Stability analysis of degenerate gradient flows via the WKB approximation

In this note we present a powerful new approach to the analysis of a class of linear, degenerate gradient flow systems that frequently arise in adaptive control and system identification. This paper has three main contributions: 1) a stability theorem utilizing a non-integral variant of the persistence of excitation (PE) conditions on the input signal; 2) upper and lower bounds (also using the non-integral PE conditions) which are shown to be superior to those derived in a classical paper on degenerate flow; and 3) construction of a one-term asymptotic approximation that is shown to perform remarkably well when compared to the numerically integrated solution. At the heart of our results is an extension of the WKB method which we name the Iterative Tracking Diagonalization (ITD) procedure. It yields a condition sufficient to ensure exponential stability of the origin. The WKB method utilizes an asymptotic expansion which relies on the existence of a time scale hierarchy. If the time scale separation parameter is sufficiently small, a few iteration steps suffice to derive an accurate estimate for the time constants of exponential stability of the norm of the parameter error vector. An important feature of our stability theorem, bounds and approximations, is that they all involve an analytical treatment of time dependences.