Pitch Stability and Control Analysis of Flying Wing Aircraft  

Joshua A. Sullivan  

lindsayo  

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Table of Contents  

NOMENCLATURE   3  

I.   INTRODUCTION   3  

II.   RELEVANT  THEORY   4  

III.   THEORETICAL  ANALYSIS   5  A.   STABILITY  AND  CONTROL  DERIVATIVES  FROM  XFLR-­‐5   5  B.   STATE-­‐SPACE  MODELS  AND  CONTROLLER  IMPLEMENTATION   6  C.   PERFORMANCE  SIMULATION   9  

IV.   EXPERIMENTAL  ANALYSIS   10  A.   WITHOUT  SAS  IMPLEMENTED   10  B.   WITH  SAS  IMPLEMENTED   11  

V.   CONCLUSION   11  

ACKNOWLEDGMENTS   12  

APPENDIX   13  

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Pitch Stability and Control Analysis of Flying Wing Aircraft

Joshua A. Sullivan University of California-San Diego Department of Mechanical and Aerospace Engineering

Pitch stability derivatives determined from software simulations are used to model the

longitudinal pitch behavior of a flying wing type aircraft. With no controller

implementation, this configuration is found to display marginal stability in pitch modes. So

as to design a more robust control structure, a stability augmentation system is enacted to

counteract disturbances to the otherwise marginally stable behavior. Root Locus design is

implemented to shape the dynamic response of the aircraft to an elevator command input

via proportional and derivative signal gains. The result is a step response that settles to a

reasonable pitch angle related to the elevator command, and rejects longitudinal

disturbances during flight. Finally, state-space models of both the stability-augmented

system and the original system are simulated so as to compare with real-time data collected

during flight tests.

Nomenclature [A] = State/System matrix [B] = Input matrix [C] = Output matrix [D] = Feedforward matrix b = Aircraft wingspan cBar = Mean geometric chord Cmq = Pitching moment coefficient Cmδe = Elevator effectiveness coefficinet D(s) = Controller transfer function G(s) = Aircraft dynamics transfer function IMU = Inertial Measurement Unit Iyy = Mass moment of inertia (y-axis) Ke = Elevator input command gain Kq = Derivative gain Kθ = Proportional gain LE/TE = Leading Edge/Trailing Edge Mde = Elevator effectiveness derivative

Mq = Pitching moment stability derivative OLHP = Open Left-Hand Plane (s-domain) q = Pitch Rate qBar = Dynamic pressure qDot = Rate of change of Pitch Rate S = Aircraft wing area SAS = Stability Augmentation System u = Control/Input vector V = Nominal aircraft speed x = State vector xDot = Derivative of State vector XFLR-5 = Aircraft analysis software y = Output vector δe = Elevator deflection δec = Elevator deflection command θ = Pitch Euler angle θDot = Rate of change of Pitch Euler angle

I. Introduction

HE analysis of pitch modal behavior is of particular importance to a flying wing aircraft because there is no

stabilizing tail to produce counteractive moments in response to flight disturbances. Furthermore, common flying

wing configurations share a common feature of having much longer wingspan than body length. This means that

T

Pg. 4

there is substantially less rotational inertia about the pitch axis, leading to increased sensitivity in the longitudinal

direction. Of the five general aircraft dynamic modes, the Short Period mode and Phugoid mode capture aircraft

pitch behavior. Phugoid modes have much slower natural frequencies and smaller damping ratios, while Short

Period modal behavior is often characterized by fast, highly damped pitching. For this particular application, only a

modified Short Period modal analysis is conducted because of its relevance to smaller, less maneuverable aircraft.

The first objective of this analysis is to model the aircraft dynamics via software simulation using XFLR-5 and

Stability/Control derivaties. From here, it becomes possible to design a Pitch SAS that tunes the aircraft dynamic

response to a favorable level. Finally, with a state-space model composed of the control and dynamics models , the

aircraft performance can be simulated using a 4th Order Runge Kutta iteration scheme. This simulated response will

give insight into the efficacy of the particular controller, as well as allow for comparison with actual flight data

measured with the onboard Inertial Measurement Unit.

II. Relevant Theory

With known aircraft dimensions, XFLR-5 is used to simulate elevator deflection at the trim condition, thus

giving Stability and Control Derivatives. The two derivatives of interest are the Pitching Moment Coefficient and

the Elevator Effectiveness Coefficient. From there, the following relationships can be used to find Pitch

Dimensional Derivatives:

€

Mq =q c 2S2VIyy

Cmq (1)

€

Mδe =q c SIyy

Cmδe (2)

The Short Period pitch mode dynamics, modified to exclude downward velocity considerations, are given by:

€

δ ˙ q δ ˙ θ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

=Mq 01.0 0⎡

⎣ ⎢

⎤

⎦ ⎥ δqδθ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

+Mδe

0⎧ ⎨ ⎩

⎫ ⎬ ⎭ δe

(3)

The state variables are pitch rate and pitch Euler angle. The variable of prime interest is the pitch Euler angle. The

Pitch SAS implements proportional and derivative feedback gains such that:

€

δe = Kq Kθ[ ] δqδθ⎧ ⎨ ⎩

⎫ ⎬ ⎭

+Keδec (4)

Using this control law in the original dynamics model yields the final state-space model:

€

δ ˙ q δ ˙ θ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

=Mq + MδeKq MδeKθ

1.0 0⎡

⎣ ⎢

⎤

⎦ ⎥ δqδθ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

+Mδe

0⎧ ⎨ ⎩

⎫ ⎬ ⎭ δec

(5)

Which is of general state-space form:

€

˙ x = A[ ] x + B[ ] u y = C[ ] x + D[ ] u (6)

Pg. 5

For general Root Locus design, the system transfer function should be rearranged such that stable poles and a system

gain can be realized. The general form for the Root Locus plot is:

€

1+K num(s)den(s)

⎛

⎝ ⎜

⎞

⎠ ⎟ = 0

(7)

III. Theoretical Analysis

The following section outlines all analysis done to model the aircraft modal behavior and stability. The work

shown in this section will be complemented by experimental results presented in later sections.

A. Stability and Control Derivatives from XFLR-5

A detailed model of the aircraft was created in the simulation and analysis software XFLR-5 to evaluate the one

stability and one control derivative needed for the control model. Elevon control surfaces added to the trailing edge

of the wing were deflected from -10º to +10º in

pure elevator motion, and the stability and modal

behavior of the aircraft was recorded. Figure 1

shows the aircraft model with deflected elevators

and pressure color contours to indicate

aerodynamic performance (i.e lift and induced

drag). Upon full completion of the simulation, a

weighted average of the necessary coefficients

was conducted to find the nominal values that

would be used to evaluate the dynamic model of

the aircraft, and to simulate all further behavior.

The weighted average was conducted by

assuming that the majority of the flight would take place within ± 5º elevator deflection. Table 1 displays all the

relevant resulting parameters from the XFLR-5 simulation. The final two values of Table 1 were evaluated using

Eq. (1) and Eq. (2) respectively. These values will be used later in the state-space models described by Eq. (3) and

Eq. (5).

Figure 1. XFLR-5 Model of Flying Wing.

Table 1. Relevant Stability and Control Coefficients, and Dimensional Derivatives

Cmq -0.7515 Cmδe -0.2472

Mq (1/s) -1.2607 Mδe(1/s2) -21.6243

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B. State-Space Models and Controller Implementation

With the appropriate parameters of Table 1, the aircraft dynamics described in the state-space model of Eq. (3)

were transformed to a plant transfer function, which is denoted G(s). This transfer function describes the

relationship between elevator deflection and the state variable of interest- pitch angle. G(s) is given by:

€

G s( ) =δθ s( )δe s( )

= KeMδe

s2 −Mqs

(8)

From this transfer function, it is evident that

there is one pole at s = 0 and one pole at s = +Mq.

As previously mentioned, the pole at zero causes

the plant to have marginal stability. While not

entirely unstable, a marginally stable system is on

the brink of reaching instability should the aircraft

experience a disturbance. Hence, marginal

stability is not sufficient for appropriate aircraft

SAS design. For simplicity, the elevator input

scaling gain, Ke, is held at a value of 1.0, so that

an elevator input command should produce an

equal change in pitch angle. The Root Locus plot

of the plant, shown in Fig. 2, displays the

marginally stable pole at zero. Furthermore, Fig.3

demonstrates a simulated response of the aircraft

plant to a unit step input to the elevator. Looking

at the step response, it’s clear that a unit step input

to the elevators causes an undamped and rapid

nose-down pitch of the aircraft. This behavior

reveals that the aircraft is indeed unable to reject

step disturbances without becoming unstable, and

thus indicates a need for feedback control.

To implement the Pitch SAS controller,

proportional and derivative gains will be used to

modify the state variable of interest- pitch angle.

The derivative of pitch angle, however, is simply

pitch rate; thus, both arguments for the controller

gains are readily measured from the onboard

Inertial Measurement Unit. Equation (4) displays this relationship, where the elevator deflection, δe has now been

replaced by the elevator deflection command given by the pilot, δec.

Figure 2. Root Locus plot of aircraft plant.

Figure 3. Step Response of aircraft plant.

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A system block diagram, composed of both the

system plant G(s), and the feedback controller D(s),

is shown in Fig. 4. Notice how this block diagram

satisfies the appropriate mathematical models given

by Eq. (3) and Eq. (4). Imposing Eq. (4) into Eq. (3)

results in the modified state-space model given by

Eq. (5), which now demonstrates the dynamic

relationship between pitch Euler angle output, and

elevator input command. Just as before with the

aircraft plant state-space model, the modified Pitch

SAS dynamics and plant can be modeled together. Looking at Fig. 4 again, the top branch is clearly the aircraft

dynamics plant. Below that are the two branches that make up the feedback controller. Therefore, the controller has

the following form:

€

D s( ) = Kqs+Kθ (9)

For a positive feedback loop such as this, the system transfer function is given by the following expression:

€

H s( ) =G s( )

1−D s( )G s( )

(10)

Since the objective of this analysis is to tune the controller gains via Root Locus design, it became imperative to

consider the closed-loop poles of the system transfer function, H(s), recalling the standard Root Locus form given by

Eq. (7). The following expression was used to conduct a Root Locus test:

€

1−D s( )G s( ) = 0→1+Kq

−KeMδe s+Kθ

Kq

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

s2 −Mqs

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

= 0

(11)

Equation 11 is of the exact form as Eq. (7), and

hence a Root Locus plot using the bracketed

expression can be used to find the derivative gain Kq.

Recall that for simplicity, Ke is set at 1.0. Notice how

the ratio Kθ / Kq can be used to place a zero in the

OLHP, providing sufficient control to move the

originally vertical Root Locus branches further into

the OLHP. Mathematically, this will then allow for a

gain to be selected that has much better damping

qualities than the original plant model alone. The

Figure 5. Root Locus plot of controlled system.

Figure 4. Control block diagram for Pitch SAS.

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resulting Root Locus plot for the now feedback controlled system is given in Fig. 5. Notice how, when compared to

the purely vertical branches of the Root Locus plot in Fig. 2, the new Root Locus branches are pulled leftward into

the OLHP by the presence of the zero placement. This zero placement is what allows for the aircraft dynamics to be

altered to produce more a responsive and operable pitch control interface.

Using MATLAB’s Root Locus package, a gain can be selected that graphically satisfies stability requirements

and damping needs. Ideally, damping would be very high, but this usually requires high gains that will saturate the

mechanical systems or lead to very high settling

times. Therefore, it’s prudent to find gains that

provide sufficient damping, while allowing for the

system to settle reasonably fast. These responsive

behaviors can again be viewed by considering the

system step response. Therefore, after

implementing a Root Locus pole/gain selection, all

controller gains were determined and the feedback

loop was implemented. The step response shown in

Fig. 6 is the resulting system behavior. Notice now

how the system damps with very little overshoot,

and settles to a value of approximately -1.0 very

quickly. Recall that the elevator command gain, Ke, was set to 1.0, thus indicating that a positive

elevator unit step input (TE down) should lead to a nose-down pitch (negative convention) of equal magnitude.

Therefore, the fact that the step response settles to roughly -1.0 indicates this very behavior. Should it be more

useful to have the aircraft pitch more or less than the elevator

deflection, the gain Ke can be changed to accommodate this

requirement. Table 2 contains the resulting controller gains, as

well as the system damping ratio and natural frequency. Both

the damping and frequency are within reasonable range for a

Short Period modal analysis, indicating solid controller design.

In regards to disturbance rejection, the system’s impulse response can give excellent insight into how the system

responds to sudden disturbances of severe, but fleeting magnitude. Ideally, the impulse plot should show the system

response leveling out at approximately zero. This indicates that the system is able to reject impulse disturbances,

and that it is able to do so with zero steady-state error. Figure 7 is the simulated impulse response of the aircraft

with Pitch SAS enabled. Just as desired, the system response settles at zero, indicating full impulse disturbance

rejection and one-to-one user input tracking. The impulse response does show the rather severe pitching behavior

that occurs right at the onset of the disturbance, but considering that an impulse is an idealized input with infinite

Figure 6. Controlled system step response.

Table 2. Resultant Controller Parameters

Kq 0.251

Kθ 1.004

Ke 1.0

ζ 0.73

ωn (rads/s) 4.28

Pg. 9

magnitude, the fact that the aircraft still returns to

zero steady-state error is indicative of stable control.

Utilizing MATLAB to run the Root Locus design

process allowed for quick iteration through various

controller gain choices to find the ones that resulted

in the best system responses. The next step in

evaluating the theoretical performance of the

aircraft and Pitch SAS was to run a state-space

performance evaluation, to model how the system

might actually appear on logged flight data. This is

the topic of the next section.

C. Performance Simulation

With a full dynamic model complete, it was then possible to run a state-space simulation of the aircraft

performance. The models- one without Pitch SAS and one with Pitch SAS- were created using 4th Order Runge

Kutta iterations with the state-space models given in Eq. (3) and Eq. (5), respectively. For each model, a time span

of 20 seconds is considered. During that span, a +10º rudder input (TE down) was commanded for 5 seconds,

followed by zero input, and then followed by a -10º rudder input (TE up) for 5 seconds. Both pitch Euler angle and

pitch rate were simulated, since both of these measurements are available from the IMU measurements. Figure 8

shows the pitch rate response, and Fig. 9 shows the pitch Euler angle response. Note that the vertical axes on both

simulations have two different sets of scalings, one for the response without the Pitch SAS enacted and one for the

response with the Pitch SAS enabled. This is because, without the controller activated, the responses were often of

much higher magnitudes than those being controlled by the SAS.

Figure 7. Impulse response of controlled system.

Figure 8. Pitch rate simulation.

Figure 9. Pitch angle simulation.

Pg. 10

Looking at the pitch rate behavior, the sharp changes in stabilized pitch rate at the onset of the elevator command

are mostly due to the discontinuous nature of a step input. Obviously, the simulated pitch rates quickly settle to zero

whie the constant step command is given- an ideal behavior to witness. On the other hand, the unstabilized pitch

rate continues to increase to much larger magnitudes during the entire step command than that displayed on the

stabilized vertical axis. It is only when the command is changed that the pitch rate begins to lessen and return to

zero. Similarly, the pitch angle magnitude with deactivated SAS rapidly increases during the constant step

command, indicating inherent aircraft pitch instability. It is the the active SAS pitch angle response that is of

particular interest. As indicated in the simulation in Fig. 9, for a +10º elevator command, the aircraft holds at pitch

Euler Angle of -10º for the entire step command. Once the command is released, the pitch angle returns to zero,

with only slight overshoot and settling time. Similarly, a -10º elevator command holds the aircraft at +10º pitch

angle. This one-to-one relationship between elevator command and aircraft pitch attitude is due to the elevator

command gain, Ke, being set to 1.0, as well as the controller gains being selected to yield a step response that settles

at approximately -1.0.

IV. Experimental Analysis

The final objective of the procedure was to implement the controller into the actual aircraft operational software.

The onboard processor is an Arduino UNO board, connected with an IMU that contains a 3-axis rate gyro chip, a 3-

axis accelerometer chip, and a 3-axis magnetometer chip. The processor is connected to a receiver which picks up

pilot input commands sent through a transmitter. The Pitch SAS simply follows the math displayed in Eq. (4),

namely that the elevator deflection is the summation of the pilot command with the controller-gain-scaled

measurements of pitch angle and pitch rate. Data is being logged with an OpenLog using a microSD memory card.

A. Without SAS Implemented

Beginning with the baseline

performance of the aircraft, flight data

was taken with no stability controller

programmed onto the onboard

processor. Figure 10 shows the

resulting pitch angle and elevator

angle data from a complete non-

stabilized flight. It is evident from the

plot that there is not the mirrored

relationship between elevator input

and pitch angle output that was seen in

the state-space models. The

sporadically fluctuating elevator inputs

Figure 10. Real-time pitch angle and elevator input data without

SAS controller.

Pg. 11

shown in the plot indicate that the pilot was constantly having to adjust the aircraft to maintain flight. Ultimately,

this is due to the inherently unstable configuration of the aircraft plant dynamics discussed in prior sections.

Without a feedback loop to utilize real-time measurements, the control is completely dependent on the pilot.

B. With SAS Implemented

On the other hand, the pitch

stabilized system should certainly

react with behavior that is very close

to the one-to-one response shown by

the step response of Fig. 6. Therefore,

it would be expected that a plot of

elevator command vs. time

superimposed on a plot of pitch angle

measurement vs. time would produce a

graph with approximate symmetry

about the x-axis. This is precisely

what is seen in Figure 11, which plots

the real-time logged data of a flight

utilizing Pitch Stability Augmentation.

In this configuration, a pilot input to

the onboard receiver gets scaled by Ke, which is assumed to be 1.0, and then added to scaled measurements of pitch

angle and pitch rate. This final summation is then the input into the elevator servos, which implement the control

law and produce the plotted behavior.

V. Conclusion

Overall, it was determined via software simulation using XFLR-5 that the flying wing configuration under

consideration was inherently marginally stable in pitch behavior. A mathematical model for the pitch dynamics was

formulated, and the Root Locus of that model was plotted. This plot provided the starting point for a controller

implementation that involved using proportional and derivative gains to place a zero in the OLHP and drive the Root

Locus branches leftward. This opened up the possibility of choosing controller gains that would allow for maximum

damping with minimum overshoot and settling time. The result was a stable step response and impulse response,

that then yielded stable and predictable state-space simulations. In actual flight, the non-stabilized aircraft did not

display overtly drastic and devastating instability, but rather required the pilot to be continuously altering elevator

control to counteract disturbances. The stability augmented system, however, served its purpose of stabilizing

around a particular pilot input, and moderately rejected disturbances. All in all, it is evident that the Pitch Stability

Augmentation System stabilized the originally unstable flight dynamics of the flying wing, proving the merits of

implementing proportional-derivative feedback controllers to reduce pitch instability.

Figure 11. Real-time pitch angle and elevator input data with SAS

controller activated.

Pg. 12

Acknowledgments

Firstly, I would like acknowledge my excellent teammates Gamer Kesheshe, Tim Wheeler, and Karcher Morris,

without whose work I would have never been able to complete this analysis in a timely fashion. I would also like to

thank Dr. Anderson for teaching every bit of theory needed to complete the analysis, and for having the patience to

remind me of it every time I had a question.

Pg. 13

Appendix

A. MATLAB code

a. Execution Script

clear all close all clc %---Conversion Factors ---% in2m = 0.0254; rads2deg = 180/pi; %---Aircraft Parameters ---% %--------------------------% % Stabilit/Control Coefficients Cmq = -0.751493333; Cmde = -0.247204286; % Dimensional/Performance ct = 6.875*in2m; cr = 10.9375*in2m; b = 30*in2m; Iyy = 0.01; lamda = ct/cr; S = .5*b*cr*(1+lamda); V = 6; rho = 1.225; cBar = (2/3)*cr*((1 + lamda + (lamda*lamda))/(1 + lamda)); qBar = .5*rho*V*V; %---Stability and Control Derivatives ---% %----------------------------------------% Mq = (qBar*cBar*cBar*S*Cmq)/(2*V*Iyy); Mde = (qBar*cBar*S*Cmde)/Iyy; %---State-Space/Transfer Function Models ---% %-------------------------------------------% % Plant A_Plant = [Mq 0; 1 0]; B_Plant = [Mde;0]; C_Plant = [0 1]; D_Plant = 0; [numPlant, denPlant] = ss2tf(A_Plant,B_Plant,C_Plant,D_Plant); Plant = tf(numPlant, denPlant); figure(1) sgrid on;

Pg. 14

rlocus(-Plant); title('Plant, G(s), Root Locus') print('-dpng','-r300','PlantRL') [yStepG, tStepG] = step(Plant); figure(2) plot(tStepG,yStepG) grid on title('Plant Step Response','FontSize',16) xlabel('Time, sec','FontSize',14) ylabel('\delta\theta, rads','FontSize',14) % Implement/Tune Controller % Kq = 0.2508; KTheta_Kq = 4; L = tf(-[Mde (Mde*KTheta_Kq)],[1 -Mq 0]); figure(3) sgrid on rlocus(L) [Kq,Poles] = rlocfind(L); KTheta = KTheta_Kq*Kq; Controller = tf([Kq KTheta],1); % System System = feedback(Plant, Controller, +1); [yImp,tImp] = impulse(System); figure(4) plot(tImp,yImp) grid on title('System Impulse Response','FontSize',16) xlabel('Time, sec','FontSize',14) ylabel('\delta\theta, rads','FontSize',14) [yStep,tStep] = step(System); figure(5) plot(tStep,yStep) grid on title('System Step Response','FontSize',16) xlabel('Time, sec','FontSize',14) ylabel('\delta\theta, rads','FontSize',14) %---Simulation Model---% %----------------------% x_1(1,1) = 0; x_1(2,1) = 0; x_2(1,1) = 0; x_2(2,1) = 0; u(1,1) = 0; dt = 0.1;

Pg. 15

time = 0:dt:25; for i = 1:length(time) - 1 de_cmd = 0.0; if (time(i) >= 5.0) de_cmd = 10/rads2deg; end if (time(i) >= 10.0) de_cmd = 0; end if (time(i) >= 15.0) de_cmd = -10/rads2deg; end if (time(i) >= 20.0) de_cmd = 0; end u(1,i) = de_cmd; r1_sim1 = dt*PitchMode(x_1(:,i),u(1,i),0,Mq,Mde,Kq,KTheta); r2_sim1 = dt*PitchMode(x_1(:,i) + (0.5*r1_sim1),u(1,i),0,Mq,Mde,Kq,KTheta); r3_sim1 = dt*PitchMode(x_1(:,i) + (0.5*r2_sim1),u(1,i),0,Mq,Mde,Kq,KTheta); r4_sim1 = dt*PitchMode(x_1(:,i) + r3_sim1,u(1,i),0,Mq,Mde,Kq,KTheta); x_1(:,i+1) = x_1(:,i) + ((1/6)*(r1_sim1 + (2*r2_sim1) + (2*r3_sim1) + r4_sim1)); r1_sim2 = dt*PitchMode(x_2(:,i),u(1,i),1,Mq,Mde,Kq,KTheta); r2_sim2 = dt*PitchMode(x_2(:,i) + (0.5*r1_sim2),u(1,i),1,Mq,Mde,Kq,KTheta); r3_sim2 = dt*PitchMode(x_2(:,i) + (0.5*r2_sim2),u(1,i),1,Mq,Mde,Kq,KTheta); r4_sim2 = dt*PitchMode(x_2(:,i) + r3_sim2,u(1,i),1,Mq,Mde,Kq,KTheta); x_2(:,i+1) = x_2(:,i) + ((1/6)*(r1_sim2 + (2*r2_sim2) + (2*r3_sim2) + r4_sim2)); end u(1,i+1) = 0; figure(6) subplot(2,1,1) [AX1,H1_1, H1_2] = plotyy(time,x_1(1,:)*rads2deg,time,x_2(1,:)*rads2deg); grid on xlabel('Time, sec','FontSize',14) ylabel('Pitch rate, q, deg/sec','FontSize',14,'Color','k') title('Aircraft Pitch Rates, q','FontSize',16) legend('Without Pitch SAS','With Pitch SAS','Location','NorthWest') set(AX1(2),'ytick',[-30 -20 -10 0 10 20 30]); subplot(2,1,2) plot(time,u(1,:)*rads2deg);

Pg. 16

grid on xlabel('Time, sec','FontSize',14) ylabel('Elevator Command, deg','FontSize',14) axis([0 time(1,end) -15 15]) figure(7) subplot(2,1,1) [AX2,H2_1, H2_2] = plotyy(time,x_1(2,:)*rads2deg,time,x_2(2,:)*rads2deg); grid on xlabel('Time, sec','FontSize',14) ylabel('Pitch Euler Angle, \theta, deg','FontSize',14,'Color','k') title('Aircraft Pitch Angles, \theta','FontSize',16) legend('Without Pitch SAS','With Pitch SAS','Location','Best') set(AX2(2),'ytick',[-40 -30 -20 -10 0 10 20 30 40]); subplot(2,1,2) plot(time,u(1,:)*rads2deg) grid on xlabel('Time, sec','FontSize',14) ylabel('Elevator Command, deg','FontSize',14) axis([0 time(1,end) -15 15])

b. State-Space Dynamics Function “ PitchMode ”

function xDot = PitchMode(x,u,StabilizedBoolean,Mq,Mde,Kq,KTheta) if exist('Kq','var') ~= 1 Kq = 0.0; end if exist('KTheta','var') ~= 1 KTheta = 0.0; end if StabilizedBoolean == 0 % The simulation is seeking to model plant q = x(1,:); Theta = x(2,:); d_e = u; qDot = (Mq*q) + (Mde*d_e); ThetaDot = q; xDot = [qDot;ThetaDot]; end if StabilizedBoolean == 1 % The simulation is seeking to model SAS q = x(1,:); Theta = x(2,:); d_ec = u; qDot = ((Mq + (Mde*Kq))*q) + (Mde*KTheta*Theta) + (Mde*d_ec); ThetaDot = q; xDot = [qDot;ThetaDot]; end end