Minimum H∞-norm regulation of linear discrete-time systems and its relation to linear quadratic discrete games

A solution is derived for the H_∞-optimization problem that arises in multivariable optimal discrete-time regulation when the controller has full access to the state vector. Two different approaches to the solution of the problem are presented. The first, which is more direct, minimizes the H_∞-norm of the transfer function matrix from the input disturbance signal to a linear combination of the states and the plant input. The second approach applies known results from the theory of difference games. It is found that the solution by the first approach yields a constant state feedback controller whose gain matrix is identical to the one used by the minimizer in the corresponding difference game