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@Arnfinn sorry I missed your comments so far back. Yes I have Ioannou & Sun's book. Petros was my advisor at USC and I know Jing through the ACC. Both cost functions can be shown to be stable through Lyapunov analysis, but apart from stability, still struggling to learn why one choice might be better than the other for other reasons than stability. Nothing in Petros's book discusses this unfortunately.
@robertbristow-johnson since my post I've given more thought myself and you are right, choice of mapping does matter. If the poles are complex and you map using a forward rectangular approximation, indeed the poles can shift outside the unit circle. I'm still not sure, but I don't think aliasing ever occurs. I believe any sense of filtering just ceases when the poles far exceed sampling frequency.
I purposely didn't mention method to keep the question general, but for the example and my simple I used a backward rectangular approximation for integration for discretation
@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.
@Fat32 Yes I agree with your last statement. And thanks for pointing out the conceptual asymmetry. I agree there as well. And in the case of the impulse response you are also trying to wash out signal (input or output) noise that corrupts the measurement. For signal deconvolution you have to somehow deal with un-modeled dynamics.
@robertbristow-johnson yes - basically that's it. So if I filter $z$ and $phi$ with the same filter - but such that it does not step on the dynamics of the model, then I should always get the same $\theta$ regardless of the filter poles - right?
@robertbristow-johnson No - least squares. I have a linear parametric model $z = \phi \theta^T$ and I'm estimating the parameter vector $\theta$ using pseudoinverse.
One thing I've never been sure about. People commonly use the kinematic model, regardless of the actual physics involved as the KF model. So what does this mismatch in modeling lead to? Could that explain the issue you see?
