3896:
1312:
4638:
3609:
5087:
96:
A deeper statement of the separation principle is that the LQG controller is still optimal in a wider class of possibly nonlinear controllers. That is, utilizing a nonlinear control scheme will not improve the expected value of the cost function. This version of the separation principle is a special
124:
LQG optimality does not automatically ensure good robustness properties. The robust stability of the closed loop system must be checked separately after the LQG controller has been designed. To promote robustness some of the system parameters may be assumed stochastic instead of deterministic. The
1058:
2003:
4291:
3891:{\displaystyle {\hat {\mathbf {x} }}_{i+1}=A_{i}{\hat {\mathbf {x} }}_{i}+B_{i}{\mathbf {u} }_{i}+L_{i+1}\left({\mathbf {y} }_{i+1}-C_{i+1}\left\{A_{i}{\hat {\mathbf {x} }}_{i}+B_{i}\mathbf {u} _{i}\right\}\right),\qquad {\hat {\mathbf {x} }}_{0}=\mathbb {E} }
4793:
4835:
2593:
4249:
2802:
problem (LQR). These problems are dual and together they solve the linear–quadratic–Gaussian control problem (LQG). So the LQG problem separates into the LQE and LQR problem that can be solved independently. Therefore, the LQG problem is called
3138:
267:
116:
the number of states of the LQG controller. This problem is more difficult to solve because it is no longer separable. Also, the solution is no longer unique. Despite these facts numerical algorithms are available to solve the associated
93:, and constitutes a linear dynamic feedback control law that is easily computed and implemented: the LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension.
3213:
1307:{\displaystyle {\dot {\hat {\mathbf {x} }}}(t)=A(t){\hat {\mathbf {x} }}(t)+B(t){\mathbf {u} }(t)+L(t)\left({\mathbf {y} }(t)-C(t){\hat {\mathbf {x} }}(t)\right),\quad {\hat {\mathbf {x} }}(0)=\mathbb {E} \left,}
348:
3970:
2789:
2244:
1015:
1392:
603:
901:
4113:
2901:
3598:
3282:
1803:
2362:
4633:{\displaystyle P_{i+1}=A_{i}\left(P_{i}-P_{i}C_{i}^{\mathrm {T} }\left(C_{i}P_{i}C_{i}^{\mathrm {T} }+W_{i}\right)^{-1}C_{i}P_{i}\right)A_{i}^{\mathrm {T} }+V_{i},\qquad P_{0}=\mathbb {E} .}
2692:
5082:{\displaystyle S_{i}=A_{i}^{\mathrm {T} }\left(S_{i+1}-S_{i+1}B_{i}\left(B_{i}^{\mathrm {T} }S_{i+1}B_{i}+R_{i}\right)^{-1}B_{i}^{\mathrm {T} }S_{i+1}\right)A_{i}+Q_{i},\quad S_{N}=F.}
4011:
1500:
4649:
2097:
3324:
1614:
69:
Under these assumptions an optimal control scheme within the class of linear control laws can be derived by a completion-of-squares argument. This control law which is known as the
2394:
3530:
2794:
Observe the similarity of the two matrix
Riccati differential equations, the first one running forward in time, the second one running backward in time. This similarity is called
2638:
1660:
825:
5116:
tends to infinity the discrete-time LQG controller becomes time-invariant. In that case the matrix
Riccati difference equations may be replaced by their associated discrete-time
4827:
4283:
2147:
2280:
1795:
1566:
1533:
1428:
518:
5178:
1047:
4124:
2963:
2932:
1722:
1691:
485:
454:
5146:
5114:
3013:
tends to infinity the LQG controller becomes a time-invariant dynamic system. In that case the second matrix
Riccati differential equation may be replaced by the associated
3011:
2987:
2386:
1460:
954:
542:
423:
399:
375:
3238:
926:
5379:
Zigic D.; Watson L.T.; Collins E.G.; Haddad W.M.; Ying S. (1996). "Homotopy methods for solving the optimal projection equations for the H2 reduced order model problem".
105:, as long as the system dynamics are linear, the optimal control separates into an optimal state estimator (which may no longer be a Kalman filter) and an LQR regulator.
3038:
154:
5406:
Collins Jr. E.G; Haddad W.M.; Ying S. (1996). "A homotopy algorithm for reduced-order dynamic compensation using the Hyland-Bernstein optimal projection equations".
5348:
Van
Willigenburg L.G.; De Koning W.L. (1999). "Optimal reduced-order compensators for time-varying discrete-time systems with deterministic and white parameters".
2798:. The first matrix Riccati differential equation solves the linear–quadratic estimation problem (LQE). The second matrix Riccati differential equation solves the
66:
criterion. Output measurements are assumed to be corrupted by
Gaussian noise and the initial state, likewise, is assumed to be a Gaussian random vector.
5474:
17:
3146:
1017:
of the cost function becomes negligible and irrelevant to the problem. Also to keep the costs finite the cost function has to be taken to be
5314:
Van
Willigenburg L.G.; De Koning W.L. (2000). "Numerical algorithms and issues concerning the discrete-time optimal projection equations".
108:
In the classical LQG setting, implementation of the LQG controller may be problematic when the dimension of the system state is large. The
5526:
273:
3905:
2700:
2155:
959:
125:
associated more difficult control problem leads to a similar optimal controller of which only the controller parameters are different.
98:
1320:
5434:
5366:
5332:
128:
It is possible to compute the expected value of the cost function for the optimal gains, as well as any other set of stable gains.
62:. The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic
833:
4016:
3029:
LQG control problem is similar to the one in continuous-time, the description below focuses on the mathematical equations.
2813:
1998:{\displaystyle {\dot {P}}(t)=A(t)P(t)+P(t)A^{\mathrm {T} }(t)-P(t)C^{\mathrm {T} }(t){\mathbf {} }W^{-1}(t)C(t)P(t)+V(t),}
3538:
3243:
1797:. These five matrices determine the Kalman gain through the following associated matrix Riccati differential equation:
2289:
547:
5672:
5578:
5222:
2646:
5635:
Athans M. (1971). "The role and use of the stochastic Linear-Quadratic-Gaussian problem in control system design".
5194:
85:
states that the state estimator and the state feedback can be designed independently. LQG control applies to both
102:
4788:{\displaystyle {\mathbf {} }K_{i}=(B_{i}^{\mathrm {T} }S_{i+1}B_{i}+R_{i})^{-1}B_{i}^{\mathrm {T} }S_{i+1}A_{i}}
3978:
2588:{\displaystyle -{\dot {S}}(t)=A^{\mathrm {T} }(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B^{\mathrm {T} }(t)S(t)+Q(t),}
1465:
2011:
59:
3287:
1571:
5691:
118:
5266:
Tryphon T. Georgiou and Anders
Lindquist (2013). "The Separation Principle in Stochastic Control, Redux".
3335:
2601:
1619:
611:
5701:
121:
which constitute necessary and sufficient conditions for a locally optimal reduced-order LQG controller.
4801:
4257:
4244:{\displaystyle {\mathbf {} }L_{i}=P_{i}C_{i}^{\mathrm {T} }(C_{i}P_{i}C_{i}^{\mathrm {T} }+W_{i})^{-1},}
2105:
5121:
5117:
3014:
2799:
2252:
1727:
1538:
1505:
1400:
490:
78:
5151:
1020:
2937:
2906:
1696:
1665:
459:
428:
5127:
5095:
2992:
2968:
2367:
1441:
935:
523:
404:
380:
356:
5696:
51:
5475:"The optimal projection equations for reduced-order discrete-time modeling estimation and control"
3221:
3133:{\displaystyle {\mathbf {x} }_{i+1}=A_{i}\mathbf {x} _{i}+B_{i}\mathbf {u} _{i}+\mathbf {v} _{i},}
909:
425:
the vector of measured outputs available for feedback. Both additive white
Gaussian system noise
262:{\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)+\mathbf {v} (t),}
1052:
The LQG controller that solves the LQG control problem is specified by the following equations:
4829:
is determined by the following matrix
Riccati difference equation that runs backward in time:
5568:
4285:
is determined by the following matrix
Riccati difference equation that runs forward in time:
5607:
5489:
101:
which states that even when the process and output noise sources are possibly non-Gaussian
82:
8:
90:
5611:
5493:
487:
affect the system. Given this system the objective is to find the control input history
5293:
5275:
5189:
5120:. These determine the time-invariant linear–quadratic estimator and the time-invariant
5596:"The effects of proportional steering strategies on the behavior of controlled clocks"
5361:
5327:
5668:
5574:
5549:
5218:
5092:
If all the matrices in the problem formulation are time-invariant and if the horizon
5297:
5640:
5615:
5541:
5505:
5497:
5453:
5445:
5415:
5388:
5357:
5323:
5285:
5247:
956:
may be either finite or infinite. If the horizon tends to infinity the first term
5662:
145:
63:
47:
3284:
represent discrete-time Gaussian white noise processes with covariance matrices
5620:
5595:
929:
31:
5392:
5685:
5644:
5553:
5545:
5449:
5289:
5238:
Anders Lindquist (1973). "On Feedback Control of Linear Stochastic Systems".
3026:
1435:
74:
55:
5378:
5510:
5458:
131:
The LQG controller is also used to control perturbed non-linear systems.
86:
5265:
3208:{\displaystyle \mathbf {y} _{i}=C_{i}\mathbf {x} _{i}+\mathbf {w} _{i}.}
5435:"The optimal projection equations for fixed order dynamic compensation"
2388:
through the following associated matrix Riccati differential equation:
134:
5501:
5419:
5251:
343:{\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+\mathbf {w} (t),}
5472:
5347:
5313:
3965:{\displaystyle \mathbf {u} _{i}=-K_{i}{\hat {\mathbf {x} }}_{i}.\,}
2784:{\displaystyle {\mathbf {} }K(t)=R^{-1}(t)B^{\mathrm {T} }(t)S(t).}
2239:{\displaystyle {\mathbf {} }L(t)=P(t)C^{\mathrm {T} }(t)W^{-1}(t).}
5280:
1010:{\displaystyle {\mathbf {x} }^{\mathrm {T} }(T)F{\mathbf {x} }(T)}
5405:
1387:{\displaystyle {\mathbf {u} }(t)=-K(t){\hat {\mathbf {x} }}(t).}
77:(a linear–quadratic state estimator (LQE)) together with a
73:
controller, is unique and it is simply a combination of a
896:{\displaystyle F\geq 0,\quad Q(t)\geq 0,\quad R(t)>0,}
5432:
1535:
using the past measurements and inputs. The Kalman gain
377:
represents the vector of state variables of the system,
5124:
in discrete-time. To keep the costs finite instead of
4108:{\displaystyle {\hat {\mathbf {x} }}_{i}=\mathbb {E} }
5154:
5130:
5098:
4838:
4804:
4652:
4294:
4260:
4127:
4019:
3981:
3908:
3612:
3541:
3338:
3290:
3246:
3224:
3149:
3041:
2995:
2971:
2940:
2909:
2896:{\displaystyle {\mathbf {} }A(t),B(t),C(t),Q(t),R(t)}
2816:
2703:
2649:
2604:
2397:
2370:
2292:
2255:
2158:
2108:
2014:
1806:
1730:
1699:
1668:
1622:
1574:
1541:
1508:
1468:
1444:
1403:
1323:
1061:
1023:
962:
938:
912:
836:
614:
550:
526:
493:
462:
431:
407:
383:
359:
276:
157:
50:
problems, and it can also be operated repeatedly for
605:
such that the following cost function is minimized:
135:
Mathematical description of the problem and solution
3326:, respectively, and are independent of each other.
112:(fixed-order LQG problem) overcomes this by fixing
5172:
5140:
5108:
5081:
4821:
4787:
4632:
4277:
4243:
4107:
4005:
3964:
3890:
3592:
3524:
3318:
3276:
3232:
3207:
3132:
3005:
2981:
2957:
2926:
2895:
2783:
2686:
2632:
2587:
2380:
2356:
2286:matrix. This matrix is determined by the matrices
2274:
2238:
2141:
2091:
1997:
1789:
1716:
1685:
1654:
1608:
1560:
1527:
1494:
1454:
1422:
1386:
1306:
1041:
1009:
948:
920:
895:
819:
597:
544:may depend linearly only on the past measurements
536:
512:
479:
448:
417:
393:
369:
342:
261:
5212:
3593:{\displaystyle F\geq 0,Q_{i}\geq 0,R_{i}>0.\,}
3277:{\displaystyle \mathbf {v} _{i},\mathbf {w} _{i}}
5683:
5473:Bernstein D.S.; Davis L.D.; Hyland D.C. (1986).
5367:Associated software download from Matlab Central
5333:Associated software download from Matlab Central
5237:
2357:{\displaystyle {\mathbf {} }A(t),B(t),Q(t),R(t)}
1438:represented by the first equation. At each time
598:{\displaystyle {\mathbf {y} }(t'),0\leq t'<t}
5573:. Englewood Cliffs: Prentice Hall. p. 27.
3329:The quadratic cost function to be minimized is
2687:{\displaystyle {\mathbf {} }S(t),0\leq t\leq T}
5567:Green, Michael; Limebeer, David J. N. (1995).
3032:The discrete-time linear system equations are
456:and additive white Gaussian measurement noise
5566:
5482:Journal of Guidance, Control, and Dynamics
5408:Journal of Guidance, Control, and Dynamics
99:separation principle of stochastic control
5634:
5619:
5509:
5457:
5279:
5215:Introduction to Stochastic Control Theory
4512:
4045:
4006:{\displaystyle {\hat {\mathbf {x} }}_{i}}
3961:
3864:
3589:
3346:
2031:
1732:
1271:
914:
802:
622:
5593:
5231:
1662:associated to the white Gaussian noises
1495:{\displaystyle {\hat {\mathbf {x} }}(t)}
5660:
5527:"Guaranteed Margins for LQG Regulators"
4013:corresponds to the predictive estimate
3240:represents the discrete time index and
2092:{\displaystyle P(0)=\mathbb {E} \left.}
14:
5684:
5637:IEEE Transactions on Automatic Control
5628:
5534:IEEE Transactions on Automatic Control
5442:IEEE Transactions on Automatic Control
5343:
5341:
5309:
5307:
5268:IEEE Transactions on Automatic Control
3319:{\displaystyle \mathbf {} V_{i},W_{i}}
1609:{\displaystyle {\mathbf {} }A(t),C(t)}
5594:Matsakis, Demetrios (March 8, 2019).
5524:
5259:
5206:
5466:
5426:
5372:
3603:The discrete-time LQG controller is
3525:{\displaystyle J=\mathbb {E} \left,}
2633:{\displaystyle {\mathbf {} }S(T)=F.}
1655:{\displaystyle \mathbf {} V(t),W(t)}
820:{\displaystyle J=\mathbb {E} \left,}
5433:Hyland D.C; Bernstein D.S. (1984).
5399:
5338:
5304:
24:
5654:
5006:
4936:
4863:
4822:{\displaystyle {\mathbf {} }S_{i}}
4753:
4684:
4618:
4475:
4406:
4363:
4278:{\displaystyle {\mathbf {} }P_{i}}
4206:
4166:
3483:
3439:
3370:
2751:
2540:
2431:
2196:
2142:{\displaystyle P(t),0\leq t\leq T}
2066:
1912:
1876:
1767:
973:
758:
702:
640:
139:
25:
5713:
2903:and the noise intensity matrices
2275:{\displaystyle {\mathbf {} }K(t)}
1790:{\displaystyle \mathbb {E} \left}
1561:{\displaystyle {\mathbf {} }L(t)}
1528:{\displaystyle {\mathbf {x} }(t)}
1423:{\displaystyle {\mathbf {} }L(t)}
513:{\displaystyle {\mathbf {u} }(t)}
401:the vector of control inputs and
18:Linear-quadratic-Gaussian control
5381:International Journal of Control
5217:. Vol. 58. Academic Press.
5173:{\displaystyle {\mathbf {} }J/N}
4643:The feedback gain matrix equals
4595:
4576:
4546:
4527:
4086:
4071:
4054:
4025:
3987:
3942:
3911:
3874:
3844:
3815:
3784:
3728:
3690:
3658:
3618:
3501:
3472:
3457:
3428:
3382:
3358:
3264:
3249:
3192:
3177:
3152:
3117:
3102:
3077:
3045:
3020:
2059:
2042:
1760:
1743:
1701:
1670:
1511:
1473:
1462:this filter generates estimates
1362:
1326:
1282:
1248:
1216:
1183:
1147:
1111:
1068:
1042:{\displaystyle {\mathbf {} }J/T}
993:
966:
788:
752:
732:
696:
661:
634:
553:
496:
464:
433:
410:
386:
362:
324:
307:
278:
243:
226:
197:
162:
27:Linear optimal control technique
5587:
5560:
5518:
5059:
4497:
3838:
2958:{\displaystyle \mathbf {} W(t)}
2927:{\displaystyle \mathbf {} V(t)}
1717:{\displaystyle \mathbf {w} (t)}
1686:{\displaystyle \mathbf {v} (t)}
1243:
871:
849:
480:{\displaystyle \mathbf {w} (t)}
449:{\displaystyle \mathbf {v} (t)}
46:is one of the most fundamental
5664:Optimal Control and Estimation
5141:{\displaystyle {\mathbf {} }J}
5109:{\displaystyle {\mathbf {} }N}
4730:
4670:
4624:
4599:
4550:
4516:
4226:
4172:
4102:
4065:
4049:
4029:
3991:
3946:
3885:
3868:
3848:
3788:
3662:
3622:
3511:
3423:
3006:{\displaystyle {\mathbf {} }T}
2982:{\displaystyle {\mathbf {} }t}
2952:
2946:
2921:
2915:
2890:
2884:
2875:
2869:
2860:
2854:
2845:
2839:
2830:
2824:
2775:
2769:
2763:
2757:
2742:
2736:
2717:
2711:
2663:
2657:
2618:
2612:
2579:
2573:
2564:
2558:
2552:
2546:
2531:
2525:
2509:
2503:
2497:
2491:
2482:
2476:
2470:
2464:
2455:
2449:
2443:
2437:
2419:
2413:
2381:{\displaystyle {\mathbf {} }F}
2351:
2345:
2336:
2330:
2321:
2315:
2306:
2300:
2269:
2263:
2230:
2224:
2208:
2202:
2187:
2181:
2172:
2166:
2118:
2112:
2078:
2072:
2053:
2047:
2024:
2018:
1989:
1983:
1974:
1968:
1962:
1956:
1950:
1944:
1924:
1918:
1903:
1897:
1888:
1882:
1867:
1861:
1852:
1846:
1840:
1834:
1825:
1819:
1779:
1773:
1754:
1748:
1711:
1705:
1680:
1674:
1649:
1643:
1634:
1628:
1603:
1597:
1588:
1582:
1568:is computed from the matrices
1555:
1549:
1522:
1516:
1489:
1483:
1477:
1455:{\displaystyle {\mathbf {} }t}
1417:
1411:
1378:
1372:
1366:
1355:
1349:
1337:
1331:
1293:
1287:
1264:
1258:
1252:
1232:
1226:
1220:
1209:
1203:
1194:
1188:
1173:
1167:
1158:
1152:
1142:
1136:
1127:
1121:
1115:
1104:
1098:
1089:
1083:
1072:
1004:
998:
985:
979:
949:{\displaystyle {\mathbf {} }T}
881:
875:
859:
853:
799:
793:
783:
777:
771:
765:
743:
737:
727:
721:
715:
709:
672:
666:
653:
647:
569:
558:
537:{\displaystyle {\mathbf {} }t}
507:
501:
474:
468:
443:
437:
418:{\displaystyle {\mathbf {y} }}
394:{\displaystyle {\mathbf {u} }}
370:{\displaystyle {\mathbf {x} }}
334:
328:
317:
311:
303:
297:
288:
282:
253:
247:
236:
230:
222:
216:
207:
201:
193:
187:
178:
172:
13:
1:
5362:10.1016/S0005-1098(98)00138-1
5328:10.1016/s0947-3580(00)70917-4
5200:
5195:Witsenhausen's counterexample
1616:, the two intensity matrices
87:linear time-invariant systems
60:additive white Gaussian noise
3233:{\displaystyle \mathbf {} i}
921:{\displaystyle \mathbb {E} }
119:optimal projection equations
7:
5661:Stengel, Robert F. (1994).
5316:European Journal of Control
5183:
5118:algebraic Riccati equations
932:. The final time (horizon)
91:linear time-varying systems
10:
5718:
5213:Karl Johan Astrom (1970).
5122:linear–quadratic regulator
3015:algebraic Riccati equation
2800:linear–quadratic regulator
79:linear–quadratic regulator
5444:. AC-29 (11): 1034–1037.
5393:10.1080/00207179208934308
2694:the feedback gain equals
110:reduced-order LQG problem
36:linear–quadratic–Gaussian
5645:10.1109/TAC.1971.1099818
5621:10.1088/1681-7575/ab0614
5546:10.1109/TAC.1978.1101812
5450:10.1109/TAC.1984.1103418
5290:10.1109/TAC.2013.2259207
52:model predictive control
5525:Doyle, John C. (1978).
5240:SIAM Journal on Control
4118:The Kalman gain equals
2149:the Kalman gain equals
5639:. AC-16 (6): 529–552.
5174:
5142:
5110:
5083:
4823:
4789:
4634:
4279:
4245:
4109:
4007:
3966:
3892:
3594:
3526:
3422:
3320:
3278:
3234:
3209:
3134:
3007:
2983:
2959:
2928:
2897:
2785:
2688:
2634:
2589:
2382:
2358:
2276:
2240:
2143:
2093:
1999:
1791:
1718:
1687:
1656:
1610:
1562:
1529:
1496:
1456:
1424:
1388:
1308:
1043:
1011:
950:
922:
897:
821:
599:
538:
514:
481:
450:
419:
395:
371:
344:
263:
148:linear dynamic system
5570:Linear Robust Control
5175:
5143:
5111:
5084:
4824:
4790:
4635:
4280:
4246:
4110:
4008:
3967:
3893:
3595:
3527:
3396:
3321:
3279:
3235:
3210:
3135:
3008:
2984:
2960:
2929:
2898:
2786:
2689:
2635:
2590:
2383:
2359:
2277:
2241:
2144:
2094:
2000:
1792:
1719:
1688:
1657:
1611:
1563:
1530:
1497:
1457:
1425:
1389:
1309:
1044:
1012:
951:
923:
898:
822:
600:
539:
515:
482:
451:
420:
396:
372:
345:
264:
5152:
5148:one has to consider
5128:
5096:
4836:
4802:
4650:
4292:
4258:
4125:
4017:
3979:
3906:
3610:
3539:
3336:
3288:
3244:
3222:
3147:
3039:
2993:
2969:
2938:
2907:
2814:
2701:
2647:
2602:
2395:
2368:
2290:
2253:
2156:
2106:
2012:
1804:
1728:
1697:
1666:
1620:
1572:
1539:
1506:
1466:
1442:
1401:
1321:
1059:
1021:
960:
936:
910:
834:
612:
548:
524:
520:which at every time
491:
460:
429:
405:
381:
357:
274:
155:
83:separation principle
5692:Control loop theory
5667:. New York: Dover.
5612:2019Metro..56b5007M
5494:1986JGCD....9..288B
5011:
4941:
4868:
4758:
4689:
4480:
4411:
4368:
4211:
4171:
3488:
3444:
3375:
2643:Given the solution
2102:Given the solution
692:
5702:Stochastic control
5190:Stochastic control
5170:
5138:
5106:
5079:
4995:
4925:
4852:
4819:
4785:
4742:
4673:
4630:
4464:
4395:
4352:
4275:
4241:
4195:
4155:
4105:
4003:
3962:
3888:
3590:
3522:
3470:
3426:
3355:
3316:
3274:
3230:
3205:
3130:
3003:
2979:
2955:
2924:
2893:
2781:
2684:
2630:
2585:
2378:
2354:
2272:
2236:
2139:
2089:
1995:
1787:
1714:
1683:
1652:
1606:
1558:
1525:
1492:
1452:
1434:of the associated
1420:
1384:
1304:
1039:
1007:
946:
918:
893:
817:
678:
595:
534:
510:
477:
446:
415:
391:
367:
340:
259:
5274:(10): 2481–2494.
4602:
4553:
4032:
3994:
3949:
3851:
3791:
3665:
3625:
2965:do not depend on
2410:
1816:
1480:
1369:
1255:
1223:
1118:
1080:
1075:
169:
16:(Redirected from
5709:
5678:
5649:
5648:
5632:
5626:
5625:
5623:
5591:
5585:
5584:
5564:
5558:
5557:
5531:
5522:
5516:
5515:
5513:
5479:
5470:
5464:
5463:
5461:
5439:
5430:
5424:
5423:
5403:
5397:
5396:
5376:
5370:
5365:
5345:
5336:
5331:
5311:
5302:
5301:
5283:
5263:
5257:
5255:
5235:
5229:
5228:
5210:
5179:
5177:
5176:
5171:
5166:
5158:
5157:
5147:
5145:
5144:
5139:
5134:
5133:
5115:
5113:
5112:
5107:
5102:
5101:
5088:
5086:
5085:
5080:
5069:
5068:
5055:
5054:
5042:
5041:
5032:
5028:
5027:
5026:
5010:
5009:
5003:
4994:
4993:
4985:
4981:
4980:
4979:
4967:
4966:
4957:
4956:
4940:
4939:
4933:
4918:
4917:
4908:
4907:
4889:
4888:
4867:
4866:
4860:
4848:
4847:
4828:
4826:
4825:
4820:
4818:
4817:
4808:
4807:
4794:
4792:
4791:
4786:
4784:
4783:
4774:
4773:
4757:
4756:
4750:
4741:
4740:
4728:
4727:
4715:
4714:
4705:
4704:
4688:
4687:
4681:
4666:
4665:
4656:
4655:
4639:
4637:
4636:
4631:
4623:
4622:
4621:
4615:
4611:
4610:
4609:
4604:
4603:
4598:
4593:
4586:
4585:
4580:
4579:
4566:
4562:
4561:
4560:
4555:
4554:
4549:
4544:
4537:
4536:
4531:
4530:
4515:
4507:
4506:
4493:
4492:
4479:
4478:
4472:
4463:
4459:
4458:
4457:
4448:
4447:
4438:
4437:
4429:
4425:
4424:
4423:
4410:
4409:
4403:
4394:
4393:
4384:
4383:
4367:
4366:
4360:
4351:
4350:
4338:
4337:
4323:
4322:
4310:
4309:
4284:
4282:
4281:
4276:
4274:
4273:
4264:
4263:
4250:
4248:
4247:
4242:
4237:
4236:
4224:
4223:
4210:
4209:
4203:
4194:
4193:
4184:
4183:
4170:
4169:
4163:
4154:
4153:
4141:
4140:
4131:
4130:
4114:
4112:
4111:
4106:
4101:
4100:
4089:
4080:
4079:
4074:
4068:
4063:
4062:
4057:
4048:
4040:
4039:
4034:
4033:
4028:
4023:
4012:
4010:
4009:
4004:
4002:
4001:
3996:
3995:
3990:
3985:
3971:
3969:
3968:
3963:
3957:
3956:
3951:
3950:
3945:
3940:
3936:
3935:
3920:
3919:
3914:
3897:
3895:
3894:
3889:
3884:
3883:
3878:
3877:
3867:
3859:
3858:
3853:
3852:
3847:
3842:
3834:
3830:
3829:
3825:
3824:
3823:
3818:
3812:
3811:
3799:
3798:
3793:
3792:
3787:
3782:
3778:
3777:
3763:
3762:
3744:
3743:
3732:
3731:
3719:
3718:
3700:
3699:
3694:
3693:
3686:
3685:
3673:
3672:
3667:
3666:
3661:
3656:
3652:
3651:
3639:
3638:
3627:
3626:
3621:
3616:
3599:
3597:
3596:
3591:
3582:
3581:
3563:
3562:
3531:
3529:
3528:
3523:
3518:
3514:
3510:
3509:
3504:
3498:
3497:
3487:
3486:
3480:
3475:
3466:
3465:
3460:
3454:
3453:
3443:
3442:
3436:
3431:
3421:
3410:
3392:
3391:
3386:
3385:
3374:
3373:
3367:
3362:
3361:
3349:
3325:
3323:
3322:
3317:
3315:
3314:
3302:
3301:
3292:
3283:
3281:
3280:
3275:
3273:
3272:
3267:
3258:
3257:
3252:
3239:
3237:
3236:
3231:
3226:
3214:
3212:
3211:
3206:
3201:
3200:
3195:
3186:
3185:
3180:
3174:
3173:
3161:
3160:
3155:
3139:
3137:
3136:
3131:
3126:
3125:
3120:
3111:
3110:
3105:
3099:
3098:
3086:
3085:
3080:
3074:
3073:
3061:
3060:
3049:
3048:
3012:
3010:
3009:
3004:
2999:
2998:
2988:
2986:
2985:
2980:
2975:
2974:
2964:
2962:
2961:
2956:
2942:
2933:
2931:
2930:
2925:
2911:
2902:
2900:
2899:
2894:
2820:
2819:
2790:
2788:
2787:
2782:
2756:
2755:
2754:
2735:
2734:
2707:
2706:
2693:
2691:
2690:
2685:
2653:
2652:
2639:
2637:
2636:
2631:
2608:
2607:
2594:
2592:
2591:
2586:
2545:
2544:
2543:
2524:
2523:
2436:
2435:
2434:
2412:
2411:
2403:
2387:
2385:
2384:
2379:
2374:
2373:
2363:
2361:
2360:
2355:
2296:
2295:
2281:
2279:
2278:
2273:
2259:
2258:
2245:
2243:
2242:
2237:
2223:
2222:
2201:
2200:
2199:
2162:
2161:
2148:
2146:
2145:
2140:
2098:
2096:
2095:
2090:
2085:
2081:
2071:
2070:
2069:
2063:
2062:
2046:
2045:
2034:
2004:
2002:
2001:
1996:
1943:
1942:
1930:
1929:
1917:
1916:
1915:
1881:
1880:
1879:
1818:
1817:
1809:
1796:
1794:
1793:
1788:
1786:
1782:
1772:
1771:
1770:
1764:
1763:
1747:
1746:
1735:
1723:
1721:
1720:
1715:
1704:
1692:
1690:
1689:
1684:
1673:
1661:
1659:
1658:
1653:
1624:
1615:
1613:
1612:
1607:
1578:
1577:
1567:
1565:
1564:
1559:
1545:
1544:
1534:
1532:
1531:
1526:
1515:
1514:
1501:
1499:
1498:
1493:
1482:
1481:
1476:
1471:
1461:
1459:
1458:
1453:
1448:
1447:
1429:
1427:
1426:
1421:
1407:
1406:
1393:
1391:
1390:
1385:
1371:
1370:
1365:
1360:
1330:
1329:
1313:
1311:
1310:
1305:
1300:
1296:
1286:
1285:
1274:
1257:
1256:
1251:
1246:
1239:
1235:
1225:
1224:
1219:
1214:
1187:
1186:
1151:
1150:
1120:
1119:
1114:
1109:
1082:
1081:
1076:
1071:
1066:
1064:
1048:
1046:
1045:
1040:
1035:
1027:
1026:
1016:
1014:
1013:
1008:
997:
996:
978:
977:
976:
970:
969:
955:
953:
952:
947:
942:
941:
927:
925:
924:
919:
917:
902:
900:
899:
894:
826:
824:
823:
818:
813:
809:
792:
791:
764:
763:
762:
761:
755:
736:
735:
708:
707:
706:
705:
699:
691:
686:
665:
664:
646:
645:
644:
643:
637:
625:
604:
602:
601:
596:
588:
568:
557:
556:
543:
541:
540:
535:
530:
529:
519:
517:
516:
511:
500:
499:
486:
484:
483:
478:
467:
455:
453:
452:
447:
436:
424:
422:
421:
416:
414:
413:
400:
398:
397:
392:
390:
389:
376:
374:
373:
368:
366:
365:
349:
347:
346:
341:
327:
310:
281:
268:
266:
265:
260:
246:
229:
200:
171:
170:
165:
160:
21:
5717:
5716:
5712:
5711:
5710:
5708:
5707:
5706:
5697:Optimal control
5682:
5681:
5675:
5657:
5655:Further reading
5652:
5633:
5629:
5592:
5588:
5581:
5565:
5561:
5529:
5523:
5519:
5502:10.2514/3.20105
5477:
5471:
5467:
5437:
5431:
5427:
5420:10.2514/3.21633
5404:
5400:
5377:
5373:
5346:
5339:
5312:
5305:
5264:
5260:
5252:10.1137/0311025
5236:
5232:
5225:
5211:
5207:
5203:
5186:
5162:
5156:
5155:
5153:
5150:
5149:
5132:
5131:
5129:
5126:
5125:
5100:
5099:
5097:
5094:
5093:
5064:
5060:
5050:
5046:
5037:
5033:
5016:
5012:
5005:
5004:
4999:
4986:
4975:
4971:
4962:
4958:
4946:
4942:
4935:
4934:
4929:
4924:
4920:
4919:
4913:
4909:
4897:
4893:
4878:
4874:
4873:
4869:
4862:
4861:
4856:
4843:
4839:
4837:
4834:
4833:
4813:
4809:
4806:
4805:
4803:
4800:
4799:
4779:
4775:
4763:
4759:
4752:
4751:
4746:
4733:
4729:
4723:
4719:
4710:
4706:
4694:
4690:
4683:
4682:
4677:
4661:
4657:
4654:
4653:
4651:
4648:
4647:
4617:
4616:
4605:
4594:
4592:
4591:
4590:
4581:
4575:
4574:
4573:
4572:
4568:
4567:
4556:
4545:
4543:
4542:
4541:
4532:
4526:
4525:
4524:
4523:
4519:
4511:
4502:
4498:
4488:
4484:
4474:
4473:
4468:
4453:
4449:
4443:
4439:
4430:
4419:
4415:
4405:
4404:
4399:
4389:
4385:
4379:
4375:
4374:
4370:
4369:
4362:
4361:
4356:
4346:
4342:
4333:
4329:
4328:
4324:
4318:
4314:
4299:
4295:
4293:
4290:
4289:
4269:
4265:
4262:
4261:
4259:
4256:
4255:
4229:
4225:
4219:
4215:
4205:
4204:
4199:
4189:
4185:
4179:
4175:
4165:
4164:
4159:
4149:
4145:
4136:
4132:
4129:
4128:
4126:
4123:
4122:
4090:
4085:
4084:
4075:
4070:
4069:
4064:
4058:
4053:
4052:
4044:
4035:
4024:
4022:
4021:
4020:
4018:
4015:
4014:
3997:
3986:
3984:
3983:
3982:
3980:
3977:
3976:
3952:
3941:
3939:
3938:
3937:
3931:
3927:
3915:
3910:
3909:
3907:
3904:
3903:
3879:
3873:
3872:
3871:
3863:
3854:
3843:
3841:
3840:
3839:
3819:
3814:
3813:
3807:
3803:
3794:
3783:
3781:
3780:
3779:
3773:
3769:
3768:
3764:
3752:
3748:
3733:
3727:
3726:
3725:
3724:
3720:
3708:
3704:
3695:
3689:
3688:
3687:
3681:
3677:
3668:
3657:
3655:
3654:
3653:
3647:
3643:
3628:
3617:
3615:
3614:
3613:
3611:
3608:
3607:
3577:
3573:
3558:
3554:
3540:
3537:
3536:
3505:
3500:
3499:
3493:
3489:
3482:
3481:
3476:
3471:
3461:
3456:
3455:
3449:
3445:
3438:
3437:
3432:
3427:
3411:
3400:
3387:
3381:
3380:
3379:
3369:
3368:
3363:
3357:
3356:
3354:
3350:
3345:
3337:
3334:
3333:
3310:
3306:
3297:
3293:
3291:
3289:
3286:
3285:
3268:
3263:
3262:
3253:
3248:
3247:
3245:
3242:
3241:
3225:
3223:
3220:
3219:
3196:
3191:
3190:
3181:
3176:
3175:
3169:
3165:
3156:
3151:
3150:
3148:
3145:
3144:
3121:
3116:
3115:
3106:
3101:
3100:
3094:
3090:
3081:
3076:
3075:
3069:
3065:
3050:
3044:
3043:
3042:
3040:
3037:
3036:
3023:
2997:
2996:
2994:
2991:
2990:
2973:
2972:
2970:
2967:
2966:
2941:
2939:
2936:
2935:
2910:
2908:
2905:
2904:
2818:
2817:
2815:
2812:
2811:
2750:
2749:
2745:
2727:
2723:
2705:
2704:
2702:
2699:
2698:
2651:
2650:
2648:
2645:
2644:
2606:
2605:
2603:
2600:
2599:
2539:
2538:
2534:
2516:
2512:
2430:
2429:
2425:
2402:
2401:
2396:
2393:
2392:
2372:
2371:
2369:
2366:
2365:
2294:
2293:
2291:
2288:
2287:
2257:
2256:
2254:
2251:
2250:
2215:
2211:
2195:
2194:
2190:
2160:
2159:
2157:
2154:
2153:
2107:
2104:
2103:
2065:
2064:
2058:
2057:
2056:
2041:
2040:
2039:
2035:
2030:
2013:
2010:
2009:
1935:
1931:
1928:
1927:
1911:
1910:
1906:
1875:
1874:
1870:
1808:
1807:
1805:
1802:
1801:
1766:
1765:
1759:
1758:
1757:
1742:
1741:
1740:
1736:
1731:
1729:
1726:
1725:
1700:
1698:
1695:
1694:
1669:
1667:
1664:
1663:
1623:
1621:
1618:
1617:
1576:
1575:
1573:
1570:
1569:
1543:
1542:
1540:
1537:
1536:
1510:
1509:
1507:
1504:
1503:
1472:
1470:
1469:
1467:
1464:
1463:
1446:
1445:
1443:
1440:
1439:
1405:
1404:
1402:
1399:
1398:
1361:
1359:
1358:
1325:
1324:
1322:
1319:
1318:
1281:
1280:
1279:
1275:
1270:
1247:
1245:
1244:
1215:
1213:
1212:
1182:
1181:
1180:
1176:
1146:
1145:
1110:
1108:
1107:
1067:
1065:
1063:
1062:
1060:
1057:
1056:
1031:
1025:
1024:
1022:
1019:
1018:
992:
991:
972:
971:
965:
964:
963:
961:
958:
957:
940:
939:
937:
934:
933:
913:
911:
908:
907:
835:
832:
831:
787:
786:
757:
756:
751:
750:
749:
731:
730:
701:
700:
695:
694:
693:
687:
682:
660:
659:
639:
638:
633:
632:
631:
630:
626:
621:
613:
610:
609:
581:
561:
552:
551:
549:
546:
545:
528:
527:
525:
522:
521:
495:
494:
492:
489:
488:
463:
461:
458:
457:
432:
430:
427:
426:
409:
408:
406:
403:
402:
385:
384:
382:
379:
378:
361:
360:
358:
355:
354:
323:
306:
277:
275:
272:
271:
242:
225:
196:
161:
159:
158:
156:
153:
152:
146:continuous-time
142:
140:Continuous time
137:
48:optimal control
44:control problem
28:
23:
22:
15:
12:
11:
5:
5715:
5705:
5704:
5699:
5694:
5680:
5679:
5673:
5656:
5653:
5651:
5650:
5627:
5586:
5579:
5559:
5540:(4): 756–757.
5517:
5488:(3): 288–293.
5465:
5425:
5414:(2): 407–417.
5398:
5387:(1): 173–191.
5371:
5337:
5303:
5258:
5246:(2): 323–343.
5230:
5223:
5204:
5202:
5199:
5198:
5197:
5192:
5185:
5182:
5180:in this case.
5169:
5165:
5161:
5137:
5105:
5090:
5089:
5078:
5075:
5072:
5067:
5063:
5058:
5053:
5049:
5045:
5040:
5036:
5031:
5025:
5022:
5019:
5015:
5008:
5002:
4998:
4992:
4989:
4984:
4978:
4974:
4970:
4965:
4961:
4955:
4952:
4949:
4945:
4938:
4932:
4928:
4923:
4916:
4912:
4906:
4903:
4900:
4896:
4892:
4887:
4884:
4881:
4877:
4872:
4865:
4859:
4855:
4851:
4846:
4842:
4816:
4812:
4796:
4795:
4782:
4778:
4772:
4769:
4766:
4762:
4755:
4749:
4745:
4739:
4736:
4732:
4726:
4722:
4718:
4713:
4709:
4703:
4700:
4697:
4693:
4686:
4680:
4676:
4672:
4669:
4664:
4660:
4641:
4640:
4629:
4626:
4620:
4614:
4608:
4601:
4597:
4589:
4584:
4578:
4571:
4565:
4559:
4552:
4548:
4540:
4535:
4529:
4522:
4518:
4514:
4510:
4505:
4501:
4496:
4491:
4487:
4483:
4477:
4471:
4467:
4462:
4456:
4452:
4446:
4442:
4436:
4433:
4428:
4422:
4418:
4414:
4408:
4402:
4398:
4392:
4388:
4382:
4378:
4373:
4365:
4359:
4355:
4349:
4345:
4341:
4336:
4332:
4327:
4321:
4317:
4313:
4308:
4305:
4302:
4298:
4272:
4268:
4252:
4251:
4240:
4235:
4232:
4228:
4222:
4218:
4214:
4208:
4202:
4198:
4192:
4188:
4182:
4178:
4174:
4168:
4162:
4158:
4152:
4148:
4144:
4139:
4135:
4104:
4099:
4096:
4093:
4088:
4083:
4078:
4073:
4067:
4061:
4056:
4051:
4047:
4043:
4038:
4031:
4027:
4000:
3993:
3989:
3973:
3972:
3960:
3955:
3948:
3944:
3934:
3930:
3926:
3923:
3918:
3913:
3900:
3899:
3887:
3882:
3876:
3870:
3866:
3862:
3857:
3850:
3846:
3837:
3833:
3828:
3822:
3817:
3810:
3806:
3802:
3797:
3790:
3786:
3776:
3772:
3767:
3761:
3758:
3755:
3751:
3747:
3742:
3739:
3736:
3730:
3723:
3717:
3714:
3711:
3707:
3703:
3698:
3692:
3684:
3680:
3676:
3671:
3664:
3660:
3650:
3646:
3642:
3637:
3634:
3631:
3624:
3620:
3601:
3600:
3588:
3585:
3580:
3576:
3572:
3569:
3566:
3561:
3557:
3553:
3550:
3547:
3544:
3533:
3532:
3521:
3517:
3513:
3508:
3503:
3496:
3492:
3485:
3479:
3474:
3469:
3464:
3459:
3452:
3448:
3441:
3435:
3430:
3425:
3420:
3417:
3414:
3409:
3406:
3403:
3399:
3395:
3390:
3384:
3378:
3372:
3366:
3360:
3353:
3348:
3344:
3341:
3313:
3309:
3305:
3300:
3296:
3271:
3266:
3261:
3256:
3251:
3229:
3216:
3215:
3204:
3199:
3194:
3189:
3184:
3179:
3172:
3168:
3164:
3159:
3154:
3141:
3140:
3129:
3124:
3119:
3114:
3109:
3104:
3097:
3093:
3089:
3084:
3079:
3072:
3068:
3064:
3059:
3056:
3053:
3047:
3022:
3019:
3002:
2978:
2954:
2951:
2948:
2945:
2923:
2920:
2917:
2914:
2892:
2889:
2886:
2883:
2880:
2877:
2874:
2871:
2868:
2865:
2862:
2859:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2823:
2792:
2791:
2780:
2777:
2774:
2771:
2768:
2765:
2762:
2759:
2753:
2748:
2744:
2741:
2738:
2733:
2730:
2726:
2722:
2719:
2716:
2713:
2710:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2641:
2640:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2596:
2595:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2542:
2537:
2533:
2530:
2527:
2522:
2519:
2515:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2451:
2448:
2445:
2442:
2439:
2433:
2428:
2424:
2421:
2418:
2415:
2409:
2406:
2400:
2377:
2353:
2350:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2320:
2317:
2314:
2311:
2308:
2305:
2302:
2299:
2282:is called the
2271:
2268:
2265:
2262:
2247:
2246:
2235:
2232:
2229:
2226:
2221:
2218:
2214:
2210:
2207:
2204:
2198:
2193:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2100:
2099:
2088:
2084:
2080:
2077:
2074:
2068:
2061:
2055:
2052:
2049:
2044:
2038:
2033:
2029:
2026:
2023:
2020:
2017:
2006:
2005:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1941:
1938:
1934:
1926:
1923:
1920:
1914:
1909:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1878:
1873:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1824:
1821:
1815:
1812:
1785:
1781:
1778:
1775:
1769:
1762:
1756:
1753:
1750:
1745:
1739:
1734:
1713:
1710:
1707:
1703:
1682:
1679:
1676:
1672:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1584:
1581:
1557:
1554:
1551:
1548:
1524:
1521:
1518:
1513:
1491:
1488:
1485:
1479:
1475:
1451:
1430:is called the
1419:
1416:
1413:
1410:
1395:
1394:
1383:
1380:
1377:
1374:
1368:
1364:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1333:
1328:
1315:
1314:
1303:
1299:
1295:
1292:
1289:
1284:
1278:
1273:
1269:
1266:
1263:
1260:
1254:
1250:
1242:
1238:
1234:
1231:
1228:
1222:
1218:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1185:
1179:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1149:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1117:
1113:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1079:
1074:
1070:
1038:
1034:
1030:
1006:
1003:
1000:
995:
990:
987:
984:
981:
975:
968:
945:
930:expected value
916:
904:
903:
892:
889:
886:
883:
880:
877:
874:
870:
867:
864:
861:
858:
855:
852:
848:
845:
842:
839:
828:
827:
816:
812:
808:
805:
801:
798:
795:
790:
785:
782:
779:
776:
773:
770:
767:
760:
754:
748:
745:
742:
739:
734:
729:
726:
723:
720:
717:
714:
711:
704:
698:
690:
685:
681:
677:
674:
671:
668:
663:
658:
655:
652:
649:
642:
636:
629:
624:
620:
617:
594:
591:
587:
584:
580:
577:
574:
571:
567:
564:
560:
555:
533:
509:
506:
503:
498:
476:
473:
470:
466:
445:
442:
439:
435:
412:
388:
364:
351:
350:
339:
336:
333:
330:
326:
322:
319:
316:
313:
309:
305:
302:
299:
296:
293:
290:
287:
284:
280:
269:
258:
255:
252:
249:
245:
241:
238:
235:
232:
228:
224:
221:
218:
215:
212:
209:
206:
203:
199:
195:
192:
189:
186:
183:
180:
177:
174:
168:
164:
141:
138:
136:
133:
56:linear systems
54:. It concerns
32:control theory
26:
9:
6:
4:
3:
2:
5714:
5703:
5700:
5698:
5695:
5693:
5690:
5689:
5687:
5676:
5674:0-486-68200-5
5670:
5666:
5665:
5659:
5658:
5646:
5642:
5638:
5631:
5622:
5617:
5613:
5609:
5606:(2): 025007.
5605:
5601:
5597:
5590:
5582:
5580:0-13-102278-4
5576:
5572:
5571:
5563:
5555:
5551:
5547:
5543:
5539:
5535:
5528:
5521:
5512:
5511:2027.42/57880
5507:
5503:
5499:
5495:
5491:
5487:
5483:
5476:
5469:
5460:
5459:2027.42/57875
5455:
5451:
5447:
5443:
5436:
5429:
5421:
5417:
5413:
5409:
5402:
5394:
5390:
5386:
5382:
5375:
5368:
5363:
5359:
5355:
5351:
5344:
5342:
5334:
5329:
5325:
5322:(1): 93–100.
5321:
5317:
5310:
5308:
5299:
5295:
5291:
5287:
5282:
5277:
5273:
5269:
5262:
5253:
5249:
5245:
5241:
5234:
5226:
5224:0-486-44531-3
5220:
5216:
5209:
5205:
5196:
5193:
5191:
5188:
5187:
5181:
5167:
5163:
5159:
5135:
5123:
5119:
5103:
5076:
5073:
5070:
5065:
5061:
5056:
5051:
5047:
5043:
5038:
5034:
5029:
5023:
5020:
5017:
5013:
5000:
4996:
4990:
4987:
4982:
4976:
4972:
4968:
4963:
4959:
4953:
4950:
4947:
4943:
4930:
4926:
4921:
4914:
4910:
4904:
4901:
4898:
4894:
4890:
4885:
4882:
4879:
4875:
4870:
4857:
4853:
4849:
4844:
4840:
4832:
4831:
4830:
4814:
4810:
4780:
4776:
4770:
4767:
4764:
4760:
4747:
4743:
4737:
4734:
4724:
4720:
4716:
4711:
4707:
4701:
4698:
4695:
4691:
4678:
4674:
4667:
4662:
4658:
4646:
4645:
4644:
4627:
4612:
4606:
4587:
4582:
4569:
4563:
4557:
4538:
4533:
4520:
4508:
4503:
4499:
4494:
4489:
4485:
4481:
4469:
4465:
4460:
4454:
4450:
4444:
4440:
4434:
4431:
4426:
4420:
4416:
4412:
4400:
4396:
4390:
4386:
4380:
4376:
4371:
4357:
4353:
4347:
4343:
4339:
4334:
4330:
4325:
4319:
4315:
4311:
4306:
4303:
4300:
4296:
4288:
4287:
4286:
4270:
4266:
4238:
4233:
4230:
4220:
4216:
4212:
4200:
4196:
4190:
4186:
4180:
4176:
4160:
4156:
4150:
4146:
4142:
4137:
4133:
4121:
4120:
4119:
4116:
4097:
4094:
4091:
4081:
4076:
4059:
4041:
4036:
3998:
3958:
3953:
3932:
3928:
3924:
3921:
3916:
3902:
3901:
3880:
3860:
3855:
3835:
3831:
3826:
3820:
3808:
3804:
3800:
3795:
3774:
3770:
3765:
3759:
3756:
3753:
3749:
3745:
3740:
3737:
3734:
3721:
3715:
3712:
3709:
3705:
3701:
3696:
3682:
3678:
3674:
3669:
3648:
3644:
3640:
3635:
3632:
3629:
3606:
3605:
3604:
3586:
3583:
3578:
3574:
3570:
3567:
3564:
3559:
3555:
3551:
3548:
3545:
3542:
3535:
3534:
3519:
3515:
3506:
3494:
3490:
3477:
3467:
3462:
3450:
3446:
3433:
3418:
3415:
3412:
3407:
3404:
3401:
3397:
3393:
3388:
3376:
3364:
3351:
3342:
3339:
3332:
3331:
3330:
3327:
3311:
3307:
3303:
3298:
3294:
3269:
3259:
3254:
3227:
3202:
3197:
3187:
3182:
3170:
3166:
3162:
3157:
3143:
3142:
3127:
3122:
3112:
3107:
3095:
3091:
3087:
3082:
3070:
3066:
3062:
3057:
3054:
3051:
3035:
3034:
3033:
3030:
3028:
3027:discrete-time
3021:Discrete time
3018:
3016:
3000:
2976:
2949:
2943:
2918:
2912:
2887:
2881:
2878:
2872:
2866:
2863:
2857:
2851:
2848:
2842:
2836:
2833:
2827:
2821:
2808:
2806:
2801:
2797:
2778:
2772:
2766:
2760:
2746:
2739:
2731:
2728:
2724:
2720:
2714:
2708:
2697:
2696:
2695:
2681:
2678:
2675:
2672:
2669:
2666:
2660:
2654:
2627:
2624:
2621:
2615:
2609:
2598:
2597:
2582:
2576:
2570:
2567:
2561:
2555:
2549:
2535:
2528:
2520:
2517:
2513:
2506:
2500:
2494:
2488:
2485:
2479:
2473:
2467:
2461:
2458:
2452:
2446:
2440:
2426:
2422:
2416:
2407:
2404:
2398:
2391:
2390:
2389:
2375:
2348:
2342:
2339:
2333:
2327:
2324:
2318:
2312:
2309:
2303:
2297:
2285:
2284:feedback gain
2266:
2260:
2233:
2227:
2219:
2216:
2212:
2205:
2191:
2184:
2178:
2175:
2169:
2163:
2152:
2151:
2150:
2136:
2133:
2130:
2127:
2124:
2121:
2115:
2109:
2086:
2082:
2075:
2050:
2036:
2027:
2021:
2015:
2008:
2007:
1992:
1986:
1980:
1977:
1971:
1965:
1959:
1953:
1947:
1939:
1936:
1932:
1921:
1907:
1900:
1894:
1891:
1885:
1871:
1864:
1858:
1855:
1849:
1843:
1837:
1831:
1828:
1822:
1813:
1810:
1800:
1799:
1798:
1783:
1776:
1751:
1737:
1708:
1677:
1646:
1640:
1637:
1631:
1625:
1600:
1594:
1591:
1585:
1579:
1552:
1546:
1519:
1502:of the state
1486:
1449:
1437:
1436:Kalman filter
1433:
1414:
1408:
1381:
1375:
1352:
1346:
1343:
1340:
1334:
1317:
1316:
1301:
1297:
1290:
1276:
1267:
1261:
1240:
1236:
1229:
1206:
1200:
1197:
1191:
1177:
1170:
1164:
1161:
1155:
1139:
1133:
1130:
1124:
1101:
1095:
1092:
1086:
1077:
1055:
1054:
1053:
1050:
1036:
1032:
1028:
1001:
988:
982:
943:
931:
890:
887:
884:
878:
872:
868:
865:
862:
856:
850:
846:
843:
840:
837:
830:
829:
814:
810:
806:
803:
796:
780:
774:
768:
746:
740:
724:
718:
712:
688:
683:
679:
675:
669:
656:
650:
627:
618:
615:
608:
607:
606:
592:
589:
585:
582:
578:
575:
572:
565:
562:
531:
504:
471:
440:
337:
331:
320:
314:
300:
294:
291:
285:
270:
256:
250:
239:
233:
219:
213:
210:
204:
190:
184:
181:
175:
166:
151:
150:
149:
147:
144:Consider the
132:
129:
126:
122:
120:
115:
111:
106:
104:
100:
94:
92:
88:
84:
80:
76:
75:Kalman filter
72:
67:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5663:
5636:
5630:
5603:
5599:
5589:
5569:
5562:
5537:
5533:
5520:
5485:
5481:
5468:
5441:
5428:
5411:
5407:
5401:
5384:
5380:
5374:
5353:
5349:
5319:
5315:
5271:
5267:
5261:
5243:
5239:
5233:
5214:
5208:
5091:
4797:
4642:
4253:
4117:
3974:
3602:
3328:
3217:
3031:
3024:
2809:
2804:
2795:
2793:
2642:
2283:
2248:
2101:
1724:and finally
1431:
1396:
1051:
928:denotes the
905:
352:
143:
130:
127:
123:
113:
109:
107:
97:case of the
95:
70:
68:
43:
39:
35:
29:
5356:: 129–138.
2249:The matrix
1432:Kalman gain
1397:The matrix
103:martingales
89:as well as
81:(LQR). The
5686:Categories
5600:Metrologia
5350:Automatica
5201:References
3025:Since the
58:driven by
5554:0018-9286
5281:1103.3005
4988:−
4891:−
4735:−
4600:^
4588:−
4551:^
4539:−
4432:−
4340:−
4231:−
4095:−
4030:^
3992:^
3947:^
3925:−
3849:^
3789:^
3746:−
3663:^
3623:^
3565:≥
3546:≥
3416:−
3398:∑
2989:and when
2805:separable
2729:−
2679:≤
2673:≤
2518:−
2486:−
2408:˙
2399:−
2217:−
2134:≤
2128:≤
1937:−
1892:−
1814:˙
1478:^
1367:^
1344:−
1253:^
1221:^
1198:−
1116:^
1078:˙
1073:^
863:≥
841:≥
680:∫
579:≤
167:˙
5298:12623187
5184:See also
586:′
566:′
114:a priori
5608:Bibcode
5490:Bibcode
2796:duality
5671:
5577:
5552:
5296:
5221:
4798:where
4254:where
906:where
353:where
34:, the
5530:(PDF)
5478:(PDF)
5438:(PDF)
5294:S2CID
5276:arXiv
3218:Here
2810:When
5669:ISBN
5575:ISBN
5550:ISSN
5219:ISBN
3975:and
3584:>
2364:and
1693:and
885:>
590:<
64:cost
5641:doi
5616:doi
5542:doi
5506:hdl
5498:doi
5454:hdl
5446:doi
5416:doi
5389:doi
5358:doi
5324:doi
5286:doi
5248:doi
71:LQG
40:LQG
30:In
5688::
5614:.
5604:56
5602:.
5598:.
5548:.
5538:23
5536:.
5532:.
5504:.
5496:.
5484:.
5480:.
5452:.
5440:.
5412:19
5410:.
5385:56
5383:.
5354:35
5352:.
5340:^
5318:.
5306:^
5292:.
5284:.
5272:58
5270:.
5244:11
5242:.
4115:.
3587:0.
3017:.
2934:,
2807:.
1049:.
42:)
5677:.
5647:.
5643::
5624:.
5618::
5610::
5583:.
5556:.
5544::
5514:.
5508::
5500::
5492::
5486:9
5462:.
5456::
5448::
5422:.
5418::
5395:.
5391::
5369:.
5364:.
5360::
5335:.
5330:.
5326::
5320:6
5300:.
5288::
5278::
5256:.
5254:.
5250::
5227:.
5168:N
5164:/
5160:J
5136:J
5104:N
5077:.
5074:F
5071:=
5066:N
5062:S
5057:,
5052:i
5048:Q
5044:+
5039:i
5035:A
5030:)
5024:1
5021:+
5018:i
5014:S
5007:T
5001:i
4997:B
4991:1
4983:)
4977:i
4973:R
4969:+
4964:i
4960:B
4954:1
4951:+
4948:i
4944:S
4937:T
4931:i
4927:B
4922:(
4915:i
4911:B
4905:1
4902:+
4899:i
4895:S
4886:1
4883:+
4880:i
4876:S
4871:(
4864:T
4858:i
4854:A
4850:=
4845:i
4841:S
4815:i
4811:S
4781:i
4777:A
4771:1
4768:+
4765:i
4761:S
4754:T
4748:i
4744:B
4738:1
4731:)
4725:i
4721:R
4717:+
4712:i
4708:B
4702:1
4699:+
4696:i
4692:S
4685:T
4679:i
4675:B
4671:(
4668:=
4663:i
4659:K
4628:.
4625:]
4619:T
4613:)
4607:0
4596:x
4583:0
4577:x
4570:(
4564:)
4558:0
4547:x
4534:0
4528:x
4521:(
4517:[
4513:E
4509:=
4504:0
4500:P
4495:,
4490:i
4486:V
4482:+
4476:T
4470:i
4466:A
4461:)
4455:i
4451:P
4445:i
4441:C
4435:1
4427:)
4421:i
4417:W
4413:+
4407:T
4401:i
4397:C
4391:i
4387:P
4381:i
4377:C
4372:(
4364:T
4358:i
4354:C
4348:i
4344:P
4335:i
4331:P
4326:(
4320:i
4316:A
4312:=
4307:1
4304:+
4301:i
4297:P
4271:i
4267:P
4239:,
4234:1
4227:)
4221:i
4217:W
4213:+
4207:T
4201:i
4197:C
4191:i
4187:P
4181:i
4177:C
4173:(
4167:T
4161:i
4157:C
4151:i
4147:P
4143:=
4138:i
4134:L
4103:]
4098:1
4092:i
4087:u
4082:,
4077:i
4072:y
4066:|
4060:i
4055:x
4050:[
4046:E
4042:=
4037:i
4026:x
3999:i
3988:x
3959:.
3954:i
3943:x
3933:i
3929:K
3922:=
3917:i
3912:u
3898:,
3886:]
3881:0
3875:x
3869:[
3865:E
3861:=
3856:0
3845:x
3836:,
3832:)
3827:}
3821:i
3816:u
3809:i
3805:B
3801:+
3796:i
3785:x
3775:i
3771:A
3766:{
3760:1
3757:+
3754:i
3750:C
3741:1
3738:+
3735:i
3729:y
3722:(
3716:1
3713:+
3710:i
3706:L
3702:+
3697:i
3691:u
3683:i
3679:B
3675:+
3670:i
3659:x
3649:i
3645:A
3641:=
3636:1
3633:+
3630:i
3619:x
3579:i
3575:R
3571:,
3568:0
3560:i
3556:Q
3552:,
3549:0
3543:F
3520:,
3516:]
3512:)
3507:i
3502:u
3495:i
3491:R
3484:T
3478:i
3473:u
3468:+
3463:i
3458:x
3451:i
3447:Q
3440:T
3434:i
3429:x
3424:(
3419:1
3413:N
3408:0
3405:=
3402:i
3394:+
3389:N
3383:x
3377:F
3371:T
3365:N
3359:x
3352:[
3347:E
3343:=
3340:J
3312:i
3308:W
3304:,
3299:i
3295:V
3270:i
3265:w
3260:,
3255:i
3250:v
3228:i
3203:.
3198:i
3193:w
3188:+
3183:i
3178:x
3171:i
3167:C
3163:=
3158:i
3153:y
3128:,
3123:i
3118:v
3113:+
3108:i
3103:u
3096:i
3092:B
3088:+
3083:i
3078:x
3071:i
3067:A
3063:=
3058:1
3055:+
3052:i
3046:x
3001:T
2977:t
2953:)
2950:t
2947:(
2944:W
2922:)
2919:t
2916:(
2913:V
2891:)
2888:t
2885:(
2882:R
2879:,
2876:)
2873:t
2870:(
2867:Q
2864:,
2861:)
2858:t
2855:(
2852:C
2849:,
2846:)
2843:t
2840:(
2837:B
2834:,
2831:)
2828:t
2825:(
2822:A
2779:.
2776:)
2773:t
2770:(
2767:S
2764:)
2761:t
2758:(
2752:T
2747:B
2743:)
2740:t
2737:(
2732:1
2725:R
2721:=
2718:)
2715:t
2712:(
2709:K
2682:T
2676:t
2670:0
2667:,
2664:)
2661:t
2658:(
2655:S
2628:.
2625:F
2622:=
2619:)
2616:T
2613:(
2610:S
2583:,
2580:)
2577:t
2574:(
2571:Q
2568:+
2565:)
2562:t
2559:(
2556:S
2553:)
2550:t
2547:(
2541:T
2536:B
2532:)
2529:t
2526:(
2521:1
2514:R
2510:)
2507:t
2504:(
2501:B
2498:)
2495:t
2492:(
2489:S
2483:)
2480:t
2477:(
2474:A
2471:)
2468:t
2465:(
2462:S
2459:+
2456:)
2453:t
2450:(
2447:S
2444:)
2441:t
2438:(
2432:T
2427:A
2423:=
2420:)
2417:t
2414:(
2405:S
2376:F
2352:)
2349:t
2346:(
2343:R
2340:,
2337:)
2334:t
2331:(
2328:Q
2325:,
2322:)
2319:t
2316:(
2313:B
2310:,
2307:)
2304:t
2301:(
2298:A
2270:)
2267:t
2264:(
2261:K
2234:.
2231:)
2228:t
2225:(
2220:1
2213:W
2209:)
2206:t
2203:(
2197:T
2192:C
2188:)
2185:t
2182:(
2179:P
2176:=
2173:)
2170:t
2167:(
2164:L
2137:T
2131:t
2125:0
2122:,
2119:)
2116:t
2113:(
2110:P
2087:.
2083:]
2079:)
2076:0
2073:(
2067:T
2060:x
2054:)
2051:0
2048:(
2043:x
2037:[
2032:E
2028:=
2025:)
2022:0
2019:(
2016:P
1993:,
1990:)
1987:t
1984:(
1981:V
1978:+
1975:)
1972:t
1969:(
1966:P
1963:)
1960:t
1957:(
1954:C
1951:)
1948:t
1945:(
1940:1
1933:W
1925:)
1922:t
1919:(
1913:T
1908:C
1904:)
1901:t
1898:(
1895:P
1889:)
1886:t
1883:(
1877:T
1872:A
1868:)
1865:t
1862:(
1859:P
1856:+
1853:)
1850:t
1847:(
1844:P
1841:)
1838:t
1835:(
1832:A
1829:=
1826:)
1823:t
1820:(
1811:P
1784:]
1780:)
1777:0
1774:(
1768:T
1761:x
1755:)
1752:0
1749:(
1744:x
1738:[
1733:E
1712:)
1709:t
1706:(
1702:w
1681:)
1678:t
1675:(
1671:v
1650:)
1647:t
1644:(
1641:W
1638:,
1635:)
1632:t
1629:(
1626:V
1604:)
1601:t
1598:(
1595:C
1592:,
1589:)
1586:t
1583:(
1580:A
1556:)
1553:t
1550:(
1547:L
1523:)
1520:t
1517:(
1512:x
1490:)
1487:t
1484:(
1474:x
1450:t
1418:)
1415:t
1412:(
1409:L
1382:.
1379:)
1376:t
1373:(
1363:x
1356:)
1353:t
1350:(
1347:K
1341:=
1338:)
1335:t
1332:(
1327:u
1302:,
1298:]
1294:)
1291:0
1288:(
1283:x
1277:[
1272:E
1268:=
1265:)
1262:0
1259:(
1249:x
1241:,
1237:)
1233:)
1230:t
1227:(
1217:x
1210:)
1207:t
1204:(
1201:C
1195:)
1192:t
1189:(
1184:y
1178:(
1174:)
1171:t
1168:(
1165:L
1162:+
1159:)
1156:t
1153:(
1148:u
1143:)
1140:t
1137:(
1134:B
1131:+
1128:)
1125:t
1122:(
1112:x
1105:)
1102:t
1099:(
1096:A
1093:=
1090:)
1087:t
1084:(
1069:x
1037:T
1033:/
1029:J
1005:)
1002:T
999:(
994:x
989:F
986:)
983:T
980:(
974:T
967:x
944:T
915:E
891:,
888:0
882:)
879:t
876:(
873:R
869:,
866:0
860:)
857:t
854:(
851:Q
847:,
844:0
838:F
815:,
811:]
807:t
804:d
800:)
797:t
794:(
789:u
784:)
781:t
778:(
775:R
772:)
769:t
766:(
759:T
753:u
747:+
744:)
741:t
738:(
733:x
728:)
725:t
722:(
719:Q
716:)
713:t
710:(
703:T
697:x
689:T
684:0
676:+
673:)
670:T
667:(
662:x
657:F
654:)
651:T
648:(
641:T
635:x
628:[
623:E
619:=
616:J
593:t
583:t
576:0
573:,
570:)
563:t
559:(
554:y
532:t
508:)
505:t
502:(
497:u
475:)
472:t
469:(
465:w
444:)
441:t
438:(
434:v
411:y
387:u
363:x
338:,
335:)
332:t
329:(
325:w
321:+
318:)
315:t
312:(
308:x
304:)
301:t
298:(
295:C
292:=
289:)
286:t
283:(
279:y
257:,
254:)
251:t
248:(
244:v
240:+
237:)
234:t
231:(
227:u
223:)
220:t
217:(
214:B
211:+
208:)
205:t
202:(
198:x
194:)
191:t
188:(
185:A
182:=
179:)
176:t
173:(
163:x
38:(
20:)
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