I'm working on a linear time-varying system and need to discretize the system like:
\begin{equation} \dot{x}=\overbrace{\begin{bmatrix} 0 &v(t) &0\\ 0& 0 &v(t)\\ 0& 0 &0\\ \end{bmatrix}}^{A^C}x+\overbrace{\begin{bmatrix}0\\0\\1\end{bmatrix}}^{B^c}u \end{equation}
with $v(t)=v_{tk}+a_{tk}(t-tk)+\frac{1}{2}j_{tk}(t-tk)^2$, $j_{tk}$ ist the constant during $t\in[tk,tk+1]$
I have to find the $A^{D}$ and $B^{D}$, but I don't know how to handle with that. The normal solution is: \begin{equation} {A}_d={\phi}_k(T_s), {B}_d=\int_{0}^{T_s}{\phi}_k(\tau){B}_cd\tau\\ \end{equation} $\phi_k$=I+$A_c$t+$\frac{1}{2!}A_{c}^2t^2+...$; but I'm not sure it's going to be work and how to discretize the LTI problem. Pls help ..
Any help is much appreciated, Thanks!
The "normal" solution is only for time-invariant systems (as you said), but the idea is the same. So, find the solution at time $t+T$ as $$\begin{align} x(t+T) &= \phi(t+T,t_0) x(t_0) + \int_{t_0}^{t+T} \phi(t+T,\tau) B(\tau) u(\tau) d\tau \\ &= \phi(t+T,t) \phi(t,t_0) x(t_0) + \phi(t+T,t) \int_{t_0}^{t} \phi(t,\tau) B(\tau) u(\tau) d\tau + \int_{t}^{t+T} \phi(t+T,\tau) B(\tau) u(\tau) d\tau \\ &= \phi(t+T,t) x(t) + \int_{t}^{t+T} \phi(t+T,\tau) B(\tau) u(\tau) d\tau \end{align}$$ Assuming Zero-Order Hold and since $B$ is time invariant, we can rewrite these as $$x_{k+1} = A_k x_k + B_k u_k$$ where $$A_k := \phi(t_k+T,t_k) ~~ \text{and} ~~ B_k := \left( \int_{t_k}^{t_k+T} \phi(t_k+T, \tau) d\tau \right) B$$ In general, we cannot go further than that, but in your specific case you can obtain $\phi(\cdot)$ explicitly.