Smooth stabilization and optimal H2 design

In this paper we propose two smooth optimization methods, one that can stabilize a system, and the other that can perform a stabilization as well as solve the optimal $H_2$-norm design problem. For both methods, we make use of the smoothed spectral abscissa, a stabilization measure which originates from the inversion of an $H_2$-norm type function, and that behaves as a smooth approximation of the spectral abscissa. In this way, we can set up an optimization framework in which a stabilizing point can efficiently be found. Taking advantage of its computation via Lyapunov equations, we derive computationally attractive formulae for the first-order and second-order derivatives of this smooth objective, which allows for the use of standard gradient- or Hessian-based optimization techniques. A second optimization framework, also involving the smoothed spectral abscissa, can be designed to deal with the $H_2$-norm synthesis. This method has the advantage that it is not necessary to find a stable point for the system on beforehand, as the stabilization is done simultaneously with the actual minimization of the $H_2$-norm. We apply the discussed methods to the class of systems with low-order, or fixed-order, feedback laws, where the number of controller parameters is smaller than the dimension of the plant.status: publishe