I am to derive a recursive version of the following cost function and examine for which choice of D can we have a estimator windup

$V(\theta) = \frac{1}{2}\sum_{t=1}^n(y(t)-\phi(t)^T\theta)^2 + \theta^TD\theta$.

I do not really see if my answer is correct and was wondering if anyone can confirm if this is correct or not. Here is how I have done it. First I find the normal equations.

$\frac{dV(\theta)}{d\theta} = \frac{1}{2}\sum_{t=1}^n(-2y(t)\phi(t)^T+2\phi(t)^T\phi(t)\theta)+2D\theta$ = $0$

$ \sum_{t=1}^ny(t)\phi(t)^T=(\sum_{t=1}^n\phi(t)^T\phi(t)+2D)\theta$ <- normal equations

Then I want to derive a recursive form.

$\hat{\theta_n} = (\sum_{t=1}^n\phi_t^T\phi_t+2D)^{-1}\sum_{t=1}^n\phi_t^Ty_t$

I say that $R_n = \sum_{t=1}^n\phi_t^T\phi_t = R_{n-1} + \phi_n^T\phi_n$

$\hat{\theta_n} = (R_n+2D)^{-1}\sum_{t=1}^n(\phi_t^Ty_t)$

$(R_n+2D)\hat{\theta_n} = \sum_{t=1}^n\phi_t^Ty_t = \sum_{t=1}^{n-1}\phi_t^Ty_t+\phi_n^Ty_n=(R_{n-1}+2D)\hat\theta_{n-1}+\phi_n^Ty_n = (R_n-\phi_n^T\phi_n+2D)\hat\theta_{n-1}+\phi_n^Ty_n = (Rn+2D)\hat{\theta}_{n-1}+\phi_n^T(y_n-\phi_n\hat\theta_{n-1})$

Thus we get

$\hat{\theta_n} = \hat{\theta}_{n-1}+(R_n+2D)^{-1}\phi_n^T(y_n-\phi_n\hat{\theta}_{n-1})$

Does this look correct?

If so, I should now determine when a value of D causes estimator windup.

As I have understod it, you should examine when $\phi_t = 0$ and see when $P_t = (R_n+2D)^{-1} = P_{t-1} $ grows for a value of D but I can't see when that is the case?

Appreciate any help given!



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