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 transfer 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.