🧪 Digital Control Simulation — First Order System + Saturation

This repository provides a tutorial-oriented simulation of a digital PI control loop applied to a first-order system identified via non-parametric methods.
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.

🎯 Goals of the Tutorial

• Model a first-order process identified experimentally (non-parametric fit)
• Discretize the plant using Zero-Order Hold (ZOH)
• Implement a discrete PI controller using incremental form
• Add saturation limits to emulate real actuator constraints
• Compare reference tracking and control signal behavior

🧩 System Model

A first-order model without delay was identified experimentally from step-response data:

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

In the included example:

$$ G_p(s)= \frac{20}{50s + 1} $$

This model is discretized with sampling period
$$T_s = 0.1 s$$

using Zero-Order Hold.

Matlab

Ts = 0.1;
gp  = tf(20,[50 1]);           % First-order plant
gpd = c2d(gp,Ts,'zoh');        % Discrete plant
[num,den] = tfdata(gpd,'v');

Python

Ts = 0.1
gp  = ctrl.tf([20], [50, 1])              # First-order plant
gpd = ctrl.c2d(gp, Ts, method='zoh')      # Discrete plant (ZOH)
num, den = ctrl.tfdata(gpd)   # returns nested lists/arrays
num = np.asarray(num, dtype=float).squeeze()
den = np.asarray(den, dtype=float).squeeze()
b1 = float(num[1])
a1 = float(den[1])

The discrete model implemented is:

Matlab

y(k) = num(2)*u1 - den(2)*y1;

Python

y[k] = b1 * u1 - a1 * y1

⚙️ 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: 0% ≤ 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 —
a hard saturation is enforced:

Matlab

if u > 100
    u=100;
end
if u< 0
    u=0;
end

Python

if u > 100:
    u = 100
if u < 0:
    u = 0

Below are example plots generated with the script:

🤝 Support projects

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

📜 License

MIT License