Abstract
The algebraic Riccati equation, denoted by ’ARE’ in the paper, is one of the most important equation in the post modern control theory, playing important role for solving H 2 and H ∞ optimal control problems. The solution of ARE is given in the form of a matrix, and a typical procedure of computing the solution uses eigenvalues and eigenvectors of matrix H, where H is a matrix determined by a given system. With the aid excellent numerical packages such as “LAPACK” for matrix computations, the procedure is quite efficient for the numerical systems (the systems without unknown parameters).
This paper considers a system with an unknown parameter k. In this case, the numerical procedure cannot be applied without fixing parameter k to a constant value. Let us consider some symbolic method to compute the solution of ARE which leaves parameter k symbolic. Letting entries of the solution matrix be unknown variables, ARE can be viewed as a set of m algebraic equations with m variables and parameter k, where m is the number of entries of the unknown matrix. Computing Groebner basis of the algebraic equations with lexicographic ordering, we obtain a polynomial whose roots are the solution of ARE (i.e. the defining polynomial of ARE). Although this method with Groebner basis gives us the defining polynomial of ARE, it is not practical. The method easily collapses when the size of a given system is large because of its heavy numerical complexities. This paper presents a practical algorithm to compute the defining polynomial. The proposed algorithm uses polynomial interpolations, and is easily parallelizable, implying that it is advantageous under multi-CPU environments. Numerical experiments indicate that even in the single CPU environments, the proposed algorithm is much more practical than that with Groebner basis.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abdallah, C., Dorato, P., Yang, W., Liska, R., Steinberg, S.: Application of Quantifier Elimination Theory to Control System Design. In: Proc. of 4th IEEE Mediterranean Symposium of Control and Automation, Maleme, Crete, pp. 340–345 (1996)
Anai, H., Yanami, H.: SyNRAC: A maple-package for solving real algebraic constraints. In: Sloot, P.M.A., Abramson, D., Bogdanov, A.V., Gorbachev, Y.E., Dongarra, J.J., Zomaya, A.Y. (eds.) ICCS 2003. LNCS, vol. 2657, Springer, Heidelberg (2003)
Dorato, P., Yang, W., Abdallah, C.: Robust Multi-Objective Feedback Design by Quantifier Elimination. J. Symbolic Computation 24, 153–159 (1997)
Hong, H., Liska, R., Steinberg, S.: Testing Stability by Quantifier Elimination. J. Symbolic Computation 24, 161–187 (1997)
Kitamoto, T.: On the computation of H ∞ norm of a system with a parameter. The IEICE Trans. Funda. (Japanese Edition) J89-A(1), 25–39 (2006)
Kitamoto, T., Yamaguchi, T.: Parametric Computation of H ∞ Norm of a System. In: Proc. SICE-ICCAS 2006, Busan, Korea (2006)
Kanno, M., Smith, M.C.: Validated numerical computation of the \(\mathcal{L}_\infty\)-norm for linear dynamical systems. J. of Symbolic Computation 41(6), 697–707 (2006)
Zhou, K., Doyle, J., Glover, K.: Robust and Optimal Control. Prentice-Hall. Inc, New Jersey (1996)
Nishimura, T., Kano, H.: Matrix Riccati Equations in Control Theory Asakura-syoten (in Japanese), Tokyo (1996)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kitamoto, T., Yamaguchi, T. (2007). On the Computation of the Defining Polynomial of the Algebraic Riccati Equation. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2007. Lecture Notes in Computer Science, vol 4770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75187-8_18
Download citation
DOI: https://doi.org/10.1007/978-3-540-75187-8_18
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-75186-1
Online ISBN: 978-3-540-75187-8
eBook Packages: Computer ScienceComputer Science (R0)