I have heard that LQR controllers can not only have estimation, but prediction too.
Let's say that we have a discrete state space system:
$$ x(k+1) = Ax(k) + Bu(k) \\ y(k) = Cx(k) + Du(k) $$
If we count $k = 1, 2, 3, \dots , n$
Then our state space model will be: $$ x(2) = Ax(1) + Bu(1) \\ y(1) = Cx(1) + Du(1) \\ x(3) = Ax(2) + Bu(2) \\ y(2) = Cx(2) + Du(2) \\ x(4) = Ax(3) + Bu(3) \\ y(3) = Cx(3) + Du(3) \\ \vdots \\ x(n+1) = Ax(n) + Bu(n) \\ y(n) = Cx(n) + Du(n) $$
If $$x(2) = Ax(1) + Bu(1) \\ y(1) = Cx(1) + Du(1)$$
Then we kan say that :
$$x(3) = A[Ax(1) + Bu(1)] + Bu(2) \\ y(2) = C[Ax(1) + Bu(1)] + Du(2) $$
And we simplify:
$$x(3) = A^2x(1) + ABu(1) + Bu(2) \\ y(2) = CAx(1) + CBu(1) + Du(2) $$
Then we add another one: $$x(4) = Ax(3) + Bu(3) \\ y(3) = Cx(3) + Du(3) $$
Then we kan say that :
$$x(4) = A[A^2x(1) + ABu(1) + Bu(2)] + Bu(3) \\ y(3) = C[A^2x(1) + ABu(1) + Bu(2)] + Du(3) $$
We can simplify
$$x(4) = A^3x(1) + A^2Bu(1) + ABu(2) + Bu(3) \\ y(3) = CA^2x(1) + CABu(1) + CBu(2) + Du(3) $$
So my conclusion is: If I remember the inputs $u(k)$ for 10 samples. I can predict the future values of states $x(k+1)$? Is this the method for predict future states from a LQR controller? Or am I totaly wrong?