Control Systems Lecture: Transfer Functions

Introduction

• Audience: Control students and veteran engineers
• Purpose: Appreciation for viewers and an invitation for feedback

What is a Transfer Function?

• Basic Definition: Laplace transform of the impulse response of a linear and time-invariant (LTI) system with zero initial conditions.
• Simplified View: A black box model that transforms input signals to output signals, modeling real physical processes.

Example to Explain Transfer Functions

• Scenario: Writing your name on a chalkboard from across the room using a complex contraption.
• System Components: Flexible sticks, claws, remote control, linear actuators, chalk
• Block Diagram Representation: Each component as a black box or transfer function

Purpose of Transfer Functions

• Simplification: Characters the behavior of each part of the system and combine to understand overall system behavior.

Introduction to the Laplace Transform

• Purpose: Maps a function from the time domain to the S domain.
• Usage: Common transformations are available in software packages like Matlab or in tables.

Common Laplace Transforms

• Dirac Delta function: Laplace transform is 1
• X(t): Laplace transform is X(s)
• First Derivative X' (t): Laplace transform is sX(s) - x(0)
• Second Derivative X''(t): Laplace transform is s²X(s) - sx(0) - x'(0)

LTI Systems and Impulse Response

• Impulse Response: Subjecting an LTI system to a Dirac Delta function results in the impulse response.
• Arbitrary Input: Decomposed into an infinite number of impulses; the response summed via convolution integral.
• Convolution Integral: Simplified by the Laplace transform; turns convolution into multiplication.

Example: Harmonic Oscillator

• System Components: Mass (m), Spring Constant (k), Forcing function (U(t))
• Equation of Motion: macceleration + kposition = U(t)
• Solution Using Laplace Transform:
  • Left Side: m(s²X(s)) + kX(s)
  • Right Side: 1 (Laplace transform of Dirac Delta function)
  • Result: X(s) = 1/(ms² + k)
• Time Domain: Inverse Laplace transform yields 1/sqrt(km) * sin(sqrt(k/m) * t)

Response to Different Inputs

• Ramp Function U(t) = t:
  • Laplace Transform: 1/s²
  • Response: 1/s² * 1/(ms² + k) (simplified multiplication, avoiding convolution integral)

Practical Application

• Component Transfer Functions: Combine individual transfer functions (like springs, dampers, actuators, etc.)
• Overall System: Multiply the transfer functions in the S domain for the holistic transfer function.

Future Applications and Topics

• Open Loop/Closed Loop Systems: Importance of transfer functions.
• Automation: Moving from manual to automatic control systems.

Takeaway

• Key Benefits: Transfer functions simplify the complex integrations needed in control system design and analysis. Future topics will further explore their applications.