Abstract
The aim of this paper is to provide a new, direct, stabilizing-controller-oriented approach to the classical LQ problem with an infinite time horizon. In our approach, the LQ problem is formulated as a parametric optimization problem of a specific type, and then analysed by the methods presented in our earlier papers. The results simplify the well-known theory due to Zabczyk (1976) and Curtain & Pritchard (1978: §4.4). The spectral factorization method is applied to construct the LQ controller. According to the recent results of Callier & Winkin (1990, 1991), its complete construction is possible without invoking a Lyapunav operator equation, provided that the uncontrolable subspace is contained in the unobservable subspace. A weakened version of this result is given and applied to two examples of the synthesis of an LQ controller. LQ problems formulated in both our examples are reducible to a finite-dimensional LQ problems for which no requirements regarding approximate controllability are needed. Therefore, our investigations remove the gap between the results due to Korytowski (1993) and the spectral-factorization approach.
Published Version
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