Block-structured Uncertainty Research Articles

Robust synthesis for linear parameter varying systems using integral quadratic constraints

A robust synthesis algorithm is developed for a class of uncertain, linear parameter varying (LPV) systems. The uncertain system is described as an interconnection of a nominal LPV system and a block structured uncertainty. The nominal part is a “gridded” LPV system with state matrices that are arbitrary functions of the parameter. The input/output behavior of the uncertainty is described by integral quadratic constraints (IQCs). The robust synthesis problem leads to a non-convex optimization. The proposed algorithm is a coordinate-wise descent similar to the well-known DK iteration for μ synthesis. It alternates between an LPV synthesis step and an IQC analysis step. Both steps can be efficiently solved as semidefinite programs. The derivation of the synthesis algorithm is less obvious for LPV systems as compared to its LTI counterpart due to the lack of a valid frequency response interpretation. The main contribution is the construction of the iterative synthesis algorithm using time domain dissipation inequalities and a scaled system analogous to that appearing in μ synthesis. It is shown that the proposed algorithm ensures that the robust performance is non-increasing at each iteration step. The effectiveness of the proposed method is demonstrated on a simple numerical example.

Structured matrix norms for robust stability and performance with block-structured uncertainty

In this paper we introduce new lower and upper robust stability bounds for structured uncertainty involving arbitrary spatial norms. Specifically, we consider a norm-bounded block-structured uncertainty characterization wherein the defining spatial norm is not necessarily the maximum singular value. This new uncertainty characterization leads to the notion of structured matrix norms for characterizing the allowable size of the nominal transfer function for robust stability. The lower and upper bounds are specialized to specific matrix norms including Holder, unitarily invariant, and induced norms to provide conditions for robust stability with several different characterizations of plant uncertainty. One of the key advantages of the proposed approach over the structured singular value is the reduction is computational complexity gained by directly addressing a given uncertainty characterization without having to transform it to a spectral-norm type characterization. Finally, we introduce a performance block within the structured matrix norm framework to address robust performance in the face of structured uncertainty.

Structured matrix norms for real and complex block-structured uncertainty

New upper bounds for robust stability are developed for block-structured real and complex uncertainty involving arbitrary spatial norms. Specifically, norm-bounded, block-structured uncertainty is considered wherein the defining norm is not necessarily the maximum singular value. In the case where the defining norm on the uncertainty characterization is set equal to the spectral norm, the resulting upper bounds generalize upper bounds for mixed-μ by permitting the treatment of nondiagonal real uncertain blocks as well as accounting for internal matrix structure in the uncertainty. The overall framework provides for a considerably simplified proof of the standard mixed-μ bound.

On the Stabilization of Uncertain Linear Systems via Bound Invariant Lyapunov Functions

This paper considers the problem of stabilizing a class of systems with an arbitrarily large uncertainty bounding set. In particular, the stabilization of linear uncertain systems via a bound-invariant Lyapunov function is examined. Necessary and sufficient conditions for the existence of a bound invariant Lyapunov function are derived for systems with block structured uncertainty and a class of systems with scalar linear uncertainty.