Robust stability conditions for a family of linear discrete-time systems subjected to uncertainties

Robust stability conditions are established for a family of linear discrete-time systems subjected to uncertainties. The traditional approach, which involves the construction of a common quadratic Lyapunov function for the entire family of systems with uncertainty, often leads to the problem of conservatism. In this connection, constructing the parametric quadratic Lyapunov functions seems promising. The main tools of the proposed approach are the apparatus of linear matrix inequalities and presented modification of the well-known Petersen's lemma. A simple approach to finding the radius of robust quadratic stability of the considered family is proposed in the paper as well. The corresponding optimization problems have the form of semi-definite programming and one-dimensional minimization, which could be easily solved numerically. The effectiveness of the proposed approach is demonstrated via numerical example. The results obtained can be generalized to the design problems for linear discrete-time systems subjected to uncertainties, to other robust statements, and to the case of exogenous disturbances.