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Verified calculation of the solution of algebraic Riccati equation. (English) Zbl 0951.65040

Csendes, Tibor (ed.), Developments in reliable computing. SCAN-98 conference, 8th international symposium on Scientific computing, computer arithmetic and validated numerics. Budapest, Hungary, September 22-25, 1998. Dordrecht: Kluwer Academic Publishers. 105-118 (1999).
Summary: We describe a new method to calculate verified solutions of the matrix Riccati equation with interval coefficients. Such an equation has to be solved when we want to find the steady state solutions of matrix Riccati differential equations with constant coefficients which arise in the theory of automatic control and linear filtering.
Given the Riccati polynomial \(P(X)\) we use the FrĂ©chet-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 \(P(X)= \Omega\).
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.
For the entire collection see [Zbl 0933.00041].

MSC:

65F30 Other matrix algorithms (MSC2010)
93C15 Control/observation systems governed by ordinary differential equations
93B52 Feedback control
15A24 Matrix equations and identities
65G20 Algorithms with automatic result verification

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