Computational methods for medium scale stiff Lyapunov differential equations

Matrix version of the backward differentiation formulas were used for solving medium scale stiff Lyapunov differential equations with known closed-form solutions. The purpose was to investigate whether the backward differentiation formulas will produce accurate solutions for stiff Lyapunov differential equations of medium sizes. The sizes of the equations under consideration in this paper range from 10 times 10 to 200 times 200. An application of this computational method is for obtaining the controllability grammian of a medium scale state-space system. The matrix backward differentiation formulas were encoded as MATLAB^reg script files with variable step-size feature, which is necessary for efficiently solving stiff equations in the steady state. Effort was put into preserving the structure of the Lyapunov differential equations. They were manipulated into algebraic Lyapunov equations so that numerically stable methods can be applied to obtain the solutions reliably. The findings of this investigation were that accurate solutions of stiff Lyapunov differential equations could be obtained by the matrix backward differentiation formulas and that the step-size of integration could be increased continually without loss of accuracy when the transient components of the solution had faded away.