Aircraft Dynamics Example

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UNIVERSITY OF CALIFORNIA, SANTA CRUZ

BOARD OF STUDIES IN COMPUTER ENGINEERING

CMPE 240: INTRODUCTION TO LINEAR DYNAMICAL SYSTEMS

Gabriel Hugh Elkaim Fall 2005

Aircraft Dynamics Example

• _{longitudinal aircraft dynamics}

• _{wind gust & control inputs} • _{linearized dynamics}

• _{steady-state analysis} • _{eigenvalues & modes} • _{impulse matrices}

1 Longitudinal aircraft dynamics

PSfrag replacements _{body axis}

horizontal

θ

variables are (small) deviations from operating point or trim conditions state (components):

• _u_{: velocity of aircraft along body axis}

• _v_{: velocity of aircraft perpendicular to body axis}

(down is positive)

• _θ_{: angle between body axis and horizontal}

(up is positive)

• _q_{= ˙}_θ_{: angular velocity of aircraft (pitch rate)}

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2 Inputs

disturbance inputs:

• _u_w_{: velocity of wind along body axis}

• _v_w_{: velocity of wind perpendicular to body axis}

control or actuator inputs:

• _δ_e_{: elevator angle (}_δ_e _>_{0 is down)} • _δ_t_{: thrust}

3 Linearized dynamics

for 747, level flight, 40000 ft, 774 ft/sec,      ˙ u ˙ v ˙ q ˙ θ      =      −_.₀₀₃ _.₀₃₉ ₀ −_.₃₂₂ −_.₀₆₅ −_.₃₁₉ ₇_.₇₄ ₀ .020 −_.₁₀₁ −_.₄₂₉ ₀ 0 0 1 0           u−_u_w v−_v_w q θ      +      .01 1 −_.₁₈ −_.₀₄ −₁_.₁₆ _.₅₉₈ 0 0      " δe δt #

• _{units: ft, sec, crad (= 0}_._01rad≈₀_.₅₇◦₎

• _{matrix coefficients are called} _{stability derivatives}

outputs of interest:

• _{aircraft speed} _u _{(deviation from trim)} • _{climb rate ˙}_h₌−_v_{+ 7}_.₇₄_θ

4 Steady-state analysis

DC gain from (uw, vw, δe, δt) to (u,h˙): H(0) =−_CA−1_B ₌ " 1 0 27.2 −₁₅_.₀ 0 −₁ −₁_.₃₄ ₂₄_.₉ #

gives steady-state change in speed & climb rate due to wind, elevator & thrust changes solve for control variables in terms of wind velocities, desired speed & climb rate

" δe δt # = " .0379 0.0229 .0020 .0413 # " u−_u_w ˙ h+vw #

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• _{level flight, increase in speed is obtained mostly by increasing elevator (i.e., downwards)} • _{constant speed, increase in climb rate is obtained by increasing thrust and increasing}

elevator (i.e., downwards)

(thrust on 747 gives strong pitch up torque)

5 Eigenvalues and modes

eigenvalues are

−₀_.₃₇₅₀±₀_.₈₈₁₈_j, −₀_.₀₀₀₅±₀_.₀₆₇₄_j • _{two complex modes, called} _short-period _and _{phugoid, respectively} • _{system is stable (but lightly damped)}

• _{hence step responses converge (eventually) to DC gain matrix}

eigenvectors are xshort =      0.0005 −₀_.₅₄₃₃ −₀_.₀₈₉₉ −₀_.₀₂₈₃      ±_j      0.0135 0.8235 −₀_.₀₆₇₇ 0.1140      , x_phug =      −₀_.₇₅₁₀ −₀_.₀₉₆₂ −₀_.₀₁₁₁ 0.1225      ±_j      0.6130 0.0941 0.0082 0.1637     

6 Short-period mode

y(t) = CetA

(<_x_short_{) (pure short-period mode motion)}

0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 0 2 4 6 8 10 12 14 16 18 20 −1 −0.5 0 0.5 1 PSfrag replacements u ( t ) ˙h( t )

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• _{only small effect on speed} _u • _period≈_{7 sec, decays in} ≈_{10 sec}

6.1 Phugoid mode

y(t) = CetA

(<_x_phug_{) (pure phugoid mode motion)}

0 200 400 600 800 1000 1200 1400 1600 1800 2000 −2 −1 0 1 2 0 200 400 600 800 1000 1200 1400 1600 1800 2000 −2 −1 0 1 2 PSfrag replacements u ( t ) ˙h( t )

• _{affects both speed and climb rate} • _period≈_{100 sec; decays in} ≈_{5000 sec}

7 Dynamic response to wind gusts

impulse response matrix from (uw, vw) to (u,h˙) (gives response to short wind bursts)

over time period [0,20]:

0 5 10 15 20 −0.1 0 0.1 0 5 10 15 20 −0.1 0 0.1 0 5 10 15 20 −0.5 0 0.5 0 5 10 15 20 −0.5 0 0.5 PSfrag replacements h11 h12 h21 h22

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over time period [0,600]: 0 200 400 600 −0.1 0 0.1 0 200 400 600 −0.1 0 0.1 0 200 400 600 −0.5 0 0.5 0 200 400 600 −0.5 0 0.5 PSfrag replacements h11 h12 h21 h22

8 Dynamic response to actuators

impulse response matrix from (δe, δt) to (u,h˙)

over time period [0,20]:

0 5 10 15 20 −2 −1 0 1 2 0 5 10 15 20 −2 −1 0 1 2 0 5 10 15 20 −5 0 5 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 PSfrag replacements h11 h12 h21 h22

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0 200 400 600 −2 −1 0 1 2 0 200 400 600 −2 −1 0 1 2 0 200 400 600 −3 −2 −1 0 1 2 3 0 200 400 600 −3 −2 −1 0 1 2 3 PSfrag replacements h11 h12 h21 h22

9 MATLAB code

% 747 longitudinal axis example for 263

% 40000 feet steady, level flight, 774 ft/sec % from bryson p151

% x = u, w, q, theta

A = [ -0.003 0.039 0 -.322; -0.065 -0.319 7.74 0; 0.020 -.101 -.429 0 0 0 1 0];

%input = u_w, w_w, delta_e, delta_t

Bw= -A(:,[1,2]); Bc= [0.01 1; -.18 -.04; -1.16 .598; 0 0]; B = [Bw, Bc];

% output: u, climb rate = -w + 7.74 theta C = [ 1 0 0 0; 0 -1 0 7.74]; H0 = -C*inv(A)*B;

H01 = H0(:,[3,4]); % DC gain matrix from delta_e delta_t to % speed, climb rate

% modal analysis

[V,Gam]=eig(A); xshort = real(V(:,1)); xphug = real(V(:,3)); xshort = xshort/norm(xshort); xphug = xphug/norm(xphug);

Nsamp = 100; %number of time samples

yshort=zeros(2,Nsamp); t=linspace(0,20,Nsamp); for i=1:Nsamp, yshort(:,i)=C*expm(t(i)*A)*xshort; end

figure(3) subplot(2,1,1) plot(t,yshort(1,:)’); axis([0 20 -1 1]) ylabel(’u’) subplot(2,1,2) plot(t,yshort(2,:)’); axis([0 20 -1 1]) ylabel(’hdot’)

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print -deps aircraft_short

yshort=zeros(2,Nsamp); t=linspace(0,2000,Nsamp); for i=1:Nsamp, yphug(:,i)=C*expm(t(i)*A)*xphug; end

figure(4) subplot(2,1,1) plot(t,yphug(1,:)’); axis([0 2000 -2 2]) ylabel(’u’) subplot(2,1,2) plot(t,yphug(2,:)’); axis([0 2000 -2 2]) ylabel(’hdot’)

print -deps aircraft_phug

% now do responses to various impulses figure(1)

h1=zeros(2,Nsamp); h2=zeros(2,Nsamp); t=linspace(0,20,Nsamp); for i=1:Nsamp,

h1(:,i)=C*expm(t(i)*A)*B(:,1); % impulse response from u_w h2(:,i)=C*expm(t(i)*A)*B(:,2); % imp resp from v_w

end

subplot(2,2,1) plot(t,h1(1,:)’); axis([0 20 -.1 0.1]) ylabel(’h11’) subplot(2,2,2) plot(t,h2(1,:)’); axis([0 20 -.1 0.1]) ylabel(’h12’) subplot(2,2,3) plot(t,h1(2,:)’); axis([0 20 -.5 0.5]) ylabel(’h21’) subplot(2,2,4) plot(t,h2(2,:)’); axis([0 20 -.5 0.5]) ylabel(’h22’) print -deps aircraft_gust1

% now do same plots over longer time scale

figure(2); t=linspace(0,600,Nsamp); for i=1:Nsamp,

h1(:,i)=C*expm(t(i)*A)*B(:,1); % impulse response from u_w h2(:,i)=C*expm(t(i)*A)*B(:,2); % imp resp from v_w

end

subplot(2,2,1) plot(t,h1(1,:)’); axis([0 600 -.1 0.1]) ylabel(’h11’) subplot(2,2,2) plot(t,h2(1,:)’); axis([0 600 -.1 0.1]) ylabel(’h12’) subplot(2,2,3) plot(t,h1(2,:)’); axis([0 600 -.5 0.5]) ylabel(’h21’) subplot(2,2,4) plot(t,h2(2,:)’); axis([0 600 -.5 0.5]) ylabel(’h22’) print -deps aircraft_gust2

% now do same things, but for actuator inputs figure(1)

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h1=zeros(2,Nsamp); h2=zeros(2,Nsamp); t=linspace(0,20,Nsamp); for i=1:Nsamp,

h1(:,i)=C*expm(t(i)*A)*B(:,3); % impulse response from delta_e h2(:,i)=C*expm(t(i)*A)*B(:,4); % imp resp from delta_t

end

subplot(2,2,1) plot(t,h1(1,:)’); axis([0 20 -2 2]) ylabel(’h11’) subplot(2,2,2) plot(t,h2(1,:)’); axis([0 20 -2 2]) ylabel(’h12’) subplot(2,2,3) plot(t,h1(2,:)’); axis([0 20 -5 5]) ylabel(’h21’) subplot(2,2,4) plot(t,h2(2,:)’); axis([0 20 0 3]) ylabel(’h22’) print -deps aircraft_act1

% now do same plots over longer time scale

figure(2); t=linspace(0,600,Nsamp); for i=1:Nsamp,

h1(:,i)=C*expm(t(i)*A)*B(:,3); % impulse response from delta_e h2(:,i)=C*expm(t(i)*A)*B(:,4); % imp resp from delta_t

end

subplot(2,2,1) plot(t,h1(1,:)’); axis([0 600 -2 2]) ylabel(’h11’) subplot(2,2,2) plot(t,h2(1,:)’); axis([0 600 -2 2]) ylabel(’h12’) subplot(2,2,3) plot(t,h1(2,:)’); axis([0 600 -3 3]) ylabel(’h21’) subplot(2,2,4) plot(t,h2(2,:)’); axis([0 600 -3 3]) ylabel(’h22’) print -deps aircraft_act2