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CHAPTER 6

CALCULATION OF TUNING PARAMETERS FOR

VIBRATION CONTROL USING LabVIEW

6.1 INTRODUCTION

Vibration control of rotating machinery is tougher and a challenging

challengerical technical problem. There are two major categories of vibration

control techniques for rotating machinery such as direct control using

vibration isolators and balancing control using vibration dampers. This

chapter presents the simulation of a PID control scheme and calculation of

tuning parameters. These parameters have been derived by using certain fuzzy

rules. In this research, obtaining the tuning parameters were formulated by

using a change in error for different continuous control mode gains.

6.2 FUZZY BASED PID CONTROLLER

Fuzzy logic system and its implementation in control algorithms are

illustrated in Figure 6.1. Fuzzification converts the crisp values into fuzzy

variables and defuzzification converts the fuzzy variables into crisp values.

This concept is being configured with membership functions, rule base, error

and change in error mapping, inference engine for executing computational

algorithm.

Alessandro Ferrero et al (2010) described fuzzy inference system to

indicate about uncertainties of few methods. Jia Ma et al (2009) discussed

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about controller for vibration isolation. Lianqing et al (2009) have also

discussed about control method for vibration isolation.

The following sections in this thesis describe a fuzzy logic system to

calculate tuning parameters or gains for PID controller.

A common fuzzy logic based PID controller is depicted in

Figure 6.1.

Figure 6.1 Fuzzy logic based PID controller

The basic model of the process to be controlled by a PID is a system

of n-th order (linear or nonlinear) with the state Equation (6.1) and (6.2)

x(n) = f(x , x` , …… , x(n-1) , t , u) (6.1)

y = g(x) (6.2)

FUZZIFICATION

Inference Engine

DEFUZZIFICATION

RULE

BASE

Error

Change in

Error

Kp Ki Kd

PID CONTROLLER

CONTROLLER O/P

Error

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with

x - process state vector x = ( x1,x2,…..,xn) = (x,x`,…..,x(n-1))T

y - process output

u - control variable

t - time parameter

f, g - linear or nonlinear functions

The basic idea of a PID-controller is to choose the control law by

considering Equations (6.3), (6.4), (6.5) and (6.6)

error e = x – xd (6.3)

change of error = x` - x

`d (6.4)

integral of error = . (6.5)

uPID = KP.e + KD.e` + KI. . , (6.6)

where ‘xd’ is the desired value (set-point).

For a linear process the control parameters KP, KD and KI are

designed in such a way that the closed-loop control is stable. Gustavo et al

(2003) discussed about gain and phase margins of PI controller design. The

corresponding analysis can be done by means of the knowledge of process

parameters (e.g. mass, damper, spring of a mechanical system) taking into

account special performance criteria. In case of non linear processes which

can be linearized around the operating point, conventional PID-controllers

also work successfully.

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PID-controllers with constant parameters in the whole working area

are robust but not optional. In this case, tuning of the PID-parameters has to

be performed.

A conventional PID-controller is better where,

The process is either linear or can be piecewise linearized.

The process can be stabilized taking into account selected

performance criteria.

6.3 DESIGN REQUIREMENT

Rao and Sreenivas (2003) studied the dynamic behaviour of

misaligned rotor system to perform harmonic analysis to detect the dominant

harmonic between two critical speeds. With the derivation of Gibbon, results

were analysed to find axial forces due to harmonics and hence it became

possible to predict the presence of misalignment.

The resultant method is based on harmonics study only and hence

the proposed methods resolve for simple controller.

David York et al (2011) presented a vibration isolator with single

DOF and vibration control mount investigation. The analysis and

experimental results were compared and for parametric identification and

system identification.

Though the study was made for single DOF, vibration is tougher to

control instantaneously. Fixed gain values were used in the control scheme. In

the proposed control auto tuning, parameters are computed with respect to the

instantaneous requirement of the controller.

Since Fuzzy logic is a method of rule-based decision making used

for expert systems and process control, it has been selected in the proposed

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research to obtain tuning parameters. To implement real-time decision making

or control a physical syste, the controller output of a PID controller is the

summation of the proportional, integral, and derivative actions. These are

controlled by respective gains, by considering the actual controller output

limited to the range specified for control output.

6.4 DESIGN ALGORITHM

In this research, it has been simulated with a fuzzy based PID

controller in LabVIEW for controlling the vibration. The fuzzy-PID has two

inputs, three output for fuzzy and four inputs and one output for PID.

For vibration control applications, since the set point or desired

output is drastically varying with larger and random amplitudes, a continuous

disturbance is found in the system.

PID controllers are found suitable in this aspect to bear with the

disturbance which is abrupt in nature. But tuning the controller for obtaining

quick settling, it is necessary to adopt an algorithm to correlate with the

change in error. Based on the following algorithm presented in this proposed

methodology, tuning parameters are calculated for PID controller. The

algorithm has been implemented in LabVIEW.

Step 1: Input the controller with sensory units

Step 2: Perceive the present values of Kp, Ki, Kd

Step 3: Compute the error

Step 4: Compute the change in error

Step 5: Execute the error mapping by membership function

Step 6: Determine the new values of Kp, Ki, Kd

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6.5 CONTROL FLOWCHART

The control flow chart is illustrated in 6.2 for calculating the

required tuning parameters for the PID Controller.

Figure 6.2 Flowchart for estimating tuning parameters

START

INPUT TO THE SYSTEM FROM SENSORS

COMPUTE ‘e’

REFER THE PRESENT GAIN VALUES

Kp, Ki, Kd

EXECUTE THE ERROR MAPPING BASED

TRIANGULAR MEMBERSHIP FUNCTION TO

MAP THE RANGE

COMPUTE ‘ e’

STOP

DETERMINE THE NEW VALUES FOR

Kp, Ki, Kd

IS e = 0 ? NO

YES

IS e = 0 ?NO

YES

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6.6 TUNING PARAMETER CALCULATION FOR PID

CONTROLLER

In the proposed method, calculation of tuning parameter or different

gain values required for achieving control action is being calculated. This

procedure is simulated using LabVIEW as illustrated in the succeeding

sections.

The error and the change in error are given to the fuzzy system

which calculates the appropriate Kp, Ki, Kd parameters that will be given to

the PID controllers.

6.6.1 Determining Triangular Membership Function of Error

The triangular membership function is used for all input and output

mapping. The error mapping is done using triangular membership function as

indicated in Figure 6.3. The error range is set from -10 to 10. The error is

always expected to lie around 0 (i.e. as minimum as possible) and the range

is mapped from -10 to -4, -5 to -1, -1.5 to 1.5, 1 to 5, 4 to 10.

The error is said to be critical if it lies between -10 to -4 or 10 to 4.

So a corresponding action has to be taken to reduce it. The error in middle

membership function must be maintained. The area between -5 to -4, -1.5 to -

1, 1.5 to 1, 4 to 5 are dual mapped by two membership functions. These

values l choose any value of the two membership functions.

When a linguistic variable is created to represent an input or output

variable, it decides the linguistic terms or categories of values. Linguistic

variables usually have an odd number of linguistic terms, with a middle

linguistic term and symmetric linguistic terms at each extreme. In most of the

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applications, three to seven linguistic terms are sufficient for categorizing the

values of a system.

Figure 6.3 Triangular membership function of error

6.6.2 Depicting Triangular Membership Function Change in Error

The change in error mapping is done using triangular membership

function. The error range is set from -10 to 10. The change in error is always

expected to lie around 0 (i.e., as minimum as possible) and the range is

mapped from -10 to -2, -5 to -1, -1.5 to 1.5, 1 to 5, 2 to 10.

The error is said to be critical if it lies between -10 to -2 or 10 to 2.

So a corresponding action has to be taken to reduce it. The change in error, as

shown in Figure 6.4, must be maintained in the middle membership function.

The area between -5 to -2, -1.5 to -1, 1.5 to 1, 2 to 5 are dual mapped by two

membership functions. These values will choose any value of the two

membership function.

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Figure 6.4 Triangular membership function change in error

6.6.3 Proportional Gain (Kp) Calculation

The membership function for Kp is mapped in three membership

function from 0 to 8 as indicated in Figure 6.5. The value of 0 to 4 is chosen if

the value has to be low, the value 3 to 7 is chosen if the value has to be

medium and the value 8 to 8 is chosen if the value has to be high.

Figure 6.5 Triangular membership function for KP

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6.6.4 Integral gain (Ki) CALCULATION

The membership function for Ki is mapped in three membership

function from 0 to 0.7. The value of 0 to 0.4 is chosen, as referred in

Figure 6.6, if the value has to be low, the value .3 to

Figure 6.6 Triangular membership function for Ki

6.6.5 Derivative Gain (Kd) Calculation

The value 0.8 is chosen if the value has to be medium and the value

0.8 to 0.7 is chosen if the value has to be high. The membership function for

Kd is mapped, as indicated in Figure 6.7, in three membership function from 0

to 0.05.

The value of 0 to .02 is chosen if the value has to be low, the value

.01 to .03 is chosen if the value has to be medium and the value 0.03 to 0.05 is

chosen if the value is to high.

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Figure 6.7 Triangular membership function for Kd

6.6.6 Kd Mapping with Error and Change in Error

The graph in the right indicates the Kd value that the fuzzy logic

controller will choose for the value of error and change in error, as illustrated

in Figure 6.8. The error and change in error can be varied to note the Kd value

that will be chosen, its weight and invoked rule.

Normally rules will describe, in words, the relationships between

input and output linguistic variables based on their linguistic terms. A rule

base is the set of rules for a fuzzy system. Based on this theory controller

gains are mapped with ‘error’ and ‘change in error’ and this the rule based

mapping is done for gains.

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Figure 6.8 Kd parameter with error and change in error

6.6.7 Kp Mapping with Error and Change in Error

The graph in the right indicates the Kp value that the fuzzy logic

controller will choose for the value of error and change in error, as presented

in Figure 6.9. The error and change in error can be varied to note the Kpvalue

that will be chosen, its weight and invoked rule.

Figure 6.9 Kp parameter with error and change in error

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6.6.8 Ki Mapping with Error and Change in Error

The graph in the right indicates the Ki value that the fuzzy logic

controller will choose for the value of error and change in error, as displayed

in Figure 6.10. The error and change in error can be varied to note the Ki

value that will be chosen, its weight and invoked rule.

Figure 6.10 Ki parameter with error and change in error

6.7 RULES FOR FUZZY CONTROLLER

The following rules are framed for setting the rule base for the

Fuzzy PID controller:

1. IF ‘Error’ is ‘very high’ AND ’change in error’ is ‘very high’

THEN ‘Kp’ is ‘high’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘medium’

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2. IF ‘Error’ is ‘very high’ AND ’change in error’ is ‘medium’

THEN ‘Kp’ is ‘high’ Also ‘Ki’is ’medium’ ALSO ‘Kd’ is

‘medium’

3. IF ‘Error’ is ‘very high’ AND ’change in error’ is ‘low’ THEN

‘Kp’ is ‘medium’ Also ‘Ki’ is ’medium’ ALSO ‘Kd’ is

‘medium’

4. IF ‘Error’ is ‘very high’ AND ’change in error’ is ‘high’ THEN

‘Kp’ is ‘high’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘medium’

5. IF ‘Error’ is ‘very high’ AND ’change in error’ is ‘very low’

THEN ‘Kp’ is ‘medium’ Also ‘Ki’is ’medium’ ALSO ‘Kd’ is

‘medium’

6. IF ‘Error’ is ‘high’ AND ’change in error’ is ‘very high’

THEN ‘Kp’ is ‘medium’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is

‘medium’

7. IF ‘Error’ is ‘high’ AND ’change in error’ is ‘high’ THEN ‘Kp’

is ‘medium’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘medium’

8. IF ‘Error’ is ‘high’ AND ’change in error’ is ‘medium’ THEN

‘Kp’ is ‘medium’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘medium’

9. IF ‘Error’ is ‘high’ AND ’change in error’ is ‘low’ THEN ‘Kp’

is ‘medium’ Also ‘Ki’is ’medium’ ALSO ‘Kd’ is ‘medium’

10. IF ‘Error’ is ‘high’ AND ’change in error’ is ‘very low’ THEN

‘Kp’ is ‘medium’ Also ‘Ki’is ’medium’ ALSO ‘Kd’ is

‘medium’

11. IF ‘Error’ is ‘medium’ AND ’change in error’ is ‘very high’

THEN ‘Kp’ is ‘low’ Also ‘Ki’ is ’medium’ ALSO ‘Kd’ is ‘low’

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12. IF ‘Error’ is ‘medium’ AND ’change in error’ is ‘high’ THEN

‘Kp’ is ‘low’ Also ‘Ki’ is ’medium’ ALSO ‘Kd’ is ‘low’

13. IF ‘Error’ is ‘medium’ AND ’change in error’ is ‘medium’

THEN ‘Kp’ is ‘low’ Also ‘Ki’ is ’low’ ALSO ‘Kd’ is ‘low’

14. IF ‘Error’ is ‘medium’ AND ’change in error’ is ‘low’ THEN

‘Kp’ is ‘low’ Also ‘Ki’ is ’low’ ALSO ‘Kd’ is ‘medium’

15. IF ‘Error’ is ‘medium’ AND ’change in error’ is ‘very low’

THEN ‘Kp’ is ‘low’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘medium’

16. IF ‘Error’ is ‘low’ AND ’change in error’ is ‘very high’ THEN

‘Kp’ is ‘low’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘low’

17. IF ‘Error’ is ‘low’ AND ’change in error’ is ‘high’ THEN ‘Kp’

is ‘low’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘low’

18. IF ‘Error’ is ‘low’ AND ’change in error’ is ‘medium’ THEN

‘Kp’ is ‘low’ Also ‘Ki’ is ’low’ ALSO ‘Kd’ is ‘low’

19. IF ‘Error’ is ‘low’ AND ’change in error’ is ‘low’ THEN ‘Kp’

is ‘low’ Also ‘Ki’is ’medium’ ALSO ‘Kd’ is ‘low’

20. IF ‘Error’ is ‘low’ AND ’change in error’ is ‘very low’ THEN

‘Kp’ is ‘low’ Also ‘Ki’is ’medium’ ALSO ‘Kd’ is ‘low’

21. IF ‘Error’ is ‘very low’ AND ’change in error’ is ‘very high’

THEN ‘Kp’ is ‘low’ Also ‘Ki’, is ’low’ ALSO ‘Kd’ is ‘low’

22. IF ‘Error’ is ‘very low’ AND ’change in error’ is ‘high’ THEN

‘Kp’ is ‘low’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘low’

23. IF ‘Error’ is ‘very low’ AND ’change in error’ is ‘medium’

THEN ‘Kp’ is ‘low’ Also ‘Ki’is ’low’ ALSO ‘Kd’ is ‘low’

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24. IF ‘Error’ is ‘very low’ AND ’change in error’ is ‘low’ THEN

‘Kp’ is ‘low’ Also ‘Ki’is ’medium’ ALSO ‘Kd’ is ‘low’

25. IF ‘Error’ is ‘very low’ AND ’change in error’ is ‘very low’

THEN ‘Kp’ is ‘low’ Also ‘Ki’ is ’medium’ ALSO ‘Kd’ is ‘low’

The above said rules have been used to derive auto tuning parameter

of PID controller. These parameters Kp, Ki, Kd will be determined

instantaneously with respect to the present status of the variable to be

controlled. LabVIEW front panel for the above controller has been shown in

Figure 8.10.

6.8 CALCULATED TUNING PARAMETERS FROM THE

PROPOSED ALGORITHM

Derived tuning parameters from the proposed algorithm are

tabulated in Table 6.1. These values indicate a small instantaneous change in

the values to adopt the continuous changes in the system. This fetches the

advantages as an adaptive auto tuning of gains over the fixed value

controllers.

Table 6.1 Adaptive Tuning parameters for PID controller

S.No Kp Ki Kd

1 7.4393 0.009361 0.001498

2 7.5944 0.01944 0.00152

3 7.5122 0.009494 0.00089

4 7.4492 0.00899 0.0015

5 7.4552 0.00923 0.00921

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Figure 6.11 and 6.12 depicted as graphs, present the Kp, Ki and Kd

values over a stipulated number of iterations in the system. It is found to be

continuously adopting within the range.

Figure 6.11 Kp Estimation

Figure 6.12 Ki , Kd Estimation

7.35

7.4

7.45

7.5

7.55

7.6

7.65

Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5

Kp

Kp

0

0.005

0.01

0.015

0.02

0.025

Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5

Ki

Kd

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6.9 SIMULATION ON CALCULATED CONTROLLER GAINS

WITH PIDCONTROLLER

Tuning parameters calculated in the above said algorithm can be

implemented in a real time system. In this proposed methodology, it is

simulated with a randomly simulated value as shown in the front panel, in

Figure 6.13..

This LabVIEW front panel has two modes for manually entering

tuning parameters and tuning the PID controller with autotuned parameters.

Set Point (SP), PV (Process Variable) and MV(Manipulated Variable) are

graphically shown for their responses.

Besides, in a separate numeric indicator, three tuning parameters

have also been displayed. The total loop is configured in an iterative loop

(WHILE structure in LabVIEW) to control the execution without affecting the

hardware interface.

Figure 6.13 LabVIEW front panel for PID controller with tuning

parameters

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6.10 ENHANCEMENT COMPARISON OF PROPOSED

CONTROL SCHEME

Rao and Sreenivas (2003) studied the misaligned rotor system and

control estimation was done by considering harmonics present in the signal.

The proposed method is well organised with 6 sensors to perceive the

complete signal with spectral density and it proposed a continuous mode

control scheme.

David York et al (2011) dealt with a vibration isolator with fixed

gain controller. The proposed control scheme describes a commonly specified

controller with auto tuning parameters for taking care of instantaneous

‘Change in Error’ and ‘Error’ in the system. It adjusts the gains in an adaptive

fashion.

6.11 MAJOR VIBRATION CONTROL COMPONENT FOR

ROTATING MACHINERY

The next section enlists the components for vibration control and

they serve suggestions.

6.11.1 Vibration Absorbers

Under certain conditions, the amplitude of vibration of the mass that

is being excited can be reduced to zero, while the second mass continues to

vibrate. If a particular system has a having large vibration under its

excitation, this vibration can be eliminated by coupling a properly designed

auxiliary spring-mass system to the main system. This forms the principle of

undamped dynamic vibration absorber where the excitation is finally

transmitted to the auxiliary system, bringing the main system to rest.

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Absorber is extremely effective at one speed only and thus is

suitable only for constant speed machines. A damped dynamic vibration

absorber can take care of the entire frequency range of excitation but at the

cost of reduced effectiveness. This will not be treated in this work as it is not

within its scope.

Various other types of absorbers have been used under various other

conditions. These are subsequently introduced in the following sub-sections.

6.11.2 Vibration Isolation

The principal of vibration isolation is an ideal case since the motion

of the block was considered in one direction only. In practice, the exciting

forces, in a machine produce motions in all directions and it becomes

necessary to consider the coupling between different modes for an effective

study. Chida et al (2004) discussed about vibration isolation controller by

frequency shapping and Giuseppe et al (2008) explained about dampers. Jia

Pengxiao et al (2010) explained about fuzzy control for vibration isolation.

Lin Yan et al (2012) narrated about vibration isolation study. Song et al

(2009) presented about magnetic suspension isolator. Yue Wenhui et al

(2011) correlated the role of computer based instrument and Zhan Xueiping

et al (2011) described about damper characteristics.

This isolation system has been designed to produce a symmetry in

two planes. Such a system is not only simpler to analyse but it also gives

isolation characteristics.

The system is symmetrical about both the vertical planes x-y and

y-z. The transmissibility characteristics for this mode are identical for the

single degree of freedom system.

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6.12 SUMMARY

Though the vibration in rotating machinery is tougher to control

mechanically, a control scheme has been suggested in this research. This

chapter depicted the methodology and rule base used in fuzzy inference

system to calculate tuning parameters in a dynamic fashion. LabVIEW based

program has been highlighted to invoke the way of control scheme with

tuning parameters with respect to the instantaneous vibration signals.