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Linear Quadratic Control Revisited: A View Through Semidefinite Programming

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Book cover Modeling, Control and Optimization of Complex Systems

Part of the book series: The International Series on Discrete Event Dynamic Systems ((DEDS,volume 14))

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Abstract

We present a unified approach to both deterministic and stochastic linear-quadratic (LQ) control via the duality theory of semi-definite programming (SDP). This new framework allows the control cost matrix to be singular or even indefinite (in the stochastic setting), a useful feature in applications such as the optimal portfolio selection of financial assets. We show that the complementary duality condition of the SDP is necessary and sufficient for the existence of an optimal LQ control under certain stability conditions. When the complementary duality does hold, an optimal state feedback control is constructed explicitly in terms of the solution to the primal SDP. Furthermore, if thestrictcomplementarity holds, then a new optimal feedback control, which is always stabilizing, is generated via the dual SDP. On the other hand, for cases where the complementary duality fails and the LQ problem has no attainable optimal solution, we develop an e-approximation scheme that achieves asymptotic optimality.

Research supported by Hong Kong RGC Grant CUHK4175/00E

Research undertaken while on leave at Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, Hong Kong; supported in part by NSF under Grant ECS-97-05392

tResearch supported in part by RGC Earmarked Grant CUHK4181/00E

Research supported in part by RGC Earmarked Grants CUHK 4054/98E and CUHK 4435/99E

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Author information

Authors and Affiliations

  1. IEOR Department, Columbia University, New York, NY, 10027

    David D. Yao

  2. Dept. of Systems Engineering and Eng Mgmt, The Chinese University, Shatin, Hong Kong

    Shuzhong Zhang

  3. Dept. of Systems Engineering and Eng Mgmt, The Chinese University, Shatin, Hong Kong

    Xun Yu Zhou

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  1. David D. Yao
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  2. Shuzhong Zhang
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  3. Xun Yu Zhou
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Editor information

Editors and Affiliations

  1. University of Massachusetts, Amherst

    Weibo Gong

  2. University of Wisconsin, Madison

    Leyuan Shi

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Yao, D.D., Zhang, S., Zhou, X.Y. (2003). Linear Quadratic Control Revisited: A View Through Semidefinite Programming. In: Gong, W., Shi, L. (eds) Modeling, Control and Optimization of Complex Systems. The International Series on Discrete Event Dynamic Systems, vol 14. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1139-7_9

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  • DOI: https://doi.org/10.1007/978-1-4615-1139-7_9

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4613-5411-6

  • Online ISBN: 978-1-4615-1139-7

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