calc_bode

This skill builds a state-space model from a mechanical system defined by mass (M), stiffness (K), and damping (C) matrices, and generates a frequency response (Bode Plot) using the python-control library.

Workflow

1. System Matrix Acquisition:

   • Extract the $M$ and $K$ matrices from a physical model or linearization tool.
   • Define the damping matrix $C$ as needed (typically proportional to stiffness or using modal damping).

2. State-Space Construction:

   • Define the state vector as $x = [q, v]^T$, where $q$ is displacement and $v$ is velocity.
   • Transform the system's equation of motion $M \ddot{q} + C \dot{q} + K q = F u$ into the standard state-space form $\dot{x} = Ax + Bu$:
   • $A = \begin{bmatrix} 0 & I \ -M^{-1}K & -M^{-1}C \end{bmatrix}$
   • $B = \begin{bmatrix} 0 \ M^{-1}F \end{bmatrix}$

3. Application of python-control:

   • Import the library: import control as ct.
   • Create the system object: sys = ct.ss(A, B, C, D).

4. IO Selection (for MIMO systems):

   • Use sys_siso = sys[output_idx, input_idx] to select the desired input/output channel.

5. Plotting and Analysis:

   • Generate the Bode plot: ct.bode_plot(sys_siso, ...).
   • Utilize flags such as Hz=True, dB=True, and deg=True.
   • Check stability if necessary via sys.poles().

Precautions

• In systems including inverted pendulums, such as the KAGRA top stage, open-loop poles may exist in the unstable region before applying stabilization control.
• Ensure frequency ranges are specified in angular frequency (rad/s) such as np.logspace(np.log10(start_hz * 2*np.pi), np.log10(end_hz * 2*np.pi), n_points), while specifying Hz=True in bode_plot.