Equivalence of robust stabilization and robust performance via feedback

TL;DR

This paper demonstrates that robust performance in linear control systems can be analyzed through robust stability criteria using feedback transformations, with special attention to dynamic controllers and the importance of refined conditions.

Contribution

It clarifies the correct application of the performance-to-stabilization reduction principle for dynamic feedback controllers, incorporating rank and LMI conditions for accurate robust performance analysis.

Findings

Robust performance can be derived from robust stability criteria via feedback.

Naive application of the reduction principle can lead to uninteresting solutions.

Refined criteria involving rank conditions yield correct robust performance results.

Abstract

One approach to robust control for linear plants with structured uncertainty as well as for linear parameter-varying (LPV) plants (where the controller has on-line access to the varying plant parameters) is through linear-fractional-transformation (LFT) models. Control issues to be addressed by controller design in this formalism include robust stability and robust performance. Here robust performance is defined as the achievement of a uniform specified $L^{2}$-gain tolerance for a disturbance-to-error map combined with robust stability. By setting the disturbance and error channels equal to zero, it is clear that any criterion for robust performance also produces a criterion for robust stability. Counter-intuitively, as a consequence of the so-called Main Loop Theorem, application of a result on robust stability to a feedback configuration with an artificial full-block uncertainty operator added in feedback connection between the error and disturbance signals produces a result on robust performance. The main result here is that this performance-to-stabilization reduction principle must be handled with care for the case of dynamic feedback compensation: casual application of this principle leads to the solution of a physically uninteresting problem, where the controller is assumed to have access to the states in the artificially-added feedback loop. Application of the principle using a known more refined dynamic-control robust stability criterion, where the user is allowed to specify controller partial-state dimensions, leads to correct robust-performance results. These latter results involve rank conditions in addition to Linear Matrix Inequality (LMI) conditions.