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Direct z-Domain Digital Controller 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. - PowerPoint PPT Presentation
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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 < cCancel RHP pole but restrict design.3. Replace z+1 with z (bounded at folding Frequency) ⇒ more controller distortion.
Direct z-DesignAdvantage: 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 6.12Design 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 6.13Design a controller for the analog plant toobtain: Ts < 1 s, ζ = 0.7
SolutionObtain 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 6.13Design 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.