mrac pid documentation

7/25/2019 Mrac Pid Documentation

1/33

AUTOMATIC TUNING OF PID CONTROLLER USING

MODEL REFERENCE ADAPTIVE CONTROL

The mini Project Report submitted in partial fulfillment of the requirements

For the award of the Degree in Bachelor of Technology

In

(Electrical and Electronics Engineering

By

B.!" #$%F&'$($)* +.,-ID&&P #$%F&'$((*

/.PR-+I #$%F&'$(01* 2.3RT"I3 #$%F&'$((4*

5nder the esteemed 2uidance of

Mr!G!R!S!Naga "#$ar% M!Tec&%

ssistant Professor

De'art$ent o Electrical and Electronics Engineering

VIGNAN)S LARA INSTITUTE OF TEC*NOLOG+ , SCIENCE

(Ailiated to -a.a&arlal Ne&r# Tec&nological Uni/ersit0% "a1inada

Vadla$#di 2 344 456

G#nt#r District% And&ra Prades&

475584754!


2/33

VIGNAN)S LARA INSTITUTE OF TEC*NOLOG+ ,

SCIENCE

(Ailiated to -a.a&arlal Ne&r# Tec&nological Uni/ersit0% "a1inada

Vadla$#di 2 344 456

G#nt#r District% And&ra Prades&

De'art$ent o Electrical and Electronics Engineering

C E R T I F I C A T E

This is to certify that this dissertation entitled as 6AUTOMATICTUNING OF PID CONTROLLER USING MODEL REFERENCE ADAPTIVE

CONTROL9is the bonafide wor7 of8 B.sha8 /.Prana9i8 +.,anideep8 2.3arthi7 submitted to

the Department of &lectrical and &lectronics &ngineering8 VIGNAN9S LARA INSTITUTE

OF TEC*NOLOG+ AND SCIENCE in partial fulfillment of the requirements for the

award of the Degree in Bachelor of Technology in &lectrical and &lectronics &ngineering.

Project 2uide "ead of the Department

2.R.!.-aga 3umar8 sst.Prof. &lectrical and &lectronics &ngineering


3/33

D E C L A R A T I O N

:e8 the students of +ignan;s


4/33

AC"NO;LEDGEMENT

:e would li7e to e=press our deep gratitude to our belo9ed director

V!MAD*USUD*ANA RAO% for his cooperation in completing our project.

:e found immense pleasure in e=pressing our gratitude and hearty things to

A!N!VEN"ATES;ARLU%"ead of &lectrical and &lectronics &ngineering department for

his timely help throughout the project schedule and the course of study.

:e wouldli7e to e=press our sincere than7s to ourbelo9ed guide G!R!S!Naga

"#$ar. for his acti9e participation and e=cellent guidance at e9ery stage and for highdynamic and moti9ate encouragement in successfully completing this project.

Finally8 we e=press our sincere than7s to all the faculty and staff members and

also another man who are directly or indirectly in9ol9ed and support in completing our

project.

B. !"

/. PR-+I

+. ,-ID&&P

2. 3RT"I3


5/33

I-D&>

bstract


6/33

0.(.( Auter loop

0.0 ?omponents

0.0.' Reference model

0.0.( ?ontroller

0.0.0 dapti9e mechanism

0.4 ?onclusion

?"PT&R @ 4 PID ?A-TRA


7/33

1.1.' ,athematical analysis of D? motor

1.C ?onclusion

?"PT&R @ C !I,5


8/33


9/33


10/33

B!TR?T

This project discusses the application of ,odel Reference dapti9e ?ontrol concepts

to the automatic tuning of PID controllers. The effecti9eness of the proposed method is

shown through simulated applications. ,odel Reference dapti9e ?ontrol algorithm is

proposed. This algorithm preser9es all the best features of ,R? such as


11/33


12/33

n autotuning method is proposed using ,odel Reference dapti9e

?ontrol #,R?* concepts and the ,IT rule. ,any adapti9e control techniques can be used

to pro9ide automatic tuning. In such applications the adaptation loop is simply switched for a

period of time when tuning is required. Perturbation signals are normally added to impro9e

parameter estimation. The adapti9e controller is run until the performance is satisfactoryH then

the adaptation loop is disconnected8 and the system is left running with fi=ed controller

parameters.

'.1 ?A-?


13/33

I!TI-2 !K!T&,E

The timeEhonoredGieglerE-ichols tuning rule #"Z-N rule"*8 as introduced in

the '4$s8 had a large impact in ma7ing PID feedbac7 controls acceptable to controlengineers. PID was 7nown8 but applied only reluctantly because of stability concerns. :ith

the GieglerE-ichols rule8 engineers finally had a practical and systematic way of tuning PID

loops for impro9ed performance.

(.0 DI!D+-T2&! AF &>I!TI-2 !K!T&,E

The disad9antages of e=isting GieglerE-ichols method are as follows.
http://www.mstarlabs.com/control/znrule.html#Ref2http://www.mstarlabs.com/control/znrule.html#Ref2http://www.mstarlabs.com/control/znrule.html#Ref2


14/33

'. Further fine tuning is needed.

(. ?ontroller settings are aggressi9e8 resulting in large o9ershoot and oscillatory responses.

0. Poor performance for processes with a dominant delay.

4. ?losed loop 9ery sensiti9e to parameter 9ariations.

1. Parameters of the step response may be hard to determine due to measurement noise

(.4 PRAPA!&D !K!T&,E

The adapti9e control process is one that continuously and automatically

measures the dynamic beha9ior of plant8 compares it with the desired output and uses the

difference to 9ary adjustable system parameters or to generate an actuating signal in such a

way so that optimal performance can be maintained regardless of the system changes. The

nature of the adaption mechanism for controlling the system performance is greatly affected

by the 9alue of adaption gain. s compared to the fi=ed gain controllers8 the adapti9e

controllers are 9ery effecti9e to handle the situations where the parameter 9ariations and

en9ironmental changes are frequent.

(.1 ?A-?


15/33

B


16/33

Controller

ReferenceModel Adjustment

MechanismPlant

ControllerParam

eters

y

m

od

el

u y

p

l

a

n

t

u

c

Fig 0.' bloc7 diagram of ,R?

0.0 ?A,PA-&-T!

The 9arious bloc7s in the ,odel Reference dapti9e ?ontroller are

'. Reference model

(. ?ontroller

0. daption mechanism

0.0.'. Reference modelE

It is used to specify the ideal response of the adapti9e control system to

e=ternal command. It should reflect the performance specifications in control tas7s. The ideal

beha9ior specified by the reference model should be achie9able for the adapti9e control

system.

0.0.(. ?ontrollerE

It is usually parameteried by a number of adjustable parameters. The control

law is linear in terms of the adjustable parameters #linear parameteriation*. dapti9e

controller design normally requires linear parameteriation in order to obtain adaptation

mechanism with guaranteed stability and trac7ing con9ergence. The 9alues of these control

parameters are mainly dependent on adaptation gain which in turn changes the control

algorithm of adaptation mechanism.


17/33

0.0.0 daptation mechanismE

dapti9e control in9ol9es modifying the control law used by the controller tocope with the fact that the parameters of the system being controlled change drastically due to

change in en9ironmental conditions or in system itself. This technique is based on the

fundamental characteristic of adaptation of li9ing organism. The adapti9e control process is

one that continuously and automatically measures the dynamic beha9ior of plant8 compares it

with the desired output and uses the difference to 9ary adjustable system parameters or to

generate an actuating signal in such a way so that optimal performance can be maintained

regardless of system changes. -ature of adaptation mechanism for controlling the system

performance is greatly affected by the 9alue of adaptation gain. It is obser9ed that for the

lower order system wide range of adaptation gain can be used to study the performance of the

system. s the order of the system increases the applicable range of adaptation gain becomes

narrow. The rule which is used for this application is ,IT rule. !imulation is done in

,T


18/33

?"PT&R E 4

PID ?A-TRA


19/33

The integral term accelerates the mo9ement of the process towards setpoint and eliminates

the residual steadyEstate error that occurs with a pure proportional controller. "owe9er8 since

the integral term responds to accumulated errors from the past8 it can cause the present 9alue

to o9ershootthe setpoint 9alue

4.4 D&RI+TI+&

The deri9ati9eof the process error is calculated by determining the slope of

the error o9er time and multiplying this rate of change by the deri9ati9e gain Kd. The

magnitude of the contribution of the deri9ati9e term to the o9erall control action is termed the

deri9ati9e gain8Kd. The deri9ati9e term is gi9en by

The deri9ati9e term slows the rate of change of the controller output. Deri9ati9e control

is used to reduce the magnitude of the o9ershoot produced by the integral component and

impro9e the combined controllerEprocess stability. "owe9er8 the deri9ati9e term slows

the transient responseof the controller. lso8 differentiation of a signal amplifies noise

and thus this term in the controller is highly sensiti9e to noise in the error term8 and can

cause a process to become unstable if the noise and the deri9ati9e gain are sufficientlylarge.

4.1 PID ?A-TRA


20/33

change. The weighted sum of these three actions is used to adjust the process 9ia a control

element such as the position of acontrol 9al9e8 or the power supplied to a heating element.

Fig 4.' bloc7 diagram of PID controller

The mathematical equation for the PID controller is gi9en is

u ( t)=Kpep ( t)+Ki ei (T) dT+Kd(ded ( t)

dt )In the absence of 7nowledge of the underlying process8 a PID controller is the best controller.

By tuning the three parameters in the PID controller algorithm8 the controller can pro9ide

control action designed for specific process requirements. The response of the controller can

be described in terms of the responsi9eness of the controller to an error8 the degree to which

the controller o9ershootsthe setpoint and the degree of system oscillation.

4.C ?A-?


21/33

!P&&D ?A-TRA< AF D? ,ATAR

1.' I-TRAD5?TIA-

This chapter gi9es the information about the mathematical analysis of the ,IT

rule8 tuning of the PID controller and the modeling of the D? motor

1.( ,T"&,TI?< -K!I!E

The idea behind the ,odel Reference dapti9e ?ontrol is to create a closed

loop controller with parameters that can be updated to change the response of the system to

match a desired model. There are many different methods for designing such a controller.

:hen designing an ,R? using the ,IT rule8 the designer chooses

The reference model8 the controller structure and the tuning gains for the

adjustment mechanism.

1.(.' ,IT ruleE

modelplant yye =

,R? begins by defining the trac7ing error8 e. This is

simply the difference between the plant output and the reference model output.

,athematically it is defined as

:here e M trac7ing error8

yplant M output of the plant

ymodelM output of the model

*#(

'*# ( eJ =

From this error a cost function of theta #/#theta** can be formed. / is gi9en as a

function of theta8 with theta being the parameter that will be adapted inside the controller. The


22/33

choice of cost function will later determine how the parameters are updated. Below8 a typical

cost function is ta7en.

To find out how to update the parameter theta8 an equation needs to be formed for the change

in theta. If the goal is to minimie the cost related to error8 it is sensible to mo9e in the

direction of the negati9e gradient of /. The change in / is assumed to be proportional to the

change in theta. Thus8 the deri9ati9e of theta is equal to the negati9e change in /. The result

for the cost function chosen abo9e is gi9en as


23/33

:here u#t*M input of the plant

7p M proportional constant

7i M integral constant

7dM deri9ati9e constant

uc M input to the controller

The transfer function of the closed loop plant along with the controller is gi9en by

The transfer function of the reference plant is gi9en by

The error is the difference between the plant output and the reference model output which is

defined mathematically as

:here e M error

y M output of the plant

ymM output of the model

The parameters of the controller which are to be tuned are

1.4 DPTI+& ,&?"-I!,E

The ,IT rule is a gradient scheme that aims to minimie the squared error e (

The updated PID controller parameters


24/33

1.1 ,AD&


25/33

+aM input 9oltage in 9olts

&bM bac7 emf in 9olts

T M torque in -Em

If M field current in amperes

/ M moment of inertia in

b M damping friction

1.1.' ,athematical nalysisE

For normal operation8 the de9eloped torque must be equal to the friction and the inertia. Theequation for the de9eloped torque is gi9en by

The armature circuit8

The bac7 emf is gi9en by

The torque de9eloped by the motor is


26/33

The 9arious parameters and their 9alues of the gi9en D? motor is as follows.

Parameter !ymbol +alue

rmature Resistance Ra $.C

rmature Inductance


27/33

?"PT&R E C

!I,5


28/33

Fig C.' !imulation diagram of PID controller

C.0 R&!5


29/33

0 50 100 150 200 250 300 350 400

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

TIME (Sec)

Amplitude

MRAC PID CONTROLLER FOR SQUARE INPUT

Plant output

Referene output

Fig C.( Autput of PID controller for square wa9e input

The 9ariation of the different parameters #3p8 3i8 3d* for the square wa9e input is gi9en as


30/33

0 100 200 300 400 500 600 700 800 900 1000-1

-0.5

0

0.5

Kp

Parameter variations of Square wave input

0 100 200 300 400 500 600 700 800 900 1000-2

-1

0

Ki

Kp

0 100 200 300 400 500 600 700 800 900 1000-5

0

5x 10

-3

Time (sec)

Kd

Kd

Ki

Fig C.0 Autput of 9arious parameters#7p8 7i8 7d*


31/33

The 9ariation of different parameters for the sinusoidal input is gi9en by

0 10 20 30 40 50 60 70 80 90 100-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

MRAC PID Controller for Sinusoidal Input

Time (Sec)

Amplitude

Reference Output

Plant output

Fig 6.4 Out put of PID controller for sine a!e input


32/33

"he !ariation of di#erent parameters for s$uare a!e input is gi!en as%

0 10 20 30 40 50 60 70 80 90 100-1

0

1

Parameter variations for sinusoidal input

Kp

0 10 20 30 40 50 60 70 80 90 100-2

-1

0

Ki

0 10 20 30 40 50 60 70 80 90 100-5

0

5x 10

-3

Time (Sec)

Kd

Kp

Ki

Kd

Fig 6.& Autput of 9arious parameters#7p8 7i8 7d*

6.4 CO'C()*IO'%+

In this chapter the simulation diagram and the results for 9arious parameters

are shown.

C,AP"-R /

CO'C()*IO'


33/33

The automatic tuning of PID controllers has been in9estigated using

,odel Reference dapti9e ?ontrol concepts and the ,IT rule. !imple adaptation laws for the

controller parameters ha9e been presented assuming that the process under control can be

appro=imated by a first order transfer function. The de9eloped adaptation rules ha9e been

applied to a first order system with a second order reference model. Furthermore8 the

proposed technique has been applied to a simulated D? motor8 which has nonElinear

dynamics. The results obtained show the effecti9eness of the technique. The resulting

performance could be impro9ed by a better choice of the length of the adaptation period. The

,IT rule is 7nown to ha9e its disad9antages. First8 the speed of adaptation depends on the

9alues of the command signal. This problem is often dealt with by using a normalied

adaptation rule. !econd8 the ,IT rule does not guarantee the stability of the nominal system.

The

mrac pid documentation

Documents