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Representation of LTI Systems
- Prof. Seungchul Lee
Transfer Function
- Equation of motion
- Laplace Transform
- Block Diagram
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Representation of LTI Systems Prof. Seungchul Lee Industrial AI - - PowerPoint PPT Presentation
Representation of LTI Systems Prof. Seungchul Lee Industrial AI - - PowerPoint PPT Presentation
Representation of LTI Systems Prof. Seungchul Lee Industrial AI Lab. Transfer Function Equation of motion Laplace Transform Block Diagram 2 Example 3 State Space Representation 4 Three Representations of LTI Systems 5 Three
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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
- State Space can be represented:
- Solving for 𝑌(𝑡) in the first equation Laplace transformed
- Substituting equation 𝑌(𝑡) into second equation Laplace transformed yields
Laplace Transform
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Converting from State Space to a Transfer Function
- We call the matrix 𝐷 𝑡𝐽 − 𝐵 −1𝐶 + 𝐸 the transfer function matrix
- Note
– The output in time
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Laplace Transform of Matrix Exponential
- Series expansion of 𝐽 − 𝐷 −1
- Series expansion of 𝑡𝐽 − 𝐵 −1
- Inverse Laplace transform of 𝑡𝐽 − 𝐵 −1
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Laplace Transform of Matrix Exponential
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Transformation of State-Space
- State space representations are not unique because we have a lot of freedom in choosing the state
vector.
– Selection of the state is quite arbitrary, and not that important
- In fact, given one model, we can transform it to another model that is equivalent in terms of its input-
- utput properties
- To see this, define model of 𝐻1(𝑡) as
- Now introduce the new state vector 𝑨 related to the first state 𝑦 through the transformation 𝑦 = 𝑈𝑨
- 𝑈 is an invertible (similarity) transform matrix
The new model of 𝐻1(𝑡)
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Same Transfer Function ?
- Consider the two transfer functions
- Does 𝐻1 𝑡 = 𝐻2(𝑡) ?
- So the transfer function is not changed by putting the state-space model through a similarity
transformation
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Decoupled LTI System
- If 𝑈 = 𝑇, transformation to diagonal matrix
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Converting a Transfer Function to State Space
- How to convert the transfer function to state space?
- We can redraw block diagram like the below
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Converting a Transfer Function to State Space
- Reverse Laplace transform
- Choose state variable:
– A convenient way to choose state variables is to choose the output, 𝑧(𝑢), and its (𝑜 − 1) derivatives as the state variables
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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
- 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
- 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
- The solution is given:
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Model Conversion in MATLAB
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State Space ↔ Transfer Function
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Summary
- LTI Systems
– In time – In Laplace (or Frequency) – In state space
- MATLAB Implementation
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