Robust stability of linear stationary systems: Analysis by parameter-dependent Lyapunov quadratic functions

The robust stability of linear stationary systems with uncertainty defined by a polyhedron of general-type matrices is studied. Sufficient conditions for this problem are derived in terms of the Lyapunov functions of the class of quadratic ferns dependent affinely on uncertainty. Necessary and sufficient conditions for the robust stability are known only for the particular cease of a system defined as an interval polynomial. The existence of Lyapunov functions with a negative-definite derivative is algebraically verified. Elsewhere this problem is solved by the S-procedure and new sufficient conditions for this algebraic problem are derived. These conditions combine the S-procedure with the necessary and sufficient conditions for a quadratic form to be sign definite under special quadratic constraints. The new conditions are less conservative than those derived on, the basis of the S-procedure.