Robust Parallel Algorithms for Solution of Riccati Equation
Publication: Journal of Aerospace Engineering
Volume 10, Issue 3
Abstract
Robust and efficient parallel-vector algorithms are presented for the solution of the Riccati equations encountered in optimal control problems on shared-memory multiprocessor machines. The algorithms have been implemented on a Cray YMP 8/8128 and applied to three large problems resulting from a continuous bridge structure, a 21-story space truss structure, and a 12-story space moment-resisting building structure. Efficiency of the algorithms is presented in terms of millions of floating point operations per second (MFLOPS) and the speedup. The MFLOPS for the largest example resulting from the 12-story space frame structure is a high 206.0. The speedup due to parallel processing only (for the same example), using seven processors, is 6.33. When vectorization is combined with parallel processing a very significant speedup of 54.4 is obtained using seven processors. The algorithms developed in this research find applications in the complex integrated control/structural optimization problem. Further, the writers are currently using them to develop large adaptive/smart structures.
Get full access to this article
View all available purchase options and get full access to this article.
References
1.
Adeli, H. (1992a). Supercomputers in engineering analysis. Marcel Dekker, Inc., New York, N.Y.
2.
Adeli, H. (1992b). Parallel processing the computational mechanics. Marcel Dekker, Inc., New York, N.Y.
3.
Adeli, H., and Saleh, A.(1997). “Optimal control of adaptive/smart bridge structures.”J. Struct. Engrg., ASCE, 123(2), 1–9.
4.
Armstrong, E. S. (1978). “ORACLS—a system for linear quadratic Gaussian control law design.”NASA Tech. Paper 1106, Nat. Aeronautics and Space Admin. (NASA), Washington, D.C.
5.
Armstrong, E. S. (1980). ORACLS—a design system for linear multivariable control. Marcel Dekker, Inc., New York, N.Y.
6.
Arnold, W. F., and Laub, A. J.(1984). “Generalized eigenproblem algorithms and software for algebraic Riccati equations.”Proc., IEEE, 72(12), 1746–1754.
7.
Byers, R.(1987). “Solving the algebraic Riccati equation with the matrix sign function.”Linear Algebra and Its Applications, 85, 267–279.
8.
Friedland, B. (1986). Control system design: an introduction to state space methods, McGraw-Hill Inc., New York, N.Y.
9.
Gardiner, J. D. (1997). “A stabilized matrix sign function algorithm for solving algebraic Riccati equations.”SIAM, J. Scientific Computing.
10.
Gardiner, J. D., and Laub, A. J.(1986). “A generalization of the matrix-sign-function solution for the algebraic Riccati equations.”Int. J. Control, 44(3), 823–832.
11.
Hewer, G. A. (1971). “An iterative technique for the computation of the steady state gains for the discrete optimal regulator.”IEEE Trans. on Automatic Control, AC-16(4), 382–383.
12.
Johnson, C. D. (1971). “Accommodation of external disturbances in linear regulators and servomechanism problems.”IEEE Trans. on Automatic Control, AC-16(6), 635–643.
13.
Khot, N. S. (1994). “Optimization of controlled structures.”Advances in design optimization, H. Adeli, ed., Chapman & Hall, Ltd., London, England, 266–296.
14.
Kim, S. W., Park, P., and Kwan, W. H.(1993). “Lower bounds for the trace of the solution of the discrete algebraic Riccati equation.”IEEE Trans. on Automatic Control, 38(2), 312–314.
15.
Kleinman, D. L. (1968). “On an iterative technique for Riccati equation computations.”IEEE Trans. on Automatic Control, AC-13(1), 114–115.
16.
Laub, A. J. (1979). “A Schur method for solving algebraic Riccati equations.”IEEE Trans. on Automatic Control, AC-24(6), 913–921.
17.
Meirovich, L. (1990). Dynamics and control of structures. John Wiley & Sons, Inc., New York, N.Y.
18.
Morse, S., and Wonham, W. H. (1971). “Status of noninteracting control.”IEEE Trans. on Automatic Control, AC-16(6), 568–581.
19.
Paige, C., and Loan, C. V.(1981). “A Schur decomposition for Hamiltonian matrices.”Linear Algebra and Its Applications, 50, 11–32.
20.
Potter, J. W.(1966). “Matrix quadratic solutions.”SIAM J. Appl. Math., 14(3), 496–501.
21.
Rhodes, L. B. (1971). “A Tutorial introduction to estimation and filtering.”IEEE Trans. on Automatic Control, AC-16(6), 688–706.
22.
Saleh, A., and Adeli, H.(1994a). “Microtasking, macrotasking, and autotasking for structural optimization.”J. Aerosp. Engrg., ASCE, 7(2), 156–174.
23.
Saleh, A., and Adeli, H.(1994b). “Parallel algorithms for structural and control optimization of structures.”J. Aerosp. Engrg., ASCE, 7(3), 297–314.
24.
Saleh, A., and Adeli, H.(1996). “Parallel algorithms for eigenvalue problem of an unsymmetric matrix.”J. Aerosp. Engrg., ASCE, 9(3), 70–79.
25.
Sandell, N. R. Jr. (1974). “On Newton's method for Riccati equation solution.”IEEE Trans. on Automatic Control, AC-19(3), 254–255.
Information & Authors
Information
Published In
Journal of Aerospace Engineering
Volume 10 • Issue 3 • July 1997
Pages: 126 - 133
Copyright
Copyright © 1997 American Society of Civil Engineers.
History
Published online: Jul 1, 1997
Published in print: Jul 1997
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.