Assume that we have a discrete state space model:
$$x(k+1) = Ax(k) + Bu(k)\\ y(k) = Cx(k) + Du(k)$$
And we want to use optimization to minimize this cost function.
$$J =\sum_{k=0}^{n}(x_k^TQx_k + u_k^TRu_k)$$
Where the states $x_k$ can be found from the augmented state space model.
$$\begin{bmatrix} x_0\\ x_1\\ x_2\\ x_3\\ \vdots\\ x_{n} \end{bmatrix} = \begin{bmatrix} A\\ A^2\\ A^3\\ A^4\\ \vdots\\ A^{n-1} \end{bmatrix}x_0+\begin{bmatrix} B & 0 & 0 & 0 & \dots & 0\\ AB & B & 0 & 0 & \dots & 0\\ A^2B & AB & B & 0 & \dots& 0\\ A^3B & A^2B & AB & B & \dots& 0\\ \vdots & \vdots & \vdots &\ddots &\ddots & 0\\ A^{n-2}B & A^{n-3}B & A^{n-4}B & A^{n-5}B & \dots & B \end{bmatrix} \begin{bmatrix} u_0\\ u_1\\ u_2\\ u_3\\ \vdots\\ u_n \end{bmatrix}$$
But what if I have a nonlinear state space model:
$$x(k+1) = A(x(k), u(k)) + B(x(k), u(k))\\ y(k) = C(x(k), u(k)) + D(x(k), u(k))$$
How can I create the augmented state space model, if I want a perfect linearization?
Because, when I linearize a state space model:
$$\dot x = f(x, u)$$
I need to linezarise the model in (0,0) by using first talyor series.
$$f_L(x,a) = f(a) + \dot f(a,u) (x-a)$$
Where $f(a) = 0$ and $\delta = (x-a)$
So the new linear state space model will be:
$$\dot x \delta = Ax\delta + B u\delta$$
Where $A = \dot f(a,u)$.
But what if I not use first taylor series? How can I then create the nonlinear augmented state space model?
Can I create the nonlinear augmented state space model like this:
$$\begin{bmatrix} x_0\\ x_1\\ x_2\\ x_3\\ \vdots\\ x_{n} \end{bmatrix} = \begin{bmatrix} \bar A\\ \bar A^2\\ \bar A^3\\ \bar A^4\\ \vdots\\ \bar A^{n-1} \end{bmatrix}+\begin{bmatrix} B & 0 & 0 & 0 & \dots & 0\\ AB & B & 0 & 0 & \dots & 0\\ A^2B & AB & B & 0 & \dots& 0\\ A^3B & A^2B & AB & B & \dots& 0\\ \vdots & \vdots & \vdots &\ddots &\ddots & 0\\ A^{n-2}B & A^{n-3}B & A^{n-4}B & A^{n-5}B & \dots & B \end{bmatrix} $$
Where $$\bar A = A(x(k_0), u(k))$$ $$A = A(x(k_0), u(k))$$ $$B = B(x(k_0), u(k))$$
?