Generation of Exact QFT Bounds for Plants With Affinely Dependent Uncertainties

Abstract

This paper presents an efficient method for the generation of exact QFT bounds for robust sensitivity reduction and gain‐phase margin specifications for plants with affinely dependent uncertainties. It is shown that, for a plant with m affinely dependent uncertainties, the exact QFT bounds for robust sensitivity reduction and gain‐phase margin specifications at a given frequency and controller phase can be computed by solving m2m‐1 bivariate polynomial inequalities corresponding to the edges of the parameter domain box. Moreover, the solution set for each bivariate polynomial inequality can be computed by solving for the real roots of one fourth‐order and six second‐order polynomials. This avoids the unfavorable trade‐off between the computational burden and the accuracy of QFT bounds that has arisen in the application of many existing QFT bound generation algorithms. Numerical examples are given to illustrate the proposed method and its computational superiority.

Generation of QFT bounds for robust tracking specifications for plants with affinely dependent uncertainties

“…From the pivoting procedure listed above, it is obvious that unlike the algorithm in 23 which generates QFT bounds by controller phase sweeping, the algorithm proposed in this paper generates QFT bounds by tracing their boundaries in the Nichols chart. The pivoting procedure 25 is also used in 24.…”

“…The comparison of the efficiency of the proposed and existing algorithms is also given. Consider the generation of the QFT bound for the robust tracing specification γ = 1.03 at ω = 1.2 for the plant taken from 23 as follows: where The chosen nominal plant G p 0 ( s ) corresponding to the parameter vector q 0 = (0, 0, 0) T is The accuracy parameters h 1 = 5 ∘ and h 2 = 0.05dB are chosen for the pivoting procedure. Let Searching upward from z 0 , we obtained that have different vertex labels L ( v ) = 1 and L ( v ) = 0.…”

“…In other words, the difficulty of computing QFT bounds can be apparently reduced if the plant templates can be exactly characterized. Motivated by the fact that at a fixed frequency, the plant template boundary of a plant with affinely dependent uncertainties is included in the set of images of the edges of the parameter domain box 21, 22, Yang 23, 24 developed algorithms for computing, except for robust tracking specification, the QFT bounds of plants with affinely dependent uncertainties without having to approximate the plant templates by finite numbers of points. Since the approximation of the plant templates by finite numbers of points is not required in both of the two algorithms, the unfavorable trade‐off between the computational burden and the accuracy of QFT bounds is completely avoided.…”

“…Since the approximation of the plant templates by finite numbers of points is not required in both of the two algorithms, the unfavorable trade‐off between the computational burden and the accuracy of QFT bounds is completely avoided. The two algorithms in 23, 24 for computing the QFT bounds of plants with affinely dependent uncertainties are efficient and accurate. However, they are not applicable to the generation of QFT bounds for robust tracking specifications since the robust tracking specification inequality cannot be reformulated to the form required by the two algorithms in 23, 24.…”