A unified approach of the measurement of solution bounds of the continuous and discrete algebraic Lyapunov equations

In this paper, an integrated study for measuring the solutions of the continuous and discrete Lyapunov equations is developed. A unified algebraic Lyapunov equation (UALE) will be given first. Then, by using a bilinear transformation, the mentioned equation can be transformed into the quasi-standard form of the discrete algebraic Lyapunov equation (DALE). Thus, those approaches appeared in the literature for solving the estimation problem of the solution of the Lyapunov equations may be used to solve the same problem for the UALE. By extending mainly the approach proposed in [20] for the continuous algebraic Lyapunov equation (CALE) associated with linear algebraic techniques, several upper and lower matrix bounds of the solution of the UALE are developed. Then, matrix bounds of the solutions of the CALE and DALE can be obtained directly by these obtained matrix bounds. Algorithms for getting better bounds of the solutions of the CALE and DALE are also given. By comparisons, the obtained solution bounds of the CALE and DALE are less restrictive or easier to be calculated than those parallel results appeared in the literature. Finally, numerical examples have given to demonstrate the merits of the proposed approach.