Orthonormal rational functions via the jury table and their applications
by Xiaodong Zhao
Ph.D. Electrical and Electronic Engineering
x, 130 leaves : ill. ; 30 cm
Various orthogonal functions play important roles in science and engineering. In this thesis, we focus on orthonormal rational functions and their applications in control theory. 2...[ Read more ]
Various orthogonal functions play important roles in science and engineering. In this thesis, we focus on orthonormal rational functions and their applications in control theory.
Jury table is a tool in the classical discrete-time control theory used to test the stability of a polynomial. The Jury table can be used to construct an orthonormal basis in the space of strictly proper rational functions with a common denominator. We show that these functions are exactly the same as those constructed from the Gram-Schmidt orthonormalization of the standard basis. Relations between these functions and balanced realization of inner functions are also given.
These orthonormal functions can be used for many other purposes. We first use them to calculate the Η2 norm of a stable system.
Hankel operator is studied and two different methods to find its matrix representation under the orthonormal basis are given. Applying singular value decomposition on this matrix, we study the Hankel singular values and the Schmidt pairs of the Hankel operator.
Using the compressed Hankel matrix, we solve the optimal and suboptimal Nehari problems. A simple and direct polynomial solution is also given. We then get the solutions to the optimal and suboptimal Hankel norm approximation problem. Finally, we study the robust stabilization problem.
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