Robustness analysis and controller synthesis using stability multipliers and scalings.
Sparks, Andrew George
1995
Abstract
Robust stability analysis of and controller synthesis for systems with real parameter uncertainty is considered using stability multipliers from absolute stability theory with scaling to reduce conservatism. First, a simplified proof of the multivariable Popov criterion is given for the case of one-sided, sector-bounded real parameter uncertainty. A loop-shifting transformation is used to extend the Popov criterion to two-sided, sector-bounded uncertain matrices, which is specialized to norm-bounded matrices by considering symmetric sectors. This criterion is rendered less conservative by including scaling matrices and the resulting scaled Popov criterion is used to derive a frequency domain upper bound for the structured singular value for real parameter uncertainty. The scaled Popov criterion is then used to derive an upper bound for the peak of the structured singular value over frequency by relating the strict positive realness of a transfer function to the feasibility of a linear matrix inequality involving the system's state space realization. Numerical examples are given to illustrate the frequency domain and peak upper bounds. Next, multipliers and scalings having more general frequency dependence than those of the scaled Popov criterion are considered. Sufficient conditions for robust stability involving rational stability multipliers and scalings are presented for systems with sector- and norm-bounded, block-structured uncertainty. The frequency-dependent multipliers and scalings render the new robustness criterion less conservative than the scaled Popov criterion, which is a special case of the new criterion with a multiplier that is an affine function of frequency and scaling that is independent of frequency. An upper bound for the peak structured singular value over frequency is then derived. Numerical examples provide a comparison of the peak upper bound utilizing two rational parameterizations of the frequency-dependent multiplier and scaling. Finally, the scaled Popov criterion is used to derive an upper bound for the worst-case ${\cal H}\sb2$ norm over the set of real parameter perturbations. This upper bound provides the basis for a robust controller synthesis procedure that guarantees asymptotic stability of the closed-loop system for all real parameter perturbations in the specified set and minimizes the upper bound, where the parameter uncertainty may appear simultaneously in the state, input, and output matrices of the plant's state space realization. Necessary conditions for the control gains that guarantee asymptotic stability of the closed-loop system and minimize this upper bound are derived using a Lagrange multiplier formulation. Numerical examples demonstrate the trade-off between nominal and robust performance of the scaled Popov controllers.Other Identifiers
(UMI)AAI9610244
Subjects
Engineering, Aerospace Engineering, Electronics and Electrical Engineering, System Science
Types
Thesis
Metadata
Show full item recordCollections
Remediation of Harmful Language
The University of Michigan Library aims to describe library materials in a way that respects the people and communities who create, use, and are represented in our collections. Report harmful or offensive language in catalog records, finding aids, or elsewhere in our collections anonymously through our metadata feedback form. More information at Remediation of Harmful Language.
Accessibility
If you are unable to use this file in its current format, please select the Contact Us link and we can modify it to make it more accessible to you.