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 Representations of Linear Systems 1) Time domain 2) Frequency domain 3) State space 6

Time and Frequency Domains • In linear system, convolution operation can be converted to product operation through Laplace transform 7

Converting from State Space to a Transfer Function • State Space can be represented: Laplace Transform • Solving for 𝑌(𝑡) in the first equation Laplace transformed • Substituting equation 𝑌(𝑡) into second equation Laplace transformed yields 8

Converting from State Space to a Transfer Function • We call the matrix 𝐷 𝑡𝐽 − 𝐵 −1 𝐶 + 𝐸 the transfer function matrix • Note – The output in time 9

Laplace Transform of Matrix Exponential • Series expansion of 𝐽 − 𝐷 −1 • Series expansion of 𝑡𝐽 − 𝐵 −1 • Inverse Laplace transform of 𝑡𝐽 − 𝐵 −1 10

Laplace Transform of Matrix Exponential 11

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- output properties • To see this, define model of 𝐻 1 (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 𝐻 1 (𝑡) 12

Same Transfer Function ? • Consider the two transfer functions • Does 𝐻 1 𝑡 = 𝐻 2 (𝑡) ? • So the transfer function is not changed by putting the state-space model through a similarity transformation 13

Decoupled LTI System • If 𝑈 = 𝑇 , transformation to diagonal matrix 14

Converting a Transfer Function to State Space • How to convert the transfer function to state space? • We can redraw block diagram like the below 15

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 16

Converting a Transfer Function to State Space • Draw this into a block diagram 17

MATLAB Implementation 18

Step Response • Start with a step response example • The solution is given: 19

Step Response 20

Step Response 21

Step Response 22

Impulse Response • Now think about the impulse response • The solution is given: 23

Impulse Response 24

Impulse Response 25

Response to a General Input • Response to a general input • The solution is given: 26

Model Conversion in MATLAB 27

State Space ↔ Transfer Function 28

Summary • LTI Systems – In time – In Laplace (or Frequency) – In state space • MATLAB Implementation 29