794:
58:
Another instance of the separation principle arises in the setting of linear stochastic systems, namely that state estimation (possibly nonlinear) together with an optimal state feedback controller designed to minimize a quadratic cost, is optimal for the stochastic control problem with output
211:
649:
403:
97:
578:
463:
518:
638:
789:{\displaystyle {\begin{bmatrix}{\dot {x}}\\{\dot {e}}\\\end{bmatrix}}={\begin{bmatrix}A-BK&BK\\0&A-LC\\\end{bmatrix}}{\begin{bmatrix}x\\e\\\end{bmatrix}}.}
308:
277:
246:
92:
71:. More generally, under suitable conditions and when the noise is a martingale (with possible jumps), again a separation principle applies and is known as the
28:, states that under some assumptions the problem of designing an optimal feedback controller for a stochastic system can be solved by designing an optimal
78:
The separation principle also holds for high gain observers used for state estimation of a class of nonlinear systems and control of quantum systems.
72:
320:
1047:
853:
68:
39:
The first instance of such a principle is in the setting of deterministic linear systems, namely that if a stable
981:
Tryphon T. Georgiou and Anders
Lindquist (2013). "The Separation Principle in Stochastic Control, Redux".
529:
414:
1103:
33:
478:
64:
36:
for the system. Thus the problem can be broken into two separate parts, which facilitates the design.
586:
59:
measurements. When process and observation noise are
Gaussian, the optimal solution separates into a
1098:
206:{\displaystyle {\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)\end{aligned}}}
869:
Tyrone Duncan and Pravin
Varaiya (1971). "On the solutions of a stochastic control system".
284:
253:
222:
8:
824:
1053:
1008:
990:
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1043:
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1032:
Proceedings of the 37th IEEE Conference on
Decision and Control (Cat. No.98CH36171)
1000:
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919:
886:
878:
52:
40:
29:
17:
81:
1092:
1027:
1004:
939:
Anders
Lindquist (1973). "On Feedback Control of Linear Stochastic Systems".
60:
1039:
55:. The separation principle does not hold in general for nonlinear systems.
1083:
Computational
Aspects of Linear Control (Numerical Methods and Algorithms)
1071:
804:
1028:"A separation principle for the control of a class of nonlinear systems"
980:
32:
for the state of the system, which feeds into an optimal deterministic
891:
952:
882:
51:(LTI system hereafter), then the combined observer and feedback is
44:
995:
868:
905:
82:
Proof of separation principle for deterministic LTI systems
398:{\displaystyle {\dot {\hat {x}}}=(A-LC){\hat {x}}+Bu+Ly\,}
823:. Thus the stability of the observer and feedback are
762:
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652:
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968:Stochastic Control of Partially Observable Systems
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271:
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205:
843:
26:principle of separation of estimation and control
1090:
938:
1070:Proof can be found in this math.stackexchange
965:
643:Now we can write the closed-loop dynamics as
1025:
862:
310:represents the internal state of the system.
908:"Information states for stochastic systems"
899:
73:separation principle in stochastic control
994:
959:
923:
890:
846:Introduction to Stochastic Control Theory
626:
569:
506:
451:
394:
932:
1034:. Vol. 1. IEEE. pp. 855–860.
1091:
983:IEEE Transactions on Automatic Control
314:We can design an observer of the form
974:
837:
86:Consider a deterministic LTI system:
906:M.H.A. Davis and P. Varaiya (1972).
573:{\displaystyle {\dot {e}}=(A-LC)e\,}
458:{\displaystyle u(t)=-K{\hat {x}}\,.}
1026:Atassi, A.N.; Khalil, H.K. (1998).
13:
14:
1115:
513:{\displaystyle e=x-{\hat {x}}\,.}
279:represents the output signal, and
69:linear-quadratic-Gaussian control
848:. Vol. 58. Academic Press.
633:{\displaystyle u(t)=-K(x-e)\,.}
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970:. Cambridge University Press.
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925:10.1016/0022-247X(72)90281-8
248:represents the input signal,
49:linear time-invariant system
7:
912:J. Math. Anal. Applications
24:, more formally known as a
10:
1120:
844:Karl Johan Astrom (1970).
65:linear-quadratic regulator
1005:10.1109/TAC.2013.2259207
1040:10.1109/cdc.1998.760800
941:SIAM Journal on Control
815:together with those of
966:A. Bensoussan (1992).
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303:{\displaystyle x(t)}
285:
272:{\displaystyle y(t)}
254:
241:{\displaystyle u(t)}
223:
93:
22:separation principle
1081:Brezinski, Claude.
819: −
811: −
408:and state feedback
67:. This is known as
47:are designed for a
43:and a stable state
1104:Stochastic control
807:are just those of
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1085:. Springer, 2002.
989:(10): 2481–2494.
801:triangular matrix
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468:Define the error
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799:Since this is a
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953:10.1137/0311025
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883:10.1137/0309026
871:SIAM J. Control
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1099:Control theory
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947:(2): 323–343.
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18:control theory
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1049:0-7803-4394-8
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855:0-486-44531-3
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61:Kalman filter
56:
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50:
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37:
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31:
27:
23:
19:
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1021:
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313:
215:
85:
77:
57:
38:
25:
21:
15:
918:: 384–402.
825:independent
805:eigenvalues
1093:Categories
892:1808/16692
831:References
34:controller
1058:126270534
996:1103.3005
741:−
712:−
684:˙
668:˙
618:−
606:−
555:−
540:˙
501:^
492:−
446:^
434:−
371:^
353:−
338:˙
333:^
107:˙
1013:12623187
45:feedback
41:observer
30:observer
1056:
1046:
1011:
852:
803:, the
216:where
63:and a
53:stable
1054:S2CID
1009:S2CID
991:arXiv
523:Then
1044:ISBN
850:ISBN
20:, a
1036:doi
1001:doi
949:doi
920:doi
887:hdl
879:doi
16:In
1095::
1052:.
1042:.
1030:.
1007:.
999:.
987:58
985:.
945:11
943:.
916:37
914:.
910:.
885:.
873:.
827:.
821:LC
813:BK
472::
75:.
1073:.
1060:.
1038::
1015:.
1003::
993::
955:.
951::
928:.
922::
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889::
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875:9
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817:A
809:A
784:.
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760:[
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744:L
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723:B
718:K
715:B
709:A
703:[
698:=
693:]
681:e
665:x
656:[
628:.
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621:e
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597:t
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591:u
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558:L
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546:=
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498:x
489:x
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470:e
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419:u
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344:=
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113:(
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