LONGITUDINAL AUTOPILOT FOR DELTA WING UAV

Ahmad Ali Ansari ,Jazib Hassan & Talha MirAeronautics and Astronautics Department

Institute of Space TechnologyIslamabad Expressway, Islamabad 44000

AbstractThe paper focuses on the evaluation of mathematical model of longitudinal mode and controller (i.e. Autopilot) of UAV (unmanned air vehicle). This paper deals with the development of the equations of motion of UAV and then its transient response is evaluated. For controller designing, transfer function is determined, closed and open loop responses are analyzed. On the basis of responses P, PI and PID controller is designed. At the end, simulation analysis using MATLAB is presented and comparison between the controllers is done. The work, presented in research paper is kept general so that it can easily be applied on aircrafts having different dynamic characteristics. All the data regarding UAV and the aerodynamics considerations are based upon a research and under development UAV at institute of space technology.

Key Words: UAV, PID Controller, Stability, MATLAB, Simulation

i. Introduction

UAV (Un-Manned Air Vehicle) is becoming popular since the advent of 21st century. An unmanned aerial vehicle (UAV; also known as a remotely piloted vehicle or RPV, or Unmanned Aircraft System (UAS)) is an aircraft that is flown by a pilot or a navigator (now called Combat Systems Officer) depending on the different Air Forces; however, without a human crew on board the aircraft .5 This is the main advantage of a UAV as large portion of its application involve military. One of the main usage of a UAV is to perform reconnaissance operations UAV’s are remotely controlled but the use of automatic controllers with certain objectives is becoming popular in recent years.This paper involves derivation of EOM’s of UAV which are then linearized and used to develop transfer function of control surface used to maintain longitudinal stability.

After that Controller is designed to control longitudinal displacement efficiently. Most efficient controller is then selected by observing time response of controllers using MATLAB.

ii. Aircraft control surfacesControl of modern aircraft is a complicated process and requires many considerations. Maneuverability and stability being one of the basic aircraft requirements, is extensively achieved through means which are fail safe and which can let the pilot focus on other aspects of flying.There are mainly six DOF (Degrees of freedom) for an aircraft. Three are translational and other three rotational.Control in these DOF’s is achieved through use of control surfaces installed on different portions of aircraft. Refer Fig. 1

Figure 1. Control Surfaces (http://www.aviatorwebsite.com)

For conventional aircraft there are three basic control surfaces of an aircraft.

i. Ailerons

ii. Elevatorsiii. Rudder

Ailerons are used for Lateral, Elevators for Longitudinal and Rudder for Directional control. Dynamically control is divided into separate Longitudinal and simultaneous Lateral and Directional Control (Coupling Effect).For a delta wing, However elevator and ailerons are replaced by Elevons. In aeronautics, canard (French for duck) is an airframe configuration of fixed-wing aircraft in which the forward surface is smaller than the rearward, the former being known as the "canard", while the latter is the main wing. In contrast a conventional aircraft has a small horizontal stabilizer behind the main wing.5 Refer fig.2

Figure 2. Delta Wing Canard Configuration

In most cases a negative control surface displacement (i.e. elevator up) results in positive response in pitch displacement and vice versa. But in canard configuration, it is opposite. Any down movement of control surface of canard results in positive displacement in pitch. This is because of aft position of C.G. with reference to canard.Elevator is our primary interest here as it is used to control moment in longitudinal axis called pitch. Elevator is displaced downwards for pitch up and it is displaced upwards for pitch down.

iii. Equations of Motions Longitudinal equations of motions are a

mathematical model of aircraft aerodynamics. They

involve all necessary aerodynamic considerations required to make an accurate autopilot.

Generally EOM’s (Equations of motion) are nonlinear: which are linearized and then used for calculation purposes.

The main assumptions involve in the process of reducing complexity of EOM’s are Mass of aircraft remains constant during analysis the aircraft is a rigid body the earth is inertial reference by using small perturbations about equilibrium

small angle approximation is made

By using above assumptions the equations of longitudinal motion are found out to be1

iv. Evaluation of Coefficients

Values of stability coefficients were found out using DATCOM (For Details See Appendix C) and manual calculations. The important stability derivatives and their corresponding values are given as in tableTable 1. Values of Stability Derivatives

Symbol Value Symbol ValueC xu

-0.050 C zα-3.806

C xα0.1683 C zq

-4.774

Cw-0.197 Cmα̇

-0.379

C zu-0.394 Cmα

-0.257

C zα̇-0.298 Cmq

-6.076

Flight condition details are given in Appendix A.

v. Calculation of Longitudinal Transfer function for Elevator Displacement

Taking Laplace transform of EOM’s using zero initial conditions and substituting values we get

(2.2284 s+0.05 )’ u (s )−0.16835 ‘ α (s)+0.197 θ(s)=00.394 ’ u(s)+(2.28535 s+3.806)’α (s)−(2.2073 s)θ(s)=0.6842δ e(0.0167 s+0.257)‘ α (s)+(0.006802 s2+0.0269 s)θ (s)=0.87089 δ e

Evaluation of short period and phugoid response:

For obtaining transient response, setting determinant of above equations equal to zero

|2.2284 s+0.05 −0.16835 0.1970.394 2.2285 s+3.806 −2.2073 s

0 0.0167 s+0.257 0.0068 s2+0.026 s|=0

0.0338 s4+0.186 s3+0.2908 s2+1.2706 s+0.0284=0( s2+8.094 s+44.163 ) ( s2+0.02206 s+0.01337 )=0Solving the equation we obtain

a. For short period approximation

s2+8.094 s+44.163=0Comparing with standard second degree equation:

s2+2ξ ωn s+ωn2

We getξ=0.609

ωn=6.6454 rad /sTime required for oscillations to damp to one-half amplitude is found to be

T 1/2=0.171 s

b. For phugoid approximation

s2+0.02206 s+0.01337=0Comparing with standard second degree equation, we get

ξ=0.095399ωn=0.11563 rad /s

Time required for oscillations to damp to one-half amplitude is found to be

T 1/2=1.04 min

For transfer function

|2.2284 s+0.05 −0.16835 0.1970.394 2.2285 s+3.806 −2.2073 s

0 0.0167 s+0.257 0.0068 s2+0.026 s||' u ( s)δ e

' α ( s )δ e

θ ( s)δ e

|=| 00.6842

0.87089|Thus:Our required transfer function can be found from

θ(s)δ e

=|2.2284 s+0.05 −0.16835 0

0.394 2.2285 s+3.806 0.68420 0.0167 s+0.257 0.87089|

∇

θ(s)δ e

= 4.299 s2+7.091 s+0.21470.03377 s4+0.2742 s3+1.499 s2+0.03656 s+0.01995

vi. System Response

Time response of transfer function of elevator (θ(s)δe

¿ is found out through MATLAB is now,

Figure 3. Open loop response

Transfer function of elevator servo motor is

Gm=k a

Ʈ τ s+1Assuming most common values, we get

Gm=13.33

s+6.67 where

The closed loop block diagram of complete system is

Transfer function of aircraft with elevator servomotor is

k a=2Ʈτ=0.15 s

Gs=Gm∗θ(s)

δ e

Gs=57.3106 s2+94.5217 s+2.8620

0.0338 s5+0.4993 s4+3.327 s3+10.0294 s2+0.2673 s+0.1330

Step response of closed loop transfer function is found to be

Figure 4 Closed loop step response

Frequency response of closed loop transfer function is

Figure 5 Open Loop Bode Diagram

vii. Controller Design

Time response of transfer function shows a divergent behavior which means system is unstable. Refer fig 4.

By giving a sinusoidal input to system transfer function a sinusoidal time response was obtained, therefore Ziegler-Nicholas method for PID Controller design could be applied.2

Root locus of system is

Figure 6 Root Locus

Root locus depicts that there exist some part of root locus at right side of s-plane. This shows that system is conditionally stable. The critical values of Gain and natural frequency are found from manual calculations by applying Routh Stability Criteria. The calculated values coincide with the graphical results obtained through root locus.These values are

K cr=0.528ωcr=8.99 rad /s

Pcr=2 πωcr

=0.6989 sec

Ziegler Nicholas tuning rule based on Critical gain and critical period values is as;2

Table 2 Ziegler Nicholas TableType of controller

Kp Ti Td

P 0.5Kcr=0.264

∞ 0

PI 0.45Kcr=0.237

Pcr

1.2=0.528

0

PID 0.6Kcr=0.317

0.5Pcr=0.349

0.125Pcr=0.087

A compact block diagram in simulink is given below.

To obtain responses of P, PI and PID Controllers values of gains were manipulated. Ref Table 1The response of P-controller is

Figure 7. P Controller Response

The response of PI Controller is

Figure 8 Time Response of PI Controller

The response of PID Controller

P controller has steady state error, which increases as time increases, which is undesirable. It has comparatively less overshoot but undershoot is high.

PI Controller has no steady error. It’s settling time is less i.e. system converges to the given input at much lesser time which is desirable.

PID Controller Converges the output to input at comparatively less time than PI controller but the disadvantage is that it has high overshoot. In case of aircrafts we cannot afford high overshoot.

viii. Conclusion:

Different type of controllers can be used for aircraft attitude control. For example transient response can be improved by using lead compensator, steady state response by lag compensator or both can be achieved by simply utilizing concept of lead lag compensator. These are classical approaches of controls. Modern control theory can also be applied for deigning controller. But the flaw is, system becomes more complicate and difficult to understand. On the other hand PID is simple to design as well as feasible to implement on the aircraft. Because of its simplicity PID controller was the main focus.

Comparing above three results, most optimum results are given by PI-Controller. The characteristic values of system with PI Controller are:

Max .Overshoot=44 %SettlingTime=t s=4 s

ξ=0.27

ωn=3.693rad

s

The method used for development of controller is kept general so that controller can be modified for aircraft with different dynamic characteristics. This allows for a wide range of application of among different variants of same aircraft and even different aircrafts.

Selection among Controllers is done by simulating mathematical model on MATLAB and then picking out the controller which not only gives the best results but also completes the objective of design of aircraft. So, aircraft is longitudinally stabilized and controlled and the stress level on pilot or navigator is reduced.

Acknowledgement:Our most sincere thanks to our seniors Mr. Hamza

Ahmad Mir and Mr. Irfan Zafar for providing us with the data of their under development UAV. We would also like to show our humble gratitude towards our instructors Dr. Attiqe –Ur- Rehman and Mr. Sher Afgan for their continued support and guidance. And most of all our deepest love and servitude o our parents without whom prayers and support we would have never reached this point.

References:

[1] Jhon H. Blacklock, Automatic Control of Aircraft and Missiles, second edition (New York: A Wiley-Interscience Publication, 1991)

[2] Katsuhiko Ogata, Modern Control Engineering, fourth edition (University of Minnesota)

[3] John D. Anderson,Jr., Fundamentals of Aerodynamics, fourth edition (New Delhi: McGraw-Hill Publishing Company Limited, 2007)

[4] James D. Lang, Aircraft Performance, Stability and Control, Vol I (United States Air Force Academy, Department of Aeronautics)

[5] Crane, Dale: Dictionary of Aeronautical Terms, third edition, page 86. Aviation Supplies & Academics, 1997. ISBN 1-56027-287-2

Appendix A

PRO-E Design of Reference Delta Wing UAV

Mission Statement:

Advanced remotely piloted killer drone which can loiter, detect, select and attack a range of targets

Sizing:

Gross Take-off Weight = 340 lbsFuselage Length = 7.5 ft

Wing Specifications

Canard Vertical Tail

Appendix B

M File (MATLAB Coding) for evaluating different transfer Functions

%writing Matrix [A B C; D E F; 0 G H]A = [0 2.2284 .05];B = [0 0 -.16835];C = [0 0 .197];D = [0 0 .394];E = [0 2.2285 3.806];F = [0 -2.2073 0];G = [0 .0167 .257];H = [.0068 .0269 0];%Evaluating Transient Response (DET=0)a = conv(A,conv(E,H));b = conv(A,conv(G,F));c = conv(B,conv(D,H));d = conv(D,conv(G,C));longitudinal = (a+d)-(b+c)solve('(.033768*s^4)+(.274158*s^3)+(1.498*s^2)+(.036563*s)+(.0199478)=0')%Writing Equation of periods%Short Period (zeta=0.609 wn=6.6454 T1/2=0.171sec)conv([1 4.0484-5.27j],[1 4.0484+5.27j])%Long Period (zeta=0.095399 wn=0.11563 T1/2=1.04min)conv([1 0.0110311-0.11511j],[1 0.0110311+0.11511j])%Evaluating Numerator of Transfer Function for Pitch Displacement [I J 0; K L M; 0 N P]I = [0 2.2284 .05];J = [0 0 -.16835];K = [0 0 .394];L = [0 2.2285 3.806];M = [0 0 .6842];N = [0 .0167 .257];P = [0 0 .87089];%Finding Det(Numerator)e = conv(I,conv(P,L));f = conv(I,conv(M,N));g = conv(K,conv(J,P));numerator = (e)-(f+g)%Writing Pitch Displacement Transfer Functionsys = tf(numerator,longitudinal)rlocus(sys)step(sys)%Writing Transfer Function of Servo MotorGm=tf([13.33],[1 6.667])%Writing overall Transfer Function of Model[n,d]=tfdata(sys,'v');[n1,d1]=tfdata(Gm,'v');num=conv(n,n1);den=conv(d,d1);Gs=tf(num,den)rlocus(Gs)

Appendix CDATCOM’s Output for stability Derivatives of Reference DeltaWing UAV