Efficient matrix-valued algorithms for solving stiff Riccati differential equations

In the time-varying case, a classical approach which has been widely used to compute the solution of the Riccati matrix equation of, for example, size n*n, is to unroll the matrices into vectors and integrate the resulting system of n/sup 2/ vector differential equations directly. If the system of vectorized differential equations is stiff, the cost (computation time and storage requirements) of applying the popular backward differentiation formulas (BDFs) to the stiff equations will be very high for large n because a linear system of algebraic equations of size n/sup 2/*n/sup 2/ must be solved at each time step. New matrix-valued algorithms based on a matrix generalization of the BDFs are proposed for solving stiff Riccatti differential equations. The amount of work required to compute the solution per time step is only O(n/sup 3/) flops by using the matrix-valued algorithms, whereas the classical approach requires O(n/sup 6/) flops per time step. >

[1]  Alan J. Laub,et al.  Efficient algorithms for solving stiff matrix-valued riccati differential equations , 1988 .

[2]  A. Hindmarsh,et al.  Large ordinary differential equation systems and software , 1982, IEEE Control Systems Magazine.

[3]  T. E. Hull,et al.  Comparing numerical methods for stiff systems of O.D.E:s , 1975 .

[4]  W. Reid,et al.  Riccati Differential Equations , 1975, IEEE Transactions on Systems, Man, and Cybernetics.

[5]  Lawrence F. Shampine,et al.  A User’s View of Solving Stiff Ordinary Differential Equations , 1979 .

[6]  Demetrios G. Lainiotis,et al.  Partitioned Ricatti solutions and integration-free doubling algorithms , 1976 .

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

[8]  D. W. Rand,et al.  Nonlinear superposition principles: a new numerical method for solving matrix Riccati equations , 1984 .

[9]  Ilan Rusnak.  Almost analytic representation for the solution of the differential matrix Riccati equation , 1988 .

[10]  C. Kenney,et al.  Numerical integration of the differential matrix Riccati equation , 1985 .

[11]  R. L. Anderson,et al.  Superposition principles for matrix Riccati equations , 1983 .

[12]  C. W. Gear,et al.  The automatic integration of stiff ordinary differential equations , 1968, IFIP Congress.

[13]  C F Curtiss,et al.  Integration of Stiff Equations. , 1952, Proceedings of the National Academy of Sciences of the United States of America.

[14]  D. Vaughan,et al.  A negative exponential solution for the matrix Riccati equation , 1969 .

[15]  P. Kokotovic.  Applications of Singular Perturbation Techniques to Control Problems , 1984 .

[16]  G. Golub,et al.  A Hessenberg-Schur method for the problem AX + XB= C , 1979 .

[17]  M. Sorine,et al.  Superposition laws for solutions of differential matrix Riccati equations arising in control theory , 1985 .

[18]  Alan J. Laub,et al.  Schur techniques for riccati differential equations , 1982 .

[19]  Roy B. Leipnik.  A canonical form and solution for the matrix Riccati differential equation , 1985, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[20]  Demetrios Lainiotis.  Generalized Chandrasekhar algorithms: Time-varying models , 1975, 1975 IEEE Conference on Decision and Control including the 14th Symposium on Adaptive Processes.

[21]  Edward J. Davison,et al.  The numerical solution of the matrix Riccati differential equation , 1973 .

[22]  R. E. Kalman,et al.  A user's manual for the Automatic Synthesis Program /program C/ , 1966 .

[23]  A. Laub,et al.  Improving the efficiency of matrix operations in the numerical solution of large implicit systems of linear differential equations , 1987 .

[24]  B. Lindberg.  A Stiff System Package Based on the Implicit Midpoint Method with Smoothing and Extrapolation , 1974 .