1942:
1450:
1937:{\displaystyle {\begin{array}{ccccc}{\boldsymbol {A^{T}}}{\boldsymbol {W}}_{o}+{\boldsymbol {W}}_{o}{\boldsymbol {A}}&=&\int _{0}^{\infty }{\boldsymbol {A^{T}}}e^{{\boldsymbol {A^{T}}}\tau }{\boldsymbol {C^{T}C}}e^{{\boldsymbol {A}}\tau }d\tau &+&\int _{0}^{\infty }e^{{\boldsymbol {A^{T}}}\tau }{\boldsymbol {C^{T}C}}e^{{\boldsymbol {A}}\tau }{\boldsymbol {A}}d\tau \\&=&\int _{0}^{\infty }{\frac {d}{d\tau }}(e^{{\boldsymbol {A^{T}}}\tau }{\boldsymbol {C}}^{T}{\boldsymbol {C}}e^{{\boldsymbol {A}}\tau })d\tau &=&e^{{\boldsymbol {A^{T}}}t}{\boldsymbol {C}}^{T}{\boldsymbol {C}}e^{{\boldsymbol {A}}t}|_{t=0}^{\infty }\\&=&{\boldsymbol {0}}-{\boldsymbol {C^{T}C}}\\&=&{\boldsymbol {-C^{T}C}}\end{array}}}
5472:
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164:
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1234:
3353:{\displaystyle {\boldsymbol {x^{T}W_{o}x}}=\int _{0}^{\infty }{\boldsymbol {x}}^{T}e^{{\boldsymbol {A^{T}}}t}{\boldsymbol {C^{T}C}}e^{{\boldsymbol {A}}t}{\boldsymbol {x}}dt=\int _{0}^{\infty }\left\Vert {\boldsymbol {Ce^{{\boldsymbol {A}}t}{\boldsymbol {x}}}}\right\Vert _{2}^{2}dt}
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25:
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571:
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4266:{\displaystyle {\begin{array}{c}{\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{array}}}
3887:
1341:
1134:
4738:{\displaystyle {\boldsymbol {W}}_{o}(t_{0},t_{1})=\int _{t_{0}}^{^{t_{1}}}{\boldsymbol {\Phi }}^{T}(\tau ,t_{0}){\boldsymbol {C}}^{T}(\tau ){\boldsymbol {C}}(\tau ){\boldsymbol {\Phi }}(\tau ,t_{0})d\tau }
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1982:
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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
159:{\displaystyle {\begin{array}{c}{\dot {\boldsymbol {x}}}(t){\boldsymbol {=Ax}}(t)+{\boldsymbol {Bu}}(t)\\{\boldsymbol {y}}(t)={\boldsymbol {Cx}}(t)+{\boldsymbol {Du}}(t)\end{array}}}
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5149:{\displaystyle {\boldsymbol {W}}_{o}(t_{0},t_{1})={\boldsymbol {W}}_{o}(t_{0},t)+{\boldsymbol {\Phi }}^{T}(t,t_{0}){\boldsymbol {W}}_{o}(t,t_{0}){\boldsymbol {\Phi }}(t,t_{0})}
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994:
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254:
232:
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888:
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That is called the discrete
Observability Gramian. We can easily see the correspondence between discrete time and the continuous time case, that is, if we can check that
3051:
742:
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2537:
2008:
564:
358:
332:
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280:
2452:{\displaystyle {\boldsymbol {A^{T}}}{\boldsymbol {(W}}_{o1}-{\boldsymbol {W}}_{o2})+{\boldsymbol {(W}}_{o1}-{\boldsymbol {W}}_{o2}){\boldsymbol {A}}={\boldsymbol {0}}}
972:
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710:
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950:
845:
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3644:{\displaystyle {\begin{array}{c}{\boldsymbol {x}}{\boldsymbol {=Ax}}+{\boldsymbol {Bu}}\\{\boldsymbol {y}}={\boldsymbol {Cx}}+{\boldsymbol {Du}}\end{array}}}
677:{\displaystyle {\boldsymbol {W_{o}}}(t)=\int _{0}^{t}e^{{\boldsymbol {A}}^{T}\tau }{\boldsymbol {C}}^{T}{\boldsymbol {C}}e^{{\boldsymbol {A}}\tau }d\tau }
4787:
3987:{\displaystyle {\boldsymbol {W}}_{do}=\sum _{m=0}^{\infty }({\boldsymbol {A}}^{T})^{m}{\boldsymbol {C}}^{T}{\boldsymbol {C}}{\boldsymbol {A}}^{m}}
1436:{\displaystyle {\boldsymbol {W_{o}}}=\int _{0}^{\infty }e^{{\boldsymbol {A^{T}}}\tau }{\boldsymbol {C^{T}C}}e^{{\boldsymbol {A}}\tau }d\tau }
1229:{\displaystyle {\boldsymbol {W_{o}}}=\int _{0}^{\infty }e^{{\boldsymbol {A}}^{T}\tau }{\boldsymbol {C^{T}C}}e^{{\boldsymbol {A}}\tau }d\tau }
5536:
5508:
4401:
4344:
5489:
1129:
is positive definite, then the system is observable. The solution is called the
Observability Gramian and can be expressed as
5515:
3462:
More properties of observable systems can be found in, as well as the proof for the other equivalent statements of "The pair
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3695:
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3465:
504:
465:
2985:
2850:
2261:{\displaystyle {\boldsymbol {A^{T}}}{\boldsymbol {W}}_{o}+{\boldsymbol {W}}_{o}{\boldsymbol {A}}=-{\boldsymbol {C^{T}C}}}
1327:{\displaystyle {\boldsymbol {A^{T}}}{\boldsymbol {W}}_{o}+{\boldsymbol {W}}_{o}{\boldsymbol {A}}=-{\boldsymbol {C^{T}C}}}
1120:{\displaystyle {\boldsymbol {A^{T}}}{\boldsymbol {W}}_{o}+{\boldsymbol {W}}_{o}{\boldsymbol {A}}=-{\boldsymbol {C^{T}C}}}
5161:
4906:
4845:
5522:
5319:{\displaystyle {\boldsymbol {\Phi }}(t_{0},t_{1})={\boldsymbol {\Phi }}(t_{1},\tau ){\boldsymbol {\Phi }}(\tau ,t_{0})}
5555:
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5388:
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1949:
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3873:{\displaystyle {\boldsymbol {A^{T}}}{\boldsymbol {W}}_{do}{\boldsymbol {A}}-W_{do}=-{\boldsymbol {C^{T}C}}}
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2013:
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3113:{\displaystyle {\boldsymbol {0}}-({\boldsymbol {W}}_{o1}-{\boldsymbol {W}}_{o2})={\boldsymbol {0}}}
855:
2796:{\displaystyle e^{{\boldsymbol {A^{T}}}t}e^{{\boldsymbol {A}}t}={\frac {d}{dt}}={\boldsymbol {0}}}
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One of the many ways one can achieve such goal is by the use of the
Observability Gramian.
1238:
In the following section we are going to take a closer look at the
Observability Gramian.
462:
One can determine if the LTI system is or is not observable simply by looking at the pair
8:
4838:
Again, we have a similar method to determine if a system is or not an observable system.
3423:{\displaystyle {\displaystyle {\boldsymbol {Ce^{{\boldsymbol {A}}t}{\boldsymbol {x}}}}}}
4053:
3753:
2808:
5450:
5439:
5417:
5406:
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is observable or not. This can be done in a very similar way of the preceding cases.
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is observable" (the equivalences are much alike for the continuous time case).
17:
4828:{\displaystyle {\boldsymbol {\dot {x}}}={\boldsymbol {A}}(t){\boldsymbol {x}}}
371:
Linear Time
Invariant (LTI) Systems are those systems in which the parameters
5583:
5337:
2184:
is unique. In order to prove so, suppose we have two different solutions for
3653:
One can check that there are equivalences for the statement "The pair
5216:
and by the property of the state transition matrix that claims that:
2155:
is stable (all its eigenvalues have negative real part) to show that
5471:
3692:
We are interested in the equivalence that claims that, if "The pair
3498:
is observable" presented in section
Observability in LTI Systems.
2032:(all its eigenvalues have negative real part). This shows us that
2061:
is indeed the solution for the
Lyapunov equation under analysis.
498:. Then, we can say that the following statements are equivalent:
1246:
4445:{\displaystyle ({\boldsymbol {A}}(t),{\boldsymbol {C}}(t))}
4388:{\displaystyle ({\boldsymbol {A}}(t),{\boldsymbol {C}}(t))}
4106:
is observable. More properties and proofs can be found in.
20:, we may need to find out whether or not a system such as
5575:
Mathematica function to compute the observability
Gramian
4545:
4114:
Linear time variant (LTV) systems are those in the form:
5445:. New York, New York: Oxford University Press. p.
5412:. New York, New York: Oxford University Press. p.
5379:. New York, New York: Oxford University Press. p.
5328:
More about the Observability Gramian can be found in.
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4099:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {B}})}
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3721:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {C}})}
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3682:{\displaystyle ({\boldsymbol {A}},{\boldsymbol {C}})}
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28:
5496:. Unsourced material may be challenged and removed.
3013:{\displaystyle e^{{\boldsymbol {A}}t}\rightarrow 0}
2973:{\displaystyle |_{t=0}^{\infty }={\boldsymbol {0}}}
5318:
5209:{\displaystyle {\boldsymbol {W}}_{o}(t_{0},t_{1})}
5208:
5148:
4954:{\displaystyle {\boldsymbol {W}}_{o}(t_{0},t_{1})}
4953:
4893:{\displaystyle {\boldsymbol {W}}_{o}(t_{0},t_{1})}
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4777:{\displaystyle {\boldsymbol {\Phi }}(t,\tau )}
4109:
5441:Linear System Theory and Design Third Edition
5408:Linear System Theory and Design Third Edition
5375:Linear System Theory and Design Third Edition
5158:that can easily be seen by the definition of
4028:is positive definite, and all eigenvalues of
3728:is observable" and all the eigenvalues of
2493:{\displaystyle e^{{\boldsymbol {A^{T}}}t}}
1445:as a solution, we are going to find that:
5556:Learn how and when to remove this message
3430:is not identically zero), and that makes
954:has full column rank at every eigenvalue
3792:is stable), then the unique solution of
3501:
3384:(assuming the non-degenerate case where
2101:is a symmetric matrix, therefore, so is
1977:{\displaystyle e^{{\boldsymbol {A}}t}=0}
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4903:We have that the Observability Gramian
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4021:{\displaystyle {\boldsymbol {W}}_{dc}}
2525:{\displaystyle e^{{\boldsymbol {A}}t}}
2327:{\displaystyle {\boldsymbol {W}}_{o2}}
2295:{\displaystyle {\boldsymbol {W}}_{o1}}
2094:{\displaystyle {\boldsymbol {C^{T}C}}}
1043:is stable) and the unique solution of
4479:if and only if there exists a finite
3452:{\displaystyle {\boldsymbol {W}}_{o}}
3144:{\displaystyle {\boldsymbol {W}}_{o}}
2177:{\displaystyle {\boldsymbol {W}}_{o}}
2123:{\displaystyle {\boldsymbol {W}}_{o}}
2054:{\displaystyle {\boldsymbol {W}}_{o}}
5494:adding citations to reliable sources
5465:
5436:
5403:
5370:
3039:{\displaystyle t\rightarrow \infty }
1336:In fact, we can see that if we take
999:If, in addition, all eigenvalues of
459:are invariant with respect to time.
2133:We can use again the fact that, if
13:
4784:is the state transition matrix of
3925:
3882:is positive definite and given by
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1997:
1864:
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1611:
1519:
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14:
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5568:
4334:{\displaystyle {\boldsymbol {C}}}
4312:{\displaystyle {\boldsymbol {B}}}
4290:{\displaystyle {\boldsymbol {A}}}
4043:{\displaystyle {\boldsymbol {A}}}
3785:{\displaystyle {\boldsymbol {A}}}
3743:{\displaystyle {\boldsymbol {A}}}
3377:{\displaystyle {\boldsymbol {x}}}
2148:{\displaystyle {\boldsymbol {A}}}
2025:{\displaystyle {\boldsymbol {A}}}
1036:{\displaystyle {\boldsymbol {A}}}
1014:{\displaystyle {\boldsymbol {A}}}
989:{\displaystyle {\boldsymbol {A}}}
452:{\displaystyle {\boldsymbol {D}}}
430:{\displaystyle {\boldsymbol {C}}}
408:{\displaystyle {\boldsymbol {B}}}
386:{\displaystyle {\boldsymbol {A}}}
249:{\displaystyle {\boldsymbol {D}}}
227:{\displaystyle {\boldsymbol {C}}}
205:{\displaystyle {\boldsymbol {B}}}
183:{\displaystyle {\boldsymbol {A}}}
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55:
5481:needs additional citations for
2532:by the right, would lead us to
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4512:{\displaystyle t_{1}>t_{0}}
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123:
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91:
85:
71:
65:
51:
45:
1:
5358:
4961:have the following property:
3506:For discrete time systems as
2064:
883:{\displaystyle (n+q)\times n}
3459:a positive definite matrix.
1946:Where we used the fact that
367:Observability in LTI Systems
7:
5331:
4110:Linear Time Variant Systems
10:
5606:
1021:have negative real parts (
737:{\displaystyle nq\times n}
4538:{\displaystyle n\times n}
4050:have magnitude less than
3750:have magnitude less than
2003:{\displaystyle t=\infty }
559:{\displaystyle n\times n}
353:{\displaystyle q\times p}
327:{\displaystyle q\times n}
301:{\displaystyle n\times p}
275:{\displaystyle n\times n}
5437:Chen, Chi-Tsong (1999).
5404:Chen, Chi-Tsong (1999).
5371:Chen, Chi-Tsong (1999).
967:{\displaystyle \lambda }
5505:"Observability Gramian"
5343:Controllability Gramian
2838:{\displaystyle \infty }
686:is nonsingular for any
5320:
5210:
5150:
4955:
4894:
4829:
4778:
4739:
4539:
4513:
4473:
4452:is observable at time
4446:
4389:
4335:
4313:
4291:
4275:That is, the matrices
4267:
4100:
4064:
4044:
4022:
3988:
3929:
3874:
3786:
3764:
3744:
3722:
3683:
3645:
3492:
3453:
3424:
3378:
3354:
3154:Also, we can see that
3145:
3114:
3040:
3014:
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2839:
2819:
2797:
2526:
2494:
2453:
2328:
2296:
2270:and they are given by
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2026:
2004:
1978:
1938:
1437:
1328:
1230:
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990:
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946:
884:
841:
738:
706:
705:{\displaystyle t>0}
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492:
453:
431:
409:
387:
354:
328:
302:
276:
250:
228:
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184:
160:
5353:Hankel singular value
5321:
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5151:
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4895:
4830:
4779:
4740:
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4474:
4472:{\displaystyle t_{0}}
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4336:
4314:
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4268:
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4065:
4045:
4023:
3989:
3909:
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3787:
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3745:
3723:
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3646:
3502:Discrete Time Systems
3493:
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3015:
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2125:
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2027:
2005:
1979:
1939:
1438:
1329:
1242:Observability Gramian
1231:
1122:
1038:
1016:
991:
969:
947:
945:{\displaystyle \left}
885:
842:
840:{\displaystyle \left}
744:observability matrix
739:
707:
679:
561:
532:
493:
454:
432:
410:
388:
355:
329:
303:
277:
251:
229:
207:
185:
168:is observable, where
161:
5490:improve this article
5222:
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4279:
4120:
4074:
4054:
4032:
4000:
3888:
3798:
3774:
3754:
3732:
3696:
3657:
3512:
3466:
3434:
3388:
3366:
3362:is positive for any
3160:
3126:
3052:
3024:
2986:
2982:using the fact that
2851:
2829:
2809:
2538:
2504:
2465:
2340:
2306:
2274:
2190:
2159:
2137:
2105:
2073:
2036:
2014:
1988:
1950:
1451:
1342:
1256:
1135:
1049:
1025:
1003:
978:
958:
896:
856:
750:
719:
690:
572:
544:
505:
466:
441:
419:
397:
375:
338:
312:
286:
260:
238:
216:
194:
172:
26:
4632:
3343:
3298:
3205:
2961:
2500:by the left and by
1868:
1703:
1615:
1523:
1374:
1167:
613:
256:are, respectively,
5316:
5206:
5146:
4951:
4890:
4825:
4774:
4735:
4598:
4535:
4509:
4469:
4442:
4385:
4331:
4309:
4287:
4263:
4261:
4096:
4060:
4040:
4018:
3984:
3870:
3782:
3760:
3740:
3718:
3679:
3641:
3639:
3488:
3449:
3420:
3418:
3374:
3350:
3299:
3284:
3191:
3151:has to be unique.
3141:
3110:
3036:
3010:
2970:
2939:
2835:
2815:
2793:
2522:
2490:
2449:
2324:
2292:
2258:
2174:
2145:
2120:
2091:
2051:
2022:
2000:
1974:
1934:
1932:
1846:
1689:
1601:
1509:
1433:
1360:
1324:
1226:
1153:
1117:
1033:
1011:
986:
964:
942:
936:
880:
837:
831:
734:
702:
674:
599:
556:
527:
488:
449:
427:
405:
383:
350:
324:
298:
272:
246:
224:
202:
180:
156:
154:
5566:
5565:
5558:
5540:
4800:
4136:
4063:{\displaystyle 1}
3763:{\displaystyle 1}
2818:{\displaystyle 0}
2805:Integrating from
2698:
1717:
1248:Lyapunov equation
42:
5597:
5561:
5554:
5550:
5547:
5541:
5539:
5498:
5474:
5466:
5461:
5460:
5444:
5434:
5428:
5427:
5411:
5401:
5395:
5394:
5378:
5368:
5325:
5323:
5322:
5317:
5312:
5311:
5293:
5279:
5278:
5266:
5255:
5254:
5242:
5241:
5229:
5215:
5213:
5212:
5207:
5202:
5201:
5189:
5188:
5176:
5175:
5170:
5155:
5153:
5152:
5147:
5142:
5141:
5123:
5115:
5114:
5096:
5095:
5090:
5081:
5080:
5062:
5061:
5056:
5038:
5037:
5025:
5024:
5019:
5007:
5006:
4994:
4993:
4981:
4980:
4975:
4960:
4958:
4957:
4952:
4947:
4946:
4934:
4933:
4921:
4920:
4915:
4899:
4897:
4896:
4891:
4886:
4885:
4873:
4872:
4860:
4859:
4854:
4835:is nonsingular.
4834:
4832:
4831:
4826:
4824:
4810:
4802:
4801:
4793:
4783:
4781:
4780:
4775:
4758:
4744:
4742:
4741:
4736:
4725:
4724:
4706:
4692:
4678:
4677:
4672:
4663:
4662:
4644:
4643:
4638:
4631:
4630:
4629:
4628:
4627:
4613:
4612:
4611:
4591:
4590:
4578:
4577:
4565:
4564:
4559:
4544:
4542:
4541:
4536:
4518:
4516:
4515:
4510:
4508:
4507:
4495:
4494:
4478:
4476:
4475:
4470:
4468:
4467:
4451:
4449:
4448:
4443:
4429:
4412:
4394:
4392:
4391:
4386:
4372:
4355:
4340:
4338:
4337:
4332:
4330:
4318:
4316:
4315:
4310:
4308:
4296:
4294:
4293:
4288:
4286:
4272:
4270:
4269:
4264:
4262:
4249:
4235:
4218:
4200:
4186:
4169:
4155:
4138:
4137:
4129:
4105:
4103:
4102:
4097:
4092:
4084:
4069:
4067:
4066:
4061:
4049:
4047:
4046:
4041:
4039:
4027:
4025:
4024:
4019:
4017:
4016:
4008:
3993:
3991:
3990:
3985:
3983:
3982:
3977:
3971:
3966:
3965:
3960:
3954:
3953:
3944:
3943:
3938:
3928:
3923:
3905:
3904:
3896:
3879:
3877:
3876:
3871:
3869:
3865:
3864:
3848:
3847:
3832:
3827:
3826:
3818:
3812:
3811:
3810:
3791:
3789:
3788:
3783:
3781:
3769:
3767:
3766:
3761:
3749:
3747:
3746:
3741:
3739:
3727:
3725:
3724:
3719:
3714:
3706:
3688:
3686:
3685:
3680:
3675:
3667:
3650:
3648:
3647:
3642:
3640:
3627:
3607:
3587:
3569:
3549:
3523:
3497:
3495:
3494:
3489:
3484:
3476:
3458:
3456:
3455:
3450:
3448:
3447:
3442:
3429:
3427:
3426:
3421:
3419:
3417:
3416:
3411:
3410:
3406:
3383:
3381:
3380:
3375:
3373:
3359:
3357:
3356:
3351:
3342:
3337:
3332:
3328:
3327:
3322:
3321:
3317:
3297:
3292:
3274:
3269:
3268:
3264:
3254:
3250:
3249:
3239:
3238:
3234:
3233:
3232:
3217:
3216:
3211:
3204:
3199:
3187:
3183:
3182:
3173:
3172:
3150:
3148:
3147:
3142:
3140:
3139:
3134:
3122:In other words,
3119:
3117:
3116:
3111:
3109:
3098:
3097:
3089:
3080:
3079:
3071:
3059:
3045:
3043:
3042:
3037:
3019:
3017:
3016:
3011:
3003:
3002:
2998:
2979:
2977:
2976:
2971:
2969:
2960:
2955:
2944:
2935:
2934:
2930:
2917:
2916:
2908:
2899:
2898:
2890:
2878:
2877:
2873:
2872:
2871:
2844:
2842:
2841:
2836:
2824:
2822:
2821:
2816:
2802:
2800:
2799:
2794:
2792:
2781:
2780:
2776:
2763:
2762:
2754:
2745:
2744:
2736:
2724:
2723:
2719:
2718:
2717:
2699:
2697:
2686:
2681:
2680:
2676:
2663:
2655:
2654:
2646:
2637:
2636:
2628:
2613:
2612:
2604:
2595:
2594:
2586:
2577:
2576:
2575:
2562:
2561:
2557:
2556:
2555:
2531:
2529:
2528:
2523:
2521:
2520:
2516:
2499:
2497:
2496:
2491:
2489:
2488:
2484:
2483:
2482:
2458:
2456:
2455:
2450:
2448:
2440:
2432:
2431:
2423:
2414:
2413:
2405:
2390:
2389:
2381:
2372:
2371:
2363:
2354:
2353:
2352:
2334:. Then we have:
2333:
2331:
2330:
2325:
2323:
2322:
2314:
2301:
2299:
2298:
2293:
2291:
2290:
2282:
2267:
2265:
2264:
2259:
2257:
2253:
2252:
2236:
2231:
2230:
2225:
2216:
2215:
2210:
2204:
2203:
2202:
2183:
2181:
2180:
2175:
2173:
2172:
2167:
2154:
2152:
2151:
2146:
2144:
2129:
2127:
2126:
2121:
2119:
2118:
2113:
2100:
2098:
2097:
2092:
2090:
2086:
2085:
2069:We can see that
2060:
2058:
2057:
2052:
2050:
2049:
2044:
2031:
2029:
2028:
2023:
2021:
2009:
2007:
2006:
2001:
1983:
1981:
1980:
1975:
1967:
1966:
1962:
1943:
1941:
1940:
1935:
1933:
1929:
1925:
1924:
1905:
1901:
1897:
1896:
1883:
1872:
1867:
1862:
1851:
1845:
1844:
1840:
1830:
1825:
1824:
1819:
1813:
1812:
1808:
1807:
1806:
1775:
1774:
1770:
1760:
1755:
1754:
1749:
1743:
1742:
1738:
1737:
1736:
1718:
1716:
1705:
1702:
1697:
1682:
1672:
1667:
1666:
1662:
1652:
1648:
1647:
1637:
1636:
1632:
1631:
1630:
1614:
1609:
1587:
1586:
1582:
1572:
1568:
1567:
1557:
1556:
1552:
1551:
1550:
1535:
1534:
1533:
1522:
1517:
1501:
1496:
1495:
1490:
1481:
1480:
1475:
1469:
1468:
1467:
1442:
1440:
1439:
1434:
1426:
1425:
1421:
1411:
1407:
1406:
1396:
1395:
1391:
1390:
1389:
1373:
1368:
1356:
1355:
1354:
1333:
1331:
1330:
1325:
1323:
1319:
1318:
1302:
1297:
1296:
1291:
1282:
1281:
1276:
1270:
1269:
1268:
1235:
1233:
1232:
1227:
1219:
1218:
1214:
1204:
1200:
1199:
1189:
1188:
1184:
1183:
1178:
1166:
1161:
1149:
1148:
1147:
1126:
1124:
1123:
1118:
1116:
1112:
1111:
1095:
1090:
1089:
1084:
1075:
1074:
1069:
1063:
1062:
1061:
1042:
1040:
1039:
1034:
1032:
1020:
1018:
1017:
1012:
1010:
995:
993:
992:
987:
985:
973:
971:
970:
965:
951:
949:
948:
943:
941:
937:
933:
924:
919:
911:
889:
887:
886:
881:
846:
844:
843:
838:
836:
832:
828:
827:
816:
796:
795:
790:
777:
765:
743:
741:
740:
735:
711:
709:
708:
703:
683:
681:
680:
675:
667:
666:
662:
652:
647:
646:
641:
635:
634:
630:
629:
624:
612:
607:
586:
585:
584:
565:
563:
562:
557:
536:
534:
533:
528:
523:
515:
497:
495:
494:
489:
484:
476:
458:
456:
455:
450:
448:
436:
434:
433:
428:
426:
414:
412:
411:
406:
404:
392:
390:
389:
384:
382:
359:
357:
356:
351:
333:
331:
330:
325:
307:
305:
304:
299:
281:
279:
278:
273:
255:
253:
252:
247:
245:
233:
231:
230:
225:
223:
211:
209:
208:
203:
201:
189:
187:
186:
181:
179:
165:
163:
162:
157:
155:
142:
122:
102:
84:
64:
44:
43:
35:
5605:
5604:
5600:
5599:
5598:
5596:
5595:
5594:
5580:
5579:
5571:
5562:
5551:
5545:
5542:
5499:
5497:
5487:
5475:
5464:
5457:
5435:
5431:
5424:
5402:
5398:
5391:
5369:
5365:
5361:
5334:
5307:
5303:
5289:
5274:
5270:
5262:
5250:
5246:
5237:
5233:
5225:
5223:
5220:
5219:
5197:
5193:
5184:
5180:
5171:
5166:
5165:
5163:
5160:
5159:
5137:
5133:
5119:
5110:
5106:
5091:
5086:
5085:
5076:
5072:
5057:
5052:
5051:
5033:
5029:
5020:
5015:
5014:
5002:
4998:
4989:
4985:
4976:
4971:
4970:
4968:
4965:
4964:
4942:
4938:
4929:
4925:
4916:
4911:
4910:
4908:
4905:
4904:
4901:
4881:
4877:
4868:
4864:
4855:
4850:
4849:
4847:
4844:
4843:
4820:
4806:
4792:
4791:
4789:
4786:
4785:
4754:
4752:
4749:
4748:
4720:
4716:
4702:
4688:
4673:
4668:
4667:
4658:
4654:
4639:
4634:
4633:
4623:
4619:
4618:
4615:
4614:
4607:
4603:
4602:
4586:
4582:
4573:
4569:
4560:
4555:
4554:
4552:
4549:
4548:
4524:
4521:
4520:
4503:
4499:
4490:
4486:
4484:
4481:
4480:
4463:
4459:
4457:
4454:
4453:
4425:
4408:
4403:
4400:
4399:
4368:
4351:
4346:
4343:
4342:
4326:
4324:
4321:
4320:
4304:
4302:
4299:
4298:
4282:
4280:
4277:
4276:
4260:
4259:
4245:
4231:
4214:
4211:
4210:
4196:
4182:
4165:
4148:
4128:
4127:
4123:
4121:
4118:
4117:
4112:
4088:
4080:
4075:
4072:
4071:
4055:
4052:
4051:
4035:
4033:
4030:
4029:
4009:
4004:
4003:
4001:
3998:
3997:
3978:
3973:
3972:
3967:
3961:
3956:
3955:
3949:
3945:
3939:
3934:
3933:
3924:
3913:
3897:
3892:
3891:
3889:
3886:
3885:
3860:
3856:
3855:
3840:
3836:
3828:
3819:
3814:
3813:
3806:
3802:
3801:
3799:
3796:
3795:
3777:
3775:
3772:
3771:
3755:
3752:
3751:
3735:
3733:
3730:
3729:
3710:
3702:
3697:
3694:
3693:
3671:
3663:
3658:
3655:
3654:
3638:
3637:
3620:
3600:
3583:
3580:
3579:
3562:
3539:
3519:
3515:
3513:
3510:
3509:
3504:
3480:
3472:
3467:
3464:
3463:
3443:
3438:
3437:
3435:
3432:
3431:
3412:
3402:
3401:
3397:
3393:
3391:
3389:
3386:
3385:
3369:
3367:
3364:
3363:
3338:
3333:
3323:
3313:
3312:
3308:
3304:
3300:
3293:
3288:
3270:
3260:
3259:
3255:
3245:
3241:
3240:
3228:
3224:
3223:
3222:
3218:
3212:
3207:
3206:
3200:
3195:
3178:
3174:
3168:
3164:
3163:
3161:
3158:
3157:
3135:
3130:
3129:
3127:
3124:
3123:
3105:
3090:
3085:
3084:
3072:
3067:
3066:
3055:
3053:
3050:
3049:
3025:
3022:
3021:
2994:
2993:
2989:
2987:
2984:
2983:
2965:
2956:
2945:
2940:
2926:
2925:
2921:
2909:
2904:
2903:
2891:
2886:
2885:
2867:
2863:
2862:
2861:
2857:
2852:
2849:
2848:
2830:
2827:
2826:
2810:
2807:
2806:
2788:
2772:
2771:
2767:
2755:
2750:
2749:
2737:
2732:
2731:
2713:
2709:
2708:
2707:
2703:
2690:
2685:
2672:
2671:
2667:
2659:
2647:
2642:
2641:
2629:
2621:
2620:
2605:
2600:
2599:
2587:
2579:
2578:
2571:
2567:
2566:
2551:
2547:
2546:
2545:
2541:
2539:
2536:
2535:
2512:
2511:
2507:
2505:
2502:
2501:
2478:
2474:
2473:
2472:
2468:
2466:
2463:
2462:
2461:Multiplying by
2444:
2436:
2424:
2419:
2418:
2406:
2398:
2397:
2382:
2377:
2376:
2364:
2356:
2355:
2348:
2344:
2343:
2341:
2338:
2337:
2315:
2310:
2309:
2307:
2304:
2303:
2283:
2278:
2277:
2275:
2272:
2271:
2248:
2244:
2243:
2232:
2226:
2221:
2220:
2211:
2206:
2205:
2198:
2194:
2193:
2191:
2188:
2187:
2168:
2163:
2162:
2160:
2157:
2156:
2140:
2138:
2135:
2134:
2114:
2109:
2108:
2106:
2103:
2102:
2081:
2077:
2076:
2074:
2071:
2070:
2067:
2045:
2040:
2039:
2037:
2034:
2033:
2017:
2015:
2012:
2011:
1989:
1986:
1985:
1958:
1957:
1953:
1951:
1948:
1947:
1931:
1930:
1920:
1916:
1912:
1910:
1903:
1902:
1892:
1888:
1887:
1879:
1877:
1870:
1869:
1863:
1852:
1847:
1836:
1835:
1831:
1826:
1820:
1815:
1814:
1802:
1798:
1797:
1796:
1792:
1790:
1785:
1766:
1765:
1761:
1756:
1750:
1745:
1744:
1732:
1728:
1727:
1726:
1722:
1709:
1704:
1698:
1693:
1687:
1680:
1679:
1668:
1658:
1657:
1653:
1643:
1639:
1638:
1626:
1622:
1621:
1620:
1616:
1610:
1605:
1599:
1594:
1578:
1577:
1573:
1563:
1559:
1558:
1546:
1542:
1541:
1540:
1536:
1529:
1525:
1524:
1518:
1513:
1507:
1502:
1497:
1491:
1486:
1485:
1476:
1471:
1470:
1463:
1459:
1458:
1454:
1452:
1449:
1448:
1417:
1416:
1412:
1402:
1398:
1397:
1385:
1381:
1380:
1379:
1375:
1369:
1364:
1350:
1346:
1345:
1343:
1340:
1339:
1314:
1310:
1309:
1298:
1292:
1287:
1286:
1277:
1272:
1271:
1264:
1260:
1259:
1257:
1254:
1253:
1244:
1210:
1209:
1205:
1195:
1191:
1190:
1179:
1174:
1173:
1172:
1168:
1162:
1157:
1143:
1139:
1138:
1136:
1133:
1132:
1107:
1103:
1102:
1091:
1085:
1080:
1079:
1070:
1065:
1064:
1057:
1053:
1052:
1050:
1047:
1046:
1028:
1026:
1023:
1022:
1006:
1004:
1001:
1000:
981:
979:
976:
975:
959:
956:
955:
935:
934:
929:
926:
925:
920:
912:
907:
903:
899:
897:
894:
893:
857:
854:
853:
830:
829:
817:
809:
808:
805:
804:
798:
797:
791:
783:
782:
779:
778:
770:
767:
766:
761:
757:
753:
751:
748:
747:
720:
717:
716:
691:
688:
687:
658:
657:
653:
648:
642:
637:
636:
625:
620:
619:
618:
614:
608:
603:
580:
576:
575:
573:
570:
569:
545:
542:
541:
537:is observable.
519:
511:
506:
503:
502:
480:
472:
467:
464:
463:
444:
442:
439:
438:
422:
420:
417:
416:
400:
398:
395:
394:
378:
376:
373:
372:
369:
339:
336:
335:
313:
310:
309:
287:
284:
283:
261:
258:
257:
241:
239:
236:
235:
219:
217:
214:
213:
197:
195:
192:
191:
175:
173:
170:
169:
153:
152:
135:
115:
98:
95:
94:
77:
54:
34:
33:
29:
27:
24:
23:
12:
11:
5:
5603:
5593:
5592:
5590:Control theory
5578:
5577:
5570:
5569:External links
5567:
5564:
5563:
5478:
5476:
5469:
5463:
5462:
5455:
5429:
5422:
5396:
5389:
5362:
5360:
5357:
5356:
5355:
5350:
5348:Gramian matrix
5345:
5340:
5333:
5330:
5315:
5310:
5306:
5302:
5299:
5296:
5292:
5288:
5285:
5282:
5277:
5273:
5269:
5265:
5261:
5258:
5253:
5249:
5245:
5240:
5236:
5232:
5228:
5205:
5200:
5196:
5192:
5187:
5183:
5179:
5174:
5169:
5145:
5140:
5136:
5132:
5129:
5126:
5122:
5118:
5113:
5109:
5105:
5102:
5099:
5094:
5089:
5084:
5079:
5075:
5071:
5068:
5065:
5060:
5055:
5050:
5047:
5044:
5041:
5036:
5032:
5028:
5023:
5018:
5013:
5010:
5005:
5001:
4997:
4992:
4988:
4984:
4979:
4974:
4950:
4945:
4941:
4937:
4932:
4928:
4924:
4919:
4914:
4900:
4889:
4884:
4880:
4876:
4871:
4867:
4863:
4858:
4853:
4842:Properties of
4840:
4823:
4819:
4816:
4813:
4809:
4805:
4799:
4796:
4773:
4770:
4767:
4764:
4761:
4757:
4734:
4731:
4728:
4723:
4719:
4715:
4712:
4709:
4705:
4701:
4698:
4695:
4691:
4687:
4684:
4681:
4676:
4671:
4666:
4661:
4657:
4653:
4650:
4647:
4642:
4637:
4626:
4622:
4617:
4610:
4606:
4601:
4597:
4594:
4589:
4585:
4581:
4576:
4572:
4568:
4563:
4558:
4534:
4531:
4528:
4519:such that the
4506:
4502:
4498:
4493:
4489:
4466:
4462:
4441:
4438:
4435:
4432:
4428:
4424:
4421:
4418:
4415:
4411:
4407:
4384:
4381:
4378:
4375:
4371:
4367:
4364:
4361:
4358:
4354:
4350:
4329:
4307:
4285:
4258:
4255:
4252:
4248:
4244:
4241:
4238:
4234:
4230:
4227:
4224:
4221:
4217:
4213:
4212:
4209:
4206:
4203:
4199:
4195:
4192:
4189:
4185:
4181:
4178:
4175:
4172:
4168:
4164:
4161:
4158:
4154:
4151:
4147:
4144:
4141:
4135:
4132:
4126:
4125:
4111:
4108:
4095:
4091:
4087:
4083:
4079:
4059:
4038:
4015:
4012:
4007:
3981:
3976:
3970:
3964:
3959:
3952:
3948:
3942:
3937:
3932:
3927:
3922:
3919:
3916:
3912:
3908:
3903:
3900:
3895:
3868:
3863:
3859:
3854:
3851:
3846:
3843:
3839:
3835:
3831:
3825:
3822:
3817:
3809:
3805:
3780:
3759:
3738:
3717:
3713:
3709:
3705:
3701:
3678:
3674:
3670:
3666:
3662:
3636:
3633:
3630:
3626:
3623:
3619:
3616:
3613:
3610:
3606:
3603:
3599:
3596:
3593:
3590:
3586:
3582:
3581:
3578:
3575:
3572:
3568:
3565:
3561:
3558:
3555:
3552:
3548:
3545:
3542:
3538:
3535:
3532:
3529:
3526:
3522:
3518:
3517:
3503:
3500:
3487:
3483:
3479:
3475:
3471:
3446:
3441:
3415:
3409:
3405:
3400:
3396:
3372:
3349:
3346:
3341:
3336:
3331:
3326:
3320:
3316:
3311:
3307:
3303:
3296:
3291:
3287:
3283:
3280:
3277:
3273:
3267:
3263:
3258:
3253:
3248:
3244:
3237:
3231:
3227:
3221:
3215:
3210:
3203:
3198:
3194:
3190:
3186:
3181:
3177:
3171:
3167:
3138:
3133:
3108:
3104:
3101:
3096:
3093:
3088:
3083:
3078:
3075:
3070:
3065:
3062:
3058:
3035:
3032:
3029:
3009:
3006:
3001:
2997:
2992:
2968:
2964:
2959:
2954:
2951:
2948:
2943:
2938:
2933:
2929:
2924:
2920:
2915:
2912:
2907:
2902:
2897:
2894:
2889:
2884:
2881:
2876:
2870:
2866:
2860:
2856:
2834:
2814:
2791:
2787:
2784:
2779:
2775:
2770:
2766:
2761:
2758:
2753:
2748:
2743:
2740:
2735:
2730:
2727:
2722:
2716:
2712:
2706:
2702:
2696:
2693:
2689:
2684:
2679:
2675:
2670:
2666:
2662:
2658:
2653:
2650:
2645:
2640:
2635:
2632:
2627:
2624:
2619:
2616:
2611:
2608:
2603:
2598:
2593:
2590:
2585:
2582:
2574:
2570:
2565:
2560:
2554:
2550:
2544:
2519:
2515:
2510:
2487:
2481:
2477:
2471:
2447:
2443:
2439:
2435:
2430:
2427:
2422:
2417:
2412:
2409:
2404:
2401:
2396:
2393:
2388:
2385:
2380:
2375:
2370:
2367:
2362:
2359:
2351:
2347:
2321:
2318:
2313:
2289:
2286:
2281:
2256:
2251:
2247:
2242:
2239:
2235:
2229:
2224:
2219:
2214:
2209:
2201:
2197:
2171:
2166:
2143:
2117:
2112:
2089:
2084:
2080:
2066:
2063:
2048:
2043:
2020:
1999:
1996:
1993:
1973:
1970:
1965:
1961:
1956:
1928:
1923:
1919:
1915:
1911:
1909:
1906:
1904:
1900:
1895:
1891:
1886:
1882:
1878:
1876:
1873:
1871:
1866:
1861:
1858:
1855:
1850:
1843:
1839:
1834:
1829:
1823:
1818:
1811:
1805:
1801:
1795:
1791:
1789:
1786:
1784:
1781:
1778:
1773:
1769:
1764:
1759:
1753:
1748:
1741:
1735:
1731:
1725:
1721:
1715:
1712:
1708:
1701:
1696:
1692:
1688:
1686:
1683:
1681:
1678:
1675:
1671:
1665:
1661:
1656:
1651:
1646:
1642:
1635:
1629:
1625:
1619:
1613:
1608:
1604:
1600:
1598:
1595:
1593:
1590:
1585:
1581:
1576:
1571:
1566:
1562:
1555:
1549:
1545:
1539:
1532:
1528:
1521:
1516:
1512:
1508:
1506:
1503:
1500:
1494:
1489:
1484:
1479:
1474:
1466:
1462:
1457:
1456:
1432:
1429:
1424:
1420:
1415:
1410:
1405:
1401:
1394:
1388:
1384:
1378:
1372:
1367:
1363:
1359:
1353:
1349:
1322:
1317:
1313:
1308:
1305:
1301:
1295:
1290:
1285:
1280:
1275:
1267:
1263:
1243:
1240:
1225:
1222:
1217:
1213:
1208:
1203:
1198:
1194:
1187:
1182:
1177:
1171:
1165:
1160:
1156:
1152:
1146:
1142:
1115:
1110:
1106:
1101:
1098:
1094:
1088:
1083:
1078:
1073:
1068:
1060:
1056:
1031:
1009:
984:
963:
940:
932:
928:
927:
923:
918:
915:
910:
906:
905:
902:
879:
876:
873:
870:
867:
864:
861:
835:
826:
823:
820:
815:
812:
807:
806:
803:
800:
799:
794:
789:
786:
781:
780:
776:
773:
769:
768:
764:
760:
759:
756:
733:
730:
727:
724:
701:
698:
695:
673:
670:
665:
661:
656:
651:
645:
640:
633:
628:
623:
617:
611:
606:
602:
598:
595:
592:
589:
583:
579:
555:
552:
549:
526:
522:
518:
514:
510:
487:
483:
479:
475:
471:
447:
425:
403:
381:
368:
365:
349:
346:
343:
323:
320:
317:
297:
294:
291:
271:
268:
265:
244:
222:
200:
178:
151:
148:
145:
141:
138:
134:
131:
128:
125:
121:
118:
114:
111:
108:
105:
101:
97:
96:
93:
90:
87:
83:
80:
76:
73:
70:
67:
63:
60:
57:
53:
50:
47:
41:
38:
32:
31:
18:control theory
9:
6:
4:
3:
2:
5602:
5591:
5588:
5587:
5585:
5576:
5573:
5572:
5560:
5557:
5549:
5538:
5535:
5531:
5528:
5524:
5521:
5517:
5514:
5510:
5507: –
5506:
5502:
5501:Find sources:
5495:
5491:
5485:
5484:
5479:This article
5477:
5473:
5468:
5467:
5458:
5456:0-19-511777-8
5452:
5448:
5443:
5442:
5433:
5425:
5423:0-19-511777-8
5419:
5415:
5410:
5409:
5400:
5392:
5390:0-19-511777-8
5386:
5382:
5377:
5376:
5367:
5363:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5338:Observability
5336:
5335:
5329:
5326:
5308:
5304:
5300:
5297:
5283:
5280:
5275:
5271:
5259:
5251:
5247:
5243:
5238:
5234:
5217:
5198:
5194:
5190:
5185:
5181:
5172:
5156:
5138:
5134:
5130:
5127:
5111:
5107:
5103:
5100:
5092:
5077:
5073:
5069:
5066:
5058:
5048:
5042:
5039:
5034:
5030:
5021:
5011:
5003:
4999:
4995:
4990:
4986:
4977:
4962:
4943:
4939:
4935:
4930:
4926:
4917:
4882:
4878:
4874:
4869:
4865:
4856:
4839:
4836:
4814:
4803:
4768:
4765:
4762:
4745:
4732:
4729:
4721:
4717:
4713:
4710:
4696:
4682:
4674:
4659:
4655:
4651:
4648:
4640:
4624:
4620:
4616:
4608:
4604:
4599:
4595:
4587:
4583:
4579:
4574:
4570:
4561:
4546:
4532:
4529:
4526:
4504:
4500:
4496:
4491:
4487:
4464:
4460:
4433:
4422:
4416:
4396:
4376:
4365:
4359:
4273:
4253:
4239:
4228:
4222:
4204:
4190:
4179:
4173:
4159:
4142:
4133:
4115:
4107:
4085:
4070:, the system
4057:
4013:
4010:
3994:
3979:
3962:
3950:
3940:
3920:
3917:
3914:
3910:
3906:
3901:
3898:
3883:
3880:
3852:
3849:
3844:
3841:
3837:
3833:
3823:
3820:
3793:
3757:
3707:
3690:
3668:
3651:
3631:
3617:
3611:
3597:
3591:
3573:
3559:
3553:
3533:
3530:
3527:
3507:
3499:
3477:
3460:
3444:
3360:
3347:
3344:
3339:
3334:
3289:
3285:
3281:
3278:
3275:
3265:
3256:
3235:
3219:
3213:
3196:
3192:
3188:
3155:
3152:
3136:
3120:
3102:
3094:
3091:
3081:
3076:
3073:
3060:
3047:
3027:
3007:
2999:
2990:
2980:
2962:
2952:
2949:
2946:
2931:
2922:
2913:
2910:
2900:
2895:
2892:
2874:
2858:
2846:
2812:
2803:
2785:
2777:
2768:
2759:
2756:
2746:
2741:
2738:
2720:
2704:
2694:
2691:
2687:
2682:
2677:
2668:
2651:
2648:
2638:
2633:
2630:
2617:
2609:
2606:
2596:
2591:
2588:
2558:
2542:
2533:
2517:
2508:
2485:
2469:
2459:
2441:
2428:
2425:
2415:
2410:
2407:
2394:
2386:
2383:
2373:
2368:
2365:
2335:
2319:
2316:
2287:
2284:
2268:
2240:
2237:
2227:
2217:
2212:
2185:
2169:
2131:
2115:
2062:
2046:
1994:
1991:
1971:
1968:
1963:
1954:
1944:
1907:
1884:
1874:
1859:
1856:
1853:
1841:
1832:
1821:
1809:
1793:
1787:
1782:
1779:
1771:
1762:
1751:
1739:
1723:
1713:
1710:
1706:
1694:
1690:
1684:
1676:
1673:
1663:
1654:
1633:
1617:
1606:
1602:
1596:
1591:
1588:
1583:
1574:
1553:
1537:
1514:
1510:
1504:
1492:
1482:
1477:
1446:
1443:
1430:
1427:
1422:
1413:
1392:
1376:
1365:
1361:
1357:
1337:
1334:
1306:
1303:
1293:
1283:
1278:
1251:
1249:
1239:
1236:
1223:
1220:
1215:
1206:
1185:
1180:
1169:
1158:
1154:
1150:
1130:
1127:
1099:
1096:
1086:
1076:
1071:
1044:
997:
961:
952:
938:
900:
891:
877:
874:
868:
865:
862:
850:
847:
833:
824:
821:
818:
801:
792:
754:
745:
731:
728:
725:
722:
713:
699:
696:
693:
684:
671:
668:
663:
654:
643:
631:
626:
615:
609:
604:
600:
596:
590:
567:
553:
550:
547:
538:
516:
499:
477:
460:
364:
361:
347:
344:
341:
321:
318:
315:
295:
292:
289:
269:
266:
263:
166:
146:
132:
126:
112:
106:
88:
74:
68:
48:
39:
21:
19:
5552:
5543:
5533:
5526:
5519:
5512:
5500:
5488:Please help
5483:verification
5480:
5440:
5432:
5407:
5399:
5374:
5366:
5327:
5218:
5157:
4963:
4902:
4837:
4746:
4547:
4397:
4274:
4116:
4113:
3995:
3884:
3881:
3794:
3691:
3652:
3508:
3505:
3461:
3361:
3156:
3153:
3121:
3048:
2981:
2847:
2804:
2534:
2460:
2336:
2269:
2186:
2132:
2068:
1945:
1447:
1444:
1338:
1335:
1252:
1245:
1237:
1131:
1128:
1045:
998:
953:
892:
851:
849:has rank n.
848:
746:
714:
685:
568:
539:
501:1. The pair
500:
461:
370:
362:
167:
22:
15:
4398:The system
2010:for stable
5516:newspapers
5359:References
2065:Properties
360:matrices.
5546:July 2008
5298:τ
5291:Φ
5284:τ
5264:Φ
5227:Φ
5121:Φ
5054:Φ
4798:˙
4769:τ
4756:Φ
4733:τ
4711:τ
4704:Φ
4697:τ
4683:τ
4649:τ
4636:Φ
4600:∫
4530:×
4134:˙
3926:∞
3911:∑
3853:−
3834:−
3295:∞
3286:∫
3202:∞
3193:∫
3082:−
3061:−
3034:∞
3031:→
3005:→
2958:∞
2901:−
2833:∞
2747:−
2639:−
2597:−
2416:−
2374:−
2241:−
1998:∞
1914:−
1885:−
1865:∞
1783:τ
1772:τ
1740:τ
1714:τ
1700:∞
1691:∫
1677:τ
1664:τ
1634:τ
1612:∞
1603:∫
1592:τ
1584:τ
1554:τ
1520:∞
1511:∫
1431:τ
1423:τ
1393:τ
1371:∞
1362:∫
1307:−
1250:given by
1224:τ
1216:τ
1186:τ
1164:∞
1155:∫
1100:−
962:λ
917:λ
914:−
875:×
822:−
802:⋮
729:×
672:τ
664:τ
632:τ
601:∫
551:×
345:×
319:×
293:×
267:×
40:˙
5584:Category
5332:See also
3330:‖
3302:‖
5530:scholar
890:matrix
852:4. The
715:3. The
566:matrix
540:2. The
5532:
5525:
5518:
5511:
5503:
5453:
5420:
5387:
4747:where
5537:JSTOR
5523:books
5509:news
5451:ISBN
5418:ISBN
5385:ISBN
4497:>
4319:and
2302:and
697:>
437:and
334:and
234:and
5492:by
5447:179
5414:171
5381:156
3020:as
2825:to
1984:at
974:of
16:In
5586::
5449:.
5416:.
5383:.
4297:,
3046::
2845::
2130:.
996:.
712:.
415:,
393:,
282:,
212:,
190:,
5559:)
5553:(
5548:)
5544:(
5534:·
5527:·
5520:·
5513:·
5486:.
5459:.
5426:.
5393:.
5314:)
5309:0
5305:t
5301:,
5295:(
5287:)
5281:,
5276:1
5272:t
5268:(
5260:=
5257:)
5252:1
5248:t
5244:,
5239:0
5235:t
5231:(
5204:)
5199:1
5195:t
5191:,
5186:0
5182:t
5178:(
5173:o
5168:W
5144:)
5139:0
5135:t
5131:,
5128:t
5125:(
5117:)
5112:0
5108:t
5104:,
5101:t
5098:(
5093:o
5088:W
5083:)
5078:0
5074:t
5070:,
5067:t
5064:(
5059:T
5049:+
5046:)
5043:t
5040:,
5035:0
5031:t
5027:(
5022:o
5017:W
5012:=
5009:)
5004:1
5000:t
4996:,
4991:0
4987:t
4983:(
4978:o
4973:W
4949:)
4944:1
4940:t
4936:,
4931:0
4927:t
4923:(
4918:o
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4888:)
4883:1
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4870:0
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4862:(
4857:o
4852:W
4822:x
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4815:t
4812:(
4808:A
4804:=
4795:x
4772:)
4766:,
4763:t
4760:(
4730:d
4727:)
4722:0
4718:t
4714:,
4708:(
4700:)
4694:(
4690:C
4686:)
4680:(
4675:T
4670:C
4665:)
4660:0
4656:t
4652:,
4646:(
4641:T
4625:1
4621:t
4609:0
4605:t
4596:=
4593:)
4588:1
4584:t
4580:,
4575:0
4571:t
4567:(
4562:o
4557:W
4533:n
4527:n
4505:0
4501:t
4492:1
4488:t
4465:0
4461:t
4440:)
4437:)
4434:t
4431:(
4427:C
4423:,
4420:)
4417:t
4414:(
4410:A
4406:(
4383:)
4380:)
4377:t
4374:(
4370:C
4366:,
4363:)
4360:t
4357:(
4353:A
4349:(
4328:C
4306:B
4284:A
4257:)
4254:t
4251:(
4247:x
4243:)
4240:t
4237:(
4233:C
4229:=
4226:)
4223:t
4220:(
4216:y
4208:)
4205:t
4202:(
4198:u
4194:)
4191:t
4188:(
4184:B
4180:+
4177:)
4174:t
4171:(
4167:x
4163:)
4160:t
4157:(
4153:A
4150:=
4146:)
4143:t
4140:(
4131:x
4094:)
4090:B
4086:,
4082:A
4078:(
4058:1
4037:A
4014:c
4011:d
4006:W
3980:m
3975:A
3969:C
3963:T
3958:C
3951:m
3947:)
3941:T
3936:A
3931:(
3921:0
3918:=
3915:m
3907:=
3902:o
3899:d
3894:W
3867:C
3862:T
3858:C
3850:=
3845:o
3842:d
3838:W
3830:A
3824:o
3821:d
3816:W
3808:T
3804:A
3779:A
3770:(
3758:1
3737:A
3716:)
3712:C
3708:,
3704:A
3700:(
3677:)
3673:C
3669:,
3665:A
3661:(
3635:]
3632:k
3629:[
3625:u
3622:D
3618:+
3615:]
3612:k
3609:[
3605:x
3602:C
3598:=
3595:]
3592:k
3589:[
3585:y
3577:]
3574:k
3571:[
3567:u
3564:B
3560:+
3557:]
3554:k
3551:[
3547:x
3544:A
3541:=
3537:]
3534:1
3531:+
3528:k
3525:[
3521:x
3486:)
3482:C
3478:,
3474:A
3470:(
3445:o
3440:W
3414:x
3408:t
3404:A
3399:e
3395:C
3371:x
3348:t
3345:d
3340:2
3335:2
3325:x
3319:t
3315:A
3310:e
3306:C
3290:0
3282:=
3279:t
3276:d
3272:x
3266:t
3262:A
3257:e
3252:C
3247:T
3243:C
3236:t
3230:T
3226:A
3220:e
3214:T
3209:x
3197:0
3189:=
3185:x
3180:o
3176:W
3170:T
3166:x
3137:o
3132:W
3107:0
3103:=
3100:)
3095:2
3092:o
3087:W
3077:1
3074:o
3069:W
3064:(
3057:0
3028:t
3008:0
3000:t
2996:A
2991:e
2967:0
2963:=
2953:0
2950:=
2947:t
2942:|
2937:]
2932:t
2928:A
2923:e
2919:)
2914:2
2911:o
2906:W
2896:1
2893:o
2888:W
2883:(
2880:[
2875:t
2869:T
2865:A
2859:e
2855:[
2813:0
2790:0
2786:=
2783:]
2778:t
2774:A
2769:e
2765:)
2760:2
2757:o
2752:W
2742:1
2739:o
2734:W
2729:(
2726:[
2721:t
2715:T
2711:A
2705:e
2701:[
2695:t
2692:d
2688:d
2683:=
2678:t
2674:A
2669:e
2665:]
2661:A
2657:)
2652:2
2649:o
2644:W
2634:1
2631:o
2626:W
2623:(
2618:+
2615:)
2610:2
2607:o
2602:W
2592:1
2589:o
2584:W
2581:(
2573:T
2569:A
2564:[
2559:t
2553:T
2549:A
2543:e
2518:t
2514:A
2509:e
2486:t
2480:T
2476:A
2470:e
2446:0
2442:=
2438:A
2434:)
2429:2
2426:o
2421:W
2411:1
2408:o
2403:W
2400:(
2395:+
2392:)
2387:2
2384:o
2379:W
2369:1
2366:o
2361:W
2358:(
2350:T
2346:A
2320:2
2317:o
2312:W
2288:1
2285:o
2280:W
2255:C
2250:T
2246:C
2238:=
2234:A
2228:o
2223:W
2218:+
2213:o
2208:W
2200:T
2196:A
2170:o
2165:W
2142:A
2116:o
2111:W
2088:C
2083:T
2079:C
2047:o
2042:W
2019:A
1995:=
1992:t
1972:0
1969:=
1964:t
1960:A
1955:e
1927:C
1922:T
1918:C
1908:=
1899:C
1894:T
1890:C
1881:0
1875:=
1860:0
1857:=
1854:t
1849:|
1842:t
1838:A
1833:e
1828:C
1822:T
1817:C
1810:t
1804:T
1800:A
1794:e
1788:=
1780:d
1777:)
1768:A
1763:e
1758:C
1752:T
1747:C
1734:T
1730:A
1724:e
1720:(
1711:d
1707:d
1695:0
1685:=
1674:d
1670:A
1660:A
1655:e
1650:C
1645:T
1641:C
1628:T
1624:A
1618:e
1607:0
1597:+
1589:d
1580:A
1575:e
1570:C
1565:T
1561:C
1548:T
1544:A
1538:e
1531:T
1527:A
1515:0
1505:=
1499:A
1493:o
1488:W
1483:+
1478:o
1473:W
1465:T
1461:A
1428:d
1419:A
1414:e
1409:C
1404:T
1400:C
1387:T
1383:A
1377:e
1366:0
1358:=
1352:o
1348:W
1321:C
1316:T
1312:C
1304:=
1300:A
1294:o
1289:W
1284:+
1279:o
1274:W
1266:T
1262:A
1221:d
1212:A
1207:e
1202:C
1197:T
1193:C
1181:T
1176:A
1170:e
1159:0
1151:=
1145:o
1141:W
1114:C
1109:T
1105:C
1097:=
1093:A
1087:o
1082:W
1077:+
1072:o
1067:W
1059:T
1055:A
1030:A
1008:A
983:A
939:]
931:C
922:I
909:A
901:[
878:n
872:)
869:q
866:+
863:n
860:(
834:]
825:1
819:n
814:A
811:C
793:2
788:A
785:C
775:A
772:C
763:C
755:[
732:n
726:q
723:n
700:0
694:t
669:d
660:A
655:e
650:C
644:T
639:C
627:T
622:A
616:e
610:t
605:0
597:=
594:)
591:t
588:(
582:o
578:W
554:n
548:n
525:)
521:C
517:,
513:A
509:(
486:)
482:C
478:,
474:A
470:(
446:D
424:C
402:B
380:A
348:p
342:q
322:n
316:q
308:,
296:p
290:n
270:n
264:n
243:D
221:C
199:B
177:A
150:)
147:t
144:(
140:u
137:D
133:+
130:)
127:t
124:(
120:x
117:C
113:=
110:)
107:t
104:(
100:y
92:)
89:t
86:(
82:u
79:B
75:+
72:)
69:t
66:(
62:x
59:A
56:=
52:)
49:t
46:(
37:x
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