Direct zDomain Digital Controller Design OUTLINE Advantagesdisadvantages Design

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

Direct z-Design Advantage: No approximation. Disadvantages: 1. Controllers: typically same form as Section 6.

Design Procedures Design simplified using MATLAB. ☻ Use Procedures 5. 1 -3 with minor

Example 6. 12 Design a digital controller for the type 0 analog plant for

Solution Select T = 0. 02 s, obtain z-transfer function. • Zero e(∞) due

Example 6. 13 Design a controller for the analog plant to obtain: Ts <

Solution Obtain TF for plant, ADC and DAC (T = 0. 01 s). •

Example 6. 13 Design a controller in the z-domain for the analog plant for

Solution • Plant type 0, same as Example 6. 10, let T = 0.

PID Needed • For ζ = 0. 7, the closed-loop poles are close to

Time Response • Meets all design specifications • < 5 % overshoot with a

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Direct z-Domain Digital Controller Design

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

Digital from Analog Designs: •

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 Procedures Design 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 0 analog plant for I. zero e(∞) due to a unit step, II. ζ = 0. 7, and III. Ts ≈1 s.

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

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 add pole 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. 005 s • 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.