Laboratory Report EE582: Control Techniques

Jeswin Mathew

Electrical and Mechanical Engineering

Year 5

200901475

1. MODELLING OF A MANUFACTURING PROCESS COMPONENT

The problem being analysed is that of a conveyer system that transports goods to different areas

of production. A conveyer system comprises of a belt on rollers that is powered by electric

motors. The velocity of the conveyor is kept constant by a feedback tachometer measurement

system. The block diagram of the total process is shown in Figure 1.

FIGURE 1: BLOCK DIAGRAM OF CONVEYOR SYSTEM

The total torque applied to the shaft is shown in Equation 1; torsional effects are excluded as the

shaft is assumed to have high torsion stiffness. All initial conditions are assumed to be zero and

the torque input is classed as a step input into the system.

( ) ( )

EQUATION 1: TORQUE APPLIED

( )

( ) ( )

( )

( )

( )

( )

The input to this system is the torque applied (Ta(s)) and the output derived is the angular

displacement - ( ). According to specification, . Therefore the

final Laplace transformation of the output signal is given below:

( )

( )

( )

The process model block diagram is given in figure 2:

FIGURE 2: PROCESS BLOCK

The optical encoder within the transducer block consists of a photodiode shined on by a light

source. A circular disk is marked with a precise circular pattern that cuts the light source during

rotation. With the aid of a Schmitt Trigger, the output is converted into a square wave – a pulse is

generated every time the light source is blocked. According to specification, the transducer

produces 159.2 Pulses/radian (Gm) and the block diagram is shown below.

FIGURE 3: OPTICAL ENCODER

The actuator block containing the DC motor is field controlled. The field circuit is defined by

equation 2;

( ) ( ) ( )

EQUATION 2: FIELD CIRCUIT

The applied torque is given by equation 3. The magnetic flux is proportional to the field current

and therefore torque (Ta) can be re-written in terms of field current (If).

( ) ( ) ( ) ( ) ( )

For a maximum field current of 5A, the torque induced is equal to 10Nm. The above torque

relation can therefore be simplified and substituted back into equation 2.

( ) ( )

( )

( )

The field resistance is 2 ohms and winding inductance is 0.5 H. Applying Laplace Transform we

get,

( )

( )

( ) ( ) ( )( )

( )

( )

Combining all the system blocks together, a Simulink model (Figure 4) was implemented in

compliance with the system model shown in figure 1. The output was monitored for three different

cases: a step input voltage change, J increased by 100% and finally B by decreased 30%.

FIGURE 4: SIMULINK MODEL OF MANFUACTURING SYSTEM

1.1. Graphical Results

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

9

Time(s)

Ang

ular

vel

ocity

(Rad

s/s)

Angular Velocity

0 5 10 15 20 25 30 35 40 45 500

1

2

3

4

5

6

7

8

Time(s)

Ang

ular

vel

ocity

(Rad

s/s)

Angular Velocity

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

Time(s)

Angula

r velo

city(R

ads/s

)

Angular Velocity

FIGURE 5: ANGULAR VELOCITY AT STEP INPUT

FIGURE 6: J INCREASED BY 100% TO 0.2

FIGURE 7: B DECREASED BY 30% TO 0.35

1.2. ANALYSIS AND CONCLUSION When B was decreased, the time constant increased due to the decrement in the term -

.

Therefore the system response was improved. This is common in a first order system. Also the final velocity (SS) increased because mechanically we had reduced the friction. The torque applied did not change for both cases and changes in J did not drastically affect the system.

2. PID CONTROL The purpose of this exercise was to gain experience with Simulink and also to run demonstrations

that illustrate the properties of modelling, simulation and control.

2.1. First Order System

Several First Order Systems were implemented in Simulink as shown in figure 8, and the step

responses were compared.

The simulation was run and the DC gain was identified to be unity from the outputs seen in the

scope. The standard representation of a first order system is

( ). In this case, K is equal to one;

the derived time constants for the three systems are tabulated below.

TABLE 1: TIME CONSTANTS (DERIVED)

The system outputs were plotted in Matlab (figure 9) using the command plot (t, Y1, t, Y2, t, Y3).

The time constant is the time period required to reach 63.2 % of steady state value. Figure 5 was

examined and the results are stated below:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time(s)

Res

pons

e

First Order System Response

Y1

Y2

Y3

System 1 1 sec

System 2 2 sec System 3 4 sec

FIGURE 8: MULTIPLE FIRST ORDER SYSTEMS AND THEIR RESPONSES

FIGURE 9: PLOTTED RESPONSE OF FIRST ORDER SYSTEMS

TABLE 2: TIME CONSTANTS (FROM PLOT)

2.2. Second Order Systems

Several second order systems were implemented in Simulink as shown in figure 10 and the step

responses were examined.

FIGURE 10: SECOND ORDER SYSTEM SIMULINK MODEL AND OUTPUTS

In order to determine system parameters for the three systems, their transfer functions were re-

arranged in the standard form -

(

)

. This obviously meant dividing all three transfer

functions by 6.25 to make the third term in the denominator equal to one. The system parameters

are tabulated below – the natural frequency does not change for all the systems.

TABLE 3: SECOND ORDER SYSTEM PARAMETERS

As with the previous exercise, the function plot() was employed to compare all system

responses. Figure 7 shows the plot obtained.

2.2.1. Discussion

The damping ratio was distinct for the three systems. System one has a damping ratio that is less

than one making it underdamped as seen in figure 11, with the damped oscillations. System two

has a damping ratio equal to unity making it critically damped and system three is over damped

(Damping ratio >1). The DC gain at zero frequency, is the same for all systems.

System 1 0.632 ~1sec

System 2 0.632 ~2sec System 3 0.632 ~4sec

System K (D.C Gain) Type of Response

1 2 0.2 2.5 rads/sec Underdamped

2 2 1 2.5 rads/sec Critically Damped

3 2 3 2.5 rads/sec Over Damped

0 2 4 6 8 10 12 14 16 18 200

0.5

1

1.5

2

2.5

3

3.5

Time(s)

Resp

onse

Second Order System Response

Y1

Y2

Y3

FIGURE 11: PLOTTED RESPONSE OF SECOND ORDER SYSTEMS

2.3. On-Off Control System

An on-off control system has two possible outputs: a positive constant +Y and its negative –Y.

They are activated when the input crosses an input. An example system (figure 12) was

implemented in Simulink.

FIGURE 12: SIMULINK MODEL OF ON-OFF CONTROL SYSTEM

The relay is outputs a 1 if input is above zero; when the input falls below zero, the relay outputs a

negative one. A transport delay of one second was also implemented. The step input was first set

to 2.5 units and then to 5 units to examine the difference; the results are shown in figure 13.

FIGURE 13: ON-OFF CONTROL SYSTEM -2.5 UNITS STEP (LEFT) AND 5 UNITS STEP (RIGHT)

2.3.1. Discussion

It is evident from figure 9 that when the step input is 2.5, the system oscillates under the relay

feedback whereas; the system exhibits damped oscillation when the input is increased to five. It

was also noticed that the removal of transport delay subsequently eradicated the sustained

oscillations. The above process is called relay auto-tuning method and is used to find the ultimate

proportional gain (Kcu) that provides sustained oscillations.

2.4. On – Off Control with Dead band

The same relay feedback control loop was explored with the addition of a dead band. The relay

will only switch when the input falls between 0.5 and -0.5. So there is a region where there is no

change. Figure 10 is the model that was created in Simulink. The region where there is no

change is therefore called the dead band. Firstly both step inputs were made equal such that the

resultant input is zero. Then one input was made greater to produce a final positive input. The

results obtained are shown in figure 15.

0 5 10 15 20 25 30-6

-4

-2

0

2

4

6

Time(s)

Response

On-Off Control System

OUTPUT

CONTROL SIGNAL

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

Time(s)

Res

pons

e

On-Off Control System

OUTPUT

CONTROL SIGNAL

FIGURE 14: SIMULINK MODEL OF ON-OFF CONTROL WITH DEADBAND

FIGURE 15: EQUAL INPUT (LEFT) AND UNEQUAL INPUT (RIGHT)

2.4.1. Discussion

From figure 15 we can see that when a non-zero input is fed into the relay feedback, the

oscillation amplitude is incremented by the value of input. When both step inputs are equal, it

produces sustained oscillations as seen in the left of figure 15. In this scenario, the relay switches

to -1 when then input is below -0.5; distinct from the previous exercise. It was also noticed that

the system exhibited damped oscillation at a set point greater than 4.

2.5. Three Term PID Control

PID stands for Proportional Integral Derivative; it is a feedback loop control method that is widely

used in control systems. It attempts to minimise error between measured and desired set point.

Two models were implemented in Simulink: one with P control and the other with PI control

In the figure above, the proportional gain was increased, and the outputs were recorded for

multiple values are shown in figure 17. It can be seen from figure 12 that any increase to

proportional gain subsequently decreases output error and also the time constant; thus rendering

faster responses.

0 2 4 6 8 10 12 14 16 18 20-1.5

-1

-0.5

0

0.5

1

1.5

Time(s)

Response

On-Off Control System with Deadband

OUTPUT

CONTROL SIGNAL

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

1.5

2

2.5

3

Time(s)

Response

On-Off Control System with Deadband

OUTPUT

CONTROL SIGNAL

FIGURE 16: PID CONTROL

Figure 17: Response to Proportional and

Integral Control

The model on the right hand of figure 17 was studies by varying the integral gain and examining

output responses. After analysis, it is evident that an integral action has noticeable effects on the

damping of the system. The integral action has a greater effect in reducing the output error whilst

regulating the overshoot and settling time. We can therefore conclude that an integral control has

an influence on system stability because it lets the user control sustained oscillation

2.6. Sustained Oscillation Tuning

The Zeigler-Nicolas Tuning method is a method of PID tuning and is

performed by setting the integral and derivate action to zero and then

tuning the proportional gain (K) until sustained oscillations are

achieved. Once steady amplitude oscillation is achieved, the gain

and period are recorded. The model implemented in Simulink is

shown in figure 18.

FIGURE 18: ZEIGLER-NICOLAS

The integral gain in figure 18 was set to zero and the proportional gain was varied (beginning

from 20) until sustained oscillations were obtained. Sustained oscillations were achieved when

the proportional gain was equal to 34.7 as illustrated by the figure on the right. The period (Pu) of

oscillations was approximated at 4.15 sec. This makes the Ultimate Gain (Ku) to be 34.7/2 =

17.35.

This is the starting point for PID tuning of the system and rendered an oscillatory behaviour.

Through testing it was found that when proportional gain was increased, the system instability

also increased.

0 2 4 6 8 10 12 14 16 18 200

0.2

0.4

0.6

0.8

1

1.2

1.4

Time(s)

Resp

onse

Response to Integral Gain

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time(s)

Resp

onse

Response to Proportional Gain

K = 1

K = 1.5

K = 2

3. PID TUNING

3.1. Motor Drive Positioning System An engineer is testing a motor drive position system by inputting step

inputs. The output response obtained is shown on the right. Some

aspects in this trace that is crucially important to the PID design:

Decay Ratio; Minimizing Settling Time; Output Error; Response Time.

The response time can be tuned to a desired value with the aid of

Proportional control; whilst the steady state offset error can be

decreased by using Integral controller. A Derivative controller is not

necessary because it is a second order system: the damping ratio cannot be controlled.

Figure 19 is a plot of the system response. A first order

system, with PI control and unity feedback, was

implemented in Simulink (Figure 20) to verify this

response, based on the calculations below.

Input Function: ( )

Output :

( ) ( ) ( ( ))

( ) ( )

( )

The unity feedback was temporarily removed and the PI controller was made redundant. The open response is recorded in figure 21 and this complies with the engineer’s results.

3.2. Boiler Control System

The system response for a boiler control system was graphically provided. The specification then

asked to determine the system parameters - – of a first order system. The method used

above was also employed here as discussed below.

(

)

( ( ))

( )

It is worth noting that, based on the plot; the system has not completed its transition to steady

state condition. And also the time constant is extracted graphically. Hence the system parameters

are just an approximation. A proportional controller with a gain Kp of 4 is proposed to be

incorporated into a unity feedback loop that contains the process transfer function. The steady

state offset error (E(s)) for a reference signal (R(s)) can be calculated.

FIGURE 19: SYSTEM RESPONSE

FIGURE 20: POSITIONING SYSTEM WITH PI CONTROL

0 5 10 15 20 25 30 3540

40.05

40.1

40.15

40.2

40.25

40.3

40.35

40.4

Time(s)

De

gre

es

System Verification

FIGURE 21: SIMULATION PLOT FOR VERIFICATION

( ( )) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )( ( )) ( ) → ( ) ( ) (

( ))

( )

(

) ( )

( )

( )

( )

An offset error can be reduced by incorporating an integral controller in

parallel with the proportional controller as shown in figure 22. The

proportional controller will decrease response time but will introduce steady

state offset error. This can be combated with the Integral controller

3.3. Speed Control Loop

A PID controller must be designed for a speed control loop that regulates the speed of a

manufacturing plant. The control system must have a 5% settling time of no more than 10 min

with minimal overshoot. Procedure C was employed in creating this design.

( )

The specification requested that a step change in reference signal must produce the minimum

possible overshoot. A system is over damped when damping ratio is greater than unity, and it is

underdamped when the damping ratio is less than unity. So to preserve a trace of overshoot

whilst keeping it minimal, it is logical to make : close to unity.

( )

( )

3.4. Position Control Loop

A position control loop applied on a D.C motor actuator demonstrates how system damping can

be used to demonstrate the tuning of damping ratio with the aid of Derivative control. The time

constant of the system is 0.5 seconds and the specification is for critical damping with a natural

frequency of . The gain K is assumed to be unity.

( )

FIGURE 22: CONTROLLER STRUCTURE CHANGE

The system with a process transfer function: ( )

( ) , was implemented in Simulink. The

model that is shown in figure 23 has a parallel PD controller; the derivative control is achieved by

a Simulink PID tuner.

The simulation results are graphically shown in figure 24. As it can be seen, the derivative control

has played its part by reducing the overshoot. This was achieved by an increment in the damping

ratio of the system. The time constant of the system is still maintained at 0.5 s. This exercise was

a viable method in achieving an appreciation for the role of each term in the PID algorithm. The

integral control was not required because there is no output offset error present.

4. CONCLUSIONS PID tuning to obtain a desired response is a challenging aspect of Control Engineering. The

laboratory exercises are able to educate the student in the basics of PID tuning methods and also

extracting the values of each term of the PID. It also conveyed the procedure of turning the

mathematical model of a system into a Simulink model. Computer simulation is an appropriate

tool in verifying designs before moving to manufacturing.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.05

0.1

0.15

0.2

Time(s)

Re

sp

on

se

Am

plit

ud

e

PD Control

FIGURE 23: PD CONTROL

FIGURE 24: SIMULATION RESULTS OF PD CONTROL