A virtual perturbation method for singular systems

A virtual perturbation is introduced in a linear time-invariant system Ex(t)=Ax(t)+Bu(t ) with E singular for the purpose of formal regularization. Methods for virtually adding ∈ to get a regular state space system are presented, and a general perturbed form is derived. For the general form a necessary and sufficient convergence condition is derived. A theory of continuous triangularization of matrix functions is presented and applied to obtain an explicit solution of the perturbed system. With such a solution, the system and its asymptotic behavior can be analyzed