Persistent Excitation is Unnecessary for On-line Exponential Parameter Estimation: A New Algorithm that Overcomes this Obstacle

In this paper, we prove that it is possible to estimate online the parameters of a classical vector linear regression equation $Y=\Omega \theta$, where $Y \in \mathbb{R}^n,\;\Omega \in \mathbb{R}^{n \times q}$ are bounded, measurable signals and $\theta \in \mathbb{R}^q$ is a constant vector of unknown parameters, even when the regressor $\Omega$ is not persistently exciting. Moreover, the convergence of the new parameter estimator is global and exponential and is given for both continuous-time and discrete-time implementations. As an illustration example, we consider the problem of parameter estimation of a linear time-invariant system, when the input signal is not sufficiently exciting, which is known to be a necessary and sufficient condition for the solution of the problem with the standard gradient or least-squares adaptation algorithms.