Chapter 9 Design Via Root Locus

arfauz@utem.edu.myFakulti Kejuruteraan Pembuatan

Universiti Teknikal Malaysia Melaka

CHAPTER 9

DESIGN VIA ROOT LOCUS

INTRODUCTION

Objectives:How to use the root locus to design cascade

compensators to improve the steady state errorHow to use the root locus to design cascade

compensators to improve the transient responseHow to use the root locus to design cascade

compensators to improve both the steady state error and the transient response

How to use the root locus to design feedback compensators to improve the transient response

How to realize the designed compensators physically

INTRODUCTION

IMPROVING TRANSIENT RESPONSE

Rather than replacing the existing system with a system whose root locus intersects the desired design of point B, we can augment or compensate the system with additional poles and zeros – so that the compensated system has a root locus that goes through the desired pole location for some value of gain

Two methods – passive or active networkDisadvantages – system order can increase,

subsequently effect the desired responseOne method of compensating for transient response is to

insert differentiator in the forward path in parallel with the gain

IMPROVING STEADY STATE ERROR

Compensators are not only used to improve the transient response of a system, they are also used independently to improve steady state error characteristics

In Chapter 7 – steady state error can be improved by adding an open loop pole at the origin in forward path, thus increasing the system type and driving the associated steady state error to zero

This additional pole at the origin requires an integrator for its realization

Compensators

Configuration of Compensations

System Improvement Technique

Feeding methodProportional – feed the error forward to the plant Integral – feed the integral of the error to the plantDerivative – feed the derivative of the error to the plant

Implemented using active networks (PI/PD)Using AmplifiersExpensive

Implemented using passive networks (Lag/lead)Less expensive no additional power required

IMPROVING STEADY-STATE ERROR VIA CASCADE COMPENSATION

IMPROVING TRANSIENT RESPONSE VIA CASCADE COMPENSATION

IMPROVING STEADY-STATE ERROR AND TRANSIENT RESPONSE

PI, PD, PID CONTROLLER

The Characteristics of P, I & D Controller

Mathematical Representation of Proportional Controller

Mathematical Representation of Integral Controller

Mathematical Representation of Derivative Controller

Mathematical Representation of PI Controller

More on IDEAL INTEGRAL COMPENSATION (PI Controller)

-θ1-θ2-θ3- θ pc+θzc ≡ (2k+1)1800

PI controller

EXAMPLE:Closed-loopsystema. beforecompensation;b. after ideal integralcompensation

Root locus foruncompensatedsystem

Root locus forcompensatedsystem

Ideal integral compensated system response and theuncompensated systemresponse of previous example

LAG COMPENSATION

If use passive networks, the pole and zero are moved to the left, close to the origin

This placement usually will not increase the system type, but will yield an improvement in the static error constant over an uncompensated system

Although the ideal compensator drives the steady state error to zero, a lag compensator with a pole that is not at the origin will improve the static error constant by a factor equal to Zc/Pc

There will also minimal effect upon the transient response if the pole-zero pair of the compensator close to the origin

a. Type 1 uncompensated system;b. Type 1 compensatedsystem;c. compensatorpole-zero plot

Root locus:a. before lag compensation;b. after lag compensation

EXAMPLE

Mathematical Representation of PD Controller

More on IDEAL DERIVATIVE COMPENSATION (PD Controller)

To speed up the original system, we can add a single zero to the forward path

This zero can be represented by a compensator whose transfer function is

Gc(S) = s + Zc

This function, the sum of a differentiator and a pure gain is called ideal derivative – PD controller

PD controller

Using ideal derivativecompensation:a. uncompensated;b. compensatorzero at –2;c. compensatorzero at –3;d. compensatorzero at – 4

Zero is moved to a different position and for each compensated case, the dominant, second order poles are farther out along the 0.4 damping ratio line than the uncompensated system

However, each of the compensated case has dominant poles with the same damping ratio as the uncompensated case thus the percent overshoot is predicted to be similar for each case!

The compensated dominant, closed loop poles has more negative real parts than the uncompensated dominant, closed loop poles thus shorter settling time, Ts

On top of that smaller peak time, Tp

Predicted characteristics for the previous shown systems

Uncompensated system and ideal derivativecompensation solutions from previous table

Example: Feedback control system for Example 9.3

Root locus for uncompensatedsystem shown in Previous Example

Uncompensated and compensated system characteristics for Previous example

Compensateddominant polesuperimposed over the uncompensatedroot locus forprevious example

Evaluating the location of the compensatingzero

Root locus for thecompensated system

Uncompensated andcompensated system step responses ofprevious example

LEAD COMPENSATION

Similar to the active ideal integral that can be approximated with passive lag network, an active ideal integral can be approximated with a passive lead compensator

Pole is farther from the imaginary axis then the zero – will result in a positive angular distribution of the compensator and thus approximates an equivalent single zero

Geometry of leadcompensation

Three of the infinitepossible leadcompensator solutions

EXAMPLE

Other Example

Mathematical Representation of PID Controller

PID Controller Design

It has two zeros and a pole at the originOne zero and the pole at the origin can be

designed as the ideal integral compensator

Another zero can be designed as the ideal derivative compensator

Follow steps in Text book – page 532Evaluate – Design PD – Design PI

PID controller

LAG-LEAD COMPENSATOR DESIGN

First design the lead compensator to improve the transient response

Next, evaluate the improvement in steady state

Finally, design the lag compensator to meet the steady state error requirement

Follow steps in Text Book – Page 537Evaluate – Design Lead – Design Lag

FEEDBACK COMPENSATION

Approach 1Approach 2

PHYSICAL REALIZATION OF COMPENSATION

Active circuit RealizationPassive circuit Realization

Chapter 9 Design Via Root Locus

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