837:
456:
232:
673:
97:
330:
139:
555:
488:
595:
575:
528:
508:
320:
300:
277:
257:
154:
616:
878:
681:
688:
in MATLAB. An interpretation of the
Rosenbrock System Matrix as a Linear Fractional Transformation can be found in.
691:
One of the first applications of the
Rosenbrock form was the development of an efficient computational method for
608:
For computational purposes, a short form of the
Rosenbrock system matrix is more appropriate and given by
897:
52:
25:
451:{\displaystyle g_{ij}={\frac {\begin{vmatrix}sI-A&-b_{i}\\c_{j}&d_{ij}\end{vmatrix}}{|sI-A|}}}
871:
695:, which is based on the pivot element method. A variant of Rosenbrock’s method is implemented in the
29:
852:
902:
33:
103:
912:
754:
864:
844:
803:
692:
533:
466:
8:
807:
580:
560:
513:
493:
305:
285:
262:
242:
816:
791:
907:
811:
720:
Rosenbrock, H. H. (1967). "Transformation of linear constant system equations".
600:
Based in this representation, Rosenbrock developed his version of the PBH test.
848:
891:
227:{\displaystyle P(s)={\begin{pmatrix}sI-A&-B\\C&D\end{pmatrix}}.}
700:
680:
The short form of the
Rosenbrock system matrix has been widely used in
792:"Minimal state-space realization in linear system theory: an overview"
24:
of a linear time-invariant system is a useful representation bridging
668:{\displaystyle P\sim {\begin{pmatrix}A&B\\C&D\end{pmatrix}}.}
836:
684:, where it is also referred to as packed form; see command
239:
In the original work by
Rosenbrock, the constant matrix
631:
356:
178:
619:
583:
563:
536:
516:
496:
469:
333:
308:
288:
265:
245:
157:
106:
55:
775:Zhou, Kemin; Doyle, John C.; Glover, Keith (1995).
667:
589:
569:
549:
522:
502:
482:
450:
314:
294:
271:
251:
226:
133:
91:
889:
796:Journal of Computational and Applied Mathematics
774:
872:
789:
879:
865:
738:
719:
146:The Rosenbrock system matrix is given by
815:
282:The transfer function between the input
890:
831:
741:State-Space and Multivariable Theory
682:H-infinity methods in control theory
755:"Mu Analysis and Synthesis Toolbox"
13:
14:
924:
259:is allowed to be a polynomial in
92:{\displaystyle {\dot {x}}=Ax+Bu,}
32:form. It was proposed in 1967 by
835:
783:
768:
747:
732:
713:
441:
424:
167:
161:
1:
817:10.1016/S0377-0427(00)00341-1
706:
603:
39:
851:. You can help Knowledge by
44:Consider the dynamic system
16:In applied mathematics, the
7:
10:
929:
830:
777:Robust and Optimal Control
739:Rosenbrock, H. H. (1970).
26:state-space representation
22:Rosenbrock's system matrix
790:De Schutter, B. (2000).
134:{\displaystyle y=Cx+Du.}
30:transfer function matrix
18:Rosenbrock system matrix
699:command of Matlab and
669:
591:
571:
551:
524:
504:
484:
452:
316:
296:
273:
253:
228:
135:
93:
670:
592:
572:
552:
550:{\displaystyle c_{j}}
525:
505:
485:
483:{\displaystyle b_{i}}
453:
317:
297:
274:
254:
229:
136:
94:
693:Kalman decomposition
617:
581:
561:
534:
514:
494:
467:
331:
306:
286:
263:
243:
155:
104:
53:
34:Howard H. Rosenbrock
843:This article about
808:2000JCoAM.121..331S
898:1967 introductions
665:
656:
587:
567:
547:
520:
500:
480:
448:
417:
312:
292:
269:
249:
224:
215:
131:
89:
860:
859:
590:{\displaystyle C}
570:{\displaystyle j}
523:{\displaystyle B}
503:{\displaystyle i}
446:
315:{\displaystyle j}
295:{\displaystyle i}
272:{\displaystyle s}
252:{\displaystyle D}
65:
920:
881:
874:
867:
839:
832:
822:
821:
819:
802:(1–2): 331–354.
787:
781:
780:
779:. Prentice Hall.
772:
766:
765:
763:
761:
751:
745:
744:
736:
730:
729:
717:
674:
672:
671:
666:
661:
660:
596:
594:
593:
588:
576:
574:
573:
568:
556:
554:
553:
548:
546:
545:
529:
527:
526:
521:
509:
507:
506:
501:
489:
487:
486:
481:
479:
478:
457:
455:
454:
449:
447:
445:
444:
427:
421:
414:
413:
399:
398:
385:
384:
351:
346:
345:
321:
319:
318:
313:
301:
299:
298:
293:
278:
276:
275:
270:
258:
256:
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250:
233:
231:
230:
225:
220:
219:
140:
138:
137:
132:
98:
96:
95:
90:
67:
66:
58:
928:
927:
923:
922:
921:
919:
918:
917:
888:
887:
886:
885:
828:
826:
825:
788:
784:
773:
769:
759:
757:
753:
752:
748:
737:
733:
718:
714:
709:
655:
654:
649:
643:
642:
637:
627:
626:
618:
615:
614:
606:
582:
579:
578:
562:
559:
558:
541:
537:
535:
532:
531:
515:
512:
511:
495:
492:
491:
474:
470:
468:
465:
464:
440:
423:
422:
416:
415:
406:
402:
400:
394:
390:
387:
386:
380:
376:
371:
352:
350:
338:
334:
332:
329:
328:
307:
304:
303:
287:
284:
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264:
261:
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244:
241:
240:
214:
213:
208:
202:
201:
193:
174:
173:
156:
153:
152:
105:
102:
101:
57:
56:
54:
51:
50:
42:
12:
11:
5:
926:
916:
915:
910:
905:
903:Control theory
900:
884:
883:
876:
869:
861:
858:
857:
840:
824:
823:
782:
767:
746:
731:
711:
710:
708:
705:
678:
677:
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675:
664:
659:
653:
650:
648:
645:
644:
641:
638:
636:
633:
632:
630:
625:
622:
605:
602:
586:
566:
544:
540:
519:
499:
490:is the column
477:
473:
461:
460:
459:
458:
443:
439:
436:
433:
430:
426:
420:
412:
409:
405:
401:
397:
393:
389:
388:
383:
379:
375:
372:
370:
367:
364:
361:
358:
357:
355:
349:
344:
341:
337:
311:
291:
268:
248:
237:
236:
235:
234:
223:
218:
212:
209:
207:
204:
203:
200:
197:
194:
192:
189:
186:
183:
180:
179:
177:
172:
169:
166:
163:
160:
144:
143:
142:
141:
130:
127:
124:
121:
118:
115:
112:
109:
99:
88:
85:
82:
79:
76:
73:
70:
64:
61:
41:
38:
9:
6:
4:
3:
2:
925:
914:
911:
909:
906:
904:
901:
899:
896:
895:
893:
882:
877:
875:
870:
868:
863:
862:
856:
854:
850:
846:
841:
838:
834:
833:
829:
818:
813:
809:
805:
801:
797:
793:
786:
778:
771:
756:
750:
742:
735:
727:
723:
716:
712:
704:
702:
698:
694:
689:
687:
683:
662:
657:
651:
646:
639:
634:
628:
623:
620:
613:
612:
611:
610:
609:
601:
598:
584:
564:
542:
538:
517:
497:
475:
471:
437:
434:
431:
428:
418:
410:
407:
403:
395:
391:
381:
377:
373:
368:
365:
362:
359:
353:
347:
342:
339:
335:
327:
326:
325:
324:
323:
322:is given by
309:
289:
280:
266:
246:
221:
216:
210:
205:
198:
195:
190:
187:
184:
181:
175:
170:
164:
158:
151:
150:
149:
148:
147:
128:
125:
122:
119:
116:
113:
110:
107:
100:
86:
83:
80:
77:
74:
71:
68:
62:
59:
49:
48:
47:
46:
45:
37:
35:
31:
27:
23:
19:
913:Matrix stubs
853:expanding it
842:
827:
799:
795:
785:
776:
770:
758:. Retrieved
749:
740:
734:
725:
721:
715:
696:
690:
685:
679:
607:
599:
462:
281:
238:
145:
43:
21:
17:
15:
557:is the row
302:and output
892:Categories
728:: 541–544.
707:References
701:GNU Octave
604:Short form
40:Definition
760:25 August
743:. Nelson.
722:Proc. IEE
624:∼
435:−
374:−
366:−
196:−
188:−
63:˙
908:Matrices
845:matrices
804:Bibcode
697:minreal
463:where
847:is a
849:stub
762:2014
530:and
28:and
812:doi
800:121
726:114
686:pck
577:of
510:of
20:or
894::
810:.
798:.
794:.
724:.
703:.
597:.
279:.
36:.
880:e
873:t
866:v
855:.
820:.
814::
806::
764:.
663:.
658:)
652:D
647:C
640:B
635:A
629:(
621:P
585:C
565:j
543:j
539:c
518:B
498:i
476:i
472:b
442:|
438:A
432:I
429:s
425:|
419:|
411:j
408:i
404:d
396:j
392:c
382:i
378:b
369:A
363:I
360:s
354:|
348:=
343:j
340:i
336:g
310:j
290:i
267:s
247:D
222:.
217:)
211:D
206:C
199:B
191:A
185:I
182:s
176:(
171:=
168:)
165:s
162:(
159:P
129:.
126:u
123:D
120:+
117:x
114:C
111:=
108:y
87:,
84:u
81:B
78:+
75:x
72:A
69:=
60:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
