1. Control Systems EE 4314 Lecture 26 April 30, 2015 Spring 2015 Indika Wijayasinghe

2. Z-Transform

3. Relationship b/w z-plane and s-plane Z-planeS-plane Re Im Re Im

4. Digital Controller Design There are two techniques for finding the difference equations for the digital controller 1.Discrete equivalent: Design D(s) first, and then obtain equivalent D(z) using Tustin’s method, Matched Pole-Zero (MPZ) method. 2.Discrete design: directly obtain the difference equation without designing D(s) first. Obtain G(z) and design D(z). Difference equations D/A and hold sensor 1 r(t)u(kT)u(t)e(kT) + - r(kT) plant G(s) y(t) clock A/D T T y(kT) Digital controller

5. Design Using Discrete Equivalent Design by discrete equivalent 1.Design a continuous compensation D(s) using continuous controller design methods such as PID, lead/lag compensator. 2.Digitize the continuous compensation: D(s)  D(z) 3.Use discrete analysis, simulation or experimentation to verify the design

6. Digitization Technique: Tustin’s Method Trapezoidal integration

7. Digitization Technique: Tustin’s Method MATLAB command

8. Relationship between s and z

9. Digitization Technique: Matched Pole-Zero (MPZ) Method

10. Digitization Technique: Pole-Zero (MPZ) Method

11. Final Value Theorem

12. Digitization Technique: Matched Pole-Zero (MPZ) Method

14. >> T=1; numD=[1 0.2]; denD=[1 2]; Ds=0.81*tf(numD,denD); Dz=c2d(Ds,T,'matched') Dz = 0.3864 z - 0.3163 ----------------- z - 0.1353

15. Digitization Technique: Matched Pole-Zero (MPZ) Method

16. Digitization Technique: Modified Matched Pole-Zero (MMPZ) Method

17. Comparison of Digital Approximation Methods All the methods are quite good at lower frequencies. A minimum sampling rate of 20 times the bandwidth is recommended.

18. Discrete Design Discrete design is an exact design method and avoids the approximations inherent with discrete equivalent. The design procedures are – Finding the discrete model of the plant G(s)  G(z) – Design the compensator directly in its discrete form D(z) A practical approach is to start the design using discrete equivalents, then tune up the result using discrete design.

19. Discrete Design Pure discrete system Mixed control system

20. Discrete Root Locus Continuous system remains stable for all values of K, but the discrete system becomes oscillatory with decreasing damping ratio as z goes from 0 to -1 and eventually becomes unstable. Z-transform table

21. Relationship b/w z-plane and s-plane  n increase  increase

22. Relationship b/w z-plane and s-plane

23. Discrete Controllers Proportional Derivative Integral Lead Compensation

24. Discrete Design  Z-transform table

25. Discrete Design Becomes unstable as K increases