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Learn about Algebraic Riccati Equation (ARE) and a numerical algorithm to compute its defining polynomial, crucial for control theory. The method involves Groebner Basis for smaller degrees and arbitrary precision arithmetic for efficiency. See numerical experiments and conclusions.
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On the computation of the defining polynomial of the algebraic Riccati equation Yamaguchi Univ. Takuya Kitamoto Cybernet Systems, Co. LTD Tetsu Yamaguchi
Outline of the presentation • What is ARE (Algebraic Riccati Equation)? • Properties of ARE • Problem formulation • Algorithm description • Numerical experiments • Conclusion
Important equation for control theory (H2 optimal control, etc) Symmetric solutions (solution matrices are symmetric) are important. There are 2^n symmetric solutions. When matrices A, W, Q are numerical matrices, a numerical algorithm to compute the solutions is already known. The numerical algorithm can not be applied when matrices A, W, Q contain a parameter. Properties of ARE
Problem formulation Example:
We can compute the defining polynomial of entries of P, not P itself.
The method with Groebner Basis: Effective for only small degree n (n=2), because of its heavy numerical complexities
Conversion from floating point numbers to integers • Arbitrary precision arithmetic can be used. • Precision required is unknown.
Conversion from integers to polynomials • Polynomial interpolation can be used. • The degree of the polynomial is unknown.
Numerical experiments (2) Environments: Maple 10 on the machine with Pentium M 2.0GHz, 1.5Gbyte memory Computation time (in seconds)
Conclusion • An algorithm to compute the defining polynomial of ARE with a parameter is given. • The algorithm uses polynomial interpolations and arbitrary precision arithmetic. • Numerical experiments suggest that the algorithm is practical for the system with size n<5. • The algorithm is suitable for multi-CPU environments.
Future direction • Further improvements of efficiency is necessary. • Modular algorithm instead of floating point arithmetic can be used (provided the head coefficient is known). • Extend application of the defining polynomial.