I trying to handle delays in a model who is poorly damped but I haveing som issues to estimate its parameters due to the delay.
Assume that we got a state space model, which is poorly damped:
$$x(k+1) = Ax(k) + Bu(k) \\ y(k) = Cx(k) + Du(k)$$
We find those matricies:
$$F = \begin{bmatrix} CA\\ CA^2\\ CA^3\\ \vdots \\ CA^{N_p} \end{bmatrix} , \Phi = \begin{bmatrix} CB &0 &0 &\cdots & 0\\ CAB & CB & 0 & \cdots & 0\\ CA^2B& CAB & 0 &\cdots &0 \\ \vdots & \vdots & \vdots & \vdots &\vdots \\ CA^{N_p-1}B & CA^{N_p-2}B & CA^{N_p-3}B & \cdots & CA^{N_p-N_c}B \end{bmatrix}$$
$$\bar R = r_{\omega} I_{N_p x N_p}$$
$$\left.\begin{matrix} R_s = \begin{bmatrix} 1\\ 1\\ 1\\ 1\\ \vdots \\ 1 \end{bmatrix} \end{matrix}\right\} N_p$$
Where $r_{\omega}$ is our tuning parameters e.g 0.0001 and $r(k)$ is our reference vector, or parameter.
When we can use this formulula were the first column of $U$ is our input signal.
$$U = (\Phi^T \Phi + \bar R)^{-1} \Phi^T(R_s r(k) - Fx(k)))$$
This is Generalized Predictive Control in state space form.
An arbitary simulation results may looks like this:
Where the green is our reference, and black is our output of the model.
The first element of input $U$ looks like this:
I'm using Recursive Least Square to estimate its parameters, but the problem I got is when the model is delayed.
Question:
Is there a way to handle delays in predictive control?