Chapter 4 Discrete Equivalents

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

Method 1 : Numerical Integration

time forward backward trapezoid

Backward Difference Method ( Backward Rectangular Rule )

stable but considerable distortion 1 stable but considerable distortion

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

cannot be used in practice 1 may be unstable, cannot be used in practice

iii) Trapezoid Integration Method Tustin Transform Method Bilinear Transform Method

stable but still noticeable frequency distortion 1 stable but still noticeable frequency distortion

Remark:

iv) Bilinear Transformation Method with Frequency Pre-warping

Procedure:

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

Method 2 : Pole and Zero Mapping

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

(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 = 

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

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