An exact solution to general SISO mixed H2/H∞ problems via convex optimization

The mixed H₂/H_∞ control problem can be motivated as a nominal LQG optimal control problem, subject to robust stability constraints, expressed in the form of an H_∞ norm bound. A related modified problem consisting on minimizing an upper bound of the H₂ cost subject to H_∞ constraints was introduced by Bernstein-Haddad (1989). Although there presently exist efficient methods to solve this modified problem, the original problem remains, to a large extent, still open. In this paper we propose a method for solving general discrete-time SISO H₂/H_∞ problems. This method involves solving a sequence of problems, each one consisting of a finite-dimensional convex optimization and an unconstrained Nehari approximation problem.