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

Z-Transform

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

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

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

Digitization Technique: Tustin’s Method Trapezoidal integration

Digitization Technique: Tustin’s Method MATLAB command

Relationship between s and z

Digitization Technique: Matched Pole-Zero (MPZ) Method

Digitization Technique: Pole-Zero (MPZ) Method

Final Value Theorem

Digitization Technique: Matched Pole-Zero (MPZ) Method

>> T=1; numD=[1 0.2]; denD=[1 2]; Ds=0.81*tf(numD,denD); Dz=c2d(Ds,T,'matched') Dz = z z

Digitization Technique: Matched Pole-Zero (MPZ) Method

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

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.

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.

Discrete Design Pure discrete system Mixed control system

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

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

Relationship b/w z-plane and s-plane

Discrete Controllers Proportional Derivative Integral Lead Compensation

Discrete Design  Z-transform table

Discrete Design Becomes unstable as K increases