The transfer function for the charge amplifier is
T(s)=\frac{Gs}{\tau s+1}
and the Transfer function for the mechanical system is
T(s)=\frac{1}{ms^2+bs+k}
When we take the inverse Laplace transform of these equations multiplied together we get
\begin{matrix} \ddot{x}(t)=-\frac{k}{m}x(t)-\frac{b}{m}\dot{x}(t)+F \\ \dot{v}(t)=\frac{1}{\tau}v(t)+G\dot{x}(t) \end{matrix}
Without states being x(t), \dot{x}(t),v(t) the state-space model of the accelerometer becomes
\begin{bmatrix} \dot{x}(t)\\ \ddot{x}(t)\\ \dot{v}(t) \end{bmatrix}= \begin{bmatrix} 0&1&0\\ -\frac{k}{m}&-\frac{b}{m}&0\\ 0&G&-\frac{1}{\tau} \end{bmatrix} \begin{bmatrix} x(t)\\ \dot{x}(t)\\ v(t) \end{bmatrix}+ \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}F
y(t)= \begin{bmatrix} -\frac{k}{m}C&-\frac{b}{m}C&0 \end{bmatrix} \begin{bmatrix} x(t)\\ \dot{x}(t)\\ v(t) \end{bmatrix}+CF
- candyAccelerometerMode
General General Accelerometer State-Space Model
[[Piezoelectric Accelerometers]]
[[State-Space Model]]