Regarding the choice of cost function in adaptive control - squared error vs absolute error I did search the question database regarding this question, and although one or two questions came close, they didn't really address my specific question.

In adaptive control based on minimizing tracking error (e.g. between plant and model), the designer is free to chose a cost function. More often than not the cost is selected as a function of the squared error.

But I've found in some practical applications that I can achieve a more robust controller by using absolute error. I understand that absolute error provides a more uniform weighting on the size of the error, and I suspect the squared error tends to 'wind-up' the adaptive controller with initially large errors. But I'm not sure how to show this in a generalized way. So I have two questions regarding this:


Is there perhaps a simple analysis that can demonstrate the stability characteristics between absolute and squared error choices in the cost function?
Any references on the matter?
This is an interesting question since both squared error and absolute error are convex functions, so they are both going to give the optimal solution when minimized. My intuition is that the $\ell_2$-norm (sum of squared values of error) converges to zero more quickly than the $\ell_1$-norm (sum of absolute values of error) when the search direction is right. For example, if the error is $\epsilon = y - Ax$. If $y$ is very close to $Ax$ and $0 &lt; |y-Ax| \leq 1$, then $\epsilon^2$ is much smaller than $|\epsilon|$. Similarly, if $|y-Ax| &gt; 1$, then $\epsilon^2 &gt; |\epsilon|$. So the $\ell_2$ norm penalizes large errors more, and small errors less than the $\ell_1$ norm. This can also be understood by looking at the graphs of the two functions when $\epsilon \in \mathbb{R}$.

Of course the continuity of the derivative is another factor. The derivative of the absolute value technically does not exist at the minimum.