Direct z-Domain DigitalController Design

OUTLINE

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

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.

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).

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

PD Compensator Zero

Example 6.12

Design a digital controller for the type 0analog plant forI. zero e(∞) due to a unit step,

II. ζ = 0.7, andIII. Ts ≈1s.

SolutionSelect 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

Example 6.13

Design a controller for the analog plant to

obtain: Ts < 1 s, ζ = 0.7

Solution

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

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.

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

Root Locus for PI Control

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).

Root Locus for PID Control

Step Response for PID Control

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.