Mixed H2/H∞ control with pole placement in a class of regions

The problem of mixed H2/H∞ control with pole placement is considered for linear time-invariant systems. This is the problem of determining a controller for linear time-invariant systems which minimizes the H2-norm of a certain closed-loop transfor function subject to an H∞-norm constraint on another closed-loop transfer function and an additional constraint on the location of the closed-loop poles in the complex plane. An optimization problem is posed for the pole-constrained H2/H∞, problem in such a way that the objective function is expressed as a weighted sum of the actual H2 cost and its upper bound. A necessary condition for the optimization problem is derived via the Lagrange multiplier technique. The condition involves a set of highly coupled equations. By sacrificing the H2 performance, an alternative optimization problem is posed in order to simplify the necessary condition. An iterative algorithm for solving the coupled equations arising in the necessary conditions is proposed and numerical examples are presented.