Give the linear time invariant system: $$\dot x = Ax + Bu, \quad x(0) = x_0$$ $$y = Cx$$
How would one solve for $x_0$, $y(t)$, or $u(t)$ given 2 of the 3 unknowns?
Clarity from comments: Let's say $(A,C)$ is not completely observable as defined by the observability matrix. ie. if $A \in \mathbb{R}^{3\times 3}$ and $C \in \mathbb{R}^{1\times 3}$ then $\text{rank}(\theta)=2$, for example
For example, say I am looking for $x_0$: therefore, I know $y(t)$ and $u(t)$ which are defined over some closed interval $[0,t]$.
The state output response of an LTI system is simply: $$x(t) = e^{At}x_0 + \int_0^te^{A(t-\tau)}Bu(\tau)d\tau$$ $$y(t) = Ce^{At}x_0 + \int_0^tCe^{A(t-\tau)}Bu(\tau)d\tau$$
Taking the second equation, $y(t)$ and $u(t)$ would be given along with $A,B,C$ and $t$, so do I simply solve for $x_0$? If so, how do you isolate $x_0$? That is, do you multiply by the inverse of $Ce^{At}$ after integrating the last term and moving it to the LHS? Am I missing something?
Thanks!