Solving algebraic Riccati equations on parallel computers using Newton's method with exact line search

Abstract We investigate the numerical solution of continuous-time algebraic Riccati equations via Newton's method on serial and parallel computers with distributed memory. We apply and extend the available theory for Newton's method endowed with exact line search to accelerate convergence. We also discuss a new stopping criterion based on recent observations regarding condition and error estimates. In each iteration step of Newton's method a stable Lyapunov equation has to be solved. We propose to solve these Lyapunov equations using iterative schemes for computing the matrix sign function. This approach can be efficiently implemented on parallel computers using ScaLAPACK. Numerical experiments on an ibm sp 2 multicomputer report on the accuracy, scalability, and speed-up of the implemented algorithms.

[1]  Ji-guang Sun,et al.  Residual bounds of approximate solutions of the algebraic Riccati equation , 1997 .

[2]  B. Anderson,et al.  Optimal control: linear quadratic methods , 1990 .

[3]  Mihail M. Konstantinov,et al.  Computational methods for linear control systems , 1991 .

[4]  Chun-Hua Guo,et al.  Analysis and modificaton of Newton's method for algebraic Riccati equations , 1998, Math. Comput..

[5]  Gene H. Golub,et al.  Matrix computations , 1983 .

[6]  Jack Dongarra,et al.  ScaLAPACK Users' Guide , 1987 .

[7]  A. Laub,et al.  Rational iterative methods for the matrix sign function , 1991 .

[8]  Leiba Rodman,et al.  Algebraic Riccati equations , 1995 .

[9]  P. Benner.  Numerical solution of special algebraic Riccati equations via an exact line search method , 1997, 1997 European Control Conference (ECC).

[10]  A. Laub,et al.  Parallel algorithms for algebraic Riccati equations , 1991 .

[11]  Peter Benner,et al.  An exact line search method for solving generalized continuous-time algebraic Riccati equations , 1998, IEEE Trans. Autom. Control..

[12]  Robert A. van de Geijn,et al.  Parallelizing the QR Algorithm for the Unsymmetric Algebraic Eigenvalue Problem: Myths and Reality , 1996, SIAM J. Sci. Comput..

[13]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem: Theory and Numerical Solution , 1991 .

[14]  Irena Lasiecka,et al.  Differential and Algebraic Riccati Equations With Application to Boundary/Point Control Problems: Continuous Theory and Approximation Theory , 1991 .

[15]  J. D. Roberts,et al.  Linear model reduction and solution of the algebraic Riccati equation by use of the sign function , 1980 .

[16]  Judith Gardiner,et al.  A generalization of the matrix sign function solution for algebraic Riccati equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[17]  Ed Anderson,et al.  LAPACK Users' Guide , 1995 .

[18]  S. Hammarling.  Numerical Solution of the Stable, Non-negative Definite Lyapunov Equation , 1982 .

[19]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[20]  Jack J. Dongarra,et al.  A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures , 2002, SIAM J. Sci. Comput..

[21]  D. Kleinman.  On an iterative technique for Riccati equation computations , 1968 .

[22]  R. Byers.  Solving the algebraic Riccati equation with the matrix sign function , 1987 .

[23]  Alan J. Laub,et al.  Local convergence analysis of conjugate gradient methods for solving algebraic Riccati equations , 1992 .

[24]  A. Varga.  On stabilization methods of descriptor systems , 1995 .

[25]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[26]  G. Hewer,et al.  The sensitivity of the algebraic and differential riccati equations , 1990 .

[27]  L. Armijo.  Minimization of functions having Lipschitz continuous first partial derivatives. , 1966 .

[28]  Alan J. Laub,et al.  A collection of benchmark examples for the numerical solution of algebraic Riccati equations I: Continuous-time case , 1998 .

[29]  William Gropp,et al.  Skjellum using mpi: portable parallel programming with the message-passing interface , 1994 .

[30]  John E. Dennis,et al.  Numerical methods for unconstrained optimization and nonlinear equations , 1983, Prentice Hall series in computational mathematics.

[31]  Judith D. Gardiner.  Stabilizing control for second-order models and positive real systems , 1992 .

[32]  A. Laub.  A schur method for solving algebraic Riccati equations , 1978, 1978 IEEE Conference on Decision and Control including the 17th Symposium on Adaptive Processes.

[33]  Thilo Penzl,et al.  Numerical solution of generalized Lyapunov equations , 1998, Adv. Comput. Math..

[34]  J. Doyle,et al.  Robust and optimal control , 1995, Proceedings of 35th IEEE Conference on Decision and Control.

[35]  Alan J. Laub,et al.  Solution of the Sylvester matrix equation AXBT + CXDT = E , 1992, TOMS.

[36]  Vasile Sima,et al.  An efficient Schur method to solve the stabilizing problem , 1981 .

[37]  James Demmel,et al.  The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers , 1997, SIAM J. Sci. Comput..

[38]  A. Laub,et al.  Benchmarks for the numerical solution of algebraic Riccati equations , 1997 .

[39]  Alan J. Laub,et al.  Error bounds for Newton refinement of solutions to algebraic riccati equations , 1990, Math. Control. Signals Syst..

[40]  Alan J. Laub,et al.  On a Newton-Like Method for Solving Algebraic Riccati Equations , 1999, SIAM J. Matrix Anal. Appl..

[41]  V. Mehrmann.  The Autonomous Linear Quadratic Control Problem , 1991 .

[42]  Anthony Skjellum,et al.  Using MPI - portable parallel programming with the message-parsing interface , 1994 .

[43]  A. Laub,et al.  Generalized eigenproblem algorithms and software for algebraic Riccati equations , 1984, Proceedings of the IEEE.

[44]  Vasile Sima,et al.  Algorithms for Linear-Quadratic Optimization , 2021 .