09 elec3114

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Design via Root Locus

• How to use the root locus to design cascade compensators to
improve the steady state error
steady-state

• How to use the root locus to design cascade compensators to
improve the transient response

• How to use the root locus to des g cascade compensators to
ow o e oo ocus o design co pe sa o s o
improve both the steady-state error and the transient response

• How to realize the designed compensators physically

Dr Branislav Hredzak
Control Systems Engineering, Fourth Edition by Norman S. Nise
Copyright © 2004 by John Wiley & Sons. All rights reserved.

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Introduction
Improving transient response
• transient response can be improved with the addition of differentiation
• the compensated system will have a root locus that goes through the
desired pole location

Improvement:
- response B is faster than response
A, while the overshoot is the same


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Improving Steady-State Error
• steady-state error can be improved with the addition of integration in the
forward path.
p


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Improving Transient Response and Steady-State Error
Steady State
• By using dynamic compensators, compensating networks can be designed
that allow to meet both transient and steady state error specifications
steady-state
simultaneously


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Compensator configurations to meet transient and
steady state
steady-state error specifications

Cascade
configuration


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Improving Steady-State Error
via Cascade Compensation
i C d C ti

There are two techniques:

1. Ideal integral compensation – uses a pure i
integrator. It
reduces the steady-state error to zero

2. Lag compensation – does not use pure integration. It
p
places the pole near the origin. It does not reduce the error
p g
to zero.


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1. Ideal Integral Compensation (PI controller) to
Improve Steady-State Error
Steady State

• Steady-state error is improved by placing an open-loop pole at the origin


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Implementation of ideal integral compensator

… zero can be adjusted by
varying K2/K1

• Since the ideal integral compensator has both proportional and integral
control, it is given the alternate name PI controller


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Problem Given the system operating with a damping ratio of 0.174, show that
the addition of the ideal integral compensator reduces the steady-state error to
h ddi i f h id l i l d h d
zero for a step input without appreciably affecting transient response. The
compensating network is chosen with a pole at the origin to increase the system
type and a zero at -0.1, close to the compensator pole, so that the angular
d l h l h h l
contribution of the compensator evaluated at the original, dominant, second-
order poles is approximately zero. Thus, the original, dominant, second-order
closed-loop poles are still approximately on the new root locus.


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11


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2. Lag compensation to Improve Steady-State Error
• Does not use pure i
D integration
i
• Uses passive networks
• The pole and zero are placed to the left, close to the origin
h l d l d h l f l h i i

static error constant

new static error constant


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• If the lag compensator pole and zero are close together, the angular
contribution of the compensator to point P is approximately zero degrees.
degrees
• K is virtually the same for the uncompensated and compensated systems,
since the lengths of the vectors drawn from the lag compensator are
approximately equal and all other vectors have not changed appreciably.
• Improvement is the steady-state error is given by

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


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Problem Compensate the system, to improve the steady-state error by a factor of
10 if the system is operating with a damping ratio of 0 174
0.174.


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Solution:

Uncompensated error (f
U d (from previous example):
i l )

A tenfold improvement means a steady-state error of
p y

Let us select


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Improving Transient Response via
Cascade Compensation

There are two techniques:

1. Ideal derivative compensation – uses a pure differentiator

2. Lead
2 L d compensation – d
ti does not use pure differentiation
t diff ti ti


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1. Ideal Derivative Compensation (PD controller)
to Improve Transient Response
• the original system can be made faster by adding a single zero to the
forward path

• Disadvantage of ideal differentiation: differentiation of high frequency
noise leads to large unwanted signals

Zero at -2


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Zero at -3


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Zero at -4


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• The damping ratio is unchanged (0.4), hence the percent overshoot is the
same for all three cases
• More negative real part of dominant poles, hence shorter settling time
• Imaginary parts are larger hence smaller peak times
larger,
• Improvement in steady state error (due to increase of Kp)


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Implementation of ideal derivative compensator


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Problem Given the system, design an ideal derivative compensator
to yield a 16% overshoot, with a threefold reduction in settling time.
y , g

Solution
16% overshoot → ζ = 0.504

3.320
Ts (new) = = 1.107
3
Real part:


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Sum of the angle of open-loop poles to the design point is 275.60
Imaginary part:


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result needs to be verified by
simulation
i l ti

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2. Lead compensation to Improve Transient Response
• consists of a pole and a zero
• if the pole is farther from the imaginary axis than the zero, the angular
contribution of the compensator is still positive and thus approximates an
p p pp
equivalent single zero
• can be implemented using passive components
• less sensitive to noise
• during design we arbitrarily select either a lead compensator pole or zero


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• infinite number of lead compensators could be used to meet the
p
transient response requirement

However during the design we have to be aware of
the static error constant, the gain, second order approximation.


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Problem Design lead compensator that will reduce the settling time by a factor of
2 while maintaining 30% overshoot.
overshoot

Solution

30% overshoot → ζ = 0.358

Ts (new) = 3.972 / 2 = 1.986 s

ωd = −2.014 tan(110.980 ) = 5.252


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Let zc= - 5

The resulting angle is -172.690
hence the pole must contribute -7.310


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Second order approximation OK
pp


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lmproving Steady-State Error and
Transient Response

• First
Fi t we design for transient response and then design for steady-state
d i f t i t d th d i f t d t t
error

• If we d i an active PD controller f ll
design ti t ll followed by an active PI controller,
db ti t ll
the resulting compensator is called a proportional-plus-integral-plus-
derivative (PID) controller

• If we first design a passive lead compensator and then design a passive
lag compensator, the resulting compensator is called a lag-lead
compensator t


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1. PID Controller Design to lmprove Steady-State
Error and Transient Response


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Design procedure

1. Evaluate the performance of the uncompensated system to determine
how much improvement in transient response is required.
2. Design the PD controller to meet the transient response specifications.
The design includes the zero location and the loop gain.
3. Simulate the system to be sure all requirements have been met.
4.
4 Redesign if the simulation shows that requirements have not been
met.
5. Design the PI controller to yield the required steady-state error.
6. Determine the gains, Kl, K2, and K3.
8. Redesign if simulation shows that requirements have not been met.
d i i l i h h i h b


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2. Lag-Lead Compensator Design to lmprove
Steady State
Steady-State Error and Transient Response
Design procedure

1.
1 Evaluate th
E l t the performance of the uncompensated system to determine how
f f th t d t t d t i h
much improvement in transient response is required.
2. Design the lead compensator to meet the transient response
g p p
specifications. The design includes the zero location, pole location, and
the loop gain.
3.
3 Simulate h
Si l the system to be sure all requirements have been met.
b ll i h b
4. Redesign if the simulation shows that requirements have not been met.
5.. Evaluate the steady-state e o performance for the lead-compensated
v u e e s e dy s e error pe o ce o e e d co pe s ed
system to determine how much more improvement in steady-state error is
required.
6. Design the lag compensator to yield the required steady-state error.
8. Redesign if the simulation shows that requirements have not been met.


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Physical realization of compensation

Active circuit realization


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Passive circuit realization


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