Relations between continuous-time and discrete-time quadratic minimization

the thesis deals with some aspects of the theory of constant linear dynamical systems. The first problem considered is an extension to multi-variable discrete-time systems of the so-called Positive Real Lemma, an algebraic criterion for continuous-time linear systems to be passive. An incidental result in the proof of the criterion is that the classical bi-linear transformation of complex function theory, when applied to the state-space realizations of transfer function matrices, yields a method for transforming the quadratic minimization problems into their discrete-time counter parts. The remainder of the thesis develops the idea suggested by this: that the limiting solutions of Riccati differential equations might be obtained from the limiting solutions of Ricatti difference equations. The theory of the Riccati difference equation is discussed in the context of the discrete-time state regulator problems and extended to quadratic minimization problems having a negative minimum cost. Iterative algorithms are then developed for obtaining the limiting solutions of the Riccati equations of the continuous-time state regulator, Kalman filtering and spectral factorization problems by means of Riccati difference equations. The algorithms are numerically stable, simple and competitive in computational effort with other methods of solving the Riccati differential equations.