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Design via Root Locus

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

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

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

• How to realize the designed compensators physically• 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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IntroductionImproving 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:Improvement: - response B is faster than response A, while the overshoot is the same



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Improving Steady-State ErrorImproving 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 ErrorImproving Transient Response and Steady-State Error• By using dynamic compensators, compensating networks can be designed

that allow to meet both transient and steady-state error specificationsthat allow to meet both transient and steady state error specificationssimultaneously



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

Cascadeconfiguration



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Improving Steady-State Errori C d C tivia Cascade Compensation

There are two techniques:

i1. Ideal integral compensation – uses a pure integrator. It reduces the steady-state error to zero

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



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

• 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… zero can be adjusted by varying K2/K1

• Since the ideal integral compensator has both proportional and integralSince 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 h ddi i f h id l i l d h dthe addition of the ideal integral compensator reduces the steady-state error to

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

d l h l h h ltype and a zero at -0.1, close to the compensator pole, so that the angular 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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2. Lag compensation to Improve Steady-State ErrorD i i• Does not use pure integration

• Uses passive networks

h l d l d h l f l h i i• The pole and zero are placed to the left, close to the origin

static error constantstatic 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 degreescontribution of the compensator to point P is approximately zero 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 17410 if the system is operating with a damping ratio of 0.174.



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Solution:

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

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

Let us select



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Improving Transient Response via Cascade CompensationCascade Compensation

There are two techniques:

1. Ideal derivative compensation – uses a pure differentiator

2 L d ti d t diff ti ti2. Lead compensation – does not use pure differentiation



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1. Ideal Derivative Compensation (PD controller) to Improve Transient Responseto 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 casessame for all three cases

• More negative real part of dominant poles, hence shorter settling time

• Imaginary parts are larger hence smaller peak times• Imaginary parts are larger, hence smaller peak times

• 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 compensatorto yield a 16% overshoot, with a threefold reduction in settling time.y , g

Solution 16% overshoot 504.0 =→ ζ

107.13320.3)( ==newTs

Real part:Real part:



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



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result needs to be verified by i l ti



simulation

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2. Lead compensation to Improve Transient Response• consists of a pole and a zero• 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 ppequivalent 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 ptransient 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% overshoot2 while maintaining 30% overshoot.

Solution

358.0 overshoot %30 =→ ζ

snewTs 986.12/972.3)( ==

252.5)98.110tan(014.2 0 =−=dω



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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 OKpp



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lmproving Steady-State Error and Transient ResponseTransient Response

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

If d i ti PD t ll f ll d b ti PI t ll• If we design an active PD controller followed by an active PI controller, 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

tcompensator



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



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1. Evaluate the performance of the uncompensated system to determine

Design procedure

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 Redesign if the simulation shows that requirements have not been4. 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 K3.7. Simulate the system to be sure all requirements have been met.

d i if i l i h h i h b8. Redesign if simulation shows that requirements have not been met.



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

Design procedure

1 E l t th f f th t d t t d t i h1. Evaluate the performance of the uncompensated system to determine how much improvement in transient response is required.

2. Design the lead compensator to meet the transient response g p pspecifications. The design includes the zero location, pole location, and the loop gain.

3 Si l h b ll i h b3. Simulate the system to be sure all requirements have been met.4. Redesign if the simulation shows that requirements have not been met.5. Evaluate the steady-state error performance for the lead-compensated . v u e e s e dy s e e o 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.7. Simulate the system to be sure all requirements have been met.8. Redesign if the simulation shows that requirements have not been met.



8. Redesign if the simulation shows that requirements have not been met.

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Physical realization of compensation

Active circuit realizationActive circuit realization



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Passive circuit realization



09 elec3114

Education

integral control

john wiley sons

cascade compensators

root locus

error p p gto

response b

dynamic compensators

g cascade