Some aspects of Gramian assignment and interpolation problems in control systems

Samir M Kherat, Purdue University

Abstract

Fast algorithms, such as the Schur algorithm or the Levinson algorithm have played an important role in signal processing and spectral analysis. Two major parts of this thesis use these algorithms along with realization theory to solve a Gramian assignment problem and some mixed H$\sp2$-H$\sp\infty$ Nehari problems. The Levinson algorithm is first used to compute all assignable controllability Gramians for single input-single output discrete time controllable systems. The procedure takes advantage of the special Toeplitz form of the Gramian when the system is written in its controllable canonical form. A simple coordinate transformation allows a generalization of the procedure. Finally, we will extend the theory of covariance control to a "bilinear setting". To motivate our mixed H$\sp2$-H$\sp\infty$ approach to interpolation and control problems, an example shows that the assumption of a white noise input to the systems leads to a Wiener (or H$\sp2$) filtering problem. On the other hand, an L$\sp2$ disturbance leads to an H$\sp\infty$ filtering problem. Because an H$\sp\infty$ design and an H$\sp2$ design have complementary actions, the proposed procedures mix both constraints to yield a compromising result. This work is motivated by the nice existence result of Kaftal, Larson and Weiss. The Schur algorithm is used to compute a solution to the Kaftal-Larson-Weiss mixed H$\sp2$-H$\sp\infty$ problem in the Nehari setting. First we use the Schur algorithm to solve a mixed H$\sp2$-H$\sp\infty$ Caratheodory interpolation problem. By passing limits, we then obtain an explicit formula in terms of Hankel matrices to solve the rational H$\sp2$-H$\sp\infty$ Nehari problem. Explicit solutions in terms of minimal state space realizations are given. The Schur algorithm along with the backward shift realization is used to provide a state space approximation to the H$\sp2$-H$\sp\infty$ rational Nehari problem. The next mixed H$\sp2$-H$\sp\infty$ interpolation problems include an H$\sp2$-H$\sp\infty$ tangential Caratheodory interpolation problem, an H$\sp2$-H$\sp\infty$ tangential Nevanlinna-Pick interpolation problem and an H$\sp2$-H$\sp\infty$ tangential Hermite-Fejer interpolation problem. For all cases we will provide the particular algorithm and some explicit computational formulas in terms of state space realizations to compute the solutions. Model matching problems and disturbance rejection problems are given as some of the applications to H$\sp2$-H$\sp\infty$ control. Finally, we provide two different procedures to solve a Kaftal-Larson-Weiss type mixed H$\sp2$-H$\sp\infty$ two block rational interpolation problem. We will first present an algorithm involving a Riccati equation and some singular value decompositions to obtain the interpolant. Then, another procedure involving a Stein equation is given.

Degree

Ph.D.

Advisors

Frazho, Purdue University.

Subject Area

Aerospace materials|Electrical engineering|Mechanical engineering

COinS