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Newton’s method for solving cross-coupled sign-indefinite algebraic Riccati equations for weakly coupled large-scale systems. (English) Zbl 1118.65048
Summary: A new algorithm for solving cross-coupled sign-indefinite algebraic Riccati equations (CSAREs) for weakly coupled large-scale systems is proposed. It is shown that since the proposed algorithm is based on the Newton’s method, the quadratic convergence is attained. Moreover, the local uniqueness of the convergence solutions for the CSAREs is investigated. Finally, in order to overcome the computation of large- and sparse-matrix related to the Newton’s method, the fixed point algorithm and the alternating direction implicit method are combined.

65H10Systems of nonlinear equations (numerical methods)
15A24Matrix equations and identities
65L05Initial value problems for ODE (numerical methods)
91A23Differential games (game theory)
34A30Linear ODE and systems, general
Full Text: DOI
[1] Broek, W. A. V.D.; Engwerda, J. C.; Schumacher, J. M.: Robust equilibria in indefinite linear-quadratic differential games. J. optim. Theory appl. 119, No. 3, 565-595 (2003) · Zbl 1084.91009
[2] Engwerda, J. C.: A numerical algorithm to find soft-constrained Nash equilibria in scalar LQ-games. Int. J. Control 79, No. 6, 592-603 (2006) · Zbl 1255.91051
[3] Delacour, J. D.; Darwish, M.; Fantin, J.: Control strategies for large-scale power systems. Int. J. Control 27, No. 5, 753-767 (1978) · Zbl 0373.93003
[4] Shen, X.; Xia, Q.; Rao, M.; Gourishankar, V.: Optimal control for large-scale systems: a recursive approach. Int. J. Syst. sci. 25, No. 12, 2235-2244 (1994) · Zbl 0820.93003
[5] Gajić, Z.; Borno, I.: General transformation for block diagonalization of weakly coupled linear systems composed of N-subsystems. IEEE trans. Circ. syst. I 47, No. 6, 909-912 (2000) · Zbl 0972.93012
[6] Petrovic, B.; Gajić, Z.: The recursive solution of linear quadratic Nash games for weakly interconnected systems. J. optim. Theory appl. 56, No. 3, 463-477 (1988) · Zbl 0622.93005
[7] Gajić, Z.; Petkovski, D.; Shen, X.: Singularly perturbed and weakly coupled linear system -- A recursive approach. Lecture notes in control and information sciences 140 (1990) · Zbl 0697.93023
[8] H. Mukaidani, Numerical Computation of Nash Strategy for Large-Scale Systems, in Proc. of 2004 American Control Conference, Boston, June 2004. pp. 5641 -- 5646.
[9] Mukaidani, H.: Optimal numerical strategy for Nash games of weakly coupled large-scale systems. Dyn. continuous, discrete impulsive syst., ser. B: appl. Algorithms 13, No. 2, 249-268 (2006) · Zbl 1176.91012
[10] Mukaidani, H.: Numerical computation for H2 state feedback control of large-scale systems. Dyn. continuous, discrete impulsive syst., ser. B: appl. Algorithms 12, No. 2, 281-296 (2005) · Zbl 1083.93015
[11] Zhou, K.; Doyle, J. C.; Glover, K.: Robust and optimal control. (1996) · Zbl 0999.49500
[12] Yamamoto, T.: A method for finding sharp error bounds for Newton’s method under the Kantorovich assumptions. Numer. math. 49, No. 2 -- 3, 203-220 (1986) · Zbl 0607.65033
[13] Penzl, T.: A cyclic low rank Smith method for large sparse Lyapunov equations with applications in model reduction and optimal control. SIAM J. Sci. comput. 21, No. 4, 1401-1418 (1999) · Zbl 0958.65052
[14] Levenberg, N.; Reichel, L.: A generalized ADI iterative method. Numer. math. 66, No. 1, 215-233 (1993) · Zbl 0797.65030
[15] Golub, G.; Nash, S.; Van Loan, C.: A Hessenberg -- Schur method for the problem AX+XB=C. IEEE trans. Autom. control 24, No. 6, 909-913 (1979) · Zbl 0421.65022