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Linear Quadratic Optimal Control

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Book cover The Robust Maximum Principle

Part of the book series: Systems & Control: Foundations & Applications ((SCFA))

Abstract

This chapter deals with the optimal control design for linear models described by a linear (maybe nonstationary) ODE. The cost functional is considered both for finite and infinite horizons. Finite horizon optimal control is shown to be a linear nonstationary feedback control with a gain matrix generated by a backward differential matrix Riccati equation. For stationary models without any measurable uncontrollable inputs and an infinite horizon the optimal control is a linear stationary feedback with a gain matrix satisfying an algebraic matrix Riccati equation. The detailed analysis of this matrix equation is presented and the conditions for the parameters of a linear system are given that guarantee the existence and uniqueness of a positive-definite solution which is part of the gain matrix in the corresponding optimal linear feedback control.

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Notes

  1. 1.

    Here we follow the presentation of the material given in Zhou et al. (1996).

  2. 2.

    In the Russian technical literature this equation is known as the matrix Lurie equation (Lurie 1951).

References

  • Hautus, M.L.J., & Silverman, L.M. (1983), ‘System structure and singular control’, Linear Algebra Appl. 50, 369–402.

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  • Lurie, A.I. (1951), Some Nonlinear Problems of the Automatic Control Theory, Gostexizdat, Moscow (in Russian).

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  • Lyapunov, A.M. (1935), General Problem of a Movement Stability, ONTI, Leningrad (in Russian) (the original by 1897).

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  • Poznyak, A.S. (2008), Advanced Mathematical Tools for Automatic Control Engineers, Vol. 1: Deterministic Technique, Elsevier, Amsterdam.

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  • Zhou, K., Doyle, J.C., & Glover, K. (1996), Robust and Optimal Control, Prentice-Hall, Upper Saddle River.

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Author information

Authors and Affiliations

  1. CIMAT, Jalisco S/N, Col. Valenciana, 36240, Guanajuato, GTO, Mexico

    Vladimir G. Boltyanski

  2. Automatic Control Department, CINVESTAV-IPN, AP-14-740 Av. IPN-2508, Col. San Pedro Zacatenco, 07000, México, D.F., Mexico

    Alexander S. Poznyak

Authors
  1. Vladimir G. Boltyanski
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  2. Alexander S. Poznyak
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Corresponding author

Correspondence to Vladimir G. Boltyanski .

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© 2012 Springer Science+Business Media, LLC

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Boltyanski, V.G., Poznyak, A.S. (2012). Linear Quadratic Optimal Control. In: The Robust Maximum Principle. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8152-4_4

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