Motor-Pendulum-Control

Source code for learning DC motor position control and rotary inverted pendulum control using MATLAB/Simulink.

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0. Project Structure

Motor-Pendulum-Control/
├── docs/                         # Figures and reference materials
│   ├── DCmotor_pid_simulink.png
│   ├── DCmotor_PID_withoutFunction.png
│   ├── RotaryPendulum_simulink.png
│   └── RotaryPendulum_StateSpaceController.png
├── matlab/
│   ├── models/                   # Simulink models
│   │   ├── DCmotor_PID.slx
│   │   ├── InvertedPendulum_simonly.slx
│   │   └── MotorPD_simonly.slx
│   ├── scripts/                  # MATLAB parameter / setup scripts
│   │   ├── init_dc_motor.m
│   │   └── init_rotary_pendulum.m
│   └── results/                  # Simulation result plots
│       ├── DC_PID_Final.png
│       ├── DC_P_Control/
│       │   ├── Kp_0point1.png
│       │   ├── Kp_1point0.png
│       │   └── Kp_negative.png
│       ├── DC_PI_Control/
│       │   ├── BigKi.png
│       │   ├── SmallKi.png
│       │   └── unstableKi.png
│       ├── DC_PD_Control/
│       │   ├── case1.png
│       │   ├── case2.png
│       │   ├── case3.png
│       │   └── case4.png
│       └── RotaryPendulum/
│           ├── alpha_result.png
│           └── theta_result.png
├── .gitignore
└── README.md

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1. DC Motor Position Control (P/PI/PD/PID)

• Model files: matlab/models/DCmotor_PID.slx, matlab/models/MotorPD_simonly.slx
• Parameter script: matlab/scripts/init_dc_motor.m

System Model

Using the parameters $K$ and $\tau$ defined in init_dc_motor.m,
the first-order system from input voltage $u(t)$ to motor angular velocity $\omega(t)$ is

$$ \tau \frac{d\omega(t)}{dt} + \omega(t) = K,u(t) $$

$$ \frac{\Omega(s)}{U(s)} = \frac{K}{\tau s + 1} $$

To obtain motor position $\theta(t)$ as the output, add an integrator, giving

$$ \frac{\Theta(s)}{U(s)} = \frac{K}{s(\tau s + 1)} $$

Simulink Model

The controller and plant are implemented in a single Simulink model using standard blocks.

Simulation Results (PID Control)

P/PI/PD/PID control results are stored in the matlab/results folder.
The figure below shows an example of the final response with PID control:

Main cases:

• matlab/results/DC_P_Control: responses for different proportional gains $K_p$
• matlab/results/DC_PI_Control: stable/unstable cases for different integral gains $K_i$
• matlab/results/DC_PD_Control: several PD parameter cases

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2. Rotary Inverted Pendulum Control

• Model file: matlab/models/InvertedPendulum_simonly.slx
• Parameter script: matlab/scripts/init_rotary_pendulum.m

System Model

This project uses a linearized state-space model defined in init_rotary_pendulum.m.
Let the state be $x = [\theta,\ \dot\theta,\ \alpha,\ \dot\alpha]^\mathsf{T}$,
the input $u = V_m$, output $y = [\theta,\ \alpha]^\mathsf{T}$. Then

$$ \dot{x} = A x + B u,\qquad y = C x + D u $$

where

$$ J_\mathrm{eq} = m_p L_r^2 + J_r,\qquad J_{p,\mathrm{eq}} = J_p + \tfrac{1}{4} m_p L_p^2,\qquad c = \tfrac{1}{2} m_p L_p L_r, $$

$$ a = \frac{k_m^2}{R_m} + D_r,\qquad b = \frac{k_m}{R_m},\qquad d = \tfrac{1}{2} m_p L_p g, $$

$$ \Delta = J_\mathrm{eq} J_{p,\mathrm{eq}} - c^2 $$

The state-space matrices are

$$ A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & -\dfrac{J_{p,\mathrm{eq}} a}{\Delta} & \dfrac{d c}{\Delta} & \dfrac{D_p c}{\Delta} \\ 0 & 0 & 0 & 1 \\ 0 & \dfrac{a c}{\Delta} & -\dfrac{J_\mathrm{eq} d}{\Delta} & -\dfrac{D_p J_\mathrm{eq}}{\Delta} \end{bmatrix}, $$

$$ B = \begin{bmatrix} 0 \\ \dfrac{J_{p,\mathrm{eq}} b}{\Delta} \\ 0 \\ -\dfrac{b c}{\Delta} \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix}, \quad D = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. $$

Simulink Model and Controller Structure

Simulation Results

The folder matlab/results/RotaryPendulum contains the response plots for the angles.

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How to Use (Quick Summary)

1. Start MATLAB and Simulink.
2. Set this repository as the current MATLAB working folder.
3. Initialize parameters:
   • DC motor : run init_dc_motor.m
   • Rotary pendulum : run init_rotary_pendulum.m
4. Open the desired .slx model and run the simulation.
5. Check the simulation results under the corresponding subfolders in matlab/results.

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