I have a state space system that I want to apply LQR to, however, the parameters of the time varying state matrix ($\mathbf{A}(t)$) depend on the input vector ($\mathbf{u}(t)$). Are there any known workarounds for this situation? Something like introducing delay or a change of variables? I've written out the equation below:
$$ x(t+1) = \big[(1-u(t)w)^2 \big]x(t) + \big[w u(t) \big]u(t)$$
$w$ is just a constant. $u(t)$ and $x(t)$ are always positive.