Nonlinear averaging theorems, and the determination of parameter convergence rates in adaptive control (Q1088626)

The system of differential equations \[ (1)\quad \dot x=\epsilon f(t,x,y),\quad (2)\quad \dot y=A(x)+\epsilon g(t,x,y),\quad x(0)=x_ 0,\quad y(0)=y_ 0 \] is considered. The existence of a function \(f_{av}(x)\) is assumed such that for all t, \(T\geq 0\), \[ | \frac{1}{T}\int^{t+T}_{t}f(\tau,x,0)d\tau -f_{av}(x)| \leq \gamma (T) \] where \(\gamma\) (T)\(\to 0\) as \(T\to \infty\). The matrix A(x) is uniformly exponentially stable for all \(x\in B_ n\), and some other assumptions are made. There exists a strictly increasing function \(\psi\) (\(\epsilon)\), \(\psi (0)=0\), such that, given \(T\geq 0\), \[ | x(t)- x_{av}(t)| \leq \psi (\epsilon)b_ T \] for some \(b_ T\), \(\epsilon_ T>0\) and for all \(t\in [0,T/\epsilon]\), \(0<\epsilon <\epsilon_ T\). Here \(x_{av}(t)\) is the solution of the averaged system \[ (3)\quad \dot x_{av}=f_{av}(x_{av}),\quad x_{av}(0)=x_ 0. \] If (3) is an exponentially stable system, then the original system (1), (2) is exponentially stable for \(\epsilon >0\) sufficiently small. These results are applied, to the Narendra-Valavani model reference adaptive control algorithm [\textit{K. S. Narendra} and \textit{L. S. Valavani}, IEEE Trans. Autom. Control AC-23, 570-583 (1978; Zbl 0389.93025)]. The results of some computer experiments are given.