# Find the instance when discrete time model has a specific value

I have the following model: $$x[k+1] = Ax[k] + Bu[k]$$ $$y[k] = x[k]$$ where $x \in \mathbb{R}$ is the state, $A \in \mathbb{R}$, $u \in \mathbb{R}^4$ are the inputs and $B \in \mathbb{R}^4$ and $y[k]$ is the output.

From theory this can be solved using: $$y[k] = A^k x[0] + \sum_{i=0}^{k-1} A^{k-1-i}Bu[i]$$

How can I find the time instance $k$ when $y[k]$ has a specific value?

• @Marc The reason why, in general, you can't solve for $k$, is because for a given input and initial condition the output $y$ might only be able to obtain values from a fixed (finite) set of values. For example $x[0]=0$, $u[k]=\vec{0}$, can only obtain $y[k]=0$. Sep 20, 2016 at 1:42