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Chapter Five Algebraic Lyapunov equations with small parameters

DOI: 10.1016/s0076-5392(06)80077-9
  • Computer Science
  • Mathematics


This chapter considers two special classes of systems, and solutions arepresented for the associated algebraic Lyapunov matrix equations. Thesesystems are the singularly perturbed and weakly coupled linear systems.It has been shown that the Lyapunov equations for these systems canbe decomposed into three lower order Lyapunov equations. In orderto obtain numerical solutions for such lower-order Lyapunov equationsthe recursive methods are presented. For weakly coupled systems theserecursive methods can be performed in parallel because the reduced-orderequations are completely decoupled. Even for singularly perturbedsystems parallelism can be obtained to some degree. The most importantfeature of these recursive algorithms is that they yield the solution withan arbitrary degree of accuracy. If k represents the number of iterationsperformed then the algorithms for singularly perturbed systems give theaccuracy of (0EK)while for weakly coupled systems the accuracy is 0(E2K)

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