I have a second-order vector differential equation
$$\mathbf{M}\ddot{\mathbf{x}}(t)+\mathbf{C}\dot{\mathbf{x}}(t)+\mathbf{K}\mathbf{x}(t)=\mathbf{P}\mathbf{u}(t)$$
in which the input vector is of the form
$$\mathbf{u}(t)=[r(t)~~r(t-\tau)]^T$$ where $r(t)$ is scalar and $\tau$ is a known delay.
Such systems appear in e.g. vehicle dynamics, where the front wheelset is excited by $r(t)$ and the rear wheelset is excited by the same input, after some delay that depends on the velocity of the vehicle and the wheelbase.
Is it at all possible to bring the above system to state-space, where the input is only $r(t)$? That is,
$$\dot{\mathbf{\xi}}(t)=\mathbf{A}\mathbf{\xi}(t)+\mathbf{B}r(t)\\ \mathbf{y}(t)=\mathbf{C}\mathbf{\xi}(t)+\mathbf{D}r(t)$$
As a possible solution may provide "strange" attributes to the matrices $\mathbf{B}$ and $\mathbf{D}$, could such a system be accordingly discretized using, say, zero-order hold?
Thanks in advance.
One way to discretize the system $$ \dot{x}(t)=Ax(t)+B_0u(t)+B_1u(t-h) $$
using a zero-order hold is to consider that the delay is a multiple of the sampling period. Let $T$ be the sampling period and $N$ be an integer such that $h=NT$.
Then, a discretized version of the system is simply given by
$$\begin{bmatrix} x_{k+1}\\ \hline u_{k}\\ u_{k-1}\\ u_{k-2}\\ \vdots\\ u_{k-N+1} \end{bmatrix}=\begin{bmatrix} e^{AT} & | & 0 & 0 & \ldots & 0 & \Psi(T)B_1\\ \hline 0 & | & 0 & 0 & \ldots & 0& 0\\ 0 & | & I & 0 & \ldots & 0& 0\\ 0 & | & 0 & I & \ldots & 0& 0\\ \vdots & | & \vdots & & \ddots & \vdots & \vdots\\ 0 & | & 0 & 0 & \ldots & I & 0 \end{bmatrix}\begin{bmatrix} x_{k}\\ \hline u_{k-1}\\ u_{k-2}\\ u_{k-3}\\ \vdots\\ u_{k-N} \end{bmatrix}+\begin{bmatrix} \Psi(T)B_0\\ \hline I\\ 0\\ 0\\ \vdots\\ 0 \end{bmatrix}u_k $$ where $\Psi(T)=\int_0^Te^{As}ds$.
Let me know if this is of any help and I will update the answer accordingly.