2103:
1604:
2098:{\displaystyle {\begin{aligned}{\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A}}^{T}&=\int _{0}^{\infty }{\boldsymbol {A}}e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB}}^{T}e^{{\boldsymbol {A}}^{T}\tau }d\tau +\int _{0}^{\infty }e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB^{T}}}e^{{\boldsymbol {A}}^{T}\tau }{\boldsymbol {A}}^{T}d\tau \\&=\int _{0}^{\infty }{\frac {d}{d\tau }}\left(e^{{\boldsymbol {A}}\tau }{\boldsymbol {B}}{\boldsymbol {B}}^{T}e^{{\boldsymbol {A}}^{T}\tau }\right)d\tau \\&=\left.e^{{\boldsymbol {A}}t}{\boldsymbol {B}}{\boldsymbol {B}}^{T}e^{{\boldsymbol {A}}^{T}t}\right|_{t=0}^{\infty }\\&={\boldsymbol {0}}-{\boldsymbol {BB}}^{T}\\&={\boldsymbol {-BB}}^{T}\end{aligned}}}
3533:
3313:
854:
4465:
4932:
1019:
170:
645:
3528:{\displaystyle {\boldsymbol {x}}^{T}{\boldsymbol {W}}_{c}{\boldsymbol {x}}=\int _{0}^{\infty }{\boldsymbol {x}}^{T}e^{{\boldsymbol {A}}t}{\boldsymbol {BB}}^{T}e^{{\boldsymbol {A}}^{T}t}{\boldsymbol {x}}\,dt=\int _{0}^{\infty }\left\Vert {\boldsymbol {B}}^{T}e^{{\boldsymbol {A}}^{T}t}{\boldsymbol {x}}\right\Vert _{2}^{2}dt}
5305:
4305:
3838:
2612:
4742:
917:
4172:
1599:
1396:
23:
2425:
1492:
1291:
5117:
5457:
3692:
1116:
2494:
2963:
4062:
3603:
849:{\displaystyle {\boldsymbol {W_{c}}}(t)=\int _{0}^{t}e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB^{T}}}e^{{\boldsymbol {A}}^{T}\tau }d\tau =\int _{0}^{t}e^{{\boldsymbol {A}}(t-\tau )}{\boldsymbol {BB^{T}}}e^{{\boldsymbol {A}}^{T}(t-\tau )}d\tau }
3274:
4460:{\displaystyle {\begin{aligned}{\dot {\boldsymbol {x}}}(t)&={\boldsymbol {A}}(t){\boldsymbol {x}}(t)+{\boldsymbol {B}}(t){\boldsymbol {u}}(t)\\{\boldsymbol {y}}(t)&={\boldsymbol {C}}(t){\boldsymbol {x}}(t)\end{aligned}}}
4067:
1499:
1296:
1609:
5020:
2348:
1415:
1214:
4927:{\displaystyle {\boldsymbol {W}}_{c}(t_{0},t_{1})=\int _{t_{0}}^{t_{1}}{\boldsymbol {\Phi }}(t_{1},\tau ){\boldsymbol {B}}(\tau ){\boldsymbol {B}}^{T}(\tau ){\boldsymbol {\Phi }}^{T}(t_{1},\tau )d\tau ,}
4644:
4587:
1014:{\displaystyle {\mathcal {C}}={\begin{bmatrix}{\boldsymbol {B}}&{\boldsymbol {AB}}&{\boldsymbol {A}}^{2}{\boldsymbol {B}}&\cdots &{\boldsymbol {A}}^{n-1}{\boldsymbol {B}}\end{bmatrix}}}
3138:
5368:
1065:
4284:
3915:
3876:
3679:
611:
571:
4310:
3697:
3176:
2690:
28:
5363:
5112:
165:{\displaystyle {\begin{aligned}{\dot {\boldsymbol {x}}}(t)&={\boldsymbol {Ax}}(t)+{\boldsymbol {Bu}}(t)\\{\boldsymbol {y}}(t)&={\boldsymbol {Cx}}(t)+{\boldsymbol {Du}}(t)\end{aligned}}}
3984:
4969:
3207:
2685:
2143:
4535:
have entries that varies with time. Again, as well as in the continuous time case and in the discrete time case, one may be interested in discovering if the system given by the pair
4206:
2646:
2489:
2457:
2260:
3632:
3305:
2343:
2289:
2220:
5300:{\displaystyle {\boldsymbol {W}}_{c}(t_{0},t_{1})={\boldsymbol {W}}_{c}(t,t_{1})+{\boldsymbol {\Phi }}(t_{1},t){\boldsymbol {W}}_{c}(t_{0},t){\boldsymbol {\Phi }}^{T}(t_{1},t)}
4533:
4511:
4489:
4228:
3979:
3937:
2314:
2191:
1206:
1184:
1158:
532:
510:
488:
466:
262:
240:
218:
196:
4711:
3542:
4176:
That is called the discrete
Controllability Gramian. We can easily see the correspondence between discrete time and the continuous time case, that is, if we can check that
1060:
3202:
912:
4737:
3833:{\displaystyle {\begin{aligned}{\boldsymbol {x}}&={\boldsymbol {Ax}}+{\boldsymbol {Bu}}\\{\boldsymbol {y}}&={\boldsymbol {Cx}}+{\boldsymbol {Du}}\end{aligned}}}
2169:
640:
366:
340:
314:
288:
2607:{\displaystyle {\boldsymbol {A}}({\boldsymbol {W}}_{c1}-{\boldsymbol {W}}_{c2})+({\boldsymbol {W}}_{c1}-{\boldsymbol {W}}_{c2}){\boldsymbol {A}}^{T}={\boldsymbol {0}}}
1136:
3005:
880:
4671:
4248:
3957:
2985:
426:
406:
386:
4974:
4167:{\displaystyle {\boldsymbol {W}}_{dc}=\sum _{m=0}^{\infty }{\boldsymbol {A}}^{m}{\boldsymbol {BB}}^{T}\left({\boldsymbol {A}}^{T}\right)^{m}}
3010:
1594:{\displaystyle {\boldsymbol {W_{c}}}=\int _{0}^{\infty }e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB}}^{T}e^{{\boldsymbol {A}}^{T}\tau }d\tau }
1391:{\displaystyle {\boldsymbol {W_{c}}}=\int _{0}^{\infty }e^{{\boldsymbol {A}}\tau }{\boldsymbol {BB}}^{T}e^{{\boldsymbol {A}}^{T}\tau }d\tau }
4595:
4538:
1293:
is positive definite, the system is controllable. The solution is called the
Controllability Gramian and can be expressed as
4253:
3884:
3845:
3648:
580:
540:
3143:
2420:{\displaystyle {\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A}}^{T}=-{\boldsymbol {BB}}^{T}}
1487:{\displaystyle {\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A}}^{T}=-{\boldsymbol {BB}}^{T}}
1286:{\displaystyle {\boldsymbol {A}}{\boldsymbol {W}}_{c}+{\boldsymbol {W}}_{c}{\boldsymbol {A}}^{T}=-{\boldsymbol {BB}}^{T}}
5310:
5059:
5524:
4937:
2651:
5452:{\displaystyle {\boldsymbol {\Phi }}(t_{1},\tau )={\boldsymbol {\Phi }}(t_{1},t){\boldsymbol {\Phi }}(t,\tau )}
1111:{\displaystyle {\begin{bmatrix}{\boldsymbol {A}}-\lambda {\boldsymbol {I}}&{\boldsymbol {B}}\end{bmatrix}}}
2110:
4179:
2958:{\displaystyle e^{{\boldsymbol {A}}t}\lefte^{{\boldsymbol {A}}^{T}t}={\frac {d}{dt}}\left={\boldsymbol {0}}}
2619:
2462:
2430:
2233:
5540:
3608:
3281:
2319:
2265:
2196:
4057:{\displaystyle W_{dc}-{\boldsymbol {A}}{\boldsymbol {W}}_{dc}{\boldsymbol {A}}^{T}={\boldsymbol {BB}}^{T}}
4516:
4494:
4472:
4211:
3962:
3920:
3598:{\displaystyle \left\Vert {\boldsymbol {B}}^{T}e^{{\boldsymbol {A}}^{T}t}{\boldsymbol {x}}\right\Vert }
2297:
2174:
1189:
1167:
1141:
515:
493:
471:
449:
245:
223:
201:
179:
4676:
3269:{\displaystyle {\boldsymbol {0}}-({\boldsymbol {W}}_{c1}-{\boldsymbol {W}}_{c2})={\boldsymbol {0}}}
1027:
3181:
5555:
888:
5493:
4716:
2148:
619:
345:
319:
293:
267:
5466:
4291:
3642:
1121:
5498:
5483:
5025:
Again, we have a similar method to determine if a system is or is not a controllable system.
2990:
5512:
1400:
In the following section we are going to take a closer look at the
Controllability Gramian.
859:
4649:
537:
One can observe if the LTI system is or is not controllable simply by looking at the pair
8:
4589:
is controllable or not. This can be done in a very similar way of the preceding cases.
4233:
3942:
2970:
411:
391:
371:
5520:
5513:
1409:
1209:
5478:
3878:
is controllable” (the equivalences are much alike for the continuous time case).
173:
5488:
17:
5015:{\displaystyle {\boldsymbol {\dot {x}}}={\boldsymbol {A}}(t){\boldsymbol {x}}}
446:
Linear Time
Invariant (LTI) Systems are those systems in which the parameters
5549:
2345:
is unique. In order to prove so, suppose we have two different solutions for
3645:), as well as the proof for the other equivalent statements of “The pair
3842:
One can check that there are equivalences for the statement “The pair
3681:
5365:
and by the property of the state transition matrix that claims that:
2316:
is stable (all its eigenvalues have negative real part) to show that
3881:
We are interested in the equivalence that claims that, if “The pair
431:
One of the many ways one can achieve such goal is by the use of the
2193:(all its eigenvalues have negative real part). This shows us that
2222:
is indeed the solution for the
Lyapunov equation under analysis.
573:. Then, we can say that the following statements are equivalent:
434:
1408:
The controllability
Gramian can be found as the solution of the
4286:
is controllable. More properties and proofs can be found in
5541:
Mathematica function to compute the controllability
Gramian
4639:{\displaystyle ({\boldsymbol {A}}(t),{\boldsymbol {B}}(t))}
4582:{\displaystyle ({\boldsymbol {A}}(t),{\boldsymbol {B}}(t))}
1951:
4739:
20:, we may need to find out whether or not a system such as
4302:
Linear time variant (LTV) systems are those in the form:
3637:
More properties of controllable systems can be found in
5461:
3133:{\displaystyle \left_{t=0}^{\infty }={\boldsymbol {0}}}
1074:
936:
5371:
5313:
5120:
5062:
4977:
4940:
4745:
4719:
4679:
4652:
4598:
4541:
4519:
4497:
4475:
4308:
4279:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}
4256:
4236:
4214:
4182:
4070:
3987:
3965:
3945:
3923:
3910:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}
3887:
3871:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}
3848:
3695:
3674:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}
3651:
3611:
3545:
3316:
3284:
3210:
3184:
3146:
3013:
2993:
2973:
2693:
2654:
2622:
2497:
2465:
2433:
2351:
2322:
2300:
2268:
2236:
2199:
2177:
2151:
2113:
1607:
1502:
1418:
1299:
1217:
1192:
1170:
1144:
1124:
1068:
1030:
920:
891:
862:
648:
622:
606:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}
583:
566:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}
543:
518:
496:
474:
452:
414:
394:
374:
348:
322:
296:
270:
248:
226:
204:
182:
26:
3171:{\displaystyle e^{{\boldsymbol {A}}t}\rightarrow 0}
5451:
5358:{\displaystyle {\boldsymbol {W}}_{c}(t_{0},t_{1})}
5357:
5299:
5107:{\displaystyle {\boldsymbol {W}}_{c}(t_{0},t_{1})}
5106:
5014:
4963:
4926:
4731:
4705:
4665:
4638:
4581:
4527:
4505:
4483:
4459:
4278:
4242:
4222:
4200:
4166:
4056:
3973:
3951:
3931:
3909:
3870:
3832:
3673:
3626:
3597:
3527:
3299:
3268:
3196:
3170:
3132:
2999:
2979:
2957:
2679:
2640:
2606:
2483:
2451:
2419:
2337:
2308:
2283:
2254:
2214:
2185:
2163:
2137:
2097:
1593:
1486:
1390:
1285:
1200:
1178:
1152:
1130:
1110:
1054:
1013:
906:
874:
848:
634:
605:
565:
526:
504:
482:
460:
420:
400:
380:
360:
334:
308:
282:
256:
234:
212:
190:
164:
2763:
441:
5547:
5519:. New York, New York: Oxford University Press.
4964:{\displaystyle {\boldsymbol {\Phi }}(t,\tau )}
4297:
5515:Linear System Theory and Design Third Edition
5307:that can easily be seen by the definition of
4208:is positive definite, and all eigenvalues of
3917:is controllable” and all the eigenvalues of
1208:is stable), and the unique solution of the
5028:
2680:{\displaystyle e^{{\boldsymbol {A}}^{T}t}}
1601:as a solution, we are going to find that:
1403:
5056:We have that the Controllability Gramian
3433:
3981:is stable), then the unique solution of
3684:
3539:(assuming the non-degenerate case where
2262:is a symmetric matrix, therefore, so is
2138:{\displaystyle e^{{\boldsymbol {A}}t}=0}
5316:
5231:
5167:
5123:
5065:
5008:
4994:
4981:
4862:
4847:
4748:
4620:
4603:
4563:
4546:
4521:
4499:
4477:
4440:
4426:
4405:
4387:
4373:
4356:
4342:
4316:
4269:
4261:
4216:
4185:
4144:
4127:
4124:
4112:
4073:
4044:
4041:
4026:
4011:
4005:
3967:
3925:
3900:
3892:
3861:
3853:
3813:
3810:
3793:
3790:
3769:
3751:
3748:
3731:
3728:
3701:
3664:
3656:
3614:
3586:
3570:
3553:
3499:
3483:
3466:
3429:
3413:
3396:
3393:
3382:
3366:
3342:
3331:
3319:
3287:
3242:
3224:
3153:
3085:
3060:
3042:
3026:
2931:
2927:
2902:
2884:
2868:
2824:
2802:
2784:
2766:
2742:
2724:
2715:
2700:
2662:
2629:
2586:
2568:
2550:
2526:
2508:
2499:
2468:
2436:
2407:
2404:
2386:
2374:
2359:
2353:
2325:
2302:
2271:
2246:
2242:
2238:
2202:
2179:
2120:
2081:
2078:
2053:
2050:
1993:
1976:
1970:
1960:
1913:
1896:
1890:
1880:
1813:
1796:
1783:
1779:
1775:
1765:
1720:
1703:
1700:
1689:
1679:
1646:
1634:
1619:
1613:
1570:
1553:
1550:
1539:
1509:
1505:
1474:
1471:
1453:
1441:
1426:
1420:
1367:
1350:
1347:
1336:
1306:
1302:
1273:
1270:
1252:
1240:
1225:
1219:
1194:
1172:
1146:
1096:
1089:
1078:
999:
982:
969:
958:
950:
947:
940:
813:
800:
796:
792:
770:
725:
712:
708:
704:
694:
655:
651:
596:
588:
556:
548:
520:
498:
476:
454:
250:
228:
206:
184:
145:
142:
125:
122:
101:
83:
80:
63:
60:
34:
5548:
4201:{\displaystyle {\boldsymbol {W}}_{dc}}
2641:{\displaystyle e^{{\boldsymbol {A}}t}}
2484:{\displaystyle {\boldsymbol {W}}_{c2}}
2452:{\displaystyle {\boldsymbol {W}}_{c1}}
2255:{\displaystyle {\boldsymbol {BB^{T}}}}
1118:has full row rank at every eigenvalue
4673:if and only if there exists a finite
3627:{\displaystyle {\boldsymbol {W}}_{c}}
3605:is not identically zero). This makes
3300:{\displaystyle {\boldsymbol {W}}_{c}}
2338:{\displaystyle {\boldsymbol {W}}_{c}}
2284:{\displaystyle {\boldsymbol {W}}_{c}}
2215:{\displaystyle {\boldsymbol {W}}_{c}}
5510:
5462:
4287:
3638:
1496:In fact, we can see that if we take
1164:If, in addition, all eigenvalues of
534:are invariant with respect to time.
2294:We can use again the fact that, if
13:
4971:is the state transition matrix of
4105:
4064:is positive definite and given by
3453:
3359:
3191:
3117:
2994:
2158:
2025:
1849:
1754:
1673:
1528:
1325:
923:
14:
5567:
5534:
4528:{\displaystyle {\boldsymbol {C}}}
4506:{\displaystyle {\boldsymbol {B}}}
4484:{\displaystyle {\boldsymbol {A}}}
4223:{\displaystyle {\boldsymbol {A}}}
3974:{\displaystyle {\boldsymbol {A}}}
3932:{\displaystyle {\boldsymbol {A}}}
2309:{\displaystyle {\boldsymbol {A}}}
2186:{\displaystyle {\boldsymbol {A}}}
1201:{\displaystyle {\boldsymbol {A}}}
1179:{\displaystyle {\boldsymbol {A}}}
1153:{\displaystyle {\boldsymbol {A}}}
527:{\displaystyle {\boldsymbol {D}}}
505:{\displaystyle {\boldsymbol {C}}}
483:{\displaystyle {\boldsymbol {B}}}
461:{\displaystyle {\boldsymbol {A}}}
257:{\displaystyle {\boldsymbol {D}}}
235:{\displaystyle {\boldsymbol {C}}}
213:{\displaystyle {\boldsymbol {B}}}
191:{\displaystyle {\boldsymbol {A}}}
5430:
5403:
5373:
5265:
5203:
4984:
4942:
4883:
4820:
3262:
3212:
3126:
2951:
2600:
2075:
2041:
2687:by the right, would lead us to
5446:
5434:
5426:
5407:
5396:
5377:
5352:
5326:
5294:
5275:
5260:
5241:
5226:
5207:
5196:
5177:
5159:
5133:
5101:
5075:
5004:
4998:
4958:
4946:
4912:
4893:
4878:
4872:
4857:
4851:
4843:
4824:
4784:
4758:
4706:{\displaystyle t_{1}>t_{0}}
4633:
4630:
4624:
4613:
4607:
4599:
4576:
4573:
4567:
4556:
4550:
4542:
4450:
4444:
4436:
4430:
4415:
4409:
4397:
4391:
4383:
4377:
4366:
4360:
4352:
4346:
4331:
4325:
4273:
4257:
3904:
3888:
3865:
3849:
3823:
3817:
3803:
3797:
3779:
3773:
3761:
3755:
3741:
3735:
3717:
3705:
3668:
3652:
3591:
3547:
3504:
3460:
3255:
3219:
3188:
3162:
2797:
2755:
2719:
2581:
2545:
2539:
2503:
1049:
1037:
835:
823:
786:
774:
667:
661:
600:
584:
560:
544:
442:Controllability in LTI Systems
155:
149:
135:
129:
111:
105:
93:
87:
73:
67:
49:
43:
1:
5504:
5114:have the following property:
3689:For discrete time systems as
2225:
1055:{\displaystyle n\times (n+p)}
3634:a positive definite matrix.
3197:{\displaystyle t\to \infty }
2107:Where we used the fact that
7:
5472:
4298:Linear Time Variant Systems
368:matrices for a system with
10:
5572:
1186:have negative real parts (
907:{\displaystyle n\times np}
4732:{\displaystyle n\times n}
4230:have magnitude less than
3939:have magnitude less than
2164:{\displaystyle t=\infty }
635:{\displaystyle n\times n}
361:{\displaystyle q\times p}
335:{\displaystyle q\times n}
309:{\displaystyle n\times p}
283:{\displaystyle n\times n}
5511:Chen, Chi-Tsong (1999).
4646:is controllable at time
1131:{\displaystyle \lambda }
3000:{\displaystyle \infty }
1404:Controllability Gramian
914:controllability matrix
856:is nonsingular for any
5494:Minimum energy control
5453:
5359:
5301:
5108:
5016:
4965:
4928:
4733:
4707:
4667:
4640:
4583:
4529:
4507:
4485:
4469:That is, the matrices
4461:
4280:
4244:
4224:
4202:
4168:
4109:
4058:
3975:
3953:
3933:
3911:
3872:
3834:
3675:
3628:
3599:
3529:
3310:Also, we can see that
3301:
3270:
3198:
3172:
3134:
3001:
2981:
2959:
2681:
2642:
2608:
2485:
2453:
2427:and they are given by
2421:
2339:
2310:
2285:
2256:
2216:
2187:
2165:
2139:
2099:
1595:
1488:
1392:
1287:
1202:
1180:
1154:
1132:
1112:
1056:
1015:
908:
876:
875:{\displaystyle t>0}
850:
636:
607:
567:
528:
506:
484:
462:
422:
402:
382:
362:
336:
310:
284:
258:
236:
214:
192:
166:
5499:Hankel singular value
5484:Observability Gramian
5454:
5360:
5302:
5109:
5017:
4966:
4929:
4734:
4708:
4668:
4666:{\displaystyle t_{0}}
4641:
4584:
4530:
4508:
4486:
4462:
4281:
4245:
4225:
4203:
4169:
4089:
4059:
3976:
3954:
3934:
3912:
3873:
3835:
3685:Discrete Time Systems
3676:
3629:
3600:
3530:
3302:
3271:
3199:
3173:
3135:
3002:
2982:
2960:
2682:
2643:
2609:
2486:
2454:
2422:
2340:
2311:
2286:
2257:
2217:
2188:
2166:
2140:
2100:
1596:
1489:
1393:
1288:
1203:
1181:
1155:
1133:
1113:
1057:
1016:
909:
877:
851:
637:
608:
568:
529:
507:
485:
463:
423:
403:
383:
363:
337:
311:
285:
259:
237:
215:
193:
167:
5369:
5311:
5118:
5060:
4975:
4938:
4743:
4717:
4677:
4650:
4596:
4539:
4517:
4495:
4473:
4306:
4254:
4234:
4212:
4180:
4068:
3985:
3963:
3943:
3921:
3885:
3846:
3693:
3649:
3609:
3543:
3535:is positive for any
3314:
3282:
3208:
3182:
3144:
3140:using the fact that
3011:
2991:
2971:
2691:
2652:
2620:
2495:
2463:
2431:
2349:
2320:
2298:
2266:
2234:
2197:
2175:
2149:
2111:
1605:
1500:
1416:
1297:
1215:
1190:
1168:
1142:
1122:
1066:
1028:
918:
889:
860:
646:
620:
581:
541:
516:
494:
472:
450:
412:
408:state variables and
392:
372:
346:
320:
294:
268:
246:
224:
202:
180:
24:
4818:
3518:
3457:
3363:
3121:
2648:by the left and by
2029:
1853:
1758:
1677:
1532:
1329:
763:
687:
264:are, respectively,
5449:
5355:
5297:
5104:
5034:
5022:, is nonsingular.
5012:
4961:
4924:
4790:
4729:
4703:
4663:
4636:
4579:
4525:
4503:
4481:
4457:
4455:
4276:
4240:
4220:
4198:
4164:
4054:
3971:
3949:
3929:
3907:
3868:
3830:
3828:
3671:
3624:
3595:
3525:
3458:
3443:
3349:
3307:has to be unique.
3297:
3266:
3194:
3168:
3130:
3014:
2997:
2977:
2955:
2677:
2638:
2604:
2481:
2449:
2417:
2335:
2306:
2281:
2252:
2212:
2183:
2161:
2135:
2095:
2093:
1949:
1839:
1744:
1663:
1591:
1518:
1484:
1388:
1315:
1283:
1198:
1176:
1150:
1128:
1108:
1102:
1052:
1011:
1005:
904:
872:
846:
749:
673:
632:
603:
563:
524:
502:
480:
458:
418:
398:
378:
358:
332:
306:
280:
254:
232:
210:
188:
162:
160:
5030:
4987:
4322:
4243:{\displaystyle 1}
3952:{\displaystyle 1}
2980:{\displaystyle 0}
2967:Integrating from
2855:
1867:
1410:Lyapunov equation
1210:Lyapunov equation
421:{\displaystyle q}
401:{\displaystyle n}
381:{\displaystyle p}
40:
5563:
5530:
5518:
5458:
5456:
5455:
5450:
5433:
5419:
5418:
5406:
5389:
5388:
5376:
5364:
5362:
5361:
5356:
5351:
5350:
5338:
5337:
5325:
5324:
5319:
5306:
5304:
5303:
5298:
5287:
5286:
5274:
5273:
5268:
5253:
5252:
5240:
5239:
5234:
5219:
5218:
5206:
5195:
5194:
5176:
5175:
5170:
5158:
5157:
5145:
5144:
5132:
5131:
5126:
5113:
5111:
5110:
5105:
5100:
5099:
5087:
5086:
5074:
5073:
5068:
5021:
5019:
5018:
5013:
5011:
4997:
4989:
4988:
4980:
4970:
4968:
4967:
4962:
4945:
4933:
4931:
4930:
4925:
4905:
4904:
4892:
4891:
4886:
4871:
4870:
4865:
4850:
4836:
4835:
4823:
4817:
4816:
4815:
4805:
4804:
4803:
4783:
4782:
4770:
4769:
4757:
4756:
4751:
4738:
4736:
4735:
4730:
4712:
4710:
4709:
4704:
4702:
4701:
4689:
4688:
4672:
4670:
4669:
4664:
4662:
4661:
4645:
4643:
4642:
4637:
4623:
4606:
4588:
4586:
4585:
4580:
4566:
4549:
4534:
4532:
4531:
4526:
4524:
4512:
4510:
4509:
4504:
4502:
4490:
4488:
4487:
4482:
4480:
4466:
4464:
4463:
4458:
4456:
4443:
4429:
4408:
4390:
4376:
4359:
4345:
4324:
4323:
4315:
4285:
4283:
4282:
4277:
4272:
4264:
4249:
4247:
4246:
4241:
4229:
4227:
4226:
4221:
4219:
4207:
4205:
4204:
4199:
4197:
4196:
4188:
4173:
4171:
4170:
4165:
4163:
4162:
4157:
4153:
4152:
4147:
4136:
4135:
4130:
4121:
4120:
4115:
4108:
4103:
4085:
4084:
4076:
4063:
4061:
4060:
4055:
4053:
4052:
4047:
4035:
4034:
4029:
4023:
4022:
4014:
4008:
4000:
3999:
3980:
3978:
3977:
3972:
3970:
3958:
3956:
3955:
3950:
3938:
3936:
3935:
3930:
3928:
3916:
3914:
3913:
3908:
3903:
3895:
3877:
3875:
3874:
3869:
3864:
3856:
3839:
3837:
3836:
3831:
3829:
3816:
3796:
3772:
3754:
3734:
3704:
3680:
3678:
3677:
3672:
3667:
3659:
3633:
3631:
3630:
3625:
3623:
3622:
3617:
3604:
3602:
3601:
3596:
3594:
3590:
3589:
3584:
3583:
3579:
3578:
3573:
3562:
3561:
3556:
3534:
3532:
3531:
3526:
3517:
3512:
3507:
3503:
3502:
3497:
3496:
3492:
3491:
3486:
3475:
3474:
3469:
3456:
3451:
3432:
3427:
3426:
3422:
3421:
3416:
3405:
3404:
3399:
3390:
3389:
3385:
3375:
3374:
3369:
3362:
3357:
3345:
3340:
3339:
3334:
3328:
3327:
3322:
3306:
3304:
3303:
3298:
3296:
3295:
3290:
3278:In other words,
3275:
3273:
3272:
3267:
3265:
3254:
3253:
3245:
3236:
3235:
3227:
3215:
3203:
3201:
3200:
3195:
3177:
3175:
3174:
3169:
3161:
3160:
3156:
3139:
3137:
3136:
3131:
3129:
3120:
3115:
3104:
3100:
3099:
3098:
3094:
3093:
3088:
3077:
3073:
3072:
3071:
3063:
3054:
3053:
3045:
3034:
3033:
3029:
3006:
3004:
3003:
2998:
2986:
2984:
2983:
2978:
2964:
2962:
2961:
2956:
2954:
2946:
2942:
2941:
2940:
2936:
2935:
2934:
2919:
2915:
2914:
2913:
2905:
2896:
2895:
2887:
2876:
2875:
2871:
2856:
2854:
2843:
2838:
2837:
2833:
2832:
2827:
2816:
2812:
2811:
2810:
2805:
2796:
2795:
2787:
2778:
2777:
2769:
2754:
2753:
2745:
2736:
2735:
2727:
2718:
2708:
2707:
2703:
2686:
2684:
2683:
2678:
2676:
2675:
2671:
2670:
2665:
2647:
2645:
2644:
2639:
2637:
2636:
2632:
2613:
2611:
2610:
2605:
2603:
2595:
2594:
2589:
2580:
2579:
2571:
2562:
2561:
2553:
2538:
2537:
2529:
2520:
2519:
2511:
2502:
2491:. Then we have:
2490:
2488:
2487:
2482:
2480:
2479:
2471:
2458:
2456:
2455:
2450:
2448:
2447:
2439:
2426:
2424:
2423:
2418:
2416:
2415:
2410:
2395:
2394:
2389:
2383:
2382:
2377:
2368:
2367:
2362:
2356:
2344:
2342:
2341:
2336:
2334:
2333:
2328:
2315:
2313:
2312:
2307:
2305:
2290:
2288:
2287:
2282:
2280:
2279:
2274:
2261:
2259:
2258:
2253:
2251:
2250:
2249:
2230:We can see that
2221:
2219:
2218:
2213:
2211:
2210:
2205:
2192:
2190:
2189:
2184:
2182:
2170:
2168:
2167:
2162:
2144:
2142:
2141:
2136:
2128:
2127:
2123:
2104:
2102:
2101:
2096:
2094:
2090:
2089:
2084:
2066:
2062:
2061:
2056:
2044:
2033:
2028:
2023:
2012:
2008:
2007:
2006:
2002:
2001:
1996:
1985:
1984:
1979:
1973:
1968:
1967:
1963:
1942:
1932:
1928:
1927:
1926:
1922:
1921:
1916:
1905:
1904:
1899:
1893:
1888:
1887:
1883:
1868:
1866:
1855:
1852:
1847:
1832:
1822:
1821:
1816:
1810:
1809:
1805:
1804:
1799:
1788:
1787:
1786:
1773:
1772:
1768:
1757:
1752:
1734:
1733:
1729:
1728:
1723:
1712:
1711:
1706:
1697:
1696:
1692:
1682:
1676:
1671:
1655:
1654:
1649:
1643:
1642:
1637:
1628:
1627:
1622:
1616:
1600:
1598:
1597:
1592:
1584:
1583:
1579:
1578:
1573:
1562:
1561:
1556:
1547:
1546:
1542:
1531:
1526:
1514:
1513:
1512:
1493:
1491:
1490:
1485:
1483:
1482:
1477:
1462:
1461:
1456:
1450:
1449:
1444:
1435:
1434:
1429:
1423:
1397:
1395:
1394:
1389:
1381:
1380:
1376:
1375:
1370:
1359:
1358:
1353:
1344:
1343:
1339:
1328:
1323:
1311:
1310:
1309:
1292:
1290:
1289:
1284:
1282:
1281:
1276:
1261:
1260:
1255:
1249:
1248:
1243:
1234:
1233:
1228:
1222:
1207:
1205:
1204:
1199:
1197:
1185:
1183:
1182:
1177:
1175:
1159:
1157:
1156:
1151:
1149:
1137:
1135:
1134:
1129:
1117:
1115:
1114:
1109:
1107:
1106:
1099:
1092:
1081:
1061:
1059:
1058:
1053:
1020:
1018:
1017:
1012:
1010:
1009:
1002:
997:
996:
985:
972:
967:
966:
961:
953:
943:
927:
926:
913:
911:
910:
905:
881:
879:
878:
873:
855:
853:
852:
847:
839:
838:
822:
821:
816:
805:
804:
803:
790:
789:
773:
762:
757:
739:
738:
734:
733:
728:
717:
716:
715:
702:
701:
697:
686:
681:
660:
659:
658:
641:
639:
638:
633:
613:is controllable.
612:
610:
609:
604:
599:
591:
572:
570:
569:
564:
559:
551:
533:
531:
530:
525:
523:
511:
509:
508:
503:
501:
489:
487:
486:
481:
479:
467:
465:
464:
459:
457:
433:Controllability
427:
425:
424:
419:
407:
405:
404:
399:
387:
385:
384:
379:
367:
365:
364:
359:
341:
339:
338:
333:
315:
313:
312:
307:
289:
287:
286:
281:
263:
261:
260:
255:
253:
241:
239:
238:
233:
231:
219:
217:
216:
211:
209:
197:
195:
194:
189:
187:
171:
169:
168:
163:
161:
148:
128:
104:
86:
66:
42:
41:
33:
5571:
5570:
5566:
5565:
5564:
5562:
5561:
5560:
5546:
5545:
5537:
5527:
5507:
5479:Controllability
5475:
5429:
5414:
5410:
5402:
5384:
5380:
5372:
5370:
5367:
5366:
5346:
5342:
5333:
5329:
5320:
5315:
5314:
5312:
5309:
5308:
5282:
5278:
5269:
5264:
5263:
5248:
5244:
5235:
5230:
5229:
5214:
5210:
5202:
5190:
5186:
5171:
5166:
5165:
5153:
5149:
5140:
5136:
5127:
5122:
5121:
5119:
5116:
5115:
5095:
5091:
5082:
5078:
5069:
5064:
5063:
5061:
5058:
5057:
5054:
5051:
5044:
5037:
5007:
4993:
4979:
4978:
4976:
4973:
4972:
4941:
4939:
4936:
4935:
4900:
4896:
4887:
4882:
4881:
4866:
4861:
4860:
4846:
4831:
4827:
4819:
4811:
4807:
4806:
4799:
4795:
4794:
4778:
4774:
4765:
4761:
4752:
4747:
4746:
4744:
4741:
4740:
4718:
4715:
4714:
4697:
4693:
4684:
4680:
4678:
4675:
4674:
4657:
4653:
4651:
4648:
4647:
4619:
4602:
4597:
4594:
4593:
4562:
4545:
4540:
4537:
4536:
4520:
4518:
4515:
4514:
4498:
4496:
4493:
4492:
4476:
4474:
4471:
4470:
4454:
4453:
4439:
4425:
4418:
4404:
4401:
4400:
4386:
4372:
4355:
4341:
4334:
4314:
4313:
4309:
4307:
4304:
4303:
4300:
4268:
4260:
4255:
4252:
4251:
4235:
4232:
4231:
4215:
4213:
4210:
4209:
4189:
4184:
4183:
4181:
4178:
4177:
4158:
4148:
4143:
4142:
4138:
4137:
4131:
4123:
4122:
4116:
4111:
4110:
4104:
4093:
4077:
4072:
4071:
4069:
4066:
4065:
4048:
4040:
4039:
4030:
4025:
4024:
4015:
4010:
4009:
4004:
3992:
3988:
3986:
3983:
3982:
3966:
3964:
3961:
3960:
3944:
3941:
3940:
3924:
3922:
3919:
3918:
3899:
3891:
3886:
3883:
3882:
3860:
3852:
3847:
3844:
3843:
3827:
3826:
3809:
3789:
3782:
3768:
3765:
3764:
3747:
3727:
3720:
3700:
3696:
3694:
3691:
3690:
3687:
3663:
3655:
3650:
3647:
3646:
3618:
3613:
3612:
3610:
3607:
3606:
3585:
3574:
3569:
3568:
3567:
3563:
3557:
3552:
3551:
3550:
3546:
3544:
3541:
3540:
3513:
3508:
3498:
3487:
3482:
3481:
3480:
3476:
3470:
3465:
3464:
3463:
3459:
3452:
3447:
3428:
3417:
3412:
3411:
3410:
3406:
3400:
3392:
3391:
3381:
3380:
3376:
3370:
3365:
3364:
3358:
3353:
3341:
3335:
3330:
3329:
3323:
3318:
3317:
3315:
3312:
3311:
3291:
3286:
3285:
3283:
3280:
3279:
3261:
3246:
3241:
3240:
3228:
3223:
3222:
3211:
3209:
3206:
3205:
3183:
3180:
3179:
3152:
3151:
3147:
3145:
3142:
3141:
3125:
3116:
3105:
3089:
3084:
3083:
3082:
3078:
3064:
3059:
3058:
3046:
3041:
3040:
3039:
3035:
3025:
3024:
3020:
3019:
3015:
3012:
3009:
3008:
2992:
2989:
2988:
2972:
2969:
2968:
2950:
2930:
2926:
2925:
2924:
2920:
2906:
2901:
2900:
2888:
2883:
2882:
2881:
2877:
2867:
2866:
2862:
2861:
2857:
2847:
2842:
2828:
2823:
2822:
2821:
2817:
2806:
2801:
2800:
2788:
2783:
2782:
2770:
2762:
2761:
2746:
2741:
2740:
2728:
2723:
2722:
2714:
2713:
2709:
2699:
2698:
2694:
2692:
2689:
2688:
2666:
2661:
2660:
2659:
2655:
2653:
2650:
2649:
2628:
2627:
2623:
2621:
2618:
2617:
2616:Multiplying by
2599:
2590:
2585:
2584:
2572:
2567:
2566:
2554:
2549:
2548:
2530:
2525:
2524:
2512:
2507:
2506:
2498:
2496:
2493:
2492:
2472:
2467:
2466:
2464:
2461:
2460:
2440:
2435:
2434:
2432:
2429:
2428:
2411:
2403:
2402:
2390:
2385:
2384:
2378:
2373:
2372:
2363:
2358:
2357:
2352:
2350:
2347:
2346:
2329:
2324:
2323:
2321:
2318:
2317:
2301:
2299:
2296:
2295:
2275:
2270:
2269:
2267:
2264:
2263:
2245:
2241:
2237:
2235:
2232:
2231:
2228:
2206:
2201:
2200:
2198:
2195:
2194:
2178:
2176:
2173:
2172:
2150:
2147:
2146:
2119:
2118:
2114:
2112:
2109:
2108:
2092:
2091:
2085:
2074:
2073:
2064:
2063:
2057:
2049:
2048:
2040:
2031:
2030:
2024:
2013:
1997:
1992:
1991:
1990:
1986:
1980:
1975:
1974:
1969:
1959:
1958:
1954:
1953:
1950:
1940:
1939:
1917:
1912:
1911:
1910:
1906:
1900:
1895:
1894:
1889:
1879:
1878:
1874:
1873:
1869:
1859:
1854:
1848:
1843:
1830:
1829:
1817:
1812:
1811:
1800:
1795:
1794:
1793:
1789:
1782:
1778:
1774:
1764:
1763:
1759:
1753:
1748:
1724:
1719:
1718:
1717:
1713:
1707:
1699:
1698:
1688:
1687:
1683:
1678:
1672:
1667:
1656:
1650:
1645:
1644:
1638:
1633:
1632:
1623:
1618:
1617:
1612:
1608:
1606:
1603:
1602:
1574:
1569:
1568:
1567:
1563:
1557:
1549:
1548:
1538:
1537:
1533:
1527:
1522:
1508:
1504:
1503:
1501:
1498:
1497:
1478:
1470:
1469:
1457:
1452:
1451:
1445:
1440:
1439:
1430:
1425:
1424:
1419:
1417:
1414:
1413:
1406:
1371:
1366:
1365:
1364:
1360:
1354:
1346:
1345:
1335:
1334:
1330:
1324:
1319:
1305:
1301:
1300:
1298:
1295:
1294:
1277:
1269:
1268:
1256:
1251:
1250:
1244:
1239:
1238:
1229:
1224:
1223:
1218:
1216:
1213:
1212:
1193:
1191:
1188:
1187:
1171:
1169:
1166:
1165:
1145:
1143:
1140:
1139:
1123:
1120:
1119:
1101:
1100:
1095:
1093:
1088:
1077:
1070:
1069:
1067:
1064:
1063:
1029:
1026:
1025:
1004:
1003:
998:
986:
981:
980:
978:
973:
968:
962:
957:
956:
954:
946:
944:
939:
932:
931:
922:
921:
919:
916:
915:
890:
887:
886:
861:
858:
857:
817:
812:
811:
810:
806:
799:
795:
791:
769:
768:
764:
758:
753:
729:
724:
723:
722:
718:
711:
707:
703:
693:
692:
688:
682:
677:
654:
650:
649:
647:
644:
643:
621:
618:
617:
595:
587:
582:
579:
578:
555:
547:
542:
539:
538:
519:
517:
514:
513:
497:
495:
492:
491:
475:
473:
470:
469:
453:
451:
448:
447:
444:
413:
410:
409:
393:
390:
389:
373:
370:
369:
347:
344:
343:
321:
318:
317:
295:
292:
291:
269:
266:
265:
249:
247:
244:
243:
227:
225:
222:
221:
205:
203:
200:
199:
183:
181:
178:
177:
159:
158:
141:
121:
114:
100:
97:
96:
79:
59:
52:
32:
31:
27:
25:
22:
21:
12:
11:
5:
5569:
5559:
5558:
5556:Control theory
5544:
5543:
5536:
5535:External links
5533:
5532:
5531:
5525:
5506:
5503:
5502:
5501:
5496:
5491:
5489:Gramian matrix
5486:
5481:
5474:
5471:
5448:
5445:
5442:
5439:
5436:
5432:
5428:
5425:
5422:
5417:
5413:
5409:
5405:
5401:
5398:
5395:
5392:
5387:
5383:
5379:
5375:
5354:
5349:
5345:
5341:
5336:
5332:
5328:
5323:
5318:
5296:
5293:
5290:
5285:
5281:
5277:
5272:
5267:
5262:
5259:
5256:
5251:
5247:
5243:
5238:
5233:
5228:
5225:
5222:
5217:
5213:
5209:
5205:
5201:
5198:
5193:
5189:
5185:
5182:
5179:
5174:
5169:
5164:
5161:
5156:
5152:
5148:
5143:
5139:
5135:
5130:
5125:
5103:
5098:
5094:
5090:
5085:
5081:
5077:
5072:
5067:
5053:
5049:
5042:
5035:
5029:Properties of
5027:
5010:
5006:
5003:
5000:
4996:
4992:
4986:
4983:
4960:
4957:
4954:
4951:
4948:
4944:
4923:
4920:
4917:
4914:
4911:
4908:
4903:
4899:
4895:
4890:
4885:
4880:
4877:
4874:
4869:
4864:
4859:
4856:
4853:
4849:
4845:
4842:
4839:
4834:
4830:
4826:
4822:
4814:
4810:
4802:
4798:
4793:
4789:
4786:
4781:
4777:
4773:
4768:
4764:
4760:
4755:
4750:
4728:
4725:
4722:
4713:such that the
4700:
4696:
4692:
4687:
4683:
4660:
4656:
4635:
4632:
4629:
4626:
4622:
4618:
4615:
4612:
4609:
4605:
4601:
4578:
4575:
4572:
4569:
4565:
4561:
4558:
4555:
4552:
4548:
4544:
4523:
4501:
4479:
4452:
4449:
4446:
4442:
4438:
4435:
4432:
4428:
4424:
4421:
4419:
4417:
4414:
4411:
4407:
4403:
4402:
4399:
4396:
4393:
4389:
4385:
4382:
4379:
4375:
4371:
4368:
4365:
4362:
4358:
4354:
4351:
4348:
4344:
4340:
4337:
4335:
4333:
4330:
4327:
4321:
4318:
4312:
4311:
4299:
4296:
4275:
4271:
4267:
4263:
4259:
4239:
4218:
4195:
4192:
4187:
4161:
4156:
4151:
4146:
4141:
4134:
4129:
4126:
4119:
4114:
4107:
4102:
4099:
4096:
4092:
4088:
4083:
4080:
4075:
4051:
4046:
4043:
4038:
4033:
4028:
4021:
4018:
4013:
4007:
4003:
3998:
3995:
3991:
3969:
3948:
3927:
3906:
3902:
3898:
3894:
3890:
3867:
3863:
3859:
3855:
3851:
3825:
3822:
3819:
3815:
3812:
3808:
3805:
3802:
3799:
3795:
3792:
3788:
3785:
3783:
3781:
3778:
3775:
3771:
3767:
3766:
3763:
3760:
3757:
3753:
3750:
3746:
3743:
3740:
3737:
3733:
3730:
3726:
3723:
3721:
3719:
3716:
3713:
3710:
3707:
3703:
3699:
3698:
3686:
3683:
3670:
3666:
3662:
3658:
3654:
3621:
3616:
3593:
3588:
3582:
3577:
3572:
3566:
3560:
3555:
3549:
3524:
3521:
3516:
3511:
3506:
3501:
3495:
3490:
3485:
3479:
3473:
3468:
3462:
3455:
3450:
3446:
3442:
3439:
3436:
3431:
3425:
3420:
3415:
3409:
3403:
3398:
3395:
3388:
3384:
3379:
3373:
3368:
3361:
3356:
3352:
3348:
3344:
3338:
3333:
3326:
3321:
3294:
3289:
3264:
3260:
3257:
3252:
3249:
3244:
3239:
3234:
3231:
3226:
3221:
3218:
3214:
3193:
3190:
3187:
3167:
3164:
3159:
3155:
3150:
3128:
3124:
3119:
3114:
3111:
3108:
3103:
3097:
3092:
3087:
3081:
3076:
3070:
3067:
3062:
3057:
3052:
3049:
3044:
3038:
3032:
3028:
3023:
3018:
2996:
2976:
2953:
2949:
2945:
2939:
2933:
2929:
2923:
2918:
2912:
2909:
2904:
2899:
2894:
2891:
2886:
2880:
2874:
2870:
2865:
2860:
2853:
2850:
2846:
2841:
2836:
2831:
2826:
2820:
2815:
2809:
2804:
2799:
2794:
2791:
2786:
2781:
2776:
2773:
2768:
2765:
2760:
2757:
2752:
2749:
2744:
2739:
2734:
2731:
2726:
2721:
2717:
2712:
2706:
2702:
2697:
2674:
2669:
2664:
2658:
2635:
2631:
2626:
2602:
2598:
2593:
2588:
2583:
2578:
2575:
2570:
2565:
2560:
2557:
2552:
2547:
2544:
2541:
2536:
2533:
2528:
2523:
2518:
2515:
2510:
2505:
2501:
2478:
2475:
2470:
2446:
2443:
2438:
2414:
2409:
2406:
2401:
2398:
2393:
2388:
2381:
2376:
2371:
2366:
2361:
2355:
2332:
2327:
2304:
2278:
2273:
2248:
2244:
2240:
2227:
2224:
2209:
2204:
2181:
2160:
2157:
2154:
2134:
2131:
2126:
2122:
2117:
2088:
2083:
2080:
2077:
2072:
2069:
2067:
2065:
2060:
2055:
2052:
2047:
2043:
2039:
2036:
2034:
2032:
2027:
2022:
2019:
2016:
2011:
2005:
2000:
1995:
1989:
1983:
1978:
1972:
1966:
1962:
1957:
1952:
1948:
1945:
1943:
1941:
1938:
1935:
1931:
1925:
1920:
1915:
1909:
1903:
1898:
1892:
1886:
1882:
1877:
1872:
1865:
1862:
1858:
1851:
1846:
1842:
1838:
1835:
1833:
1831:
1828:
1825:
1820:
1815:
1808:
1803:
1798:
1792:
1785:
1781:
1777:
1771:
1767:
1762:
1756:
1751:
1747:
1743:
1740:
1737:
1732:
1727:
1722:
1716:
1710:
1705:
1702:
1695:
1691:
1686:
1681:
1675:
1670:
1666:
1662:
1659:
1657:
1653:
1648:
1641:
1636:
1631:
1626:
1621:
1615:
1611:
1610:
1590:
1587:
1582:
1577:
1572:
1566:
1560:
1555:
1552:
1545:
1541:
1536:
1530:
1525:
1521:
1517:
1511:
1507:
1481:
1476:
1473:
1468:
1465:
1460:
1455:
1448:
1443:
1438:
1433:
1428:
1422:
1405:
1402:
1387:
1384:
1379:
1374:
1369:
1363:
1357:
1352:
1349:
1342:
1338:
1333:
1327:
1322:
1318:
1314:
1308:
1304:
1280:
1275:
1272:
1267:
1264:
1259:
1254:
1247:
1242:
1237:
1232:
1227:
1221:
1196:
1174:
1162:
1161:
1148:
1127:
1105:
1098:
1094:
1091:
1087:
1084:
1080:
1076:
1075:
1073:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1022:
1008:
1001:
995:
992:
989:
984:
979:
977:
974:
971:
965:
960:
955:
952:
949:
945:
942:
938:
937:
935:
930:
925:
903:
900:
897:
894:
883:
871:
868:
865:
845:
842:
837:
834:
831:
828:
825:
820:
815:
809:
802:
798:
794:
788:
785:
782:
779:
776:
772:
767:
761:
756:
752:
748:
745:
742:
737:
732:
727:
721:
714:
710:
706:
700:
696:
691:
685:
680:
676:
672:
669:
666:
663:
657:
653:
631:
628:
625:
614:
602:
598:
594:
590:
586:
562:
558:
554:
550:
546:
522:
500:
478:
456:
443:
440:
417:
397:
377:
357:
354:
351:
331:
328:
325:
305:
302:
299:
279:
276:
273:
252:
230:
208:
186:
157:
154:
151:
147:
144:
140:
137:
134:
131:
127:
124:
120:
117:
115:
113:
110:
107:
103:
99:
98:
95:
92:
89:
85:
82:
78:
75:
72:
69:
65:
62:
58:
55:
53:
51:
48:
45:
39:
36:
30:
29:
18:control theory
9:
6:
4:
3:
2:
5568:
5557:
5554:
5553:
5551:
5542:
5539:
5538:
5528:
5526:0-19-511777-8
5522:
5517:
5516:
5509:
5508:
5500:
5497:
5495:
5492:
5490:
5487:
5485:
5482:
5480:
5477:
5476:
5470:
5468:
5464:
5459:
5443:
5440:
5437:
5423:
5420:
5415:
5411:
5399:
5393:
5390:
5385:
5381:
5347:
5343:
5339:
5334:
5330:
5321:
5291:
5288:
5283:
5279:
5270:
5257:
5254:
5249:
5245:
5236:
5223:
5220:
5215:
5211:
5199:
5191:
5187:
5183:
5180:
5172:
5162:
5154:
5150:
5146:
5141:
5137:
5128:
5096:
5092:
5088:
5083:
5079:
5070:
5048:
5041:
5033:
5026:
5023:
5001:
4990:
4955:
4952:
4949:
4921:
4918:
4915:
4909:
4906:
4901:
4897:
4888:
4875:
4867:
4854:
4840:
4837:
4832:
4828:
4812:
4808:
4800:
4796:
4791:
4787:
4779:
4775:
4771:
4766:
4762:
4753:
4726:
4723:
4720:
4698:
4694:
4690:
4685:
4681:
4658:
4654:
4627:
4616:
4610:
4590:
4570:
4559:
4553:
4467:
4447:
4433:
4422:
4420:
4412:
4394:
4380:
4369:
4363:
4349:
4338:
4336:
4328:
4319:
4295:
4293:
4289:
4265:
4250:, the system
4237:
4193:
4190:
4174:
4159:
4154:
4149:
4139:
4132:
4117:
4100:
4097:
4094:
4090:
4086:
4081:
4078:
4049:
4036:
4031:
4019:
4016:
4001:
3996:
3993:
3989:
3946:
3896:
3879:
3857:
3840:
3820:
3806:
3800:
3786:
3784:
3776:
3758:
3744:
3738:
3724:
3722:
3714:
3711:
3708:
3682:
3660:
3644:
3640:
3635:
3619:
3580:
3575:
3564:
3558:
3538:
3522:
3519:
3514:
3509:
3493:
3488:
3477:
3471:
3448:
3444:
3440:
3437:
3434:
3423:
3418:
3407:
3401:
3386:
3377:
3371:
3354:
3350:
3346:
3336:
3324:
3308:
3292:
3276:
3258:
3250:
3247:
3237:
3232:
3229:
3216:
3185:
3165:
3157:
3148:
3122:
3112:
3109:
3106:
3101:
3095:
3090:
3079:
3074:
3068:
3065:
3055:
3050:
3047:
3036:
3030:
3021:
3016:
2974:
2965:
2947:
2943:
2937:
2921:
2916:
2910:
2907:
2897:
2892:
2889:
2878:
2872:
2863:
2858:
2851:
2848:
2844:
2839:
2834:
2829:
2818:
2813:
2807:
2792:
2789:
2779:
2774:
2771:
2758:
2750:
2747:
2737:
2732:
2729:
2710:
2704:
2695:
2672:
2667:
2656:
2633:
2624:
2614:
2596:
2591:
2576:
2573:
2563:
2558:
2555:
2542:
2534:
2531:
2521:
2516:
2513:
2476:
2473:
2444:
2441:
2412:
2399:
2396:
2391:
2379:
2369:
2364:
2330:
2292:
2276:
2223:
2207:
2155:
2152:
2132:
2129:
2124:
2115:
2105:
2086:
2070:
2068:
2058:
2045:
2037:
2035:
2020:
2017:
2014:
2009:
2003:
1998:
1987:
1981:
1964:
1955:
1946:
1944:
1936:
1933:
1929:
1923:
1918:
1907:
1901:
1884:
1875:
1870:
1863:
1860:
1856:
1844:
1840:
1836:
1834:
1826:
1823:
1818:
1806:
1801:
1790:
1769:
1760:
1749:
1745:
1741:
1738:
1735:
1730:
1725:
1714:
1708:
1693:
1684:
1668:
1664:
1660:
1658:
1651:
1639:
1629:
1624:
1588:
1585:
1580:
1575:
1564:
1558:
1543:
1534:
1523:
1519:
1515:
1494:
1479:
1466:
1463:
1458:
1446:
1436:
1431:
1411:
1401:
1398:
1385:
1382:
1377:
1372:
1361:
1355:
1340:
1331:
1320:
1316:
1312:
1278:
1265:
1262:
1257:
1245:
1235:
1230:
1211:
1125:
1103:
1085:
1082:
1071:
1046:
1043:
1040:
1034:
1031:
1023:
1006:
993:
990:
987:
975:
963:
933:
928:
901:
898:
895:
892:
884:
869:
866:
863:
843:
840:
832:
829:
826:
818:
807:
783:
780:
777:
765:
759:
754:
750:
746:
743:
740:
735:
730:
719:
698:
689:
683:
678:
674:
670:
664:
629:
626:
623:
615:
592:
576:
575:
574:
552:
535:
439:
437:
436:
429:
415:
395:
375:
355:
352:
349:
329:
326:
323:
303:
300:
297:
277:
274:
271:
175:
152:
138:
132:
118:
116:
108:
90:
76:
70:
56:
54:
46:
37:
19:
5514:
5460:
5055:
5046:
5039:
5031:
5024:
4591:
4468:
4301:
4175:
3880:
3841:
3688:
3636:
3536:
3309:
3277:
2966:
2615:
2293:
2229:
2106:
1495:
1407:
1399:
1163:
536:
445:
432:
430:
174:controllable
15:
4592:The system
2171:for stable
1021:has rank n.
5505:References
5465:, p.
5463:Chen (1999
4290:, p.
4288:Chen (1999
3641:, p.
3639:Chen (1999
2226:Properties
5444:τ
5431:Φ
5404:Φ
5394:τ
5374:Φ
5266:Φ
5204:Φ
4985:˙
4956:τ
4943:Φ
4919:τ
4910:τ
4884:Φ
4876:τ
4855:τ
4841:τ
4821:Φ
4792:∫
4724:×
4320:˙
4106:∞
4091:∑
4002:−
3454:∞
3445:∫
3360:∞
3351:∫
3238:−
3217:−
3192:∞
3189:→
3163:→
3118:∞
3056:−
2995:∞
2898:−
2780:−
2738:−
2564:−
2522:−
2400:−
2159:∞
2076:−
2046:−
2026:∞
1937:τ
1924:τ
1885:τ
1864:τ
1850:∞
1841:∫
1827:τ
1807:τ
1770:τ
1755:∞
1746:∫
1739:τ
1731:τ
1694:τ
1674:∞
1665:∫
1589:τ
1581:τ
1544:τ
1529:∞
1520:∫
1467:−
1412:given by
1386:τ
1378:τ
1341:τ
1326:∞
1317:∫
1266:−
1126:λ
1086:λ
1083:−
1035:×
991:−
976:⋯
896:×
844:τ
833:τ
830:−
784:τ
781:−
751:∫
744:τ
736:τ
699:τ
675:∫
627:×
577:The pair
428:outputs.
353:×
327:×
301:×
275:×
38:˙
5550:Category
5473:See also
3592:‖
3548:‖
3505:‖
3461:‖
388:inputs,
176:, where
1062:matrix
642:matrix
435:Gramian
5523:
4934:where
5521:ISBN
4691:>
4513:and
2459:and
1024:The
885:The
867:>
616:The
512:and
342:and
242:and
5469:).
5467:176
4294:).
4292:169
3643:145
3178:as
2987:to
2145:at
1138:of
172:is
16:In
5552::
4491:,
3204::
3007::
2291:.
490:,
468:,
438:.
316:,
290:,
220:,
198:,
5529:.
5447:)
5441:,
5438:t
5435:(
5427:)
5424:t
5421:,
5416:1
5412:t
5408:(
5400:=
5397:)
5391:,
5386:1
5382:t
5378:(
5353:)
5348:1
5344:t
5340:,
5335:0
5331:t
5327:(
5322:c
5317:W
5295:)
5292:t
5289:,
5284:1
5280:t
5276:(
5271:T
5261:)
5258:t
5255:,
5250:0
5246:t
5242:(
5237:c
5232:W
5227:)
5224:t
5221:,
5216:1
5212:t
5208:(
5200:+
5197:)
5192:1
5188:t
5184:,
5181:t
5178:(
5173:c
5168:W
5163:=
5160:)
5155:1
5151:t
5147:,
5142:0
5138:t
5134:(
5129:c
5124:W
5102:)
5097:1
5093:t
5089:,
5084:0
5080:t
5076:(
5071:c
5066:W
5052:)
5050:1
5047:t
5045:,
5043:0
5040:t
5038:(
5036:c
5032:W
5009:x
5005:)
5002:t
4999:(
4995:A
4991:=
4982:x
4959:)
4953:,
4950:t
4947:(
4922:,
4916:d
4913:)
4907:,
4902:1
4898:t
4894:(
4889:T
4879:)
4873:(
4868:T
4863:B
4858:)
4852:(
4848:B
4844:)
4838:,
4833:1
4829:t
4825:(
4813:1
4809:t
4801:0
4797:t
4788:=
4785:)
4780:1
4776:t
4772:,
4767:0
4763:t
4759:(
4754:c
4749:W
4727:n
4721:n
4699:0
4695:t
4686:1
4682:t
4659:0
4655:t
4634:)
4631:)
4628:t
4625:(
4621:B
4617:,
4614:)
4611:t
4608:(
4604:A
4600:(
4577:)
4574:)
4571:t
4568:(
4564:B
4560:,
4557:)
4554:t
4551:(
4547:A
4543:(
4522:C
4500:B
4478:A
4451:)
4448:t
4445:(
4441:x
4437:)
4434:t
4431:(
4427:C
4423:=
4416:)
4413:t
4410:(
4406:y
4398:)
4395:t
4392:(
4388:u
4384:)
4381:t
4378:(
4374:B
4370:+
4367:)
4364:t
4361:(
4357:x
4353:)
4350:t
4347:(
4343:A
4339:=
4332:)
4329:t
4326:(
4317:x
4274:)
4270:B
4266:,
4262:A
4258:(
4238:1
4217:A
4194:c
4191:d
4186:W
4160:m
4155:)
4150:T
4145:A
4140:(
4133:T
4128:B
4125:B
4118:m
4113:A
4101:0
4098:=
4095:m
4087:=
4082:c
4079:d
4074:W
4050:T
4045:B
4042:B
4037:=
4032:T
4027:A
4020:c
4017:d
4012:W
4006:A
3997:c
3994:d
3990:W
3968:A
3959:(
3947:1
3926:A
3905:)
3901:B
3897:,
3893:A
3889:(
3866:)
3862:B
3858:,
3854:A
3850:(
3824:]
3821:k
3818:[
3814:u
3811:D
3807:+
3804:]
3801:k
3798:[
3794:x
3791:C
3787:=
3780:]
3777:k
3774:[
3770:y
3762:]
3759:k
3756:[
3752:u
3749:B
3745:+
3742:]
3739:k
3736:[
3732:x
3729:A
3725:=
3718:]
3715:1
3712:+
3709:k
3706:[
3702:x
3669:)
3665:B
3661:,
3657:A
3653:(
3620:c
3615:W
3587:x
3581:t
3576:T
3571:A
3565:e
3559:T
3554:B
3537:t
3523:t
3520:d
3515:2
3510:2
3500:x
3494:t
3489:T
3484:A
3478:e
3472:T
3467:B
3449:0
3441:=
3438:t
3435:d
3430:x
3424:t
3419:T
3414:A
3408:e
3402:T
3397:B
3394:B
3387:t
3383:A
3378:e
3372:T
3367:x
3355:0
3347:=
3343:x
3337:c
3332:W
3325:T
3320:x
3293:c
3288:W
3263:0
3259:=
3256:)
3251:2
3248:c
3243:W
3233:1
3230:c
3225:W
3220:(
3213:0
3186:t
3166:0
3158:t
3154:A
3149:e
3127:0
3123:=
3113:0
3110:=
3107:t
3102:]
3096:t
3091:T
3086:A
3080:e
3075:)
3069:2
3066:c
3061:W
3051:1
3048:c
3043:W
3037:(
3031:t
3027:A
3022:e
3017:[
2975:0
2952:0
2948:=
2944:]
2938:t
2932:T
2928:A
2922:e
2917:)
2911:2
2908:c
2903:W
2893:1
2890:c
2885:W
2879:(
2873:t
2869:A
2864:e
2859:[
2852:t
2849:d
2845:d
2840:=
2835:t
2830:T
2825:A
2819:e
2814:]
2808:T
2803:A
2798:)
2793:2
2790:c
2785:W
2775:1
2772:c
2767:W
2764:(
2759:+
2756:)
2751:2
2748:c
2743:W
2733:1
2730:c
2725:W
2720:(
2716:A
2711:[
2705:t
2701:A
2696:e
2673:t
2668:T
2663:A
2657:e
2634:t
2630:A
2625:e
2601:0
2597:=
2592:T
2587:A
2582:)
2577:2
2574:c
2569:W
2559:1
2556:c
2551:W
2546:(
2543:+
2540:)
2535:2
2532:c
2527:W
2517:1
2514:c
2509:W
2504:(
2500:A
2477:2
2474:c
2469:W
2445:1
2442:c
2437:W
2413:T
2408:B
2405:B
2397:=
2392:T
2387:A
2380:c
2375:W
2370:+
2365:c
2360:W
2354:A
2331:c
2326:W
2303:A
2277:c
2272:W
2247:T
2243:B
2239:B
2208:c
2203:W
2180:A
2156:=
2153:t
2133:0
2130:=
2125:t
2121:A
2116:e
2087:T
2082:B
2079:B
2071:=
2059:T
2054:B
2051:B
2042:0
2038:=
2021:0
2018:=
2015:t
2010:|
2004:t
1999:T
1994:A
1988:e
1982:T
1977:B
1971:B
1965:t
1961:A
1956:e
1947:=
1934:d
1930:)
1919:T
1914:A
1908:e
1902:T
1897:B
1891:B
1881:A
1876:e
1871:(
1861:d
1857:d
1845:0
1837:=
1824:d
1819:T
1814:A
1802:T
1797:A
1791:e
1784:T
1780:B
1776:B
1766:A
1761:e
1750:0
1742:+
1736:d
1726:T
1721:A
1715:e
1709:T
1704:B
1701:B
1690:A
1685:e
1680:A
1669:0
1661:=
1652:T
1647:A
1640:c
1635:W
1630:+
1625:c
1620:W
1614:A
1586:d
1576:T
1571:A
1565:e
1559:T
1554:B
1551:B
1540:A
1535:e
1524:0
1516:=
1510:c
1506:W
1480:T
1475:B
1472:B
1464:=
1459:T
1454:A
1447:c
1442:W
1437:+
1432:c
1427:W
1421:A
1383:d
1373:T
1368:A
1362:e
1356:T
1351:B
1348:B
1337:A
1332:e
1321:0
1313:=
1307:c
1303:W
1279:T
1274:B
1271:B
1263:=
1258:T
1253:A
1246:c
1241:W
1236:+
1231:c
1226:W
1220:A
1195:A
1173:A
1160:.
1147:A
1104:]
1097:B
1090:I
1079:A
1072:[
1050:)
1047:p
1044:+
1041:n
1038:(
1032:n
1007:]
1000:B
994:1
988:n
983:A
970:B
964:2
959:A
951:B
948:A
941:B
934:[
929:=
924:C
902:p
899:n
893:n
882:.
870:0
864:t
841:d
836:)
827:t
824:(
819:T
814:A
808:e
801:T
797:B
793:B
787:)
778:t
775:(
771:A
766:e
760:t
755:0
747:=
741:d
731:T
726:A
720:e
713:T
709:B
705:B
695:A
690:e
684:t
679:0
671:=
668:)
665:t
662:(
656:c
652:W
630:n
624:n
601:)
597:B
593:,
589:A
585:(
561:)
557:B
553:,
549:A
545:(
521:D
499:C
477:B
455:A
416:q
396:n
376:p
356:p
350:q
330:n
324:q
304:p
298:n
278:n
272:n
251:D
229:C
207:B
185:A
156:)
153:t
150:(
146:u
143:D
139:+
136:)
133:t
130:(
126:x
123:C
119:=
112:)
109:t
106:(
102:y
94:)
91:t
88:(
84:u
81:B
77:+
74:)
71:t
68:(
64:x
61:A
57:=
50:)
47:t
44:(
35:x
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