Adaptive Control Metrics via the Windowed Laplace Transform

Abstract

In this paper, we introduce and describe a new type of Joint Time-Frequency Analysis (JTFA), which we call the Windowed Laplace Transform (WLT) method. In addition to general signal processing applications, this new method can be used to extend Nyquists’s classical stability analysis criteria and stability metrics, from LTI to LTV (Linear, TimeVarying) systems a step towards analyzing the more general case of NLTV (Non-Linear, Time Varying) systems. We apply WLT to two LTV models: A simple, exactly solvable 1D toy model, and an MRAC (Model Reference Adaptive Control) innerloop aerodynamic model. The latter is linearized about its reference plant trajectory and ideal neural-network weights. In either model, we assume no physical-layer delays in the system (plant or plantplus-controller), but allow the mathematical (virtual) injection of time delays in order to formulate stability margin metrics. We identify two such metrics: the Extended Phase Margin (EPM), which extends Nyquist’s Phase Margin metric to the LTV case; and the more physically meaningful Time-Delay Margin (TDM), which is defined as the maximal virtually-injected time delay before the EPM metric is driven down to zero. These new ‘quasi-LTI’ stability metrics can be visualized by viewing frames of a time-dependent Nyquist contour as befits a JTFA approach. What is more, it is possible to derive systematically-improvable analytical approximants for our novel EPM and TDM stability margin metrics. In this paper, we analyze a numerical example for each of the two aforementioned LTV models. In subsequent publications, we will present analytical and semi-analytical approximants, proofs of several theorems guaranteeing asymptotic exponential stability, and an extension of our analysis to fully NLTV MRAC schemes all based upon the WLT method and the new EPM and TDM metrics.