950:
946:
942:
937:
934:on 2010-01-27
933:
929:
928:
922:
921:
912:
909:
906:
903:
900:
896:
893:
891:
888:
886:
883:
881:
878:
876:
873:
871:
868:
866:
863:
861:
858:
857:
848:
845:
842:
838:
835:
834:
825:
822:
819:
816:
813:
809:
806:
803:
800:
797:
794:
791:
788:
787:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
744:
741:
740:
731:
727:
724:
722:
718:
714:
711:
710:
699:
697:
692:
688:
683:
681:
677:
673:
669:
665:
655:
652:
647:
645:
641:
633:
627:
613:
592:
589:
584:
580:
576:
573:
570:
567:
557:
555:
551:
547:
542:
528:
508:
505:
502:
499:
496:
474:
470:
466:
463:
460:
455:
450:
446:
442:
437:
433:
429:
424:
419:
415:
391:
388:
383:
379:
372:
369:
366:
363:
354:
340:
320:
317:
314:
309:
302:
299:
292:
287:
283:
279:
274:
267:
264:
237:
234:
229:
225:
218:
215:
212:
209:
200:
184:
180:
157:
153:
144:
140:
139:bouncing ball
133:Bouncing ball
130:
128:
124:
120:
116:
106:
104:
100:
96:
92:
91:control graph
88:
84:
80:
76:
71:
69:
65:
61:
57:
53:
52:state machine
49:
45:
41:
37:
33:
32:hybrid system
19:
1213:
1204:
1195:
1185:, retrieved
1163:
1153:
1112:(1): 3–129,
1109:
1105:
1089:
1071:
1037:
1033:
1007:
986:(2): 28–93,
983:
979:
971:the original
948:
944:
932:the original
926:
895:Jump process
737:Reachability
713:HyEQ Toolbox
684:
670:or a hybrid
661:
648:
636:
632:verification
558:
553:
549:
543:
355:
201:
136:
129:congestion.
118:
112:
102:
98:
94:
90:
82:
78:
74:
72:
47:
39:
31:
29:
1106:SIAM Review
951:(1): 3–34,
899:probability
125:, and even
123:controllers
68:fuzzy logic
64:neural nets
1235:Categories
1187:2022-06-08
1145:References
777:JuliaReach
707:Simulation
651:algorithms
95:invariants
1126:212727046
1050:CiteSeerX
966:1813/6241
841:synthesis
726:PowerDEVS
696:PowerDEVS
672:Petri net
614:λ
590:≥
577:⊥
574:λ
571:≤
529:γ
503:γ
467:γ
464:−
367:∈
318:−
303:˙
268:˙
213:∈
56:automaton
1000:46488751
854:See also
824:S-TaLiRo
808:KeYmaera
765:HyCreate
721:Simulink
550:infinite
489:, where
333:, where
127:Internet
109:Examples
1042:Bibcode
847:SpaceEx
802:HSolver
743:Ariadne
58:, or a
1178:
1124:
1078:
1052:
1022:
998:
818:PHAVer
796:HyTech
717:MATLAB
642:, and
554:finite
119:impact
103:events
46:) and
1122:S2CID
1102:(PDF)
996:S2CID
837:SCOTS
831:Other
771:HyPro
759:Flow*
356:When
202:When
117:with
75:state
34:is a
1176:ISBN
1076:ISBN
1020:ISBN
790:C2E2
753:CORA
730:DEVS
719:and
691:DEVS
687:DEVS
666:, a
546:Zeno
506:<
500:<
235:>
87:flow
83:mode
66:and
48:jump
40:flow
1219:doi
1168:doi
1114:doi
1060:doi
1012:doi
988:doi
961:hdl
953:doi
949:138
747:C++
1237::
1174:,
1162:,
1120:,
1110:62
1108:,
1104:,
1058:,
1048:,
1038:25
1036:,
1018:,
994:,
984:29
982:,
959:,
947:,
943:,
810::
745::
698:.
682:.
646:.
593:0.
105:.
54:,
30:A
1221::
1170::
1116::
1062::
1044::
1014::
990::
963::
955::
585:1
581:x
568:0
509:1
497:0
475:2
471:x
461:=
456:+
451:2
447:x
443:,
438:1
434:x
430:=
425:+
420:1
416:x
395:}
392:0
389:=
384:1
380:x
376:{
373:=
370:D
364:x
341:g
321:g
315:=
310:2
300:x
293:,
288:2
284:x
280:=
275:1
265:x
241:}
238:0
230:1
226:x
222:{
219:=
216:C
210:x
185:2
181:x
158:1
154:x
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.