KRYLOV SUBSPACE METHODS OF HESSENBERG BASED FOR ALGEBRAIC RICCATI EQUATION

Yajun Xie; Minhua Yin; Limin Ren

doi:10.11948/20190185

2020 Volume 10 Issue 3

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Yajun Xie, Minhua Yin, Limin Ren. KRYLOV SUBSPACE METHODS OF HESSENBERG BASED FOR ALGEBRAIC RICCATI EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1047-1059. doi: 10.11948/20190185

Citation:

Yajun Xie, Minhua Yin, Limin Ren. KRYLOV SUBSPACE METHODS OF HESSENBERG BASED FOR ALGEBRAIC RICCATI EQUATION[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1047-1059. doi: 10.11948/20190185

KRYLOV SUBSPACE METHODS OF HESSENBERG BASED FOR ALGEBRAIC RICCATI EQUATION

1.
College of Technology, Fuzhou University of International Studies and Trade, Fuzhou 350202, China
2.
College of industry and commerce, Fujian Jiangxia university, Fuzhou 350108, China
3.
Department of Mathematics & Physics, Fujian Jiangxia university, Fuzhou 350108, China

Corresponding author: Email: xieyajun0525@163.com(Y. Xie)
Fund Project: The project is supported by National Natural Science Foundation of China (Grant No.11071041), Fujian Natural Science Foundation (Grant Nos.2019J01879, 2016J01005); The New Century Training Plan of Fujian Province University and The educational reform project (Grant No. 2017[52], J2018B013)

Abstract

Abstract

In this paper, we propose a class of special Krylov subspace methods to solve continuous algebraic Riccati equation (CARE), i.e., the Hessenberg-based methods. The presented approaches can obtain efficiently the solution of algebraic Riccati equation to some extent. The main idea is to apply Kleinman-Newton's method to transform the process of solving algebraic Riccati equation into Lyapunov equation at every inner iteration. Further, the Hessenberg process of pivoting strategy combined with Petrov-Galerkin condition and minimal norm condition is discussed for solving the Lyapunov equation in detail, then we get two methods, namely global generalized Hessenberg (GHESS) and changing minimal residual methods based on the Hessenberg process (CMRH) for solving CARE, respectively. Numerical experiments illustrate the efficiency of the provided methods.
- Continuous algebraic Riccati equation (CARE) /
- Krylov subspace method /
- Hessenberg-based method /
- Pivoting strategy /
- Petrov-Galerkin condition.
MSC: 65H10, 65K05, 49M15

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References

[1]	L. Amodei and J. M. Buchot, An invariant subspace method for large-scale algebraic Riccati equation, Appl. Numer. Math., 2010, 60, 1067-1082. Google Scholar
[2]	A. C. Antoulas, Approximation of large-scale dynamical systems, SIAM, Philadelphia, 2005.https://www.researchgate.net/publication/243786519_Approximation_of_Large-Scale_Dynamical_Systems Google Scholar
[3]	S. Agoujil, A. H. Bentbib, K. Jbilou, and E. M. Sadek, A minimal residual norm method for large-scale Sylvester matrix equations, Elect. Trans. Numer. Anal., 2014, 43, 45-59. Google Scholar
[4]	L. Bao, Y. Lin and Y. Wei, A modified simple iterative method for nonsymmetric algebraic Riccati equations arising in transport theory, Appl. Math. Comput., 2006, 181(2), 1499-1504. Google Scholar
[5]	P. Benner, J. Li and T. Penzl, Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems, Numer. Linear Algebra. Appl., 2008, 15, 755-777. Google Scholar
[6]	J. P. Chehab and M. Raydan, Inexact Newton's method with inner implicit preconditioning for algebraic Riccati equations, Comp. Appl. Math., 2017, 36, 955-969. DOI10.1007/s40314-015-0274-8. Google Scholar
[7]	E. Chu, H. Fan, W. Lin and C. Wang, Structure-preserving doubling algorithms for periodic discrete-time algebraic Riccati equations, Int. J. Control, 2004, 77, 767-788. Google Scholar
[8]	D. Chu, W. Lin and R. Tan, A Numerical Method for a Generalized Algebraic Riccati Equation, SIAM J. Control Optim., 2006, 45(4), 1222-1250. Google Scholar
[9]	B. N. Datta, Numeritcal methods for linear control systems design and analysis, Elservier Press, Amsterdam, 2003. Google Scholar
[10]	R. Freund, G. H. Golub and N. M. Nachtigal, Iterative solution of linear systems, Acta Numer., 1992, 1, 57-100. Google Scholar
[11]	X. Guo, W. Lin and S. Xu, A structure-preserving doubling algorithm for nonsymmetric algebraic Riccati equation, Numer. Math., 2006, 103, 393-412. Google Scholar
[12]	M. Heyouni and H. Sadok, A new implementation of the CMRH method for solving dense linear systems, J. Comput. Appl. Math., 2008, 213(2), 387-399. Google Scholar
[13]	K. Jbilou, Block Krylov subspace methods for large algebraic Riccati equations, Numer. Algorithms, 2003, 34, 339-353. Google Scholar
[14]	K. Jbilou, Low rank approximate solutions to large Sylvester matrix equations, Appl. Math. Comput., 2006, 177(1), 365-376. Google Scholar
[15]	K. Jbilou, A. Messaoudi, and H. Sadok, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math., 1999, 31(1), 49-63. Google Scholar
[16]	J. Juang, Existence of algebraic matrix Riccati equations arising in transport theory, Linear Algebra Appl., 1995, 230, 89-100. Google Scholar
[17]	L. Lu, Solution form and simple iteration of a nonsymmetric algebraic Riccati equation arising in transport theory, SIAM J. Matrix Anal. Appl. 2005, 26, 679-685.https://www.researchgate.net/publication/220656505_Solution_Form_and_Simple_Iteration_of_a_Nonsymmetric_Algebraic_Riccati_Equation_Arising_in_Transport_Theory Google Scholar
[18]	Matrix Market. http://math.nist.gov/MatrixMarket/. Google Scholar
[19]	V. Mehrmann, The autonomous linear quadratic control problem, theory and numerical solution. Lecture Notes in Control and Information Sciences, Springer, Berlin, 1991, 163. Google Scholar
[20]	T. Penzl, LYAPACK: A MATLAB toolbox for large Lyapunov and Riccati equations, model reduction problems, and linear-quadratic optimal control problems, users guide, 2000, 1(0). Available at: www.tu-chemnitz.de/sfb393/lyapack. Google Scholar
[21]	H. Sadok, CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm, Numer. Algorithm, 1999, 20, 303-321. Google Scholar
[22]	H. Sadok and D. B. Szyld, A new look at CMRH and its relation to GMRES, BIT Numer. Math., 2012, 52(2), 485-501. Google Scholar
[23]	N. Sandell, On Newton's method for Riccati equation solution, IEEE Trans. Auto. Control, 1974, 19, 254-255. Google Scholar
[24]	V. Simoncini, On two numerical methods for the solution of large-scale algebraic Riccati equations, IMA J. Numer. Anal., 2013, 1-17.https://www.researchgate.net/publication/265480867_On_two_numerical_methods_for_the_solution_of_large-scale_algebraic_Riccati_equations Google Scholar
[25]	K. Zhang and C. Gu, Flexible global generalized Hessenberg methods for linear systems with multiple right-hand sides, J. Comput. Appl. Math, 2014, 263, 312-325. Google Scholar