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Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review

  • Chapter
The Riccati Equation

Part of the book series: Communications and Control Engineering Series ((CCE))

Abstract

This review is concerned with two algebraic Riccati equations. The first is a quadratic matrix equation for an unknown n × n matrix X of the form

$$XDX + XA + A*X - C = 0,$$
(2.1)

where A, D, C are n × n complex matrices with C and D hermitian. Further hypotheses are imposed as required, although Section 2.3 contains some discussion of more general non-symmetric quadratic equations. The second equation has the fractional form

$$X = A*XA + Q - (C + B*XA)*{(R + B*XB)^{ - 1}}(C + B*XA),$$
(2.2)

where R and Q are hermitian m × m and n × n matrices, respectively, and A, B, C are complex matrices with respective sizes n × n, n × m, and m × n. The two equations are frequently referred to as the “continuous” and “discrete” Riccati equations, respectively, because they arise in physical optimal control problems in which the time is treated as a continuous variable, or a discrete variable.

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Authors
  1. Peter Lancaster
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  2. Leiba Rodman
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Editor information

Editors and Affiliations

  1. Dipartimento di Elettronica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Milano, Italy

    Sergio Bittanti

  2. Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA, 93106, USA

    Alan J. Laub

  3. Department of Mathematics, University of Groningen, P.O. Box 800, NL-9700 AV, Groningen, The Netherlands

    Jan C. Willems

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© 1991 Springer-Verlag Berlin Heidelberg

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Lancaster, P., Rodman, L. (1991). Solutions of the Continuous and Discrete Time Algebraic Riccati Equations: A Review. In: Bittanti, S., Laub, A.J., Willems, J.C. (eds) The Riccati Equation. Communications and Control Engineering Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58223-3_2

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  • DOI: https://doi.org/10.1007/978-3-642-58223-3_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63508-3

  • Online ISBN: 978-3-642-58223-3

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