Simple model of car-following as a linear system of ODE

Question

I have the following simple model of car-following without delay:

$\ddot{x}_k(t)=\lambda[\dot{x}_{k-1}(t)-\dot{x}_k(t)],k=1,2,...,n. \quad\qquad(1)$

Where $\ddot{x}_k$ is the acceleration of the follower, $\dot{x}_{k-1}(t)-\dot{x}_k(t)$ is the relative velocity between the leading car and the follower car and $\lambda$ is the sensitivity.

How can I express the equation $(1)$ as the next linear system of ODE

$\dot{\mathbf{x}}(t)=A\mathbf{x}(t)+B\mathbf{\alpha}(t).$

Where $A$ is a matrix of $n\times n$ and $B$ is a matrix of $n\times m$.

Answer 1

You get $\mathbf {\dot x}=A\mathbf x+\mathbf b$ where $A=λ(-I+S)$ with $S$ is the sub-diagonal matrix with entries $1$ on the first sub-diagonal. And $\mathbf b=λx_0(t)\mathbf e_1$.

Set $n=4$, write out the equations, especially the first one, and transcribe into matrix form to confirm that pattern.