A Factorization Algorithm with Applications to the Linear Filtering and Control Problems

This paper investigates some important control and systems applications on the light of the results brought up in the preceding parts of this study. In the first place, we generalize the Davis and Barry integral formula for the optimal feedback gain in the LQR problems. The new setting covers a wider class of cost functions and overcomes the difficulty of treating unstable systems. The new formula, equipped with the spectral factorization method proposed in Parts I and II provides a fast and efficient procedure for solving many LQR problems. As a by-product, we present as well a method for the spectral factorization of certain rational matrices. Our parallel results for the discrete time case are given in brief together with many interesting computational properties. The potentialities of our approach are demonstrated once more by providing a fast algorithm for the factorization of positive matrix polynomials.