Rational functions whose zeros and poles are constrained by conditions arising in control
We shall study three different, yet related mathematical problems. The first is given as follows. Beta problem. Given β > 0, determine if there exists a rational function H which has no poles in D so that the rational function Cz :=-z+z2 +bHz 1-zHz has no zeros and no poles in D . ^ This problem in an equivalent form has been studied by Blondel, et al., and its statement can be found in, Open Problems in Mathematical Systems and Control Theory. Previous work on this problem has yielded that there exists β* such that if β < β*, then no solution to the Beta Problem exists. Furthermore, it has been shown that 10−5 < β* < 1e2 . In this dissertation we improve upon the upper bound on β*. We prove that β* ≤ 7 − 4 3 ≈ 0.0718 by constructing a solution to a limiting case of the Beta Problem. ^ The Beta Problem is a special ease of the following problem. ^ Avoidance problem. Given a rational function P , and given nonnegative integers n and m, determine if there exists a rational function C=PQ such that i.Degree P=n,Degree Q=m. ii.C hasno zerosand nopoles inthe closedunit disk. ii.C zPz ≠-1 forall z∈D . ^ An algorithm is proposed to construct numerical solutions to the Avoidance Problem by solving an optimization problem. Numerical Experience is reported. ^ By solving the Avoidance Problem, one also solves the simultaneous stabilization problem of control theory. It is well known that “[the] simultaneous stabilization problem is recognized as one of the hard open problems in linear systems theory.” With the proposed algorithm, it is possible to construct solutions to the simultaneous stabilization problem when they exist. ^ The third problem to be studied is an interpolation problem. ^ Interpolation problem. Given z1, … , zk ∈ D with zℓ ≠ zj for ℓ ≠ j, w1, … , wk ∈ C , wℓ ≠ 0 for all ℓ = 1, … , k and nonnegative integers n, m, determine if there exist polynomials P and Q such that i.DegreeP =n,DegreeQ =m ii.P,Q arenonzero in D iii.Pz ℓ=wℓQ zℓ=0,for allℓ=1, &ldots;,k. ^ An algorithm is proposed to obtain numerical solutions to the Interpolation Problem by solving an optimization problem. Numerical experience is reported, and properties of solutions to the optimization problem are proved. ^
Rebecca L Sparks,
"Rational functions whose zeros and poles are constrained by conditions arising in control"
Dissertations and Master's Theses (Campus Access).