Numerical integration of the differential matrix Riccati equation

Two new Bernoulli substitution methods for solving the Riccati differential equation are tested numerically against direct integration of the Riccati equation, the Chandrasekhar algorithm, and the Davison-Maki method on a large set of problems taken from the literature. The first of these new methods was developed for the time-invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric Riecati equation whose solution can be given explicitly in terms of a matrix exponential of order
n
. This method is fast and accurate when the extremal solutions of the associated algebraic Riccati equation are well separated. The second new method was developed as a means of eliminating the instabilities associated with the Davison-Maki algorithm. By using reinitialization at each time step the Davison-Maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time-invariant and time-dependent problems. From the results of our study we conclude that the modified Davison-Maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the Chandrasekhar algorithm is better.