State-Space (SS) Formulation for Equations of Motion

[Linder88]

Homework Statement

The task is to write the following equations of motion as in equation (2) considering the inputs and outputs as in equation (3)
\begin{equation}
\begin{cases}
(I_b+m_bl_b^2)\ddot{\theta}_b=m_bl_bg\theta_b-m_bl_b\ddot{x}_w-\frac{K_t}{R_m}v_m+\bigg(\frac{K_eK_t}{R_m}+b_f\bigg)\bigg(\frac{\dot{x}_w}{l_w}-\dot{\theta}_b\bigg)\\
\bigg(\frac{J_w}{l_w}+l_wm_b+l_wm_w\bigg)\ddot{x}_w=-m_bl_bl_w\ddot{\theta}_b+\frac{K_t}{R_m}v_m-\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\bigg(\frac{\dot{x}_w}{l_w}-\dot{\theta}_b\bigg)
\end{cases}
\end{equation}

Homework Equations

Since this course is focused on control based State-Space (SS) models, we do now rewrite our EOM as
\begin{equation}
\begin{cases}
\dot{x}=Ax+Bu
Cx+Du
\end{cases}
\end{equation}
for oppurtune x, A, B, C and D. As for the input and output, assume for now
\begin{equation}
u=v_m\\
y=\theta_b
\end{equation}

The Attempt at a Solution

Equation (3) in (2)
\begin{equation}
\begin{cases}
\dot{x}=Ax+Bv_m\\
\theta_b=Cx+Dv_m
\end{cases}
\end{equation}

 

[Linder88]

Define $\alpha$ and $\beta$ as
\begin{equation}
\begin{cases}
\alpha=\frac{1}{\frac{J_w}{l_w}+l_wm_b+l_wm_w}\\
\beta=\frac{1}{l_b+m_bl_b^2}
\end{cases}
\end{equation}
Now substitute these two expressions into Equation (1) as in
\begin{equation}
\begin{cases}
\ddot{x}_w=\alpha\bigg(-m_bl_bl_w\ddot{\theta}_b+\frac{K_t}{R_m}v_m-\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\bigg(\frac{\dot{x}_w}{l_w}-\dot{\theta}_b\bigg)\bigg)\\
\ddot{\theta}_b=\beta\bigg(m_bl_bg\theta_b-m_bl_b\ddot{x}_w-\frac{K_t}{R_m}v_m+\bigg(\frac{K_eK_t}{R_m}+b_f\bigg)\bigg(\frac{\dot{x}_w}{l_w}-\dot{\theta}_b\bigg)\bigg)
\end{cases}
\end{equation}
Expand the parentheses
\begin{equation}
\begin{cases}
\ddot{x}_w=\alpha\bigg(-m_bl_bl_w\ddot{\theta}_b+\frac{K_t}{R_m}v_m-\bigg(\frac{K_eK_t}{R_Ml_w}\dot{x}_w-\frac{K_eK_t}{R_M}\dot{\theta}_b+\frac{b_f}{l_w}\dot{x}_w-b_f\dot{\theta}_b\bigg)\bigg)\\
\ddot{\theta}_b=\beta\bigg(m_bl_bg\theta_b-m_bl_b\ddot{x}_w-\frac{K_t}{R_m}v_m+\frac{K_eK_t}{R_Ml_w}\dot{x}_w-\frac{K_eK_t}{R_M}\dot{\theta}_b+\frac{b_f}{l_w}\dot{x}_w-b_f\dot{\theta}_b\bigg)
\end{cases}
\end{equation}
Collect coefficient in front of $x_w,\dot{x}_w,\ddot{x}_w,\theta_b,\dot{\theta}_b$ and $\ddot{\theta}_b$
\begin{equation}
\begin{cases}
\ddot{x}_w=\alpha\bigg(-\bigg(\frac{K_eK_t}{R_Ml_w}+\frac{b_f}{l_w}\bigg)\dot{x}_w+\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\dot{\theta}_b-m_bl_bl_w\ddot{\theta}_b+\frac{K_t}{R_M}v_m\bigg)\\
\ddot{\theta}_b=\beta\bigg(\bigg(\frac{K_eK_t}{R_Ml_w}+\frac{b_f}{l_w}\bigg)\dot{x}_w-m_bl_b\ddot{x}_w+m_bl_bg\theta_b-\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\dot{\theta}_b-\frac{K_t}{R_M}v_m\bigg)
\end{cases}
\end{equation}
Add the variables with second order derivatives to the left hand side
\begin{equation}
\begin{cases}
\ddot{x}_w+m_bl_bl_w\ddot{\theta}_b=\alpha\bigg(-\bigg(\frac{K_eK_t}{R_Ml_w}+\frac{b_f}{l_w}\bigg)\dot{x}_w+\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\dot{\theta}_b+\frac{K_t}{R_M}v_m\bigg)\\
\ddot{\theta}_b+m_bl_b\ddot{x}_w=\beta\bigg(\bigg(\frac{K_eK_t}{R_Ml_w}+\frac{b_f}{l_w}\bigg)\dot{x}_w+m_bl_bg\theta_b-\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\dot{\theta}_b-\frac{K_t}{R_M}v_m\bigg)
\end{cases}
\end{equation}
Express in State-Space form
$$
\begin{pmatrix}
\dot{x}_w\\
\ddot{x}_w+m_bl_bl_w\ddot{\theta}_b\\
\dot{\theta}_b\\
\ddot{\theta}_b+m_bl_b\ddot{x}_w
\end{pmatrix}
=
\begin{pmatrix}
0&1&0&0\\
0&-\alpha\bigg(\frac{K_eK_t}{R_Ml_w}+\frac{b_f}{l_w}\bigg)&0&\alpha\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\\
0&0&0&1\\
0&\beta\bigg(\frac{K_eK_t}{R_Ml_w}\bigg)&\beta m_bl_bg&-\beta\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)
\end{pmatrix}
\begin{pmatrix}
x_w\\
\dot{x}_w\\
\theta_b\\
\dot{\theta}_b
\end{pmatrix}
+
\begin{pmatrix}
0\\
\alpha\frac{K_t}{R_M}\\
0\\
-\beta\frac{K_t}{R_M}
\end{pmatrix}
v_m
$$