Structural properties of solutions of continuous-time and discrete-time matrix Lyapunov equations in controllable form

Structural properties of solutions of both continuous- and discrete-time matrix Lyapunov equations for SISO systems described by state models given in the controllable canonical form are discussed. The solutions have the so-called Xiao symmetric structure and the Toeplitz symmetric structure, respectively. The solution evaluation requires only linear dimension-reduced tasks to be considered. It is shown that the matrices and the right-hand vectors of these tasks are of very simple forms obtained by direct arrangement of the coefficients of the characteristic polynomials of the system matrices. Numerical examples are included to illustrate the solutions for stable minimal and unstable nonminimal systems