Approximations to Riccati equations having slow and fast modes

Problems associated with the optimal linear regulator when one control is penalized much less than the others, and the optimal linear estimator when one measurement is of a much higher quality than the others are considered. Both situations cause the Riccati equation's solution to have transients with radically different speeds. In order to generate solutions with radically different transient speeds, a large number of small numerical integration time steps must be used-small to capture the rapid transient and a large number to generate the slow transient. Approximations are derived to the solutions to these Riccati equations. Although the number of scalar numerical integrations required is reduced by only one, a closed-form approximation is derived for the rapidly varying part of the transient. This allows the use of large numerical integration time steps and results in a considerable decrease in computation time. Measures are derived of both the errors in the approximations and how they affect the resulting regulators and estimators. A numerical example is given comparing the solution of the Riccati equation to its approximation.