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Real-Time Robustness analysis of Quadrotor

UAV’s

Alejandro Gomez

Department of Electronics and Electrical Engineering

University of Glasgow

A thesis submitted for the degree of Master of Science

2012

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Abstract

This work gives a description of the operation of the quadrotor and states the equations that

explain its behavior. The rotor dynamics are included in the system to make it more realistic. The

controllers are designed by using the sliding mode control technique and a PI controller is

designed exclusively for the rotors. Finally, the robustness analysis is performed by using mu

analysis to determine the robust stability of the system in the presence of variations of real

parameters.




Contents

INTRODUCTION ……………………………………………………………………………………………………………………………….1

MODELLING……………………………………………………………………………………………………………………………………..3

2.1 SYSTEM DESCRIPTION………………………………………………………………………………………………………….3

2.2 ROTOR DYNAMICS……………………………………………………………………………………………………………….6

2.3 THE COMPLETE MODEL………………………………………………………………………………………………….……9

3. CONTROL STRUCTURE…………………………………………………………………………………………………………….……11

3.1 CONTROL SYSTEM……………………………………………………………………………………………………………….11

3.2 SLIDING MODE CONTROL TECHNIQUE…………………………………………………………………………………11

3.3 SLIDING MODE CONTROLLER DESIGN……..………………….………………………………………………………14

3.4 PI CONTROLLER WITH ANTI WIND UP..………………………………………………………………………………..19

3.5 SIMULATION OF THE PI CONTROLLER WITH ANTI WIND UP………………………………………………...21

3.6 SIMULATION OF THE COMPLETE SYSTEM USING SLIDING MODE CONTROL AND PI ANTI WIND UP…………………………………………………………………………………………………………………………………………..23

3.6.1 OPERATION OF THE SYSTEM WITH ROLL, PITCH AND YAW ANGLES SET TO ZERO………...23

3.6.2 OPERATION OF THE SYSTEM WITH ROLL ANGLE DIFFERENT FROM ZERO………………….…..26

3.6.3 OPERATION OF THE SYSTEM WITH PITCH ANGLE DIFFERENT FROM ZERO………………….…29

3.6.4 OPERATION OF THE SYSTEM WITH YAW ANGLE DIFFERENT FROM ZERO……………………...32

4. ROBUSTNESS ANALYSIS………………………………………………………………………………………………………………..35

4.1 ROBUSTNESS ANALYSIS OF THE QUADROTOR……………………………………………………………..………….37

5. CONCLUSIONS AND FUTURE WORK……………………………………………………………………………………………..41

REFERENCES………………………………………………………………………………………………………………………….….……..42




List of figures

Fig. 3.1 sliding surface……………………………………………………………………………………………………………….12

Fig.3.2a. Controller for altitude………………………………………………………………………………………………...13

Fig.3.2b. Controller for the roll movement……………………………………………………………………………….13

Fig.3.2c. Controller for the pitch movement…………………………………………………………………………….13

Fig.3.2d. Controller for the yaw movement……………………………………………………………………………..14

Fig. 3.3 Altitude signal with chattering……………………………………………………………………………………..16

Fig3.4 Altitude signal obtained by avoiding chattering using saturation………………………………….17

Fig. 3.5 control system for altitude and attitude……………………………………………………………………..19

Fig.3.6 Speed control loop for the motors………………………………………………………………………………20

Fig.3.7 PI with antiwind up implemented in simulink…………………………………………………………….20

Fig. 3.8 PID controller with antiwind up scheme……………………………………………………………………20

Fig. 3.9a Voltages of the rotors………………………………………………………………………………………………22

Fig. 3.9b Speed of the rotors………………………………………………………………………………………………….22

Fig. 3.10a Altitude control by using sliding mode controller. Altitude over 1m above ground. Roll, pitch and yaw angles are set to zero………………………………………………………………………………………………23

Fig. 3.10b Speed of the rotors, roll, pitch and yaw angles set to zero……………………………………24

Fig. 3.10c Voltage of the rotors, roll, pitch and yaw angles set to zero………………………………….25

Fig. 3.11b speed of the rotors with a roll angle set to 0.1 radians at 3 seconds…………………….27

Fig. 3.11c Voltages of the rotors when the roll angle set to 0.1 radians at 3 seconds…………..28

Fig. 3.12a Altitude control by using sliding mode controller. Altitude over 1m above ground when the pitch angle is set to 0.1 rad. Roll and yaw angles are set to zero…………………………………………..29

Fig. 3.12b speed of the rotors with the pitch angle set to 0.1 radians at 3 seconds………………30

Fig. 3.12c voltage of the rotors with the pitch angle set to 0.1 radians at 3 seconds…………….31

Fig. 3.13a Altitude control by using sliding mode controller. Altitude over 1m above ground when the yaw angle is set to 0.1 rad. Roll and pitch angles are set to zero…………………………………………..32




Fig. 3.13b speed of the rotors with the yaw angle set to 0.1 radians at 3 seconds. ………………33

Fig. 3.13c Voltages of the rotors when yaw is set to 0.1 radians at 3 seconds………………………34

Fig. 4.1 M-Δ structure for robustness analysis………………………………………………………………………35

Fig4.2 μ upper bound and lower bounds…………………………………………………………………..…………38

Fig. 4.3 Altitude of the quadrotor when the uncertainty parameters are perturbed…………..39

Fig. 4.4 Altitude of the quadrotor in the worst case scenario……………………………………………….40




List of tables

Table2.1. Nominal values for the Draganflyer’s parameters……………………………………………………6

Table 2.2 Motors’ parameters…………………………………………………………………………………………………7

Table 3.1 Values of ૃ and ƞ for the controllers………………………………………………………….…………..18




1

SECTION ONE

INTRODUCTION

An Unmanned aerial vehicle (UAV) is an aircraft that is able to fly without human intervention. Some UAV’s are controlled remotely, by using remote control, and some are autonomous. UAV’s are widely used in military applications; they are also used in environment where there is high chemical toxicity, or radiation to take samples without putting in risk human lives. They are useful in fields like cartography, agriculture, geology or hydrology, where the conditions of the operation environment are really difficult. In order to deal with these difficult conditions, the UAV must be robust enough. This is the reason why the robustness analysis in systems like these is important [12].

The UAV that is going to be analyzed in this report is a quadrotor. The quadrotor is a multivariable system (MIMO) with a number of uncertain parameters such as, the mass or inertia values [1] [2]. The first step is to analyze not only the dynamical model that describes the behavior of the system but also the rotors dynamics. The next step was to design the controllers.

The purpose of this project has been to design the controller and to perform the robustness analysis for the altitude control of a quadrotor helicopter by using the non linear control technique sliding mode control to design the controller and the mu analysis to do the robustness analysis [11] [12] [21].

In order to analyze this system, a mathematical model has been used. This model has been developed in [2]. Then, the design of the non linear controller is performed by using the sliding mode control technique. The model consists of non linear equations written in Matlab/Simulink to obtain the dynamical responses.

The performance of the controllers was verified using a simulink model. After testing the control system, the robustness analysis had to be done. The mu analysis was useful to perform this robustness analysis because it allows analyzing MIMO systems that presents structured uncertainties.

This work is organized in five sections. Section one presents the introduction of the work developed.

Section two has the model of the system and the system description.

In section three, the controllers are designed by using PI with anti wind up technique for the rotors exclusively and the sliding mode control for the complete system.




2

In section four, the robustness analysis for the controller designed in the section three is done by using mu analysis in an algorithm developed in matlab.

Finally, section five contains the conclusions obtained after analyzing the results.




3

SECTION TWO

MODELLING This section presents the description of the system by explaining the movements of the quad rotor. The dynamic equations are stated are detailed in the document entitled “Design and control of quad rotors with application to autonomous flying.” written in [1]. The quadrotor which is going to be analyzed is known as Draganflyer, so the values of the parameters that are included in the equations correspond to Draganflyer r/c. Besides, the model includes the rotor dynamics to make the system more realistic. Finally, the complete system will be obtained by joining the dynamics model of the quad rotor with the rotor dynamics.

2.1 SYSTEM DESCRIPTION The Draganflyer is a radio-controlled four-rotor aerial vehicle with four channels of input to control the motion in six Degrees-of-Freedom –DoF. The motion of Draganflyer can be controlled by changing the speed of the four rotors disposed in cross configuration, as in the figure 2.1.

Fig2.1 Draganflyer scheme

The manoeuvers of the Draganflyer are the following:

According to the figure 2.1, the rotor 2 and 4 rotate clockwise while the rotors 1 and 3 rotate counterclockwise. It is going to be considered that the initial position of the quad rotor is zero. The vertical lifting is gotten by varying the speed of each rotor in the same proportion.




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The quad rotor has three rotational movements. The rotational movements over x, y and z axes are called roll, pitch and yaw respectively. These movements are obtained as follows.

The roll movement is produced by varying the speed of the rotors 2 and 4. In order to get a positive roll the speed of the rotor 2 has to be increased while the speed of the rotor 4 is decreased. The negative roll is reached by increasing the speed of the rotor 4 and decreasing the speed of the rotor 2.

The pitch movement is produced by varying the speed in the rotors 1 and 3. In order to get a positive pitch the speed of the rotor 1 has to be increased while the speed of the rotor 3 is decreased. The negative roll is reached by increasing the speed of the rotor 3 and decreasing the speed of the rotor 1.

Finally, to obtain the yaw movement, it is needed to increase the speed of the rotors 1 and 3 and decrease the speed of the rotors 2 and 4 in this way a positive yaw is reached; otherwise, a negative yaw is obtained.

The model that has been chosen for this work is the model described in the thesis: “Design and control of quadrotors with application to autonomus flying.”[1] This model assumes the following:

• The structure is rigid.

• The structure is symmetrical.

• The propellers are rigid.

• Thrust and drag are directly proportional to the square of propeller’s speed.

There are a number of physical effects that act on a helicopter simultaneously, those effects are the following:

• Aerodynamic effects which are caused by the rotation of the motors (CΩ2).

• Inertial torques, caused by changes in the speed of the propellers 퐽훺̇ ()

• Gyroscopic effects, due to changes in orientation of the rigid body (Jθψ, JΩθ).

• Gravity and friction (푚푔,푘휑̇, 휃̇, 휓̇).

The following equations explain the behavior of the system.




5

푥̈ = (∅) ( ) ( ) (∅) ( )푈 (2.1)

푦̈ = (∅) ( ) ( ) (∅) ( )푈 (2.2)

푧̈ = −푔 + (∅) ( )푈 (2.3)

∅̈ = 휃̇휓̇(̇ ) − 휃훺̇ + 푈 (2.4)

휃̈ = ∅̇휓̇(̇ )− 휃훺̇ + 푈 (2.5)

휓̈ = 휃̇∅̇(̇ ) + 푈 (2.6)

The input signals are described from the equation (2.7) to (2.10).

푈 = 푏(훺 + 훺 + 훺 + 훺 ) (2.7)

푈 = 푙푏(−훺 + 훺 ) (2.8)

푈 = 푙푏(−훺 + 훺 ) (2.9)

푈 = 푏(−훺 + 훺 − 훺 + 훺 ) (2.10)

The quad rotor has twelve states, which are the following:

푋 = [푥푥̇푦푦̇푧푧̇∅∅̇휃휃̇ 휓휓̇]

where, x, y and z are the position in the x, y, and z. 푥̇, 푦̇,푎푛푑푧̇are the speed in those axes. In addition to this, ∅,휃,휓 are the pitch, roll, and yaw angles respectively, and the parameters ∅̇,휃, 휓̇̇ are the speed for pitch, roll and yaw.

The input signal 푈 is the total drag of the rotors. 푈 ,푈 ,푎푛푑푈 are the moments for pitch, roll and yaw respectively.

The angular speed for each rotor is noted as 훺 ,훺 ,훺 ,훺 , for the Draganflyer, the speed is going to vary from 0 to 278 rad/s. Other important nominal values are stated in the table 2.1

Parameter Name Value Ix Inertia of the quad rotor in x 0.006228 kgm2 Iy Inertia of the quad rotor in y 0.006225 kgm2 Iz Inertia of the quad rotor in z 0.01121 kgm2




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m Mass 0.52kg Jr Inertia of the rotor 8.66*10-7 kgm2 l Distance from the rotor to the centre of the quad rotor 0.235m d Drag 7.5*10-7 Nms2 b Drag factor 3.13*10-7 Ns2

Table2.1. Nominal values for the Draganflyer’s parameters

The values written in the table 2.1 were taken from the report “Modelo y control lqr de una aeronave de cuatro motores.” [2].

The maximum value of the speed of the motors is 278rad/s. for 12V. Considering this number, the values for l, b and d stated in the table 2.1, and the equations 2.7-2.10, it is possible to find the ranges for U1, U2, U3 and U4.

0N ≤ U1 < 9.67N

-0.57N ≤ U2 < 0.57N

-0.57N ≤ U3 < 0.57N

-0.11N ≤ U4 < 0.11N

2.2 ROTOR DYNAMICS The rotors used in the system are DC motors, so the equations for these motors will be:

퐿 = 푉 − 푅푖 − 푘푏휔1 (2.11)

= (휏 − 퐾 휔 − 휏 ) (2.12)

휏 = 퐾 푖 (2.13)

If the propeller and the gearbox are introduced, the equations written above are going to change in the following way:

퐿 = 푉 − 푅푖 − 푘푏휔2 (2.14)

= ∗

(퐾 푖 −퐾 푟휔 − 휔 ) (2.15)

= 푟 (2.16)

The parameters that are used in these equations are described in the table2.2.




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Parameter Description ω1 Speed of the motor ω2 Speed of the propeller Jt Inertia

τelec Torque of the motor Τload Torque of the load

i Current V Input voltage kb EMF constant kf Damping ratio R Resistance of the motor L Inductance of the motor n Efficiency r Gearbox ratio

Table 2.2 Motors’ parameters

If the Laplace transform is applied to the equations (2.11) and (2.12), the result will be the following.

퐿푠퐼(푠) = −푅퐼(푠) + 푉(푠)− 푘 휔1

(2.17)

퐽 푠휔1(푠) = 푘 퐼(푠) − 푘 휔1(푠)

(2.18)

From (2.17) and (2.18) the transfer function (푠)can be derived. This will be.

휔1푉

(푠) =푘

퐽 퐿푠 + 푠 퐽 푅 + 푘 퐿 + 푘 푅 + 푘 푘

Following the procedure described in [2], the transfer function written above could be approximated to the equation (2.19).

퐺(푠) =휔1푉

(푠) =푘

퐽 푅푠 + 푘 푅 + 푘 푘

(2.19)




8

Applying the Laplace transform to the equations (2.14) and (2.15), as it is stated in [2], the transfer function for the motor with the propeller included will be:

(푠) = (2.20)

where;

km = 4.3mNm/A

Jt = 7.8x10-7 kgm2

Kb = 0.45V/rad/s

kf = 2.6x10-5Nms

R = 0.7 ohm.

With these values, the equation (2.20) becomes:

퐺(푠) =7880

푠 + 357.73

(2.21)

The equation (2.21) is the transfer function of the motors used in the system.




9

2.3 THE COMPLETE MODEL

Fig.2.2 Complete model of the quadrotor implemented in simulink The block called motors is composed by three stages. The first stage described by the equations 2.7-2.10. The input signals will be the values of the forces that the system requires to work. These forces are noted as U10, U20, U30 and U40. The output signals for this stage are the speed for every motor.

The second stage consists of four control loops. In this case four motors are being used, so it is needed to control the speed of these four motors. Each loop is composed of the transfer function of the motors (equation 2.21), a controller to control the speed of the motor, which is a PI controller with antiwind [2] and a unitary feedback. The input signals in these loops will be the output signals obtained from the first stage, so the output of these loops are going to be the real speeds of the motors.

Finally, the third stage is composed by a relation between the angular speed of the motors y the forces. This relation is derived from the equations 2.7-2.10. These equations could be rewritten as in the equation (2.22)

Ui = S.D.Ωi (2.22)

where




10

푈 = [푈 푈 푈 푈 ]

훺 = [훺 훺 훺 훺 ]

푆 =

푏 0 0 00 푙푏 0 00 0 푙푏 00 0 0 푑

퐷 =

1 1 1 10 −1 0 1−1 0 1 0−1 1 −1 1

Hence, the relation that is going to be used in the third stage will be the equation (2.23)

Ωi = S-1.D-1.Ui (2.23)

Following this the output signals of the third stage are the forces noted U1, U2, U3 and U4, which are the input signals for the Draganflyer stated in the equations 2.1-2.6.

The block called Draganflyer contains the equations 2.1-2.6 that describe the dynamical model of the quadrotor.




11

SECTION THREE

3. CONTROL STRUCTURE

3.1 CONTROL SYSTEM A quad rotor has six degrees of freedom in its movements. In accordance with the equations 2.1-2.6, there are four control input signals. The action of these input signals makes the quadrotor moves forwards, backward, to the left, to the right, upwards or down. In general, the quadrotor is a non-linear and unstable system; therefore, it is a difficult system to be controlled. In order to control the system, four control equations will be designed, which are going to be connected to U1, U2, U3 and U4. The signal U1 is used to define the reference of the altitude while the signals U2, U3 and U4 give references control the roll, pitch and yaw of the quadrotor respectively. The control action must be precise all the time to minimize the risk of crash. Considering that the system is non-linear, the sliding mode control will be the control technique used for this system. In addition to this, a PI controller with anti wind has been used to control the motors.

3.2 SLIDING MODE CONTROL TECHNIQUE The main idea to control a system by using sliding mode control is to make the states of the system converge to a sliding surface and make them stay on it, as it is shown in the figure 3.1. In this way, the dynamics of the system is defined by the equations that determine this surface. Establishing these equations and making them act on the system makes possible to obtain the stabilization of the system, a precise set point following and the regulation of the variables.




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Fig. 3.1 sliding surface

The sliding surface, noted as “s” is defined by the equation 3.1

푠 = 푒̇ + 휆푒 (3.1)

where 훌 is a parameter design greater than zero. In order to reach the main idea of the sliding mode control, the control law will be defined by the equation 3.2

u = ueq + ucr. (3.2)

The equation 3.2 shows that the control law has two elements which are ueq and ucr. ueq is the equivalent control and this is the part of the controller that maintains the state of the system restricted to the sliding surface. ucr is the correction control and this is the part that make the state to converge the sliding surface and satisfy the following inequality:

푠. 푠̇ < 0

“Geometrically, this inequality means that the time derivatives of the state error vector always point toward the sliding surface when system is in reaching mode, and therefore, the system dynamics will approach to the surface dynamics in a finite time.” [4] The procedure to find ueq and ucr for the quadrotor is detailed in section 3.3. The figure 3.2a, 3.2b, 3.2c and 3.2d show the block diagram used to control the altitude, roll, pitch and yaw movements respectively.




13

Fig.3.2a. Controller for altitude

Fig.3.2b. Controller for the roll movement

Fig.3.2c. Controller for the pitch movement




14

Fig.3.2d. Controller for the yaw movement

Every block contains the equations of control for the movements of the system. The signals ControlZ, ControlPhi, Control Theta and Control Psi will be the control signals of the system.

3.3 SLIDING MODE CONTROLLER DESIGN This section presents how to obtain the equations of control of the blocks of the figures 3.2a, 3.2b, 3.2c, and 3.2d. The controller is going to have the following structure.

푢 = 푢 + 푢 Where ueq is the equivalent control, which makes the state stays in the sliding surface s = 0 and ucr that is the correction control, which makes the state converge to s = 0. In order to find ueq the following procedure was followed. The sliding surface is defined as 3.1

푠 = 푒̇ + 휆푒 e is the error defined as the difference between the measured state and the desired state. For the altitude of the quadrotor the erro would be.

푒 = 푧 − 푧 (3.3)




15

z will be the measured state and zd is the desired state.

By replacing (3.3) in (3.1) the following result is obtained.

푠 = (푧̇ − 푧̇ ) + 휆(푧 − 푧 ) (3.4) The law for the attractive surface is the derivative of (3.1).

푠̇ = 푒̈ + 휆푒̇ 푠̇ = (푧̈ − 푧̈ ) + 휆(푧̇ − 푧̇ )

From chapter two, the equation (2.3) 푧̈ is known and it is.

푧̈ = −푔 +cos(∅) cos(휓)

푚 푈

Hence

푠̇ = −푔 +cos(휙) cos(휓)

푚 푈 − 푧̈ + 휆(푧̇ − 푧̇ )

Considering U1 = uer + ucr the equation above becomes.

푠̇ = −푔+cos(휙) cos(휓)

푚 (푢 + 푢 ) − 푧̈ + 휆(푧̇ − 푧̇ )

The aim is to find ueq. In order to do this it is possible to assume that the state is on the sliding surface, so ucr and 푠̇ are 0; therefore, ueq becomes:

푢 = [푔 − 휆(푧̇ − 푧̇ ) + 푧̈ ]푚

cos(휙) cos(휓)

(3.5)

In order to find ucr, a Lyapunov function V is defined. This function must be positive-definite.

푉 =12 푠 > 0

The derivative of the function V must be negative-definite.




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푉̇ = 푠푠̇ = 푠푢 < 0 In order to make sure that the derivative of V is negative-definite, ucr should be:

푢 = −푠푖푔푛(푠).휂 (3.6)

Where s is defined by (3.4), 훌 and ƞ are design parameters which are greater than zero. In this way, adding (3.5) and (3.6) the controller for the altitude is obtained. The implementation of the controller for the altitude gives the result showed in the figure 3.3. It is seen that the output signal oscillates. This oscillation is called chattering, it can causes low control accuracy, high wear of moving mechanical parts, and high heat losses in power circuits. [8]

Fig. 3.3 Altitude signal with chattering.

In order to avoid chattering, saturation is added to the correction control, so ucr would be.

ucr = -sat(s) ƞ

____ Altitude ____ Reference




17

by applying this, the chattering is reduced as it is shown in the figure 3.4.

Fig3.4 Altitude signal obtained by avoiding chattering using saturation The same procedure is followed to get the controllers for pitch, roll and yaw.

푢 = [−휆 휙̇ − 휙̇ + 휙̈ − 휃̇휓̇(퐼 − 퐼퐼 )]퐼

푢 = [−휆 휃̇ − 휃̇ + 휃̈ − 휙̇휓̇(퐼 − 퐼퐼 )]퐼

푢 = [−휆 휓̇ − 휓̇ + 휓̈ − 휃̇휙̇(퐼 − 퐼퐼 )]퐼

(3.7)

with:

푠 = 휙̇ − 휙̇ + 휆(휙 − 휙 )





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푠 = 휃̇ − 휃̇ + 휆(휃 − 휃 ) 푠 = 휓̇ − 휓̇ + 휆(휓− 휓 )

(3.8)

The 훌 and ƞ values that were found suitable for this system are detailed in the table 3.1

Altitude Roll Pitch Yaw ૃ 68 5 5 5 ƞ 25 100 100 100

Table 3.1 Values of ૃ and ƞ for the controllers The system completely controlled is presented in the figure 3.5. The figure 3.2a named as “Altitude” is the controller for the altitude, designed through the sliding mode control method. Looking at the block the input number three, is the set point. Here it is possible to set the altitude for the quadrotor. The output of this block gives the control signal ControlZ that is going to be connected to the input U10 of the block called motors as it is shown in the figure 3.5. Signal ControlZ gives the drag needed to lift the quad rotor. Following the structure of the block motors explained in the section two, this drag, is directly related to the speed of reference for the motors according to the equations 2.7-2.10; Therefore, this value of the drag will be input signal for the first stage of the block motors and the output of this stage will give the desired speed of the motors which is the reference for the speed control loop of the motors. The motors receive this referent value of the speed and automatically the needed voltage will be applied to the motors. In this way, the motors will rotate at the speed needed, giving the enough force to lift the quadrotor at the desired altitude. The control for the speed of the motors consists of a simple control loop as it is presented in the figure 3.6. The blocks 3.2b, 3.2c and 3.2d work in a similar fashion. The signals ControlPhi, Control Theta and ControlPsi, connected to U20, U30 and U40 will give the moments needed to make the quad rotor rotate over x, y and z respectively. It means that ControlPhi sends the control signal to make the quadrotor perform the roll movement, ControlTheta controls the pitch movement and ControlPsi controls the yaw movement. Due to these rotational movements, the speed of the motors will change, but the system will always have the speed to maintain the quad rotor at the same altitude all the time.




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Fig. 3.5 control system for altitude and attitude

3.4 PI CONTROLLER WITH ANTI WIND UP The figure 3.6 is only a close-loop control composed of two main blocks. One of them is the transfer function of the motor and the other one is the controller. The controller is a PI controller with anti wind up fig 3.7. This is the second stage of the block motors. This loop has as input signal the reference value of the speed, which comes from the first stage of the block motors. The output signal will be the real value of the speed of the motors.




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Fig.3.6 Speed control loop for the motors

Fig.3.7 PI with antiwind up implemented in simulink The PI controller with antiwind up deals with the limitations of the motors, the limitation in this case is the voltage range [0-12V]. A standard Integral control still integrates when the input is saturated which produces big overshoots. In order to avoid this, a PI controller with antiwind up has been implemented. The figure 3.7 is equivalent to the figure 3.8.

Fig. 3.8 PID controller with antiwind up scheme

K∞ is calculated following the equation 3.9




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퐾 = lim 퐾(푠) (3.9) K(s) is the equation of the PI controller defined by 3.10. It means that first it is needed to calculate a standard PI controller and then convert it to PI with antiwind up. The transfer function located in the feedback way is found as it is described in the block of the figure 3.8

퐾(푠) =

( )

(3.10)

By using 3.9 in 3.10 it is found that

퐾 = 푘 And

퐾(푠) − 퐾 =( )

− = − (3.11)

kp is the proportional constant of the P controller and ki is the integral constant of the I controller. For this system kp = 0.42 and ki = 15.47.

3.5 SIMULATION OF THE PI CONTROLLER WITH ANTI WIND UP The figure 3.9a and b show the behavior of the rotors without controller, when a standard PI is connected to the system, and finally when a PI controller with anti wind up is connected.




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Fig. 3.9a Voltages of the rotors

Fig. 3.9b Speed of the rotors

____ No control ____ PI standard ____ PI antiwind up

____ No control ____ PI standard ____ PI antiwind up




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It is clearly seen in the figure 3.9a that the PI controller with anti wind up eliminates the peak produced in the transient time. When there is no control in the motor the steady state for the voltage is 8V for 278 rad/s, but a huge peak of around 278V is present in the transient time. This peak is decreased by adding a PI controller; however it is big yet. It is around 120V. The peak is totally eliminated, when a PI controller with anti wind is added to the system. Looking at the figure 3.9b is possible to say that in these three cases the speed is stable; however, the speed reaches the reference value of 278 rad/s only when there is no controller connected. When either the PI standard or the PI with anti wind is plugged in the system the speed is reduced to 264.3 rad/s.

3.6 SIMULATION OF THE COMPLETE SYSTEM USING SLIDING MODE CONTROL AND PI ANTI WIND

3.6.1 OPERATION OF THE SYSTEM WITH ROLL, PITCH AND YAW ANGLES SET TO ZERO

Fig. 3.10a Altitude control by using sliding mode controller. Altitude over 1m above ground. Roll, pitch and yaw angles are set to zero.




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The figure 3.10a presents the altitude of the quad rotor. In this case the altitude set point is 1m while the set point for the angles is zero radians. According to the graph the qua rotor reaches 1m at 1.5 seconds, in the beginning, it has a small overshoot, finding a maximum peak of around 1.06m before staying stable. The quad rotor is lifting smoothly during 1.5 seconds until it reaches the desired altitude. In this case, the altitude set point is 1m, the system reaches 1.006m in steady stable,which is considered as an acceptable value due to the position error is 0.6%. The altitude signal does not present chattering.

Fig. 3.10b Speed of the rotors, roll, pitch and yaw angles set to zero

The figure 3.10b presents the speed of the motors when the angles are set to zero. The motors must have 200 rad/s to lift the quad rotor 1m above the ground. The signal of the speed for all of the motors is distorted during one second approximately, from approximately 1.5 seconds until 2.5 seconds because of the small overshoot that is present in the altitude. It means that during this time the quadrotor reaches complete stability.




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Fig. 3.10c Voltage of the rotors, roll, pitch and yaw angles set to zero The figure 3.10c shows the voltage of the rotors. These voltages have a distortion during 1 second from approximately 1.5 to 2.5 seconds until the overshoot in the altitude disappears. During this distortion the maximum peak value for the voltage is 12 V. This is because of the saturation in the PI controller with anti wind up that avoids bigger peaks. Now, it is going to be analyzed what happen when the roll, pitch and yaw angles are different from zero.




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3.6.2 OPERATION OF THE SYSTEM WITH ROLL ANGLE DIFFERENT FROM ZERO.

Fig. 3.11a Altitude control by using sliding mode controller. Altitude over 1m above ground when the roll angle is set to 0.1 rad, pitch and yaw angles are set to zero.

The figure 3.11a describes the behaviour of the system when after 3 seconds of hovering the roll angle Phi reaches 0.1 radians. At this point, the pitch angle presents distortions, but the controller tries to keep it at zero. The yaw angle at 3 seconds has a negative peak which value is -9x10-6 radians which is a very small value that does not affect the altitude. Even though the pitch and yaw angles have distortions at 3 seconds, the altitude is stable at 1.006m, the altitude does not experience distortions at 3 seconds.




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Fig. 3.11b speed of the rotors with a roll angle set to 0.1 radians at 3 seconds.

In the figure 3.11b are represented the speed of the rotors. Now, the rotors experience two distortions. The first one, from 1.5 to 2.5 seconds is due to the overshoot present in the altitude signal. The other distortion occurs at 3 seconds because of the change in the roll angle. This second distortion lasts 0.5 seconds, from 3 to 3.5 seconds, this is the time needed to the roll angle is stable at 0.1 rad.




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Fig. 3.11c Voltages of the rotors when the roll angle set to 0.1 radians at 3 seconds

The behavior of the voltages of the rotors is described in the figure 3.11c. It is possible to note that the voltages in the rotors 1 and 3 at 3 seconds have an oscillation different from the distortion that motor 2 and 4 present at the same time. During the distortion, the rotors 1 and 3 have a voltage of 8.75V and their speed is 193 rad/s. The rotor 2 has a voltage of 8.82V and its speed is 194.2 rad/s. The rotor 4 has 8.64V and the speed is 191.8 rad/s. The speed of the rotors 2 and 4 are unbalanced and this causes that the yaw angle has a positive peak, as it stated in the section two. The speed of the rotor 2 is bigger than the speed of the rotor 4 to reach a positive roll as it is described in the section two.




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3.6.3 OPERATION OF THE SYSTEM WITH PITCH ANGLE DIFFERENT FROM ZERO

Fig. 3.12a Altitude control by using sliding mode controller. Altitude over 1m above ground when the pitch angle is set to 0.1 rad. Roll and yaw angles are set to zero.

The figure 3.12a describes the behaviour of the system when after 3 seconds of hovering the pitch angle Theta reaches 0.1 radians. At this point, the roll angle presents a neglectable positive peak of 0.5x10-10 radians during 0.5 seconds, until the pitch angle stays stable near 0.1 radians, after that the angle stays in zero. The yaw angle at 3 seconds has a positive peak which value is approximately 9x10-6 radians which is a very small value that does not affect the altitude. Even though the pitch and yaw angles have distortions at 3 seconds, the altitude is stable at 1.006m, the altitude does not experience distortions at 3 seconds.




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Fig. 3.12b speed of the rotors with the pitch angle set to 0.1 radians at 3 seconds. The figure 3.12b presents the speed of the rotors when at 3 seconds the pitch angle Theta changes from 0 to 0.1 radians. It is present a distortion at 3 seconds because of this change, but in general the speed is kept at around 202 rad/s to maintain the altitude at 1m above ground.




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Fig. 3.12c voltage of the rotors with the pitch angle set to 0.1 radians at 3 seconds.

The behavior of the voltages of the rotors is described in the figure 3.12c. It is possible to note that the voltages in the rotors 1 and 3 at 3 seconds have a distortion different from the oscillation that rotors 2 and 4 present at the same time. The rotor 1 has a voltage of 9.12V and its speed is 202.5 rad/s. The rotor 3 has a voltage of 9.26V and its speed is 202.3 rad/s. The rotors 2 and 4 have voltages of 9.19V and the speed is 202.4 rad/s. The speed of the rotors 1 and 3 are unbalanced and this causes that the roll and yaw angles have a positive peak, as was stated in the section two. The speed in the rotor1 is greater than the speed in the rotor 3 because in this case the pitch is positive.




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3.6.4 OPERATION OF THE SYSTEM WITH YAW ANGLE DIFFERENT FROM ZERO

Fig. 3.13a Altitude control by using sliding mode controller. Altitude over 1m above ground when the yaw angle is set to 0.1 rad. Roll and pitch angles are set to zero.

The figure 3.13a describes the behaviour of the system when after 3 seconds of hovering the yaw angle Psi reaches 0.1 radians. At this point, the roll and pitch angles stay at zero without distortions and the altitude is stable at 1.006m. The altitude does not experience distortions at 3 seconds.




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Fig. 3.13b speed of the rotors with the yaw angle set to 0.1 radians at 3 seconds.

The figure 3.13b presents the speed of the rotors when at 3 seconds the yaw angle Psi changes from 0 to 0.1 radians. It is present a distortion at 3 seconds because of this change, but in general the speed is kept at around 202 rad/s to maintain the altitude at 1m above ground.




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Fig. 3.13c Voltages of the rotors when yaw is set to 0.1 radians at 3 seconds The behavior of the voltages of the rotors is described in the figure 3.13c. During the the time of the second perturbation, the rotors 1 and 3 have 9.54V and 207.9 rad/s. while the rotors 2 and 4 have 9.08V and 191.1 rad/s. when the speeds of the rotors 1 and 3 are greater than the speeds of the rotors 2 and 4 a positive yaw is reached.




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SECTION FOUR

4. ROBUSTNESS ANALYSIS The controllers for the nominal system has been designed and tested. The results have shown that the system is stable for the nominal values. Now the robustness is going to be analyzed. It means that the system is going to be studied to see how much can change some physical parameters; such as, the mass or the inertia values until the system becomes unstable. This task is going to be weighted out by using μ analysis. In order to perform the robustness analysis, the system must be changed into the M-Δ formulation as in the figure 4.1.

Fig. 4.1 M-Δ structure for robustness analysis

where Δ represents the source of uncertainty. Δ is diagonal matrix where the uncertain perturbations are located. M is the transfer function from the output to the input of the perturbations [5]. M is going to be found with the equation 4.1.

푀 = [퐶(푠퐼 − 퐴) 퐵] + 퐷 (4.1)




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The matrices A, B, C and D in the equation 4.1 are obtained from a model designed in simulink. This model has decoupled into the known and the unknown parts of the system. It means that the model is designed and drawn as in the figure 4.1. For our model the uncertainty parameters will be the mass, the inertia values for x, y and z and the inertia values of the motors. These uncertainty parameters will be defined as follows.

푚 = 푚(1 + 훿 ) (4.2) 퐼 = 퐼푥(1 + 훿 ) (4.3) 퐼 = 퐼푦(1 + 훿 ) (4.4) 퐼 = 퐼푧(1 + 훿 ) (4.5) 퐽 = 퐽푟(1 + 훿 ) (4.6)

Where 푚, 퐼푥, 퐼푦, 퐼푧푎푛푑퐽푟횤 are the nominal values for the mass, inertia moments for x, y, z and the inertia of the motors respectively. Note that it has been written Jri. i is going to vary from 1 to 4 because there are four motors in the system. The parameters δm, δIx, δIy, δIz, and δJri are the uncertainties. The diagonal matrix Δ is defined by 4.7.

∆=

⎣⎢⎢⎢⎢⎢⎢⎢⎡훿 퐼 0 0 0 0 0 0 0

0 훿 0 0 0 0 0 00 0 훿 0 0 0 0 00 0 0 훿 0 0 0 00 0 0 0 훿 0 0 00 0 0 0 0 훿 0 00 0 0 0 0 0 훿 00 0 0 0 0 0 0 훿

⎦⎥⎥⎥⎥⎥⎥⎥⎤

(4.7)

Once that M and Δ have been found, the robustness analysis will be perform by using μ analysis. This μ analysis consists in finding the minimum Δ to make the system unstable. From the figure 4.1 can be seen that

푦∆ = 푀푢∆ 푦∆ = 푀∆푦∆

(퐼 − 푀∆)푦 = 0 (4.8)

From 4.8 two possibilities can be seen.

i) If (I-MΔ) ≠ 0 the only one solution for this is y = 0, which is associated with the stable system.




37

ii) If (I-MΔ) = 0 the solution for this is y ≠ 0, so it is going to possible to find infinite solutions for y, which is associated with instability.

In this way, what must be found is the minimum Δ to make det(I-MΔ) = 0. The structured singular value (SSV) or μ is used to evaluate the robust stability. If the value of μ for all frequencies is less than 1, the system is robustly stable. “In order to calculate SSV, computational tools are used. These tools are based on numerical optimization problem. This problem is solved for complex uncertainties. When the uncertainties are not complex, the upper and lower bounds are going to be used. The upper bound gives a reliable robust stability bound. The lower bound is a measure of the accuracy of the analysis and gives the minimum known perturbation norm that guarantee system instability. If both values are close, it means that the analysis has been very accurate. Far values, particularly with very low lower bounds mean that the upper bound is probably very conservative.” [15]

4.1 ROBUSTNESS ANALYSIS OF THE QUADROTOR From the model drawn in simulink, and using the algorithm written in Matlab, the upper and lower bounds for the system are indicated in the figure 4.2.




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Fig4.2 μ upper bound and lower bound

From the figure 4.2 the following can be concluded. The figure 4.2a shows that the maximum value of μ for the upper bound is 4.83, which is greater than 1. It means that the controller does not achieve robust stability. Robust stability is ensured if ‖∆‖∞ ≤ 1/μ for this case 0.2072. It means that if the uncertainty parameters are perturbed up to this amount, the system is going to be stable. This situation was verified by perturbing the uncertainty parameters at this quantity, the result of this perturbation is shown in the figure 4.3 For this perturbation, the system tends to be oscillatory; however, it tries to maintain the altitude.

____ lower bound ____ upper bound




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Fig. 4.3 Altitude of the quadrotor when the uncertainty parameters are perturbed. The worst case scenario was also extracted. The worst scenario values for the uncertainty parameters are -1. In other words, if all deltas of the matrix 4.7 are -1, the system is going to be unstable as in the figure 4.4, where it is shown that the system has no equilibrium point when the inertia values of the motors are perturbed.





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Fig. 4.4 Altitude of the quadrotor in the worst case scenario





41

SECTION FIVE

5. CONCLUSIONS AND FUTURE WORK The aim of this work has been to examine the stability for the control altitude of a quadrotor Draganflyer. The system includes the model of the motors to make the analysis more realistic. Two types of controllers were used in the system. One of those is a PI controller with antiwind up. This technique allowed controlling the speed of the motors obtaining an accurate response, which means that the output speed of the motors follows in a precise way to the reference speed value without overshoots or oscillations. This technique was able to eliminate huge peaks of voltage peaks of the motors around 278V. The other controller, used to control the altitude and the rotational movements was designed by using the sliding mode control technique. By implementing these two types of controllers, the results of the nominal stability were very satisfactory. The system was tested for low altitudes, 1m above ground in this case. The system reaches the desired altitude presenting a very small overshoot (0.6%) and the error position was null obtaining. In terms of altitude, it can be concluded that the response is fairly acceptable. In terms of rotational movements, the system reaches the desired angles without presenting overshoots or errors in the position. It means that the states of altitude and rotational movements were accurately controlled. The robustness analysis was performed by using μ analysis. The uncertainty parameters taken into account were the mass (m), the inertia moments Ix, Iy, Iz and the inertia of the motors (Jr). The results of this analysis show that the system is stable for ‖∆‖∞ = 0.2072. By perturbing the uncertainty parameters to this value, the system tends to oscillate, but it tries to be in the same altitude. The robust stability is not good enough, taking into account that the sliding mode control technique is attractive because of its robustness, probably because of the saturation used to reduce chattering in the sliding mode controller, which reduces the system robustness [8]. The suggestion for future work is try to use another method to reduce chattering or implement a robust controller by using H∞ in order to obtain robust stability.




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REFERENCES [1] Samir Bouabdallah. Design and control of quad rotors with application to autonomous flying. École Polytechnique federale de Lausanne. 2007. [2] Juan Maria Diaz Cano. Modelo y control LQR de una aeronave de cuatro motores. Universidad de Sevilla. 2007. [3] J. Colorado, A. Barrientos, Senior Member, IEEE, A. Martinez, B. Lafaverges, and J. Valente. Mini-quadrotor Attitude Control based on Hybrid Backstepping & Frenet-Serret Theory. 2010. [4] Rubén Rojas, Alfredo R. Castellano, Ramón O. Cáceres, Oscar Camacho.Una propuesta de control por modo deslizante para convertidores electrónicos de potencia. [5] Slotine, Li. Applied nonlinear control. 1991 [6] Claudia Mary, Luminita Cristiana Totu, Simon Konge Koldbæk. Modelling and control of autonomus quad-rotor. Aalborg Universitet. 2010. [7] Manuel Lopez Martinez, Manuel G. Ortega, Carlos Vivas, Francisco R. Rubio. Control no lineal robusto de una maqueta de helicóptero con rotores de velocidad variable. Universidad de Sevilla.2007. [8] Yigeng Huangfu, S. Laghrouche, Weiguo Liu, Senior Member, IEEE, and A. Miraoui, A Chattering Avoidance Sliding mode Control for PMSM Drive. 8th IEEE International Conference on Control and Automation Xiamen, China, June 9-11, 2010. [9] Francisco Morata Palacios. Controlador Fuzzy de un quadrotor. Universidad Complutense de Madrid 2009. [10] X-D. Sun & T. Clarke. APPLICATION OF HYBRID CI/HOOCONTROL TO MODERN HELICOPTERS. University of York. CONTROL'94.21-24March 1994. Conference Publication No. 389.0 IEE 1994. [11] Blaise G. Morton and Robert M. McAfoos. A mu test for robustness analysis of a real parameter variation problem. Systems and Research Center Honeywell Inc. Minneapolis, Minnesota 55413. [12] Mehmet ¨Onder Efe. Robust Low Altitude Behavior Control of a Quadrotor Rotorcraft Through Sliding Modes. Mediterranean Conference on Control and Automation July 27-29 2007 Athens-Greece.




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[13] Mohamed Faycal Khelfi, Abderrahmane Kacimi. Robust control with sliding mode for a quadrotor unmanned aerial vehicle. 978-1-4673-0158-9/12/$31.00 ©2012 IEEE. [14] Frank Hoffmann, Niklas Goddemeier, Torsten Bertram. Attitude estimation and control of a quadrocopter. The 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems October 18-22, 2010, Taipei, Taiwan. [15] A. Barrientos, and J. Colorado, Members, IEEE. Miniature Quad-rotor Dynamics Modeling & Guidance for Vision-based Target Tracking Control Task. Authorized licensed use limited to: Univ Politecnica de Madrid. Downloaded on May 03,2010 at 09:41:27 UTC from IEEE Xplore. [16] Santiago Cobreces Alvarez. Optimization and Analysis of the Current Control Loop of VSCs Connected to Uncertain Grids through LCL Filters. Universidad de Alcala 2009. [17] Halil AKÇAKAYA, H. Alpaslan YILDIZ, Gaye SALAM2, and Fuat GÜRLEYEN. SLIDING MODE CONTROL OF AUTONOMOUS UNDERWATER VEHICLE. [18] Guilherme Vianna Raffo. Modelado y control de un helicopter Quadrotor. Universidad de Sevilla 2007. [19] Vadim Utkin, Hoon Lee. CHATTERING PROBLEM IN SLIDING MODE CONTROL SYSTEMS. ¤ Dept. of Electrical Engineering, The Ohio State University, USA 2015 Neil Avenue Columbus, OH 43210 USA. [20] Karl Johan Astrom. PID control. Control system design 2002. [21] Sigurd Skogestad, Ian Postlethwaite. Multivariable feedback control analysis and design. August 29, 2001. [22] The DraganFlyer. http://www.draganfly.com


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