A Matrix WKB Approach to Feedforward and Feedback Control

We apply a powerful new analytical approximation method, recently developed by the authors, to the design and analysis of feedforward and feedback control systems. This formalism employs a matrix version of the WKB expansion, which is an asymptotic approximation method familiar in quantum mechanics and classical continuum mechanics. Our matrix WKB formalism has proven remarkably useful in approximating and characterizing the long-term dynamics of systems of ODEs (both linear and nonlinear) when there exists a time scale hierarchy. In particular, the linear error dynamics encountered in the analysis and design of controllers for multi-input multi-output systems, is typically formulated as a first-order vector differential equation involving a time-dependent matrix. To illustrate our matrix WKB approach, we consider the feedforward and feedback control of the single link manipulator. The desired trajectory is assumed to vary sinusoidally with time. For sufficiently small expansion parameters, the closed form WKB approximants can be used to determine safe controller parameters. Given a specific time scale hierarchy, we use a theorem reported previously to partition the controller parameter space into three distinct regions in which the system is, respectively: exponentially stable, exponentially unstable, and undecided. The undecided region is a narrow strip about a computable transition hypersurface. This strip can be made progressively narrower by working to a high enough order in the WKB expansion. In the limit of infinitely small expansion parameters, the transition curve tends to the actual stability-instability boundary.