On the stability of continuous-time adaptive controllers for pure delay systems
The paper addresses the problem of analyzing the stability properties of the one-dimensional retarded differential-difference equation (RDDE) that arises when an adaptive controller is applied to a system described by a pure delay, i.e. \(y(t)=ku(t-h)\), \(h>0\) and k an unknown positive gain. The RDDE in question is \[ {\tilde \theta}\dot {\;}=-\sigma {\tilde \theta}(t)-\gamma B(t){\tilde \theta}(t-h)+f(t) \] where \(\gamma >0\), \(\sigma >0\) and B(t), f(t) are uniformly bounded functions of t. Three different results are given. Firstly, under the assumption that \(B(t+T)=B(t)\) with \(h=mT\) for an integer n, necessary and sufficient conditions for stability are established. The technical device used for the proof is Floquet theory. Secondly, using averaging theory, a stability-instability boundary is derived for sufficiently small values of \(\gamma\). The result applies to B(t) almost periodic. Finally, we introduce a modified adaptation and control laws which gives a B(t) of the form \[ B(t)=K(t)y^ 2[t-h(t)]/\{1+y^ 2[t-h(t)]\}, \] with K,h: \({\mathbb{R}}_+\to {\mathbb{R}}\). It is shown that the equilibrium of the RDDE in this case is globally stable. Two alternative proofs, one based on Lyapunov functionals and the second one using Razumikhin-type theorems, are given.
one-dimensional retarded differential-difference equation
scientific article; zbMATH DE number 3987166