Some Chandrasekhar-type algorithms for quadratic regulators

Abstract

The by-now classical method for the quadratic regulator problem is based on the solution of an n × n matrix nonlinear Riccati differential equation, where n is the dimension of the state-vector. Care has to be exercised in numerical solution of the Riccati equation to ensure nonnegative-definiteness of its solution, from which the optimum m × n feedback gain matrix K(•) is calculated by a further matrix multiplication. For constant-parameter systems, we present a new algorithm that requires only the solution of n(m + p) simultaneous equations: the nm elements of the feed-back gain matrix K(•) and the np elements of a rank-p square-root of the derivative of P(•), where p is the rank of the nonnegative-definite weighting matrix Q that measures the contribution of the state trajectory to the cost functional. If n is large compared with p and m, our algorithm can provide considerable computational savings over direct solution of the Riccati equation, where n(n + 1)/2 simultaneous equations have to be solved. Also the square-root feature means that with reasonable care the automatic nonnegative-definiteness of the derivative matrix-P(•) can be carried over to P(•) itself. Similar results can be obtained for indefinite Q matrices, but with n(m + 2p) equations rather than n(m + p). The equations of our algorithm have the same form as certain famous equations introduced into astrophysics by S. Chandrasekhar, which explains our terminology. The method used in the paper can also be applied to Lyapunov differential equations, as discussed in an Appendix, and to the linear least-squares estimation of stationary processes, as discussed elsewhere.