Kybernetika 44 no. 3, 328-335, 2008

Sixty years of cybernetics: a comparison of approaches to solving the H2 control problem

Vladimír Kučera


The control problem consists of stabilizing a control system while minimizing the norm of its transfer function. Several solutions to this problem are available. For systems in form, an optimal regulator can be obtained by solving two algebraic Riccati equations. For systems described by , either Wiener-Hopf optimization or projection results can be applied. The optimal regulator is then obtained using operations with proper stable rational matrices: inner-outer factorizations and stable projections. The aim of this paper is to compare the two approaches. It is well understood that the inner-outer factorization is equivalent to solving an algebraic Riccati equation. However, why are the stable projections not needed in the state-space approach? The difference between the two approaches derives from a different construction of doubly coprime, proper stable matrix fractions used to represent the plant. The transfer-function approach takes any doubly coprime fractions, while the state-space approach parameterizes such representations and those selected then obviate the need for stable projections.