Direct z-Domain Digital Controller Design

1. Direct z-Domain DigitalController Design

2. OUTLINE • Advantages/disadvantages. • Design procedures. • Direct z-design examples.

3. Digital from Analog Designs: Advantage: Familiar design. Disadvantages: 1. Controller distortion. 2. Poles/zeros in subsets of unit circle. (s +a) bilinear transformation [z−(c−a)/(c+a)] gives RHP zeros, a < c Cancel RHP pole but restrict design. 3. Replace z+1 with z (bounded at folding Frequency) ⇒ more controller distortion.

4. Direct z-Design Advantage: No approximation. Disadvantages: 1. Controllers: typically same form as Section 6.3, but poles are not restricted to RHP. 2. z-plane is less familiar & selection of pole locations is less intuitive. 3. Stable region inside unit circle (much smaller than left half of the s-plane).

5. Design Procedures Design simplified using MATLAB. ☻ Use Procedures 5.1-3 with minor changes. ☻ Modify (5.14) (for z-domain) (5.14)

6. PD Compensator Zero

7. Example 6.12 Design a digital controller for the type 0 analog plant for I. zero e(∞) due to a unit step, II. ζ = 0.7, and III. Ts≈1s.

8. Solution Select T = 0.02 s, obtain z-transfer function. •Zero e(∞) due to step: •Use a PI controller type 1, pole at z =1 •Zero at z = 0.98, meets the design specs. •Results almost identical to Example 6.8

9. Example 6.13 Design a controller for the analog plant to obtain: Ts< 1 s, ζ = 0.7

10. Solution Obtain TF for plant, ADC and DAC (T = 0.01 s) . • PD controller: Pole-zero cancellation and add pole at origin (approx. realizable). • Controller meets transient response specs. • Like Example 6.9

11. Example 6.13 Design a controller in the z-domain for the analog plant for τ < 0.3 s, dominant pole ζ ≥ 0.7, e(∞) due to step input = 0.

12. Solution • Plant type 0, same as Example 6.10, let T = 0.005s • For e(∞) due to step input = 0, use PI control • pole at z = 1, zero at z = 0.995

13. Root Locus for PI Control

14. PID Needed • For ζ = 0.7, the closed-loop poles are close to the unit circle (much slower than specified). • Need PID controller: cancel pole closest to (not on) the unit circle. • Add pole at z = 0 (realizable controller).

15. Root Locus for PID Control

16. Step Response for PID Control

17. Time Response • Meets all design specifications • < 5 % overshoot with a fast time response • Better than Example 6.10 (digital controller via analog design). • Analog design can possibly be improved with trial and error (time consuming). • Direct design in the z-domain using MATLAB can be easier than indirect design.