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This paper addresses the problems of stabilization and H∞ control by means of state feedback parameterdependent gains applied to discrete-time linear systems whose matrices are affected by arbitrarily time-varying parameters belonging to a polytope. The solution of the proposed design conditions, written as a finite set of linear matrix inequalities at the polytope vertices, allows to obtain a parameter-dependent gain (i.e. a gain scheduled controller) as an analytical function of the parameters. The proposed strategy is different from similar approaches in the literature, that are based on discretizations of the space of parameters to determine interpolated control gains, or that assume special structures for the time-varying parameters or even suppose that some of the system matrices are fixed and time-invariant in order to have a convex design problem. Numerical examples illustrate the efficiency of the conditions given in the paper. I. INTRODUCTIONThe design of gain scheduled controllers has been an important issue in systems theory and control applications for decades (see the survey papers [1], [2] and references therein). Basically, this technique focus on determining the control gain as a function of the system time-varying parameters, supposed to be available in real time. A classical way to compute a gain scheduled controller for a given linear parameter-varying (LPV) model of a plant, that usually comes from the linearization of the nonlinear model of the plant around operating points, follows the steps: i) determine a grid in the space of parameters to choose a family of plants and design one local controller for each plant, ii) based on the values of the parameters (measured or estimated on-line), schedule the control gains using some interpolation method, iii) assess the closed-loop system stability and performance. Although the system performance can be improved by means of increasing the precision of the discretization of the space of parameters (at the price of increasing the computational burden) this approach may be unreliable, since the global stability and performance are only assessed through simulation. Another problem is that the rates of variation of the timevarying parameter are not taken into account in the design, which may lead to instability or poor performance in the case of fast time-varying parameters [3], [4].More recently, several design approaches based on Lyapunov functions attempt to provide gain scheduled controllers to cope with time-varying parameters with bounded

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