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Verified calculation of the solution of algebraic Riccati equation

In: Developments in Reliable Computing

Author

Listed:
  • Wolfram Luther

    (Gerhard-Mercator-Universität — GH Duisburg, Informatik II)

  • Werner Otten

    (Gerhard-Mercator-Universität — GH Duisburg, Informatik II)

Abstract

In this note we describe a new method to calculate verified solutions of the matrix Riccati equation (ARE) with interval coefficients. Such an equation has to be solved when we want to find the steady state solutions of matrix Riccati differential equation with constant coefficients which arises in the theory of automatic control and linear filtering. Given the Riccati polynomial P(X) we use the Fréchet-derivative at X to derive a linear equation of type CX + XD = P. Applying Brouwer’s fixed point theorem, we find an interval matrix [X] that includes a positive definite solution of the equation P(X) = Ω. First we want to give an outline of linear-quadratic control theory. Then we present results concerning the geometric structures of all solutions and enumerate linearly and quadratically convergent algorithms to find a solution used to construct the optimal feedback control for linear-quadratic optimal control problems.

Suggested Citation

  • Wolfram Luther & Werner Otten, 1999. "Verified calculation of the solution of algebraic Riccati equation," Springer Books, in: Tibor Csendes (ed.), Developments in Reliable Computing, pages 105-118, Springer.
    • Handle: RePEc:spr:sprchp:978-94-017-1247-7_8
    DOI: 10.1007/978-94-017-1247-7_8
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