# 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:

1. Is there perhaps a simple analysis that can demonstrate the stability characteristics between absolute and squared error choices in the cost function?

2. Any references on the matter?

• I'm not a controls guy so my opinion matters little. However, I wonder if squared error is typically used because of its obvious connection to least-squares based methods. One nice feature of the squared error cost function versus absolute error: the squared error cost function's derivative with respect to the error is continuous everywhere. The absolute error cost function's derivative is discontinuous near $\text{error} = 0$. I could imagine that this could have implications for stability and/or tracking error. – Jason R Sep 8 '16 at 15:27
• What form does your cost-function have? The standard least-squares cost-function would typically be $J(\theta) = \int_{0}^{t} \left( y(\tau) - \theta(t) u(\tau) \right)^{2} \textrm{d} \tau$. Are you suggesting $J(\theta) = \int_{0}^{t} \left| y(\tau) - \theta(t) u(\tau) \right| \textrm{d} \tau$ instead? – Arnfinn Sep 14 '16 at 10:40
• @Arnfinn my question - more general than specific, but the specific practical work that generated the question is on an application of model reference adaptive control where the plant is considered a scalar and the model is a first order lag. In this application the error was the difference between the output of the closed loop plant using an integrator with adjustable gain, and the model output. The cost I examined was either the square or absolute value of this error. So I guess the answer to your question - yes. But I'm using a gradient (MIT-like) minimization rather than least squares. – docscience Sep 14 '16 at 13:36
• Do you have the book by Ioannou and Sun? amazon.com/Robust-Adaptive-Control-Electrical-Engineering/dp/… – Arnfinn Sep 14 '16 at 23:03
• At any rate, this is a non-linear problem as you probably know, and the general frameworks for stability and convergence analysis would be Lyapunov stability analysis and/or the Grönwall–Bellman lemma... – Arnfinn Sep 14 '16 at 23:06