H2-optimization - Theory and applications to robust control design

$H_2$-optimization removes the stochastics from LQG optimization (Doyle, Glover, Khargonekar, and Francis, 1989.) It relies on the observation that the customary signal-based mean square criterion of LQG optimization may be re-interpreted as a system norm (in particular, the 2-norm), without direct reference to the signals that are involved. A moment's thought, however, reveals that the $H_2$-paradigm allows the consideration of design problems that the conventional LQG formulation and solution does not permit. These extended problems include quite naturally frequency dependent weighting functions and colored measurement noise Although LQG optimization has been generalized to include these "singular" problems a long time ago these results are not widely used for control system design and no standard software appears to be available for their application. Nevertheless, the extra flexibility provided by the general $H_2$-problem is quite attractive. Moreover, the $H_2$-paradigm allows treating the stochastic problem parameters such as noise intensities as the design parameters that they really are. Frequency dependent weighting functions permit to design for integrating action in a fashion that is considerably less ad hoc than is usual in the LGQ context. They also provide other loop shaping tools for robust control design such as explicit control of high-frequency roll-off. After surveying the potential applications of $H_2$-optimization some of the solution algorithms that are currently available are reviewed. The best-known solution of the standard $H_2$ problem is described by Doyle, Glover, Khargonekar, and Francis (1989) but applies to a limited class of problems only that does not extend much beyond conventional LQG. Early polynomial matrix solutions (Hunt, Sebek and Kucera, 1994) suffer from complexity. Recent versions of the polynomial matrix solution (Meinsma, 2000; Kwakernaak, this paper) and the descriptor solution (Takaba and Katayama, 1998; Kwakernaak, this paper) offer implementations that are suitable for a wide and flexible class of useful design applications. Implementations of both algorithms have been tested with the help of the Polynomial Toolbox for MATLAB (www.polyx.com). The paper concludes with several sample applications and a design example.