Abstract
Recently, a general numerical procedure has been developed for solvable systems of singular differential equationsE(t)x′(t)+F(t)x(t)=f(t). This paper shows how to exploit the structure present in many control problems to reduce the computational effort substantially. An example is worked which shows that additional reductions are possible in some cases.
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References
K. E. Brenan, Stability and Convergence of Difference Approximations for High Index Differential—Algebraic Systems with Applications in Trajectory Control, Ph.D. thesis, University of California, 1983.
[B2] K. E. Brenan, Numerical simulation of trajectory prescribed path control problems,IEEE Trans. Automat. Control,31 (1983), 266–269.
[C1] S. L. Campbell,Singular Systems of Differential Equations, Pitman, New York, 1980.
[C2] S. L. Campbell,Singular Systems of Differential Equations, II, Pitman, New York, 1982.
[C3] S. L. Campbell The numerical solution of higher index linear time varying singular systems of differential equations,SIAM J. Sci. Statist. Comput.,6, (1985), 334–348.
[C4] S. L. Campbell, Rank deficient least squares and the numerical solution of linear singular implicit systems of differential equations, inLinear Algebra and Its Role in Systems Theory, pp. 51–63, Contemporary Mathematics, Vol. 47. American Mathematical Society, Providence, RI, 1985.
[C5] S. L. Campbell, Consistent initial conditions for linear time varying singular systems, inFrequency Domain and State Space Methods for Linear Systems (C. I. Byrnes and A. Lindquist, eds.), pp. 313–318, North-Holland, Amsterdam, 1986.
[C6] S. L. Campbell, A general form for solvable linear time varying singular systems of differential equations,SIAM J. Math. Anal.,18 (1987), 1101–1115.
[CC] S. L. Campbell and K. Clark, Order and the index of singular time invariant linear systems,Systems Control Lett. 1 (1981), 119–122.
[CR] S. L. Campbell and N. J. Rose, A singular second order linear system arising in electric power systems analysis,Internat. J. Systems Sci.,13 (1981), 101–108.
K. Clark, Difference Methods for the Numerical Solution of Nonautonomous Linear Systems, Ph.D. thesis, North Carolina State University, 1986.
[G] C. W. Gear, The simultaneous numerical solution of differential-algebraic equations,IEEE Trans. Circuit Theory,18 (1971), 89–95.
C. W. Gear, G. K. Gupta, and B. Leimkuhler, Automatic integration of Euler-Lagrange equations with constraints,J. Comput. Appl. Math. (to appear).
[GV] G. H. Golub and C. F. Van Loan,Matrix Computations, Johns Hopkins University Press, Baltimore, MD, 1983.
V. B. Haas, On optimal controls having finite order of singularity, preprint.
[LP] P. Lötsted and L. R. Petzold, Numerical solution of nonlinear differential equations with algebraic constraints, I: Convergence results for backward differentiation formulas,Math. Comp.,46 (1986), 491–516.
[PL] L. R. Petzold and P. Lötstedt, Numerical solution of nonlinear differential equations with algebraic constraints, II: Practical implications,SIAM J. Sci. Statist. Comput.,7 (1986), 726–733.
[R] W. T. Reid,Riccati Differential Equations, Academic Press, New York, 1972.
[S] L. M. Silverman, Inversion of multivariable linear systems,IEEE Trans. Automat. Control,14 (1969), 270–276.
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This research was supported in part by the Air Force Office of Scientific Research under Grant No. 84-0240.
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Campbell, S.L. Control problem structure and the numerical solution of linear singular systems. Math. Control Signal Systems 1, 73–87 (1988). https://doi.org/10.1007/BF02551237
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DOI: https://doi.org/10.1007/BF02551237