Servo-SysID — Servo System Identification & PID Tuning

Author: Nour Ammar

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Overview

This project characterizes the real-world dynamic behavior of a hobby servo motor by collecting hardware step-response data using an Arduino and a potentiometer, fitting a transfer function model using MATLAB's System Identification Toolbox, and tuning a discrete PID controller using pidtune.

The motivation was to study whether a servo actually reaches its commanded angle accurately, understand the lag and overshoot in its real response, and design a feedback controller that could correct for those deviations. The PID gains were validated in closed-loop MATLAB simulation.

This is a practical controls study that can be applied to any hobby servo where positional accuracy matters.

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Workflow

Servo_Pot_Readings.ino  (Arduino)
  - Sweeps servo from 1000 µs to 2000 µs PWM (0° to 180°)
  - Reads potentiometer angle feedback via analogRead (A0)
  - Logs time (ms), PWM input (µs), angle output (°) to Serial at 50 Hz
        │
        ▼  CSV exported from Serial Monitor
        │
Servoos.csv
  - Columns: time_ms, pwm_input_us, angle_output_deg
        │
        ▼
Servo_SysID.m  (MATLAB)
  - Loads CSV, converts time to seconds
  - Prepends 300 zero samples to improve transient capture
  - Applies 5 Hz lowpass filter to angle output to remove noise
  - Creates iddata object and fits 2nd order transfer function (tfest)
  - Discretizes model using ZOH (Ts ≈ 0.021 s)
  - Tunes discrete PID controller using pidtune
  - Plots closed-loop step response

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What This Study Shows

A common assumption when using hobby servos is that servo.write(90) reliably moves the shaft to exactly 90°. In practice, servos have internal mechanical and electrical dynamics that introduce lag, overshoot, and steady-state error — especially under load.

This study measures those dynamics directly from hardware using a potentiometer as a real angle feedback sensor, identifies a mathematical model of the servo's behavior, and designs a PID controller that could close the position loop and correct for those errors in real time.

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Results

Identified Transfer Function

• Model fit accuracy: 88.80%
• Model order: 2nd order, 1 zero
• Fit method: tfest via MATLAB System Identification Toolbox

The identified continuous-time transfer function:

        0.5556s + 0.2407
G(s) = ------------------
       s² + 46.03s + 3.08e-10

Property	Value
Numerator	[0.5556, 0.2407]
Denominator	[1, 46.0310, 3.0785e-10]
Noise Variance	0.1947
Model Fit	88.80%

The near-zero constant term in the denominator (3.08e-10) indicates the servo behaves close to an integrating system at low frequencies — meaning without feedback it drifts rather than holding position, which is exactly the behavior that motivates closed-loop PID control.

Discretized & Tuned PID Gains

The transfer function was discretized using Zero-Order Hold (ZOH) at Ts = 0.0210 s, then a discrete PID controller was tuned using pidtune:

Parameter	Value
Kp	14.4956
Ki	98.1215
Kd	0.1414
Tf	0
Integration formula	Forward Euler
Derivative formula	Forward Euler
Sampling time (Ts)	0.0210 s

The high Ki value (98.12) relative to Kp (14.50) reflects the integrating nature of the plant — significant integral action is needed to correct steady-state position error.

Closed-Loop Step Response

Metric	Observed
Overshoot	~17%
Settling time	~10 seconds
Steady-state error	~0 (integral action eliminates it)

The tuned controller produces a stable closed-loop response. The overshoot and settling time are acceptable for a proof-of-concept servo position controller.

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Hardware Setup

Component	Detail
Microcontroller	Arduino (servo pin: D9, pot pin: A0)
Servo	Standard hobby servo
Feedback sensor	Potentiometer mechanically coupled to servo shaft
Sampling rate	50 Hz (20 ms delay)
PWM sweep range	1000 µs → 2000 µs (0° → 180°)
PWM step size	7 µs per sample

Potentiometer coupling: A 3D-printed piece connected the servo horn to the potentiometer shaft, allowing the pot to rotate with the servo and provide real angle feedback as a voltage reading on A0, mapped to degrees using analogRead.

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Repository Structure

Servo-SysID/
├── Servo_Pot_Readings.ino     # Arduino: servo sweep + potentiometer data logger
├── Servo_SysID.m              # MATLAB: system ID, transfer function fit, PID tuning
├── Servoos.csv                # Logged data: time, PWM input, angle output
├── step_response.png          # Closed-loop step response plot from MATLAB
└── README.md

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How to Run

Step 1 — Collect data (CSV already included)

1. Upload Servo_Pot_Readings.ino to an Arduino
2. Mechanically couple a potentiometer to the servo shaft
3. Open Serial Monitor at 115200 baud
4. Copy output to a .csv file named Servoos.csv

Step 2 — Run MATLAB system identification

1. Open Servo_SysID.m in MATLAB
2. Make sure Servoos.csv is in the same directory
3. Run the script — it will display the transfer function, PID gains, and plot the step response

Requirements: MATLAB with System Identification Toolbox and Control System Toolbox