1. Chapter 4
Discrete Equivalents

2. i) numerical integration ii) pole and zero mapping
Goal: to obtain a discrete-time controller ( filter, equalizer, compensator ) which provides transient and frequency response characteristics as close as possible to those of the original continuous-time controller
analog controller
digital controller
D/A
A/D
Three approaches:
i) numerical integration
ii) pole and zero mapping
iii) hold equivalence

3. Method 1 : Numerical Integration

5. time
forward
backward
trapezoid

6. Backward Difference Method
( Backward Rectangular Rule )

8. stable but considerable distortion1
stable but considerable distortion

9. ii) Forward Difference Method
( Forward Rectangular Rule / Euler Method )

10. cannot be used in practice1
may be unstable,
cannot be used in practice

11. iii) Trapezoid Integration Method
Tustin Transform Method
Bilinear Transform Method

12. stable but still noticeable frequency distortion1
stable but still noticeable frequency distortion

13. Remark:

15. iv) Bilinear Transformation Method
with Frequency Pre-warping

17. Procedure:

19. Remarks:
1. Approximation will be correct if
2. However, we must have if a stable filter
is to remain stable after warping

20. Method 2 : Pole and Zero Mapping

23. Method 3 : Hold Equivalent
H(s)
sampler
H(s)
hold

29. (T = 0.1, T = 1, and T = 2 ). Example in page 195
ex) The third order low-pass Butterworth filter designed to have unity pass bandwidth ( = 1 ), Use sampling periods
(T = 0.1, T = 1, and T = 2 ). Example in page 195
i) T = 0.1
bilinear = o
warped = +
backward = *
forward = 

30. ii) T = 1
bilinear = o
warped = +
backward = *
forward = 

31. iii) T = 2
bilinear = o
warped = +
backward = *
forward = 