Lecture on Linear and Time Invariant Systems (LTI)

Introduction

• LTI Systems: Class of systems that respond predictably to inputs.
• Defining Properties: Homogeneity, Superposition (Additivity), and Time Invariance.

Key Properties of LTI Systems

Linearity

1. Homogeneity

   • Scaling input by a factor results in scaling the output by the same factor.
   • Example: Doubling the input doubles the output.
   • Graphical Example: Step input of 1 produces sinusoid with amplitude 1; step input of 2 produces sinusoid with amplitude 2.

2. Superposition (Additivity)

   • Adding two inputs results in the addition of their corresponding outputs.
   • Example: Input X1 produces output Y1, and input X2 produces output Y2; thus, X1 + X2 produces Y1 + Y2.

• Combined Homogeneity and Superposition: If the input is scaled and summed, the output will also be scaled and summed by the same amounts.

Time Invariance

• System's behavior is consistent over time regardless of when the action happens.
• Example: Sending a slinky down the stairs behaves the same regardless of time.
• Mathematical Representation: Input X(t-a) produces output Y(t-a).
• Translation Invariance: Covers translation in both time and space.

Practical Relevance of LTI Systems

• Richard Feynman's Insight: Linear systems are important because we can solve them with mathematical tools.
• Real-World Systems: Many physical systems can be accurately approximated by LTI models.

Impulse Response and Convolution

• Impulse Response: The system's response to an impulse (e.g., hitting a mass with a hammer).
  • Time Invariance: Same response if impulse occurs at different times.
  • Homogeneity: Double the impulse results in double the response.
  • Superposition: Summation of individual impulse responses.
• Convolution: Summing responses to continuous inputs is called convolution.
  • S Domain Transformation: Using Laplace transform, convolution in time domain becomes multiplication in S domain.

Design Applications in Control Engineering

• Controller Design: Use of LTI models to design and tune controllers.
• Component Design: Ensuring sensors and actuators can be modeled as LTI systems for accurate control.
• Nonlinear Systems: Understanding the operating region where systems exhibit linear behavior.
  • Example: Linear region of a spring's force vs. stretch graph.

Conclusion

• Understanding LTI systems is crucial for control system design.
• Future lectures will cover linearizing nonlinear systems and dealing with time variance.

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• Q&A: Questions, comments, and future lesson suggestions are welcomed.

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