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I am trying to find the state equations for a mass spring system.

I found the transfer function to be

\[

H(s) = \frac{X_1(s)}{F(s)} = \frac{m_2s^2 + b_2s + k}

{s\big[m_1m_2s^3 + (m_2b_1 + m_1b_2)s^2 +

(k(m_1 + m_2) + b_1b_2)s + k(b_1 + b_2)\big]}

\]

I found the transfer function from

\begin{align}

m_1\ddot{x}_1 &= F - b_1\dot{x}_1 - k(x_1 - x_2)\\

m_2\ddot{x}_2 &= - b_2\dot{x}_2 - k(x_2 - x_1)

\end{align}

So I am trying to find the state matrices \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), and \(\mathbf{D}\) where

\begin{align}

\dot{\mathbf{x}} &= \mathbf{A}\mathbf{x} + \mathbf{B}F\\

\mathbf{y} &= \mathbf{C}\mathbf{x} + \mathbf{D}F

\end{align}

The transfer function is extremely complicated though. How can I obtain the state matrices?

I do know what the matrices are, but I can't find obtain them:

\begin{align}

\mathbf{A} &=

\begin{bmatrix}

0 & 1 & 0 & 0\\

-\frac{k}{m_1} & -\frac{b_1}{m_1} & \frac{k}{m_1} & 0\\

0 & 0 & 0 & 1\\

\frac{k}{m_2} & 0 & -\frac{k}{m_2} & -\frac{b_2}{m_2}

\end{bmatrix}\\

\mathbf{B} &=

\begin{bmatrix}

0\\

\frac{1}{m_1}\\

0\\

0

\end{bmatrix}\\

\mathbf{C} &=

\begin{bmatrix}

1 & 0 & 0 & 0\\

0 & 0 & 1 & 0

\end{bmatrix}\\

\mathbf{D} &= \mathbf{0}

\end{align}

I found the transfer function to be

\[

H(s) = \frac{X_1(s)}{F(s)} = \frac{m_2s^2 + b_2s + k}

{s\big[m_1m_2s^3 + (m_2b_1 + m_1b_2)s^2 +

(k(m_1 + m_2) + b_1b_2)s + k(b_1 + b_2)\big]}

\]

I found the transfer function from

\begin{align}

m_1\ddot{x}_1 &= F - b_1\dot{x}_1 - k(x_1 - x_2)\\

m_2\ddot{x}_2 &= - b_2\dot{x}_2 - k(x_2 - x_1)

\end{align}

So I am trying to find the state matrices \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), and \(\mathbf{D}\) where

\begin{align}

\dot{\mathbf{x}} &= \mathbf{A}\mathbf{x} + \mathbf{B}F\\

\mathbf{y} &= \mathbf{C}\mathbf{x} + \mathbf{D}F

\end{align}

The transfer function is extremely complicated though. How can I obtain the state matrices?

I do know what the matrices are, but I can't find obtain them:

\begin{align}

\mathbf{A} &=

\begin{bmatrix}

0 & 1 & 0 & 0\\

-\frac{k}{m_1} & -\frac{b_1}{m_1} & \frac{k}{m_1} & 0\\

0 & 0 & 0 & 1\\

\frac{k}{m_2} & 0 & -\frac{k}{m_2} & -\frac{b_2}{m_2}

\end{bmatrix}\\

\mathbf{B} &=

\begin{bmatrix}

0\\

\frac{1}{m_1}\\

0\\

0

\end{bmatrix}\\

\mathbf{C} &=

\begin{bmatrix}

1 & 0 & 0 & 0\\

0 & 0 & 1 & 0

\end{bmatrix}\\

\mathbf{D} &= \mathbf{0}

\end{align}

Last edited: