A time-domain penalty function approach to mixed H2/H ∞-control using parameter optimization methods

In this paper we consider the problem of minimum nominal H₂ -norm with H_∞-constraints for systems with multiple operating points. The performance measure is defined as a weighted sum of the corresponding nominal H₂-norms while robust stability of the individual closed-loop systems is defined in terms of a H_∞-bound for each plant condition. In this paper we define a new time-domain scalar cost function J_∞ (t_f) representing the H_∞-bounds in an overall cost function for the mixed H₂/H_∞-design. J_∞(t_f ) is, for finite time t_f, a penalty function and, for t _f→∞, a barrier function. Using J_∞(t_f), the mixed H₂/H_∞-design problem results in an unconstrained optimization problem, that, for t_f→∞, recovers the original objective of minimizing the performance measure subject to the H_∞-bounds. The resulting optimization problem is smooth and hence standard gradient-based software can be applied. The class of controllers considered includes proper and strictly proper LTI controllers with fixed structure and/or fixed order