Feedback Systems Overview

Key Concepts

• Feedback Systems: Often focused on reference signal and output relationship.
• Transfer Functions: To fully understand feedback system behavior, consider:
  • Four transfer functions for systems without feed-forward paths.
  • Six transfer functions for systems with feed-forward paths.

Importance of Multiple Transfer Functions

• Intuition: Understanding the necessity of multiple transfer functions for analyzing system properties.
• Goal: Provide insights beyond mathematics; includes practical demonstrations.

Reference to Literature

• Recommended: Works by Professor Carl Astrum for in-depth understanding.

Basic Feedback Loop Diagram

• Components:
  • Process (p): The system to control.
  • Feedback Controller (c): Controls the process.
  • Feed Forward Controller (f): Improves system response.
• Control Variable (u): Output from the controller that affects system state.
• External Input (r): Reference signal.
• Output (y): Resulting behavior of the system.

External Factors Affecting System

• External Disturbances: Physical changes impacting system state (e.g., potholes, wind).
• Noise: Measurement errors affecting our understanding of system states.
• Disturbances vs. Noise:
  • Disturbances change system states directly.
  • Noise alters our perception of the system's state.

Analyzing Sensitivities

• Key Questions:
  • System sensitivity to noise?
  • Sensitivity to disturbances?
  • Variations in process behavior?

Transfer Functions Relationships

• External Signals:
  1. Reference (r)
  2. Disturbance (d)
  3. Noise (n)
• Internal Signals:
  1. State (x)
  2. Output (y)
  3. Control Variable (u)
• Nine Relationships: Between inputs and outputs derived from transfer functions.
• Gang of Six Transfer Functions: Unique relationships when analyzing feedback systems.

Sensitivity Functions

1. Noise Sensitivity Function
2. Disturbance Sensitivity Function
3. Sensitivity Function
4. Complementary Sensitivity Function

• Understanding these functions helps in analyzing trade-offs in system design.

Loop Shaping and System Stability

• Loop Shaping: Adjusting feedback loops to meet desired design criteria.
• Stability Analysis: All transfer functions share the same denominator (1 + pc).
• Changes in one aspect affect others, necessitating balance among performance metrics.

Practical Demonstration: Kwanzaa Cube Servo 2

• Setup: Using a rotary arm with a pendulum as a disturbance.
• Experiment: Analyzing controller response when subject to oscillations from the pendulum.

Experimental Results

• Initial Test: Controller struggles against pendulum disturbance; oscillations visible in output.
• PD Controller: Tuned to highlight disturbance effects.
• Notch Filter Implementation:
  • Placed in feedback: Reduces oscillations but limits disturbance rejection.
  • Placed in feed-forward: Allows for improved reference tracking without affecting disturbance rejection.

Comparison of Configurations

• Notch filter in feed-forward path shows better performance against disturbances compared to feedback loop.
• Illustrates the importance of understanding feedback and feed-forward impacts on performance metrics.

Conclusion

• The Gang of Six transfer functions provide a methodical approach to optimize feedback systems.
• Encouragement to explore further with practical implementations in Simulink and Kwanzaa Cube Servo 2.
• Reminder to subscribe for future content and review additional resources.