H2-optimization - Theory and applications to robust control design

2 H optimization was conceived to remove 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 H norm), without direct reference to the signals that are involved. A moment's thought, however, reveals that the 2 H -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 (some of these are already considered in Kwakernaak and Sivan, 1972) 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 2 H -problem is quite attractive. Moreover, the 2H -paradigm allows to treat the stochastic problem parameters such as noise intensities as the designparameters that they really are.Frequency dependent weighting functions permit to design for integrating action in a fashion that is con-siderably less ad hoc than is usual in the LGQ context. They also provide other loop shaping tools forrobust control design such as explicit control of high-frequency roll-off.After surveying the potential applications of 2H -optimization the various solution algorithms that arecurrently available are reviewed. The best-known solution is described by Doyle, Glover, Khargonekar,and Francis (1989) but applies to a limited class of problems only that does not extend much beyondconventional LQG. Early polynomial matrix solutions (Hunt, Šebek and Kucera, 1994) suffer fromcomplexity. Recent versions of the polynomial matrix solution (Meinsma, this symposium; Kwakernaak,this paper) and the descriptor solution (Takaba and Katayama, 1998) offer implementations that aresuitable for a wide and flexible class of useful design applications. The presentation concludes with one or two design examples.ReferencesJ. C. Doyle, K. Glover, P. P. Khargonekar, and B. Francis, “State-space solutions to the standard 2H andH∞ control problems,” IEEE Trans. Aut. Control vol. 34, pp. 831–847 (1989).K. J. Hunt, M. Sebek, and V. Kucera, “Polynomial solution of the standard multivariable 2H -optimalcontrol problem,”IEEE Trans. Aut. Control vol. 39, pp. 1502–1507 (July, 1994). H. Kwakernaak and R. Sivan,Linear Optimal Control Systems. Wiley-Interscience, New York, 1972.G. Meinsma, “The standard 2H problem.” ROCOND 2000, Prague, June 21–23, 2000.K. Takaba and T. Katayama, “ 2H output feedback control for descriptor systems.” Automatica vol. 34,pp. 841–850 (July, 1998).