A new procedure for discretization and state feedback control of uncertain linear systems

This paper addresses the problem of constant sampling discretization of uncertain time-invariant continuous-time linear systems in polytopic domains. To circumvent the difficulty of dealing with the exponential of uncertain matrices, a new discretization method, based on Taylor series expansion, is proposed. The resulting discrete-time uncertain system is described in terms of homogeneous polynomial matrices with parameters lying in the unit simplex and an additive norm-bounded uncertainty which represents the discretization residual error. As a second contribution, linear matrix inequality (LMI) based conditions for the synthesis of a stabilizing state feedback control for discrete-time linear systems with polynomial dependence on the uncertain parameters and an additive norm-bounded uncertainty are proposed. Numerical experiments illustrate the discretization technique advantages of using higher orders in the Taylor series expansion to obtain more precise approximations. The examples also show that, at the price of simple line searches in a scalar parameter, and using Lyapunov functions of higher degrees, less conservative results for robust state feedback control design of discretized uncertain systems can be obtained.