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Representation of LTI Systems

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Transfer Function

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  • Equation of motion

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  • Laplace Transform

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  • Block Diagram

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Example

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State Space Representation

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Three Representations of LTI Systems

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Three Representations of Linear Systems

  1. Time domain
  2. Frequency domain
  3. State space

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Time and Frequency Domains

  • In linear system, convolution operation can be converted to product operation through Laplace transform

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Converting from State Space to a Transfer Function

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Laplace Transform

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Converting from State Space to a Transfer Function

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Laplace Transform of Matrix Exponential

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Laplace Transform of Matrix Exponential

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Transformation of State-Space

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Same Transfer Function ?

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Decoupled LTI System

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Converting a Transfer Function to State Space

  • How to convert the transfer function to state space?

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  • We can redraw block diagram like the below

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Converting a Transfer Function to State Space

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Converting a Transfer Function to State Space

  • Draw this into a block diagram

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MATLAB Implementation

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Step Response

  • Start with a step response example

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  • The solution is given:

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Step Response

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Step Response

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Step Response

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Impulse Response

  • Now think about the impulse response

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  • The solution is given:

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Impulse Response

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Impulse Response

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Response to a General Input

  • Response to a general input

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  • The solution is given:

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Model Conversion in MATLAB

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Summary

  • LTI Systems
    • In time
    • In Laplace (or Frequency)
    • In state space

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  • MATLAB Implementation

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