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Motor Transfer Function Identification

Project Overview

This project implements a system identification approach for DC motors using step response analysis. The system sends a step setpoint (75 RPM) to a motor, captures encoder data, and estimates the transfer function using Error-Output (EO) identification algorithms.

Project Structure

Motor_Transfer_Function_Identification/
├── code/                           # Source code files
├── data_cleaned/                   # Processed and filtered data
├── data_initial/                   # Raw captured data from encoder
├── result/                         # Identified transfer functions and analysis
├── motor_img_characteristics/      # Motor specification images
└── README.md                       # This file

Methodology

1. Data Collection

  • Input: Step setpoint of 75 RPM to motor controller
  • Output: Angular velocity data captured via encoder
  • Sampling: Discrete-time measurements at regular intervals
  • Storage: Raw data saved in data_initial/ folder

2. Data Processing

  • Clean and filter raw encoder data
  • Remove noise and outliers
  • Processed data stored in data_cleaned/ folder

3. System Identification

  • Apply Error-Output (EO) identification algorithm
  • Estimate second-order discrete transfer function parameters
  • Results saved in result/ folder

4. Transfer Function Form

The identified transfer function follows the standard discrete ARX model:

G(z) = Y(z)/U(z) = (b₁z⁻¹ + b₂z⁻²)/(1 + a₁z⁻¹ + a₂z⁻²)

Where:

  • z⁻¹ is the unit delay operator
  • b₁, b₂ are numerator coefficients (defining zeros)
  • a₁, a₂ are denominator coefficients (defining poles)
  • Y(z) is the output (motor speed)
  • U(z) is the input (PWM command)

Motor Dynamics Reference

Dynamic Behavior Classification

Very Fast, Responsive

Example: Small coreless drone motor

  • Characteristics: Quick response, low inertia, may have slight overshoot
  • Parameters:
    • b₁: 0.10 - 0.25
    • b₂: 0.05 - 0.15
    • a₁: -1.6 - -1.2
    • a₂: 0.60 - 0.80

Well-Tuned, Critically Damped

Example: Precision servo motor

  • Characteristics: Fastest response without overshoot, smooth operation
  • Parameters:
    • b₁: 0.15 - 0.30
    • b₂: 0.10 - 0.20
    • a₁: -1.5 - -1.1
    • a₂: 0.50 - 0.70

Slow, Overdamped

Example: Large gearmotor with high inertia

  • Characteristics: Slow, sluggish response, no overshoot
  • Parameters:
    • b₁: 0.02 - 0.08
    • b₂: 0.01 - 0.05
    • a₁: -1.9 - -1.7
    • a₂: 0.85 - 0.95

With Significant Time Delay

Example: Communication lag, slow control systems

  • Characteristics: Response pauses for several samples before rising
  • Parameters:
    • b₁: 0.01 - 0.05
    • b₂: 0.10 - 0.30
    • a₁: -1.3 - -0.9
    • a₂: 0.40 - 0.60

Oscillatory, Underdamped

Example: Elastic coupling, poor tuning

  • Characteristics: Fast response but with large overshoot and ringing
  • Parameters:
    • b₁: 0.20 - 0.40
    • b₂: 0.05 - 0.15
    • a₁: -1.1 - -0.5
    • a₂: 0.10 - 0.40

Parameter Interpretation Guide

  • Larger |b₁|: Faster initial response to a command
  • a₂ closer to 1.0: Slower system dynamics (poles near z=1)
  • a₁ highly negative: Typically indicates faster, more damped response
  • Oscillation Condition: If (a₁)² < 4(a₂), the system has complex poles and will exhibit overshoot and ringing

Usage

  1. Setup: Configure motor and encoder connections
  2. Data Collection: Run step response experiment
  3. Processing: Clean and filter collected data
  4. Identification: Execute EO algorithm to estimate transfer function
  5. Validation: Analyze model fit and system characteristics
  6. Control Design: Use identified model for controller design

Applications

  • Motor Control: Design PID controllers based on identified model
  • System Analysis: Understand motor dynamics and limitations
  • Performance Optimization: Tune system parameters for desired response
  • Predictive Modeling: Simulate motor behavior under different conditions

Notes

  • Ensure proper sampling frequency for accurate identification
  • Consider noise filtering to improve identification accuracy
  • Validate results against known motor specifications
  • Multiple experiments may be needed for robust parameter estimation

Future Work

  • Implement closed-loop controller design based on identified model
  • Compare with other identification methods (Least Squares, Subspace)
  • Extend to MIMO systems for multi-motor applications
  • Real-time parameter adaptation capabilities