Project 3 Final Report (2)

8/8/2019 Project 3 Final Report (2)

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Modelling, Feedback Control Design and Simulation

of an Industrial ApplicationDean P. Stavrou#1

#Electrical Engineering Dept., Cleveland State University

2121 Euclid Ave, Cleveland Ohio 44115-2214, USA

[email protected]

Abstract These instructions outline the process and reasoningused to design and model a feedback control system to a

positioning control system for an industrial motion application.The system must be designed so that a load can be moved 12inches in 0.3 seconds with an accuracy of 1% or better. To meet

the design requirements a negative feedback control system anda PID controller will be utilized.

I. INTRODUCTION

Modern control theory is based on the same principles from

the late 1700s. James Watt invented his centrifugal governor

in 1788. This device utilized feedback control to regulate the

amount of fuel given to an engine to maintain a constant speed.This device paved the way for the internal combustion engine

and modern control theory.

Negative feedback control is used to establish equilibrium

in a system by feeding back the systems output to the input.

This feedback is measured to the reference of the system andwill produce an error which is then adjusted by a controller.

This error is continuously adjusted to try to reduce it to its

minimum, while also keeping the system stable. Negative

feedback control will be used for the positioning control

system for an industrial motion application.

A Proportional-Integral-Derivative or PID controller is a

mechanism used for feedback control. A PID controller

corrects the error between a measured process variable and a

desired setpoint by calculating then outputting a correctiveaction that can adjust the process accordingly.

The PID controllers algorithm consists of three parameters,

the Proportional, Integral and Derivative gains. TheProportional gain determines the reaction to the current error,

the Integral gain determines the reaction based on the sum of

recent errors, and the Derivative gain determines the reaction

to the rate at which the error has been changing. The weighted

sum of these three parameters is used to adjust the process via

a control element, which in this case would be load position.

When calculating transfer functions for control systems

including a PID controller, the controller can be represented

mathematically by the equation

This can be simplified by taking the Laplace Transform

yielding

The positioning control system consists of a digitalcontroller, a DC motor drive and a load of 235 lbs which is to

be moved linearly by 12 inches in 0.3 seconds with an

accuracy of 1% or better.

II. SYSTEM PARAMETERS

The part selection, system parameters and plant model for

the positioning control system has already been completed by

Professors Donald Zeller and Jack Zeller. Their results can be

found below

A.Electrical ParametersWinding resistance and inductance: Ra = 0.4 La = 8 mH

(The transfer function of the armature voltage to the current is

Back EMF constant: Power Amplifier Gain: Current Feedback Gain:

B.Mechanical ParametersTorque Constant:

Motor Inertia: Pulley Radius: Load Weight: Total Inertia:


2/21


3/21

There are no zeros forthis transfer function.

VII. TIME CONSTANT, SETTLING TIME & OVERSHOOT FOR

For since , is much larger

than , P1is the dominant pole which meansthat P2 can be disregarded.

Therefore

Since transfer function is a second order, overdampedsystem, which means there are two real poles, there is no

overshoot.

VIII. VERIFICATION

The transfer functions for and

can be verified by

using Matlabs Simulinktool. This can be done by setting up a

simulation of the derived transfer function along with the

block diagram model ofthe actual system. Then a mux block

can be used to impose the two outputs on top of each other.

This is will show if the transfer function is a correct

approximation to the actual system.

Figure 6:Block diagram ofthe system coupled with the

derived transfer functions for and

.

A step function was used as the reference with the step time

setto 1, and the step size setto 10. By observing the graphs of

the outputs it can be seen thatthe transfer functions are indeed

an accurate model ofthe system.

Figure 7: Graph ofthe Transfer Function & Block

Diagram

Figure 8:

Graph ofthe Transfer Function & Block

Diagram

IX.CLOSING THE LOOP WITH A PID CONTROLLER

Now that the system operation has been verified, the loop

can be closed and control can begin. To simulate the

implementation of a PID controller with this system is an easy

task. Again Matlabs Simulink software will be used. The

following block diagram can be set up in Simulink and used to

simulate actual operation.

Figure 9: PID Implementation ofthe Position Control System

The design requirement forthis system is to move the load

10 inches in 0.3 seconds.

0 500 1000 1500 2000 25000

100

200

300

400

500

600

700

Position(in)

Time (ms)

Vout/Vc Output

Block Diagram

Transfer Function

0 500 1000 1500 2000 25000

1000

2000

3000

4000

5000

6000

Time (ms)

Position(in)

Xout/Vc

Block Diagram

Transfer Function

Reference

la

Xout

Vout

Step

Vc

Torque Di

turbance

Xout

Vout

la

Plant

PID

PID Controller

0

Constant

s + 800s + 2458.5s3 2132000

To X

ut Tran

erFun

ti

n

+800

+2458.52 132000

To V

ut Tran

erFun

ti

n

1.25

Tran

erF n51

Tran

erF n4

1

Tran

erF n3

2.5.02

+1Tran

erF n2

2.5.02 +1

Tran

erF n1

Step

S

pe5

S

pe2

1Gain8

13.2Gain7

80Gai n6

.075Gain5

1.49Gain4

1.25Gai n31Gain2

.075Gai n11

1.49Gain10

13.2Gain180Gain


4/21

The first step is to observe the response of the system with

no error correction. This means with the P set to 1, I and D set

to 0. Doing this will yield the following result.

Figure 10: Kp set to 1, Effectively No PID Control

It can be seen from the graph that the system response is

severely overdamped, but does achieve stability after

approximately 1 second. To correct this and meet the design

requirements the PID controllers parameters must be tuned to

the proper values.

X. MANUAL TUNING OF A PID CONTROLLER

Selection of the proper PID values can be done in two ways.

The first way is to do it mathematically. This can be done by

calculating the closed loop transfer function for the system

including the PID controller. Since the transfer function of the

system

, has already been calculated, it can be used inconjunction with the algorithm for the PID controller to attain

the transfer function.

This simplifies to,

This equation can then be further simplified by eliminating the

ki term which means setting the I parameter of the PID to zero.

Doing this yields,

This simplifies to,

There are now only two parameters that must be tuned, P and

D; this is a PD controller instead of a PID. It is very common

to not use all three parameters of the PID controller.

To further simplify this equation the D parameter can be set

to 1. Leaving only the P parameter to be calculated, this can

be done using many different mathematical methods.

The second way to select PID parameters is to tune the

parameters manually, this is the method selected for this

control system.

Initially the I parameter was set to 0; the P and D

parameters were set to 1. Simulating this, the following

response was attained.

Figure 11: PD Controller Response; KP & KD = 1

This is a much better response than the previous output

with no PID control, but this still does not meet the design

requirements of 12 inches in 0.3 seconds. To meet this

requirement the P parameter was increased by 1 until the

requirement was met. When the P parameter was set to 25, the

design requirement was met.

Figure 12: PD Controller Response, KP = 25 & KD = 1

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

16

18

P

s

t

!

(

!

)

"#

$

e ($

s)

Ne%

at#

ve&

eedback C' (

t) '

0

W#

th' 1

t PI2

C' (

t) '

0 0

e)

L'

a d V e0

'

c#

ty

0 500 1000 1500 2000 2500 30000

1

2

3

4

5

6

7

8

9

10

11

P

s

t

!

(

!

)

"#

$

e ($

s)

&

eedback C' (

t) '

0

W#

th P2

C' (

t) '

0 0

e)

L'

a d V e0

'

c#

ty

0 500 1000 1500 2000 2500 30000

2

4

6

8

10

12

" #

$

e ($

s)

P

st

!

(

!

)

&

eedback C' (

t) '

0

W#

th P2

C' (

t) '

0 0

e)

L'

a d V e0

'

c#

ty


5/21


6/21

seen visually by using a tool built into Matlab called

SISOTOOL.

The SISO Design Tool is a graphical user interface (GUI)

that allows you to analyse and tune SISO feedback control

systems. Using the SISO Design Tool, you can graphically

tune the gains and dynamics of the compensator (C) and

prefilter (F) using a mix of root locus and loop shaping

techniques.

The first step in using this tool is to set-up the transfer

function in Matlab. This can be done in various ways. Themethod used here is to define s as a transfer function andthen define H(s). Then the function can be passed into

SISOTOOL using a simple command. The coding for this

operation is very simple and, is as follows.

s = tf(s);

H = (207.05/(s*(s+3.09)))

sisotool(H)

After inputting these commands SISOTOOL loads and

displays the Root-Locus plot of the transfer function as well as

the open-loop Bode plots for gain and phase. Also using the

SISO Design Tool, design constraints can be added to provide

visual representation of where poles or zeros can be placed to

achieve the desired outcomes. Setting the constraints of

settling time (TS) less than 0.3 seconds and overshoot of lessthan 2%, the Root-Locus plot will display right and wrong

locations for pole/zero placement (Figure 17).

Figure 17: Root-Locus and Bode Plots from SISOTOOL

From Figure 15 it can been observed that a pole/zero

placement anywhere before -13 on the real axis will yield

undesired results, as well as any damping coefficient () less

than 0.78. These two values will aid in the tuning of the PID

controller to meet the design requirements.

C.PID Parameter SelectionThe PID can now be properly tuned by using the

information obtained from the SISO Design Tool. First thoughthe PID parameters must be incorporated into the plants

transfer function. Doing this will yield the closed loop transfer

function of ,

Since it has already been established that an integral gain does

not apply in this situation the transfer function simplifies to,

Now the Kd parameter can be set equal to 1 and the Kp

parameter can be set to 13. This will place a zero at -13 whichis what the Root-Locus plot from the SISO Design Tooldisplayed.

This simplifies to,

Effectively changing the poles locations to

Therefore the new transfer function is

Since the pole at -161.27 is ten times further away from the

pole at -13.9 it can be discarded as a non-dominant pole. This

will simplify the transfer function even further.

Now it is time to set the PID parameters in Matlab and

evaluate the systems response (Figure 18). Plugging in Kp =

13 and Kd = 1 into the simulation for the positioning controlsystem yields the output seen in Figure 19.

Figure 18: Matlab Block Diagram

la

Xout

ou t

simout

To

ors

ace

Vc

Tor

ue

istur

ance

Xout

Vout

la

Subs

stem

Step

PID

PIDon troller

onstan t


7/21

Figure 19: PD Controller Response, KP = 13 & KD = 1

Zooming into the plot the, it can be seen that the load does

move 12 in 0.3 seconds (Figure 20).

Figure 20: Zoomed In Plot of the System Response

Our design requirement has been met the load is moved 12

inches in 0.3 seconds with no overshoot.

XII. TRAPEZOIDAL PROFILE

In industry a basic step function is never used as a reference.

Instead a Trapezoidal Profile is used to simulate systems. The

Trapezoidal Profile can be set up in Matlab by using a 1-D

lookup table and setting up the input vector as [0 .1 .2 .3 .4]and the output vector as [0 1.5 1.5 0 0].

Figure 21: Block Diagram of the Trapezoidal Profile

Figure 22: 1D-Lookup Table Settings for TrapezoidalProfile

Figure 23: Plot of the Trapezoidal Profile

0 500 1000 1500 2000 2500-2

0

2

4

6

8

10

12

14

Pos

ition(in)

Time (ms)

Fe e

ack

ontrol

it

PD

ontroller

oa

Position

50 100 150 200 250 300 350 400 450 500 550-2

0

2

4

6

8

10

12

Position(in)

Time (ms)

Fe e

ack

ontrol

it

PD

ontroller

oa

Position

Trapezoid

simout

To Workspace

Scope

1/0.3

Normalzing

Gain

1

s

Integrator

12

Final

Position

10

Clock

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

Trapezoi

al P rofile

eference

Time (s)

Position(in)


8/21

Figure 24: Plot of a Basic Step Response

Figure 25: Step Response VS Trapezoidal Profile

Now by using the Trapezoidal Profile as the referenceinstead of the basic step function for the control system, the

control system will be just as it would be in industry.

Figure 26: Trapezoidal Profile Used As Reference

Figure 27: Trapezoidal Profile PD Controller Response,

KP = 13 & KD = 1

Figure 28: Trapezoidal Profile Vs Step Input, PD Controller

Figure 29: Zoomed In View Of Trapezoidal Profile VS

Step Input Plot

By observing Figure 29, the Trapezoidal Function does

change the system response. There is now some overshoot but

it is below the 2% design constraint. Another observance that

can be made is that both responses settle at the same time, and

this also stayed within the constraint of 0.3 settling time.

Figure 30: Trapezoidal Profile VS Step Input For Vout

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

Time (s)

Basic Step

eference

Pos

ition(in)

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

Time (ms)

Position(in)

Step

nput V s. Trapezoi

al P rofile

nput

Step

nputTrapezoi

al P rofile

la

Xout

Vout

Trapezoid

simout

To Workspace

Vc

Torque Disturbance

Xout

Vout

la

Subsystem

PID

PIDController

1/0.3

Normalzing

Gain

1

s

Integrator

12

Final

Position 0

Constant

10

Clock

0 500 1000 150 0 200 0 25000

2

4

6

8

10

12

Time (ms)

Position

(in)

PD Res ponse With Trapezoidal Profile As Reference

Load Position

0 500 1000 150 0 200 0 25000

2

4

6

8

10

12

System S tep Response V s S ystem Trapezoidal Profi le

Time (ms)

Position

(in)

Trapezoidal P rofile

Step Response

15 0 200 250 300 350 400 450 500 5 50

0

2

4

6

8

10

12

System S tep Response V s S ystem Trapezoidal Profi le

Time (ms)

Position

(in)


Step Response

0 5 0 100 150 2 00 250 300 350 400 4 50 500-20

0

20

40

60

80

100

120Trapezoidal Vout VS Step Input Vout

Time (ms)

Velocity

(in/s)


Step Input


9/21


10/21


11/21

XVI. SINUSOIDAL TORQUE DISTURBANCE

Adding in a sinusoidal torque disturbance at 10% of the

maximum torque (132 in-lbs) will change the system response

greatly depending on the frequency of the sinusoid. By

observing the loop gain transfer function of the system andthen observing the bode plot of the transfer function (Figure

41) it can be observed that the system is acting as a low pass

filter.

At any frequency, the system will remain stable and within

the accuracy requirement of 1%. As the frequency of the

sinusoid increases the system will remain stable until it hits a

high enough frequency which will cause attenuation of the

sinusoid. By observing Figures 42 through Figures 47 it can

be seen that the system will in-fact never become unstable.

The outside disturbance will only cause vibrations which canbe accounted for by using mechanical components with a high

tolerance for these vibrations.

Figure 41: Sinusoidal Torque Disturbance, Freq = 1 rad/s




Bode Diagram

Frequency (rad/sec)

-6 0

-4 0

-2 0

0

20

System: G

Frequency (rad/sec): 4 .05

Magnitude (dB): -19.3

System: G



System: G


Magnitude (dB): 13.3

Magnitude(dB)

System: G



10-1

100

101

0

45

90

13 5

18 0

Phase(deg) System: G

Phase Margin (deg): -33.8

Delay Margin (sec): 0.56

At frequency (rad/sec): 10 .2

Closed Loop Stable? Yes

200 400 600 800 1000 1200 1400 1600 1800

11.3

11.4

11.5

11.6

11. 7

11.8

11.9

12

12.1

12.2

Time(ms)

Position(in)

Sinusoidal Torque Disturbance V s No Torque Disturbance

Torque Disturbance = 0

Torque Disturbance,f = 1 rad/s

200 400 600 800 1000 1200 1400 1600 1800

11.85

11.9

11.95

12

12.05

12.1

12.15

12.2

Time(ms)

Position(in)




200 400 600 800 1000 1200 1400 1600 1800 2000

11.8

11.85

11.9

11.95

12

12.05

12.1

12.15

Time(ms)

Position(in)





12/21


Figure 46: Sinusoidal Disturbance, Freq = 1000 rad/s

Figure 47: Sinusoidal Disturbance, Freq = 1000 rad/s

XVII. NYQUIST PLOT

A Nyquist Plot can be used to verify system stability. The

first thing that must be done is to calculate the loop gain

transfer function.

A Nyquist Plot can be obtained using Matlab. First the loop

gain transfer function has to be defined. Next the nyquist

function can be used to produce the plot. The Matlab code is

as follows.s = tf(s);

N = (s*(s+3.09))/(s*(s+3.09)+207)

Nyquist(N)

This code will produce the Nyquist Plot in Figure 73.

Figure 73: Nyquist Plot of the Loop Gain Transfer Function

XVIII. STABILITY MARGINS

The stability of the closed loop system can be determined by

examining the loop gain transfer function and the associated

stability margins. From Figure 74, the phase margin is -33.6

degrees and the gain margin is infinite.

200 300 400 500 600 700

11.95

12

12.05

12.1

Time(s)

Phase(Degrees)

Sinusoidal Torque Disturbance VS No Torque Disturbance

Torque Disturbance

0

Torque Disturbance f

100 rad/s

400 600 800 1000 1200 1400 1600 1800 2000 220011.96

11.98

12

12.02

12.04

12.06

12.08

12.1

Time(s)

Phase(Degrees)

Sinusoidal Torque Disturbance VS N o Torque Disturbance

Torque Disturbance

0


1000 rad/s

500 550 600 650 700 750 800

11.999

11.9995

12

12.0005

12.001

12.0015

12.002

12.0025

12.003

Time(s)

Phase(Degrees)

Sinusoidal Torque Disturbance VS No Torque Disturbance

Torque Disturbance

0


1000 rad/s

N

quist Diagram

Real

xis

Imaginar

xis

2 1 0 1 2 3 45

4

3

2

1

0

1

2

3

4

5

S

stem z

Phase{

argin (deg)

180

Dela

{

argin (sec)

0

t frequenc

(rad/sec) InfClosed

|

oo}

Stable~

es

0 d

10 d

6 d 4 d

2 d

10 d

6 d 4 d

2 d

S

stem

z

Phase{

argin (deg)

33.8

Dela

{

argin (sec)

0.56

t frequenc

(rad/sec)

10.2

Closed|

oo}

Stable~

es


13/21

Figure 74: Stability Margins, Loop Gain Transfer Function

XIX. INERTIA CHANGE

Inertia is another system parameter that can affect the

stability of the system. This stability was examined by

increasing the inertia gain constant block in the simulation ofthe system. The inertia gain is calculated as bring 1/Jt which

initially is set to 1. The system output can be seen in Figure 48.

Figure 48: Inertia Gain = 1, Initial System Parameter

As the inertia (Jt) is increased the gain (1/Jt) decreases. As

the inertia gain is decreases the system becomes underdamped

(Figure 49 & 50).

Figure 49: Inertia Gain = 1vs Inertia Gain = .1


As the inertia gain is decreased even further, the system

will become more underdamped until it becomes unstable

(Figure 51).

Bode Diagram

Frequency (rad/sec)

-6 0

-4 0

-2 0

0

20

System: G



System: G



System: G


Magnitude (dB): 13.3

M

agnitude(dB)

System: G



10-1

100

101

0

45

90

13 5

18 0

Phase(deg) System: G

Phase Margin (deg): -33.8

Delay Margin (sec): 0.56

At frequency (rad/sec): 10 .2

Closed Loop Stable? Yes

200 300 400 500 600 700 800 900

11.94

11.96

11.98

12

12.02

12.04

12.06

12.08

12.1

12.12

Time(s)

Position(Degrees)

Effect of Increasing Inertia Gain

Inertia Gain

1

0 500 1000 1500 2000 2500 3000 3500 40000

2

4

6

8

10

12

14

16

Time(s)

Position

(Degrees)


Inertia Gain

1

Inertia Gain

.1

0 500 1000 1500 2000 2500 3000 3500 40000

5

10

15

20

25

Time(s)

Position(Degrees)


Inertia Gain

1Inertia Gain

.0 1


14/21


XX. TRANSPORT DELAY

The introduction of a transport delay in the feedback loop

of the system can cause the system to become unstable.

The transport delay is used to delay the output from being

fed back into the controller immediately making the

controller wait before it adjusts the setpoint. The first

simulation run was with a delay of 1 Pico-second (Figure

52). This did not cause much of an issue but a definite

change can be seen.

Figure 52: 1 Pico-second Transport Delay

As the transport delay becomes longer, from 1 Pico-

second to 6.5 Milli-second delays the signal begins to

oscillate. This problem can be examined in Figure 53

through Figure 55.

Figure 53: 1 Nano-second Transport Delay

Figure 54: 1 micro-second Transport Delay

Figure 55: 1 Milli-second Transport Delay

0 500 1000 1500 2000 2500 3000 3500 40000

2

4

6

8

10

12

14

Time(s)

Pos

ition(Degrees)

Effect ofIncreasing Inertia Gain

Inertia Gain = 1

Inertia Gain = .0001

500 550 600 650 700 750 800

11.9

11.95

12

12.05

12.1

12.15

Time(s)

Position

(Degrees)

Effect of Trans

ort De

ay

Trans

ort De

ay = 0

Trans

ort De

ay = 1

s

550 600 650 700 750 80011.8

11.85

11.9

11.95

12

12.05

12.1

12.15

Time(s)

Pos

ition(Degrees)

Effect of Trans

ort De

ay

Trans

ort De

ay = 0

Trans

ort De

ay = 1 ns

540 560 580 600 620 640 660 680 700 720

11.94

11.96

11.98

12

12.02

12.04

12.06

12.08

12.1

12.12

12.14

Time(s)

Position(Degrees)

Effect of Trans

ort De

ay

Trans

ort De

ay = 0

Trans

ort De

ay = 1 us

330 340 350 360 370 380 390 400 410 420 430

12.03

12.04

12.05

12.06

12.07

12.08

12.09

12.1

12.11

12.12

Time(s)

Position(Degrees)

Effect of Trans

ort De

ay

Trans

ort De

ay = 0

Trans

ort De

ay = 1 ms


15/21

Figure 55: 6.5 Milli-second Transport Delay

At 1 second transport delay the system reaches saturation

about its oscillation. The system can be deemed unstable due

the oscillations.

Figure 56: 1 second Transport Delay

XXI. RESONANT MODE

A resonant mode function will now be added to the

simulation. The resonant mode can be defined as

with = .05 and n unknown. By adding this

into the simulation and varying the natural frequency (n) the

lowest resonant frequency that the system can handle can be

determined. To determine the resonant frequency the natural

frequency (n) and damping coefficient () can be substituted

into the equation with = .05 and nvarying.It was found that the lowest frequency the system can

handle is n = 855 rad/s, which is n = 852.86 rad/s. This

value was found by performing many guess and check

simulations. Figure 57 shows the output with a natural

frequency of 100 rad/s or resonant frequency of 99.75 rad/s.

Figure 57: Resonance Frequency = 99.75 rad/s

From the figure it can be observed that the frequency needs

to be higher.

Figure 58 shows the natural frequency set at 1000 rad/s or

the resonant frequency of 997.5 rad/s. The system remains

stable and meets the desired specifications. Any natural

frequency setting higher than 1000 rad/s will allow the system

to remain stable.


Figure 59 shows the lowest natural frequency possible, 855

rad/s or 852.86 rad/s when converted to resonant frequency.

Any resonant frequency lower 852.86 rad/s will take the

system outside of the desired specifications, and eventuallycause it to become unstable as the frequency becomes small.

200 300 400 500 600 700 800 900 1000 1 100

11 .96

11 .98

12

12 .02

12 .04

12 .06

12 .08

12 .1

12 .12

Ti

(

)

iti

(

r

)

ff

t

fTr

rt

l

Tr

rt

l

= 0

Tr

rt

l

= 6.5

0 500 1000 1500 2000 2500 3000 3500

-150

-100

-50

0

50

100

150

Ti

(

)

iti

(

r

)

ff

t

fTr

rt

l

Tr

rt

l

= 0

Tr

rt

l

= 1

0 200 400 600 800 1000 1200 1400-1.5

-1

-0.5

0

0.5

1

1.5x 10

4

Ti

(

)

iti

(i

)

ff

t

fV

r

i

R

t

r

q

R

t

r

q

= 99 .75

0 200 400 600 800 1000 1200 14000

2

4

6

8

10

12

14

Ti

(

)

iti

(i

)

ff

t

fV

r

i

R

t

r

q

R

t

r

q

= 997 .5


16/21


XXII. LOOP SHAPING

Loop shaping is a method of controller design which uses a predetermined transfer function as the system controller

instead of a PID controller. For loop shaping to work

effectively the loop gain transfer function must have high gain

at low frequency and cross the 0 dB axis at -20 dB per decade

then have low gain for high frequency. An example of a

proper loop shaping bode plot can be seen in Figure 60.

Figure 60: Loop Shaping Bode Plot

To apply the loop shaping design method for this plant, the

first thing that must be done is to bode plot the open loop

transfer function of the plant. The open loop transfer function

of the plant in standard form is

and the bode plot is show in Figure 61.

Figure 61: Bode Plot of GP(s)

Now a transfer function must be derived which can morph

the Bode Plot of GP(s) into the generic loop shaping Bode Plot

of Figure 60. This can be done by starting at the low

frequency high gain and working to high frequency low gain.

One transfer function that resembles the loop shaping Bode

Plot is . The next step now is to try and modify GP(s)

into this form. This can be done by eliminating the pole at -

3.09 and multiplying by

. This will produce a GC(s)

transfer function of

. This new transfer

function is what will be used to replace the PID controller.The new Matlab simulation diagram will be of that in Figure

62. The plot of the system response can be seen in Figure 63.

Figure 62: Loop Shaping Matlab Simulation Diagram

Figure 63: Loop Shaping System Response

0 1000 2000 3000 4000 5000 6000 70000

2

4

6

8

10

12

14

Time(s)

Position(in)

Effect of Varying

esonant Fre

uency

esonant Fre

uency = 852.86

-60

-40

-20

0

20

40

60

80

System: H

Fre

uency (ra

sec): 0.113

agnitu

e (

): 57.

System: H

Fre

uency (ra

sec) : 1.14

agnitu e ( ): 21.3

System: H

Fre

uency (ra

sec): 10

agnitu

e (

) : 0 System: H

Fre

uency (ra

sec) : 117

agnitu

e (

) :-25.1

agnitu

e(

)

o

e

iagram

Fre

uency (ra

sec)

10-1

100

101

102

103

-180

-150

-120

-

0

System: H

P

ase

argin (

eg) : 78.6

e ay argin (sec): 0.137

t fre uency (ra

sec): 10

ose

oo

Sta

e? Y es

P

ase(

eg)

o

e

iagram

Fre

uency (ra

sec)

-40

-20

0

20

40

60

agnitu

e(

)System:

-

Fre

uency (ra

sec): 14.8

agnitu e ( ):-0.648

10-1

100

101

102

-180

-150

-120

-

0

System:-

P

ase

argin (

eg) : 12.3

e

ay

argin (sec): 0.015

t fre

uency (ra

sec) : 14.2

ose oo Sta e? Y es

System:-

P

ase

argin (

eg) : 12.3

e

ay

argin (sec): 0.015

t fre

uency (ra

sec) : 14.2

ose oo Sta e? Y esP

ase(

eg)

l

X

V

z

i

i

V

i

b

X

V

l

b

y

lzi

i

6

+

4

+

+

L

h

i

i

l

P

i

i

l

0 500 1000 1500 2000 25000

2

4

6

8

10

12

14

Time(s)

Position(in)

oo

S

a

ing S ystem

es

onse

S ystem

es

onse


17/21

Since the output meets the design specification the specified

controller GC(s) can be utilized.

XXIII. LOOP SHAPING AND WHITE NOISE

Now noise will be introduced into the loop shaping

simulation and compared to the effect of noise with the PID

Controller. The first simulation was performed with white

noise set to 0.01% (Figure 64). When comparing Figure 64

with Figure 38 it can be seen that they look very similar.

Figure 64: Loop Shaping 0.01% White Noise


Once again when the white noise level hits 0.15% thesystem goes out of spec and is not good anymore. This can be

seen in Figure 66.


XXIV. LOOP SHAPING WITH CONSTANT TORQUEDISTURBANCE

Adding in a constant torque disturbance at 10% of the

maximum torque (132 in-lbs), the system response does

change (Figure 67). The load would go beyond 12 inches and

the reverse and try to steady out at 12 inches.

Figure 67: Loop Shaping, 10% Constant Torque Disturbance

This could cause major problems for any precise motion

system. The maximum amount of constant torque disturbance

that this system can handle is 0%. Any amount of disturbancehigher than 0% will not cause instability but it will bring the

system out of specifications.

XXV. LOOP SHAPING AND SINUSOIDAL TORQUEDISTURBANCE

Adding in a sinusoidal torque disturbance at 10% of the

maximum torque (132 in-lbs) will change the system response

greatly depending on the frequency of the sinusoid. By

observing the loop gain transfer function of the system and

0 500 1000 1500 2000 2500 3000-2

0

2

4

6

8

10

12

14

Tim

(

)

o

ition(in)

it

N

i

ff

t O n

oop

ping

ontrol

it

N

i

= 0.01 %

0 500 1000 1500 2000 2500 3000-2

0

2

4

6

8

10

12

14

Tim

(

)

o

ition(in)

it

N

i

ff

t O n

oop

ping

ontrol

it

N

i

= 0.15 %

100 200 300 400 500 600 700 800 900 1000

11 .5

11 .6

11 .7

11 .8

11 .9

12

12 .1

12 .2

12 .3

Tim

(

)

o

ition(in)

it

N

i

ff

t O n

oop

ping

ontrol

it

N

i

= 0.15 %

100 200 300 400 500 600 700

11 .6

11 .8

12

12 .2

12 .4

12 .6

12 .8

Tim

(

)

o

ition(in)

on

t

nt Tor

u

Di

tur

n

On

oop

ping

ontrol

on

t

nt Tor

u

Di

tur

n

= 0%

n

t

nt Tor

u

Di

tur

n

= 10 %


18/21

then observing the bode plot of the transfer function (Figure

68) it can be observed that the system is acting as a low pass

filter.

Figure 68: Bode Plot of the Loop Gain Transfer Function

At any frequency, the system will remain stable and within the

accuracy requirement of 1%. As the frequency of the sinusoid

increases the system will remain stable until it hits a high

enough frequency which will cause attenuation of the sinusoid.

By observing Figure 69 it can be seen that the system will beout of specifications even at low frequency. By observing

Figure 70 and Figure 71, it can be seen that at high frequency

the system will never go unstable. The outside disturbance

will only cause vibrations which can be accounted for by

using mechanical components with a high tolerance for thesevibrations.

Figure 69: Sinusoidal Torque Disturbance, Freq = .01 rad/s

Figure 70: Sinusoidal Torque Disturbance,Freq = 1000 rad/s

Figure 71: Closer Look, Sinusoidal Torque Disturbance,Freq = 1000 rad/s

XXVI. LOOP SHAPING NYQUIST

A Nyquist Plot can be obtained using Matlab. First the

loop gain transfer function has to be defined. Next the

nyquist function can be used to produce the plot. The Matlab

code is as follows.

s = tf(s);

N = (s^3+100*s^2)/(s^3+100*s^2+6700*s+6700)

Nyquist(N)

This code will produce the Nyquist Plot in Figure 73.

-15 0

-10 0

-50

0

a

it

e(

)

e

ia

ra

re

e

cy (ra

/se c )

10-2

10-1

100

101

102

103

0

45

90

13 5

18 0

Sys te

K

P as e

a r i (

e ): - 11 9

e lay

a r

i

(se c ): 0 .0739

t fre

e cy (ra

/se c ): 57

l

se

S ta

le ? Yes

P

ase(

e

)

200 400 600 800 1000 1200 1400 1600 1800 2000

9.5

10

10 .5

11

11 .5

12

12 .5

Ti

e(s)

P

siti

(i

)

S i

s

i

al Tor

e

ist

r

a

ce

oopS

api

ontrol

oTor

e

ist

r

ance

S inusoi

al Tor

ue

istur

ance, f = .01

0 500 1000 1500 2000 2500 3000 35000

2

4

6

8

10

12

14

Ti

e(s)

Position

(in)

S inusoi

al Tor

ue

istur

ance

n

oopS

aping

ontrol

oTor

ue

istur

ance

S inusoi

al Tor

ue

istur

ance, f = 1000

605 610 615 620 625 63012 .026

12 .0265

12 .027

12 .0275

12 .028

12 .0285

12 .029

12 .0295

Ti

e(s)

Position

(in)

S inusoi

al Tor

ue

istur

ance

n

oopS

aping

ontrol

oTor

ue

istur

ance

S inusoi

al Tor

ue

istur

ance, f = 1000


19/21

Figure 75: Nyquist Plot of the Loop Gain Transfer Function

XXVII. LOOP SHAPING STABILITY MARGINS

The stability of the closed loop system can be determined

by examining the loop gain transfer function and the

associated stability margins. From Figure 76, the phase

margin is -119 degrees and the gain margin is infinite.

Figure 76: Bode Plot of the Loop Gain Transfer Function

XXVIII. EFFECTS OF CHANGING INERTIA GAIN WITH LOOPSHAPING

As in the previous simulation involving inertia gain change,

the system will respond by at first becoming under-damped.

As the inertia gain grows smaller the system response will become more and more under-damped until it becomes

unstable. This change can be observed in Figure 77.

Figure 77: Effects of Changing Inertia Gain

XXIX. TRANSPORT DELAY

The introduction of a transport delay in the feedback loop

of the system can cause the system to become unstable. The

transport delay is used to delay the output from being fed back

into the controller immediately making the controller wait

before it adjusts the setpoint. As the transport delay increases

the system response will begin to oscillate until it goes out of

control.

Figure 77: Effects of Increasing Transport Delay

XXX. RESONANT MODE WITH LOOP SHAPING METHOD

A resonant mode function will now be added to thesimulation. The resonant mode can be defined as

with = .05 and n unknown. By adding this

into the simulation and varying the natural frequency (n) the

lowest resonant frequency that the system can handle can be

determined. To determine the resonant frequency the natural

frequency (n) and damping coefficient () can be substituted

N qui t Di gr m

!

"

l # xi

Im

$

gin

$

r%

&

xi

'

- 1 -0 .5 0 0 .5 1 1 .5-1

-0 .8

-0 .6

-0 .4

-0 .2

0

0 .2

0 .4

0 .6

0 .8

10 ( B

-20 ( B

-10 (

B

- 6 ( B

- 4 ( B-2 ( B

20 ( B

10 ( B

6 (

B

4 ( B2 ( B

)

t"

m:0

1

2

"

3

rgin ((

"

g): - 18 0

D"

l 3

rgin ( "

4 ): 0

A t fr"

qu"

n 4 (r ( / "

4 ): Inf5

lo

"

(

6

oop)

t 7

l"

8

Y"

)

t"

m: 01

2

"

3

rgin ((

"

g): -11 9

D"

l

3

rgin (

"

4

): 0 .0 7 3 9

A t fr"

qu"

n 4 (r ( / "

4 ): 57

Clo "

( 6

oop)

t 7 l"

8

Y"

-15 0

-10 0

- 50

0

50

M

agnitu

9

@

(9

B)

Bo ("

Dia gram

Fr"

qu"

n4

(rad/

"

4

)

10-2

10-1

100

101

102

103

0

45

90

13 5

18 0

Sys t"

m: 01

2 as e Ma rgin ( de g): - 11 9

De lay Ma rgin (se c ): 0 .0 7 3 9

A t freque ncy (rad/se c ): 57

Close d 6

oop S tab le? Y es

Phase

(deg)

0 500 1000 1500 2000 2500-20

-10

0

10

20

30

40

50

60

70

80

Time(s)

P

A

sition(in)

B

ffec t O fCha nging Inertia G ain

Inertia G ain = 1

Inertia G ain = .1

Inertia G ain = .01

Inertia G ain = .001

Inertia G ain = .0001

1.1

.01.001

.0001

0 500 1000 1500 2000 2500 3000-200

-150

-100

-50

0

50

100

150

200

250

Time(s)

P

A

sition(in)

B

ffec t O fIncreas ing Transport D elay

Transport De lay = 0

Transport De lay = .25 s

Transport De lay = .5s

Transport De lay = .75 s

Transport De lay = 1s


20/21

into the equation with = .05 and nvarying.

Figure 78 shows the effect of increasing resonance

frequency has on the system response. The lowest resonance

frequency that the system can handle is approximately 180

rad/s. This can be seen by observing Figure 79 throughFigure 81.

Figure 78: Effects of Increasing Resonance Frequency

Figure 79: Effects of Increasing Resonance Frequency

Figure 80: r = 180 rad/s VS r = 1000 rad/s

Figure 81: r = 180 rad/s VS r = 1000 rad/s

XXXI. CONCLUSIONS

The design requirement for the position control system was

to move a load 12 inches in 0.3 seconds using manual tuning

methods for a PID Controller. This was met by implementing

negative feedback control and a PID controller. Using the

SISO Design Tool, the PID parameters were found to be KP =13, KI = 0 and KD = 1.

A Trapezoidal Profile was used to replace the basic step

input as the reference to the control system. This was done

because the Trapezoidal Profile is an industry standard and I

am sure to encounter it in the future.White Noise was introduced into the system to represent

the various vibrations and effects which the mechanical

components have on the system. This noise can cause the

system to go out of control if the proper mechanical parts are

not used to compensate or these mechanical disturbances.

Torque Disturbance can be a major issue when designing a

control system. I learned that by neglecting the Torque

disturbance the entire control system would have to be re-

evaluated to incorporate this. Another way to deal with thedisturbance is using various disturbance rejection methods,

which is something I did not explore for this control system. Nyquist Plots can be used to predict and verify system

stability. For this control system adding any poles or zeros

outside of the encirclement will keep the system stable. This

can be seen in the Nyquist Plot generated for this control

system.

I found there to be many differences between the system

being controlled by a PID and the system being controlled by

loop shaping. These differences can be seen in Table 1 below.

0 500 100 0 150 0 2 000 250 0 30 00 350 0 400 0 4 500 5000-40

-30

-20

-10

0

10

20

30

40

50

60

Time(s)

P

C

sition(in)

D

ffec t O fIncreas ingE

esF

nance Frequency

E

esF

nance Frequency = 100 rad/sRes

F

nance Frequency = 150 rad/s

ResF


ResF


ResF


0 2000 4000 6000G

000 10000 120000

2

4

6

G

10

12

14

Time(s)

P

C

sition(in)

D

ffec t O fIncreas ing ResF

nance Frequency

Re sF


Re sF


Re sF

nance Frequency = 1H

5 rad/s

Re sF

nance Frequency = 1G

0 rad/s

Re sF


0 2 0 00 40 0 0 6 0 00 I 00 0 10 0 0 0 1 2 00 00

2

4

6

I

10

12

14

Tim e(s )

P

P

sition

(in)

Q

ffec t O fIncreas ing Re sR

na nce Fre que ncy

Re sR

na nce Fre quency = 150 rad/s

Re sR

na nce Fre quency = 165 rad/s

Re sR

na nce Fre quency = 1S

5 rad/s

Re sR

na nce Fre quency = 1I

0 rad/s

Re s R na nce Fre quency = 1 0 0 0 rad/ s

3 6 0 3S

0 3I

0 3T

0 4 0 0 4 10 4 20

1 2 .0I

1 2 .0I

5

1 2 .0T

1 2 .0T

5

1 2 .1

1 2 .1 0 5

1 2 .1 1

1 2 .1 1 5

1 2 .1 2

1 2 .1 2 5

1 2 .1 3

Tim e (s )

P

P

sition

(in)

Q

ffec t O fInc rea s ing Re sR

na nce Frequency

Re sR

na nce Fre que ncy = 150 rad/s

Re sR

na nce Fre que ncy = 165 rad/s

Re sR

na nce Fre que ncy = 1S

5 rad/s

Re sR

na nce Fre que ncy = 1I

0 rad/s

Re sR

na nce Fre que ncy = 1 0 0 0 rad/ s


21/21

Parameter PID Loop Shaping

Maximum White

Noise

0.15% 0.15%

Constant

Torque

Disturbance

13.26% 0%

Sinusoidal

Torque

Disturbance

Attenuation

40 rad/s 100 rad/s

GainMargin -33..8 degrees 119 degrees

PhaseMargin

Inertia Gain

Change

.0001 .1

Maximum

Transport Delay

6.5 ms 250ms

LowestResonance

Frequency

852.86 rad /s 180 rad/s

Table 1: Comparing the Parameters of PID Design Method

Against The Parameters Found for Loop Shaping Design

Method

This project was very beneficial to me. I was able to get

hands on experience with Matlabs Simulink software and

also tuning of a PID controller. I was also able to apply the

loop shaping design method which I found much more

intuitive than playing around with various PID parameters. I

was also able to use the theoretical knowledge of control

theory and apply it in a real world application. This was also

my first experience with the IEEE standard document format.This is a benefit because the format will be used for the rest of

my engineering career and must be perfected so that I can

convey my work in a professional, intelligent, and precisemanner.

Project 3 Final Report (2)

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