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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!

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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.

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  • $\begingroup$ Thank you so much for your help! $\endgroup$ – Yunhua Hu Jan 15 at 8:57

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