State-Space (SS) models

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1. Nov 17, 2015

Linder88

1. The problem statement, all variables and given/known data
The task is to write the following equations of motion as in equation (2) considering the inputs and outputs as in equation (3)

\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}

2. Relevant equations
Since this course is focused on control based State-Space (SS) models, we do now rewrite our EOM as

\begin{cases}
\dot{x}=Ax+Bu
Cx+Du
\end{cases}

for oppurtune x, A, B, C and D. As for the input and output, assume for now

u=v_m\\
y=\theta_b

3. The attempt at a solution
Equation (3) in (2)

\begin{cases}
\dot{x}=Ax+Bv_m\\
\theta_b=Cx+Dv_m
\end{cases}

2. Nov 22, 2015

Staff: Admin

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Nov 25, 2015

Linder88

Define $\alpha$ and $\beta$ as

\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}

Now substitute these two expressions into Equation (1) as in

\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}

Expand the parentheses

\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}

Collect coefficient in front of $x_w,\dot{x}_w,\ddot{x}_w,\theta_b,\dot{\theta}_b$ and $\ddot{\theta}_b$

\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}

Add the variables with second order derivatives to the left hand side

\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}

Express in State-Space form
$$\begin{pmatrix} \dot{x}_w\\ \ddot{x}_w+m_bl_bl_w\ddot{\theta}_b\\ \dot{\theta}_b\\ \ddot{\theta}_b+m_bl_b\ddot{x}_w \end{pmatrix} = \begin{pmatrix} 0&1&0&0\\ 0&-\alpha\bigg(\frac{K_eK_t}{R_Ml_w}+\frac{b_f}{l_w}\bigg)&0&\alpha\bigg(\frac{K_eK_t}{R_M}+b_f\bigg)\\ 0&0&0&1\\ 0&\beta\bigg(\frac{K_eK_t}{R_Ml_w}\bigg)&\beta m_bl_bg&-\beta\bigg(\frac{K_eK_t}{R_M}+b_f\bigg) \end{pmatrix} \begin{pmatrix} x_w\\ \dot{x}_w\\ \theta_b\\ \dot{\theta}_b \end{pmatrix} + \begin{pmatrix} 0\\ \alpha\frac{K_t}{R_M}\\ 0\\ -\beta\frac{K_t}{R_M} \end{pmatrix} v_m$$

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