🧪 Digital Control Simulation — ARX Model + Saturation

This repository provides a tutorial-oriented simulation of a digital PI control loop applied to a discrete-time ARX model identified from experimental data.
The objective is to reproduce in simulation the same discrete behavior expected when the controller is later implemented on a microcontroller (Arduino, Teensy, ESP32, etc.), including actuator saturation.

⚠️ Note: This tutorial is for educational purposes only. It focuses on simulation and understanding discrete-time digital control with ARX models.

🎯 Goals of the Tutorial

• Model a system using a discrete-time ARX model identified experimentally
• Implement a discrete PI controller using incremental form
• Add saturation limits to emulate real actuator constraints
• Compare reference tracking and control signal behavior

🧩 ARX System Model

The discrete-time ARX model is defined as:

$$ A(z)y(k) = B(z)u(k) + e(k) $$

Where:

$$ A(z) = 1 - 1.937 z^{-1} + 1.152 z^{-2} - 0.2144 z^{-3} $$

$$ B(z) = -0.001961 z^{-3} $$

Matlab

a = [1  -1.9366 1.1523 -0.2144]; % note the sign for implementation
b = [0 0 0 -0.001961];       % B(z) with zeros for delays

Python

a = np.array([1, -1.9366, 1.1523, -0.2144], dtype=float)  # note the sign for implementation
b = np.array([0, 0, 0, -0.001961], dtype=float)           # B(z) with zeros for delays

The discrete model implemented is:

Matlab

y(k) = -a(2)*y1 - a(3)*y2 - a(4)*y3  + b(1)*u1 + b(2)*u2 + b(3)*u3+ b(4)*u4;

Python

y[k] = (-a[1] * y1 -a[2] * y2-a[3] * y3 + b[0] * u1 + b[1] * u2 + b[2] * u3 + b[3] * u4)

• Sampling period: $$T_s = 0.004 s$$
• This model was identified from experimental data using ARX method.

⚙️ Digital PI Controller (Incremental Form)

The discrete control law implemented is:

$$ u(k) = u(k-1) + K_0 e(k) + K_1 e(k-1) $$

Matlab

error  = Ref(k) - y(k);
u  = u1 + K0*error + K1*error1;

Python

error = Ref[k] - y[k]
u = u1 + K0 * error + K1 * error1

With tuning parameters derived from:

• Proportional gain: $$K_p$$
• Integral time: $$T_i$$
• Sampling time: $$T_s$$

Where

$$ K_0 = K_p + \frac{K_p}{2T_i}T_s $$

$$ K_1 = -K_p + \frac{K_p}{2T_i}T_s $$

🔒 Actuator Saturation

To emulate microcontroller behavior, the controller output is limited to a predefined range:

• Prevents unrealistic actuator commands
• Reflects PWM or DAC limits on embedded hardware
• Avoids integrator wind-up if actuator saturates

Example: -100% ≤ u(n) ≤ 100%

Without saturation, simulation results may falsely assume an ideal actuator with infinite authority, which never matches microcontroller deployments. To emulate real microcontroller behavior — such as PWM range or fixed DAC limits —

Matlab

if u > 100
    u=100;
end
if u <-100
    u=-100;
end

Python

if u > Umax:
    u = Umax
elif u < Umin:
    u = Umin

Below are example plots generated with the script:

🤝 Support projects

Support me on Patreon https://www.patreon.com/c/CrissCCL

📜 License

MIT License