Start with the simplest possible model: $$ y(t)=\theta\,u(t) $$ At first, estimating \(\theta\) looks trivial: $$ \hat{\theta}=\frac{y}{u} $$ But in practice, this becomes dangerous near zero crossings and in the presence of measurement noise. Least Squares solves this by accumulating information over time. Recursive Least Squares turns that idea into a real-time estimator. And then comes the interesting part: Instead of using the full time-varying inverse covariance \(P(t)\), many adaptive control laws replace it with a constant gain \(\gamma\). You give up part of the estimator’s statistical optimality — but gain much simpler error dynamics: $$ \dot{\phi}=-\gamma u^2(t)\phi $$ That makes the closed-loop stability analysis far more transparent. But even a perfectly stable estimator cannot learn if the system provides no information. That is where **Persistent Excitation** becomes essential: No sufficiently rich input → no new information → no guaranteed parameter convergence. In short: **Stability does not guarantee learning. Learning requires excitation.** Swipe through the carousel for the full path from direct division → Least Squares → continuous-time RLS → gradient estimation → Persistent Excitation. #ControlTheory #AdaptiveControl #SystemIdentification #RLS #LeastSquares - @governing.science