Control of Uncertainty-Affected Discrete Time Linear Systems via Convex Programming Aharon Ben-Tal ∗ Stephen Boyd † Arkadi Nemirovski ‡ November 29, 2005 Abstract In [1], we have demonstrated that robust optimization of linear finite-horizon control in a discrete time linear dynamical system affected by uncertain disturbances becomes computation- ally tractable if, rather than using natural parameterization of a linear output-based control law, one passes to a specific re-parameterization of the law, representing it as affine control law based on the so called “purified outputs”. With the traditional parameterization, the states and the controls, being affine in the initial state and the disturbances, are highly nonlinear in the parameters of the control law; with the new parameterization, the states and the controls become bi-affine, that is, affine in the initial state and the disturbances, the parameters of the control law being fixed, and affine in the parameters of the control law, the disturbances and the initial state being fixed. As a result, synthesis of a finite-horizon control law satisfying, in a robust fashion, a given system of linear constraints on the finite-horizon state-control trajectory, reduces to solving an explicit convex program and thus becomes computationally tractable. In this follow up paper we extend the above methodology to optimizing infinite-horizon control in a time-invariant linear system and illustrate our methodology on examples involving discrete time H ∞ - and L 1 -control. 1 Introduction Consider a discrete time linear dynamical system x 0 = z x t+1 = A t x t + B t u t + R t d t ,t =0, 1, ... y t = C t x t + D t d t (1.1) where x t are states, y t are outputs, u t are controls, and d t are external disturbances at time t. In [1], we have associated with system (1.1) “closed” by an arbitrary control law u t = U t (y 0 ,y 1 , ..., y t ) ∗ MIT, Boston, USA, on sabbatical leave from Technion – Israel Institute of Technology, Haifa, Israel, abental@ie.technion.ac.il † Stanford University, Stanford, USA, boyd@stanford.edu ‡ Georgia Institute of Technology, Atlanta, USA, nemirovs@isye.gatech.edu 1