Hidden Conic representation of Indefinite Linear Quadratic Control
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- 英文论文题名：Hidden Conic representation of Indefinite Linear Quadratic Control
- 论文作者：Cong Cheng ; Lixin Tang
- 英文论文作者：Cong Cheng ; Lixin Tang ; Industrial and system engineering ; Northeastern University
- 年：2017
- 作者机构：Industrial and system engineering,Northeastern University;
- 英文论文关键词：Indefinite LQ ; Conic quadratic formulation ; Tractable
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（B）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（B）
- 语种：英文
- 分类号：O231
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：5
- 文件大小：209k
- 原文格式：D
- 会议级别：全国
摘要
The traditional linear quadratic control can be solved elegantly via the classical algebraic Riccati equation. Unfortunately, this Riccati approach is rather limitation to handle the indefinite linear quadratic control, since the inverse(generalized inverse) of indefinite matrix is hard to express. The semidefinite programming(or linear matrix inequality) based approach could tackle indefinite linear quadratic control in a few cases, however, the complexity of solving the semidefinite programming is impractical for real-time control, especial in large scale problem. In this paper, we discuss two classes of indefinite linear quadratic control, the corresponding conic quadratic formulations are derived. An attractive feature of the proposed formulations is the numerical tractability, especially when compared to approaches based on semidefinite programming.
The traditional linear quadratic control can be solved elegantly via the classical algebraic Riccati equation. Unfortunately, this Riccati approach is rather limitation to handle the indefinite linear quadratic control, since the inverse(generalized inverse) of indefinite matrix is hard to express. The semidefinite programming(or linear matrix inequality) based approach could tackle indefinite linear quadratic control in a few cases, however, the complexity of solving the semidefinite programming is impractical for real-time control, especial in large scale problem. In this paper, we discuss two classes of indefinite linear quadratic control, the corresponding conic quadratic formulations are derived. An attractive feature of the proposed formulations is the numerical tractability, especially when compared to approaches based on semidefinite programming.
The traditional linear quadratic control can be solved elegantly via the classical algebraic Riccati equation. Unfortunately, this Riccati approach is rather limitation to handle the indefinite linear quadratic control, since the inverse(generalized inverse) of indefinite matrix is hard to express. The semidefinite programming(or linear matrix inequality) based approach could tackle indefinite linear quadratic control in a few cases, however, the complexity of solving the semidefinite programming is impractical for real-time control, especial in large scale problem. In this paper, we discuss two classes of indefinite linear quadratic control, the corresponding conic quadratic formulations are derived. An attractive feature of the proposed formulations is the numerical tractability, especially when compared to approaches based on semidefinite programming.
引文
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[2]Ferrante,A.,Ntogramatzidis,L.,The generalized discrete algebraic Riccati equation in linear-quadratic optimal control.Automatica,49(2):471-478,2013b.
[3]Ferrante,A.,Ntogramatzidis,L.,A reduction technique for discrete generalized algebraic and difference Riccati equations.Linear and Multilinear Algebra,62(11):1460-1474,2014.
[4]Berkovitz L.D.,Optimal Control Theory.New York:Springer,1974.
[5]Boyd S P,El-Ghaoui L,Feron E,Linear Matrix Inequalities in System and Control Theory,SAM[J].Studies in Applied Mathematics,15(6),1994.
[6]Lewis F.L.,Syrmos V.,Optimal control,New York:John Wiley&Sons,1995.
[7]Ran,A.C.M.,Trentelman,H.L.,Linear quadratic problems with indefinite cost for discrete time systems.SIAM Journal on Matrix Analysis and Applications,14(3):776-797,1993.
[8]Hassibi,B.,Sayed,A.H.,Kailath,T.,Indefinite-quadratic estimation and control,a unified approach to H2and H∞theories,Philadelphia:SIAM.,1999
[9]Bilardi,G.,Ferrante,A.,The role of terminal cost/reward in finite horizon discrete-time LQ optimal control,Linear Algebra and its Applications,425:323-344,2007.
[10]Vandenberghe,L.,Boyd,S.:Semidefinite programming.SIAM,38:49-95,1996.
[11]Yao D D,Zhang S,Zhou X Y.,A primal-dual semi-definite programming approach to linear quadratic control[J].IEEE Transactions on Automatic Control,46(9):1442-1447,2001.
[12]Nesterov,Y.,Nemirovsky,A.,Interior-Point Polynomial Methods in Convex Programming,SIAM,Philadelphia,1999.
[13]Yao D D,Zhang S,Zhou X Y.,Stochastic Linear-Quadratic Control via Semidefinite Programming[J].SIAM Journal on Control&Optimization,40(3):801-823,2001.
[14]Uhlig F.,Definite and semidefinite matrices in a real symmetic matrix pencil,Pac.J.Math.,49(2),561-568,1972.
[15]Bertsekas D.,Dynamic Programming and Optimal Control.Nashua,NH:Athena Scientific,2005.
[16]Aharon Ben-Tal,Dick den Hertog,Hidden conic quadratic representation of some nonconvex quadratic optimization problems,Math.Program.,Ser.A,143:1-29,2014.
[2]Ferrante,A.,Ntogramatzidis,L.,The generalized discrete algebraic Riccati equation in linear-quadratic optimal control.Automatica,49(2):471-478,2013b.
[3]Ferrante,A.,Ntogramatzidis,L.,A reduction technique for discrete generalized algebraic and difference Riccati equations.Linear and Multilinear Algebra,62(11):1460-1474,2014.
[4]Berkovitz L.D.,Optimal Control Theory.New York:Springer,1974.
[5]Boyd S P,El-Ghaoui L,Feron E,Linear Matrix Inequalities in System and Control Theory,SAM[J].Studies in Applied Mathematics,15(6),1994.
[6]Lewis F.L.,Syrmos V.,Optimal control,New York:John Wiley&Sons,1995.
[7]Ran,A.C.M.,Trentelman,H.L.,Linear quadratic problems with indefinite cost for discrete time systems.SIAM Journal on Matrix Analysis and Applications,14(3):776-797,1993.
[8]Hassibi,B.,Sayed,A.H.,Kailath,T.,Indefinite-quadratic estimation and control,a unified approach to H2and H∞theories,Philadelphia:SIAM.,1999
[9]Bilardi,G.,Ferrante,A.,The role of terminal cost/reward in finite horizon discrete-time LQ optimal control,Linear Algebra and its Applications,425:323-344,2007.
[10]Vandenberghe,L.,Boyd,S.:Semidefinite programming.SIAM,38:49-95,1996.
[11]Yao D D,Zhang S,Zhou X Y.,A primal-dual semi-definite programming approach to linear quadratic control[J].IEEE Transactions on Automatic Control,46(9):1442-1447,2001.
[12]Nesterov,Y.,Nemirovsky,A.,Interior-Point Polynomial Methods in Convex Programming,SIAM,Philadelphia,1999.
[13]Yao D D,Zhang S,Zhou X Y.,Stochastic Linear-Quadratic Control via Semidefinite Programming[J].SIAM Journal on Control&Optimization,40(3):801-823,2001.
[14]Uhlig F.,Definite and semidefinite matrices in a real symmetic matrix pencil,Pac.J.Math.,49(2),561-568,1972.
[15]Bertsekas D.,Dynamic Programming and Optimal Control.Nashua,NH:Athena Scientific,2005.
[16]Aharon Ben-Tal,Dick den Hertog,Hidden conic quadratic representation of some nonconvex quadratic optimization problems,Math.Program.,Ser.A,143:1-29,2014.