Basic Control Engineering

Prof. Wonhee Kim

Ch.2. Modeling in the Frequency Domain

2

Laplace Transform Review

- Laplace transform

- Inverse Laplace transform

3

Laplace Transform Review

4

Laplace Transform Review

5

Laplace Transform Review:

- Solution of a Differential Equation

Example 2.3)

6

System Modeling in Frequency Domain

r(t)

r(t)

c(t)

c(t)

From input to output: Convolution

0

*t

c g r t g r t d

7

System Modeling in Frequency Domain

C s G s R s

r(t) c(t) R(s) C(s)

Laplace

transform

C sG s

R s

Transfer function

0

*t

c g r t g r t d

8

System Modeling in Frequency Domain

R(s) C(s)

9

System Modeling in Frequency Domain

Example 2.4)

r(t) c(t) R(s) C(s)

Laplace

transform

2

2

1

2

dc tc t r t

dt

sC s C s R s

C sG s

R s s

10

System Modeling in Frequency Domain

Example 2.5)

1

2

C sG s

R s s

1

2C s G s R s R s

s

1

R ss

1 1 1/ 2 1/ 2

2 2C s G s R s

s s s s

21 1

2 2

tc t e

11

Electrical Network Transfer Functions

Electrical Network Transfer Functions

Example 2.6)

Electrical Network Transfer Functions

Example 2.6) For the capacitor:

For the resistor:

For the inductor:

Impedance:

/Ci t Cdv t dt

I s sCV s

14

Translational Mechanical System Transfer Functions

15

Rotational Mechanical System Transfer Functions

16

Translational Mechanical System Transfer Functions

Example 2.16)

17

Nonlinearities

18

Nonlinearities - Linearization

19

Nonlinearities - Linearization

20

Nonlinearities - Linearization