Multidimensional transform methods for control of uncertain time-varying systems

Sanjay Lall, Control and Dynamical Systems, Caltech

A necessary extension of robust control methods for analysis of general uncertain systems is the ability to compute robustness margins for dynamical systems subject to perturbations along specified trajectories. When nonlinearities are modelled as integral quadratic constraints (IQCs), or with sector and polytopic bounds, this problem reduces to a linear time-varying (LTV) control problem, subject to uncertainty with special structure.

Previous methods for robust analysis and synthesis of LTV systems have focussed on Riccati equations. In comparison, uncertain time-invariant problems have been formulated to take advantage of convex methods, and this has enabled analysis of IQC and other uncertainty structures. These methods have taken advantage of the frequency domain for time-invariant systems, and the corresponding mu problem structure.

In this talk, we will discuss new developments in the theory of LTV systems, focussing on the construction of a new multidimensional transform with properties similar to that of frequency. In particular, we will show that using this transformation, LTV problems are strongly connected to mu-theory. One of the advantages of this framework is that there is a formal correspondence between LTI and LTV problems, which indicates potential for direct generalization of known LTI robust control results, including model reduction and recent robust synthesis techniques.