# [SOLVED]State equations

#### dwsmith

##### Well-known member
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}

Last edited:

#### dwsmith

##### Well-known member
We can write this as one differential equation:
$m_1m_2\ddddot{w} + (m_1b_2 + m_2b_1)\dddot{w} + (k(m_1 + m_2) + b_1b_2)\ddot{w} + k(b_1 + b_2)\dot{w} = m_2\ddot{u} + b_2\dot{u} + ku$
If the LHS had only a 3rd time derivative and the RHS had only a first time derivative, I could follow the (harder) labeled example here.
But I have tried to follow that idea by setting up the q derivatives as
\begin{alignat}{2}
q_1 &= w\\
q_2 &= \dot{w}\\
q_3 &= \ddot{w}\\
q_4 &= \dddot{w} - m_2\dot{u} - b_2u\\
\dot{q}_1 &= \dot{w} &&={} q_2\\
\dot{q}_2 &= \ddot{w} &&={} q_3\\
\dot{q}_3 &= \dddot{w} &&={} q_4\\
\dot{q}_4 &= \ddddot{w} -m_2\ddot{u} - b_2\dot{u}
\end{alignat}
However, this didn't seem to work. Should the q's be setup differently? Or am I not implementing this correctly?
$- (m_2^2b_1 + m_1m_2b_2)\dot{u} - (m_2b_1b_2 + m_1b_2^2)u - (m_2b_1 + m_1b_2)q_4$
since there is a $$\dot{u}$$ present which came from
$\dddot{w}(m_2b_1 + m_1b_2) = (m_2\dot{u} + b_2u + q_4)(m_2b_1 + m_1b_2).$
In the harder example, it says "The method has failed because there is a derivative of the input on the right hand, and that is not allowed in a state space model." This cause the concern with $$\dot{u}$$ in $$\dot{q}_4$$. Thus, I am lead to believe I need a slightly different setup for this problem.