A REPORT

ON

THE DEVELOPMENT OF A MODEL AND CONTROLLER OF

A VERTICAL TAKE-OFF AND LANDING (VTOL) SYSTEM

USING LABVIEW AND SIMULINK

BY

ABARA, DANIEL NSOR STUDENT ID: 9158439

SUBMITTED TO

DR. OGNJEN MARJANOVIC

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE COURSE UNIT

EEEN60108 CONTROL FUNDAMENTALS

2014-15 SESSION

TABLE OF CONTENTS

Page

1.0 Summary 1

2.0 Introduction 1

3.0 Results and Discussion 2

4.0 Conclusion 5

References 5

Appendices 6

1

PID PI

1

+ P(s)

r = r u =

Iref u = Vm y = Im u =

Iref

1.0 SUMMARY

This report is on the development of a Proportional-Integral-Derivative (PID) controller

model for a Vertical Take-off and Landing (VTOL) system. The model was developed by

manually computing the VTOL parameters, and then using National Instruments LabVIEW

System Identification toolkit to estimate the Transfer Function for the system. This

estimated transfer function was then simulated on the real system in the laboratory and it

was observed that the estimated transfer function matched the measured signal of the

system more accurately than the manually developed model. A PID controller was designed

and used to tune the system, and the VTOL specifications were approximately satisfied with

percentage overshoot equal to 20% and peak time approximately 1s when simulated in

LabVIEW.

2.0 INTRODUCTION

This laboratory utilized National Instruments LabVIEW and Simulink to demonstrate the

development of a Proportional-Integral-Derivative (PID) controller model for a Vertical take-

off and Landing (VTOL) system. The block diagram of the control is shown in figure 1 below.

Figure 1: Cascade Control block diagram of the VTOL system

The system consists of a Proportional-Integral (PI) inner loop and a Proportional-Integral-

Derivative (PID) outer loop. The PI loop regulates the current supply to the motor based on

a desired current reference and the PID loop regulates the pitch of the VTOL body [1]. Three

groups of experiments were conducted namely, Current Control, Modelling and Flight

Control. Using the current control Virtual Instrument (VI) in LabVIEW, and a number of

equations including the one below, a model was derived for the PI current control loop to

the desired specification of natural frequency (n) of 42.5 rad/s and damping ratio () of 0.7,

with an assumed inductance (Lm) of 53.8 mH. The standard second-order characteristic

equation which is applicable to the system is as follows:

2 + 2 + 2 (1)

The VTOL trainer model parameters were obtained by carrying out the Modelling

experiments. The VTOL trainer transfer function was identified and used to validate the

model using the LabVIEW System Identification toolkit. Proportional-Derivative (PD) and

Proportional-Integral-Derivative (PID) steady-state error analysis was carried out on the

identified transfer function using the LabVIEW Flight Control VI. The goal here was to

investigate the effect of integral action in the controller. The transfer function previously

identified for the VTOL trainer system using LabVIEW System Identification toolkit, and the

motor system, was used to simulate the closed-loop VTOL system in Simulink and analysed.

2

3.0 RESULTS AND DISCUSSION

The results are presented in three sections namely Current control, Modelling and Flight

control which are the three groups of experiments which were carried out in the laboratory.

Please note that all the estimations and details of calculations are provided in Appendix B.

3.1 Current Control

The Average resistance Rm was found to be = 2.79. See Table 2 in Appendix A1. In

executing qualitative current control, when integral gain ki_c = 0, there is a steady-state

error. An overshoot is also observed from the response. The steady-state error is due to the

absence of integral gain and therefore no integral action in the PI controller, see Figure A2.1

in Appendix A2. When proportional gain kp_c = 0 and ki_c = 100, the steady state error is

removed due to the presence of integral action. However, the response is slow compared

with when kp_c was greater than 0, as integral action does not increase speed. The absence

of proportional gain reduces the speed of the response. See Figure A2.2 in Appendix A2.

To design the current controller, the PI gains kp,c and ki,c were obtained using

equations (2) and (3) and found to be approximately 0.4 and 91.18 respectively.

The proportional and integral gains can be shown to be as follows:

, = + 2 (2)

, = 2 (3)

See Table 1 in Appendix A for the VTOL parameter definitions.

The calculated gains kp,c and ki,c were simulated in the LabVIEW Current Control VI. It was

observed that the reference and measured current responses were tracking. See Figure A2.3

in Appendix A2.

3.2 Modelling

The current required to make the VTOL trainer horizontal is the equilibrium current Ieq and

was estimated to be = 1.4 A, see Figure A2.4 in Appendix A2. The thrust current-torque

constant due to torques acting on the VTOL body, Kt was also calculated using the values in

Table 3 (See appendix A1) and equation 4 below, with g = 9.8 m/s2.

+ 22 11 33 = 0 (4)

The value of Kt was found to be equal to 0.0144 N.m/A. The natural frequency is given by

equation (5) below.

= 2 (5)

The period, T for four (4) oscillations was measured from the response and used to obtain

the frequency, fn. See Table 4 in Appendix A1, and Figure A2.5 in appendix A2. The natural

frequency was found to be = 4.33 rad/s. The moment of Inertia J was then calculated using

equation (6) below and values from Table 3 in appendix A1, and was found to be equal to be

0.003 kg.m2.

= 2

=1 (6)

The stiffness K, of the VTOL trainer was calculated from the transfer function of the current

to position dynamics of the VTOL trainer given in equation (7).

3

() =

(2+

+

) (7)

It is clear that the denominator of equation (7) matches equation (1), the second-order

characteristic equation.

Therefore, 2 =

(8)

The above was used to compute K and it was found to be = 0.0562 Nm/rad. The VTOL

trainer transfer function was then obtained by substituting the values of the current-torque

constant Kt, the moment of Inertia J, the estimated viscous damping of VTOL, B and the

stiffness K, into equation (7). The following model was the result.

() = 0.0144

0.003(2+0.002

0.003+

0.0562

0.003) =

.

+.+. (9)

The transfer function obtained above was entered into LabVIEW VTOL Modelling Virtual

Instrument and simulated. See Figure A2.6 in appendix A2. It was observed that the

simulation did not match the measured signal accurately. The measured signal lagged

behind the simulated signal. After the simulation, the estimated transfer function generated

by the LabVIEW VTOL Modelling VI, which best describes the response of the system was

identified, recorded, and is given below.

() =.

.+.+ (10)

This system (equation (10)) was then simulated in LabVIEW and it was observed from the

response that the simulated signal matched the measured signal more accurately compared

to the response of the manually derived transfer function (equation (9)). See Figure A2.7 in

Appendix A2.

3.3 Flight Control

In this section, the effect of integral action in the controller was investigated. The figure

below shows the VTOL device step response to a square wave set point when using

Proportional-Derivative (PD) controller.

Figure 2: VTOL step response to square wave set point using PD Controller

4

A PD steady-state error of approximately 50% was measured from the above figure as

follows.

Set point R = 4 deg., Measured output y = 2 deg.

Error e = R y = 4 2 = 2 deg. Therefore, %Error = (2/4) x 100 = 50%.

Comparing this to the theoretical value of the PD steady-state error which was found to be

approximately 2.6 degrees, there is a difference of 0.6 degrees due to experimental errors in

obtaining data/values during the laboratory.

When a PID controller was used (addition of integral gain), it was observed that the

steady-state error reduced to 0 as the integral gain ki was incremented up to 4.0 A,

compared with when it was just a PI controller (ki = 0). See Figure A2.8 in Appendix A2. The

integral action was used to eliminate the steady-state error. This PID steady-state error of 0

also conforms to the theoretically calculated value of the PID steady-state error which was

found to be 0.

The damping ratio , and natural frequency n, required to meet a peak time of 1.0

seconds and a percentage overshoot of 20% were computed and were found to be 0.5 and

3.6 rad/s respectively. The PID gains kp, ki and kd needed to meet the VTOL trainer

specifications above were computed and found to be approximately -0.5, 2.7 and 0.8

respectively. From the result of the computation, it is clear that the value of kp is incorrect.

This is due to experimental errors during the laboratory. The system was therefore tuned

with gains of 4, 6 and 1 for kp, ki and kd respectively so as to complete the laboratory tasks,

and the result obtained is given in Figure 3. These gains were found by trial and error.

After the laboratory, the simulation of the closed-loop system in Simulink using the

gains found by trial and error was carried out and the response is shown in Figure 4.

Figure 3: Simulation of VTOL system using designed PID controller

The percentage overshoot (PO) and Peak time (tp) were estimated from the response and

were found to be 20% and 0.6 seconds respectively. From the results, the VTOL trainer

specifications are approximately satisfied. The Percentage overshoot matches the

specification accurately. The peak time however matches the VTOL specifications

approximately (0.6 1.0 seconds). The closed loop VTOL system was then simulated in

5

Simulink using the identified transfer function and the response obtained is given in figure 4

below.

Figure 4: Simulation of closed-loop VTOL system in Simulink

As shown above, the response in Simulink was different from the one observed on the real

system (figure 3). The response obtained from the real system is more oscillatory. This is

because the simulation from Simulink is just a step (pulse) signal to estimate the systems

response and cannot give us accurate information about the signal over a long time period,

while in simulating on the real system using LabVIEW, the set point input is a continuous set

of square waves (pulses) which are tracked by the measured signal and would only

terminate when current is no more supplied to the VTOL motor.

4.0 CONCLUSIONS

It can be inferred from the results obtained that the estimated transfer function better

describes the system and therefore when simulated, the response matched the reference

signal more accurately than the response of the manually developed model. The overshoot

matched the requirement of 20% accurately while the peak time met the requirement

approximately. It can also be deduced from the closed-loop simulation of the VTOL system

in Simulink, that the peak time is 0.7 which is approximately 1s which meets the VTOL

system requirements. Also, it shows that more information about a model is obtained when

simulated on the real system compared with when simulated with a step in Simulink.

REFERENCES

1. Quanser Inc. QNET VTOL Trainer for NI ELVIS: Student Workbook. Retrieved October 2nd

2014 from http://www.lehigh.edu/~inconsy/lab/me389/guidelines/QNET_VTOL_Exp08-

Position_Student.pdf

2. The University of Manchester. Laboratory Manual on Developing a model and controller

of a Vertical Take-Off and Landing (VTOL) system using LabVIEW and Simulink.

6

APPENDICES

Appendix A Tables and Figures

A1. List of Tables

Table 1: PI Current Control Design Parameters

Parameter Value Units

Rm 2.80

Lm 53.8 x 10-3 H

0.7

n 42.5 Rad/s kp,c 0.4011 V/A

ki,c 97.1762 A/(A.s)

Table 2: Exercise 1 Finding Resistance

Input Voltage (V) Measured current(A) Resistance()

4 1.4 2.85

5 1.8 2.78

6 2.15 2.79

7 2.5 2.80

8 2.9 2.76

Average Resistance: Rm, avg = 2.79

The Average resistance Rm was found to be 2.79 2.8

Table 3: VTOL Specifications

Description Symbol Value Unit

Propeller mass m1 0.09 kg

Counter-weight mass m2 0.27 kg

VTOL body m3 0.048 kg

Length from pivot to propeller centre l1 0.156 m

Length from pivot to centre of counter-weight l2 0.052 m

Length from pivot to centre of VTOL body l3 0.043 m

Total length of VTOL body lh 0.284 m

Estimated viscous damping of VTOL B 0.002 N.m/(rad/2)

Equilibrium current Ieq 1.400 A

Current-torque constant Kt 0.014 N.m/A

Moment of inertia J 3 x 10-3 kg.m2

Stiffness K 0.056 Nm/rad

Table 4: VTOL Trainer Natural Frequency

Parameter Symbol Value Units

Measured time between n oscillations Tn 5.8 s

Number of oscillations N 4.0

Natural Frequency fn 0.6897 Hz

Natural Frequency (n=2fn) n 4.33 rad/s

7

A2. List of Figures

A2.1 Current Control

Figure A2.1: Qualitative Current Control kp_c = 0.25, ki_c = 0. (Exercise 2)

Figure A2.2: Qualitative Current Control kp_c = 0, ki_c = 100 (Exercise 3)

Figure A2.3: Current Control Design kp,c = 0.4, ki,c = 97.18 (Exercise 5)

8

A2.2 Modelling

Figure A2.4: Measuring the Equilibrium Current, Ieq = 1.4 A (Exercise 6)

Figure A2.5: Measuring the natural frequency n,

T = 7-1.2=5.8s, for n=4 oscillations

f= 4/T=4/5.8=0.68Hz.

9

Figure A2.6: Simulation of the manually derived VTOL trainer Transfer Function

Figure A2.7: Simulation of the Identified transfer function estimated by LabVIEW

A2.3 Flight Control

Figure A2.8: VTOL trainer PID Controller steady-state error, ki = 4.0 A/(rad.s)

10

Appendix B Computations and Estimations

B1. Current Control Design

To the design the current controller, the PI gains kp,c and ki,c were obtained using

equations 2.2 and 2.3 (from main section) respectively as follows:

, = + 2 (2.2)

kp,c = -2.8 + (2 x 0.7 x 42.5 x 0.0538) = 0.4

And, , = 2 (2.3)

ki,c = 42.52 x 0.0538 = 97.18 (Exercise 4)

B2. Modelling

B2.1 The thrust current-torque constant, Kt was calculated as follows:

+ 22 11 33 = 0

Kt = ((0.048 x 9.8 x 0.043) + (0.09 x 9.8 x 0.156) (0.27 x 9.8 x 0.052))/1.4

Kt = 0.0144 N.m/A (Exercise 7)

B2.2 The natural frequency was calculated as follows:

= 2

And =1

T = 7.0 - 1.2 = 5.8s, for n = 4 oscillations

fn = 4/5.8 = 0.6897 Hz

Therefore, n = 2 x 3.142 x 0.6897 = 4.33 rad/s. (Exercise 8)

B2.3 The moment of Inertia J, was calculated from the expansion of equation 3.4 as follows:

= 112 + 22

2 + 332

J = (0.09 x 0.1562) + (0.27 x 0.0522) + (0.048 x 0.0432)

Therefore, moment of Inertia J about the pitch axis = 0.003 kg.m2 (Exercise 9)

B2.4 The Stiffness, K of the VTOL trainer was calculated using the following equation:

2 =

From the above, K = Jn2 = 0.003 x 4.332 = 0.0562 Nm/rad (Exercise 10)

B3. Flight Control

B3.1 The theoretical PD steady state error for the VTOL trainer with kp = 2 and kd = 1 and a

step amplitude R0 = 4 is calculated as follows:

It can be shown that PD steady state error =0

+

ess = (4 x 0.0562)/(0.0562+(0.0144 x 2)) = 2.6447 deg.

B3.2 The damping ratio was computed as follows:

= ln (

100)

1

ln (

100)2+ 2

11

= ln (20

100)

1

ln (20

100)2+ 2

= 0.4559 0.5

B3.3 The natural frequency, n was computed as follows:

=

12 =

1(0.5)2 = 3.6276 3.6 rad/s (Exercise 20)

B3.4 The PID gains were computed as follows:

= +2+

2

kp = (-0.0562+(2 x 0.5 x 3.6 x 0.003)+(3.62x0.003)/0.0144 = - 0.4528

=

2

ki = (3.62 x 0.003)/0.0144 = 2.7

= ++2

kd = (-0.002+0.003+(2 x 0.5 x 3.6 x 0.003))/0.0144 = 0.8194 (Exercise 21)

B3.5 The percentage-overshoot and peak time estimated from the PID controller response

is as follows:

= 100( 0)

0

= 100(8.47)

7= 20%

= 0

= 165.6-165.0 = 0.6 seconds