A Dialog-Oriented and Gradient-Based Stability Margin in Uncertain Systems

Abstract

Gradients and matricial gradients for optimally increasing the stability margin and the admissible uncertainty of a dynamic system are the targets of this presented article. To design a dynamic system, the gradients are used in a dialog between a system scientist and gradient-based computer support. The stability margin is derived for output state controllers, including regular state controllers. The resonant frequency and the damping factors are investigated as a direct function of the maximum admissible uncertainty. The resulting gradients are extended to observer-assisted controllers, to minimum-order observers, and to dynamic-output state controllers.

Robustness Gradients for Discrete-Time Control Systems

“…This formula (Weinmann, 2005) presents the gradient with respect to K y for the upper bound of the admissible unstructured uncertainty. …”

“…In a former article (Weinmann, 2005), continuous-time systems were taken into consideration. We do not choose eigenvalues of the closed-loop system or weighting matrices primarily and afterwards discuss the result whether the choice was adequate or not.…”

Stability Margin and Spherical Uncertainty

“…In order to improve the stability margin, i.e., to enlarge h 0 , the controller can be adapted incrementally. Therefore, the gradient with respect to K y obeys to (Weinmann 2005) …”

“…(For single-input single-output systems, h 0 corresponds to q à of Tesi et al 1995. ) The stability margin h 0 of a nominal (unperturbed) multivariable dynamic system of nth order with the coefficients matrix A cl of the closed loop results from executing the minimization (Weinmann 2005(Weinmann , 2006 …”

Stability margin gradients using the polynomial matrix representation

“…The idea of incremental change of parameters to improve the stability margin was originally presented in (Weinmann, 2005) where the state-space representation is used. The sensitivity @ @K in (Thompson, 1995) is closely related to the matricial gradient in this paper.…”