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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Hautus, M.L.J., & Silverman, L.M. (1983), ‘System structure and singular control’, Linear Algebra Appl. 50, 369–402.
Lurie, A.I. (1951), Some Nonlinear Problems of the Automatic Control Theory, Gostexizdat, Moscow (in Russian).
Lyapunov, A.M. (1935), General Problem of a Movement Stability, ONTI, Leningrad (in Russian) (the original by 1897).
Poznyak, A.S. (2008), Advanced Mathematical Tools for Automatic Control Engineers, Vol. 1: Deterministic Technique, Elsevier, Amsterdam.
Zhou, K., Doyle, J.C., & Glover, K. (1996), Robust and Optimal Control, Prentice-Hall, Upper Saddle River.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
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
Download citation
DOI: https://doi.org/10.1007/978-0-8176-8152-4_4
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-8151-7
Online ISBN: 978-0-8176-8152-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)