ON SOLVING THE LYAPUNOV LINEAR MATRIX EQUATIONS AND INEQUALITIES USING THE KRYLOV SUBSPACE METHOD
A new approach to solving the Lyapunov linear matrix equations and inequalities on the basis of a Krylov subspace method (based, in turn, on the Cayley-Hamilton identity) is offered. This method is used for solving various problems of analysis and synthesis of linear multi-input multi-output (MIMO) systems. These problems include (i) calculation of a balanced realization of transfer matrix of the linear MIMO system in the state space, (ii) reduction and decomposition of a model of this system in the state space, (iii) definition of controlled and observed subspaces, (iv) stabilization with the help of the state-elements feedback. The Krylov subspace method in a combination with the technique of calculating zero divisors of a matrix is used here for solving the Lyapunov matrix equations and inequalities. Correlation between the method and the known Lo¨ wner - Heinz inequality is established. Effectiveness of the approach is demonstrated by the methodical examples