F-18 Longitudinal Dynamics: Stability, Control and Plots

Learn about the stability and controllability of F-18 Longitudinal Dynamics system and plot magnitude and phase plots. Get insights into the equivalent compensator for the closed loop system.

Consider the F-18 Longitudinal Dynamics.............................1
For system stability................................................3
for magnitude and phase plots.......................................4
For System Controllability..........................................5
Consider the F-18 Longitudinal Dynamics
A=[-0.0239 -28.3172 0 -32.2;-0.0003 -0.3621 1 0;0 -2.2115 -0.2532 0; 0 0 1 0]
B=[-3.8114 0.001; -0.0515 0;-2.8791 0;0 0]
C=[0 -1 0 1;1 0 0 0]
...Compute the open loop system modes
...To obtain the eigenvalues and eigenvectors
[V,D]=eig(A);
disp('EigenValues:')
disp(abs(V))
disp('EigenVector:')
disp(abs(D))
poles=[-1.149;-8;-8.5;-9]; %place observer poles
K=place(A,B,abs(poles));
disp(abs(K))
[t1 t2]=eig(A-B*K)
eig(A-B*K)
pls=[-0.05+0.09i,-0.05-0.09i,-1.05+1.1i,-1.05-1.1i];
L=place(A',C',pls')'
% To obtain the eigen value
eig(A-L*C)
Ac=abs(A-B*K-L*C);
Bc=L;
Cc=abs(K);
Gcss=ss(Ac,Bc,Cc,0);
Gc=tf(Gcss);
zpk(Gc)
A =
-0.0239 -28.3172 0 -32.2000
-0.0003 -0.3621 1.0000 0
1

0 -2.2115 -0.2532 0
0 0 1.0000 0
B =
-3.8114 0.0010
-0.0515 0
-2.8791 0
0 0
C =
0 -1 0 1
1 0 0 0
EigenValues:
0.9987 0.9987 1.0000 1.0000
0.0252 0.0252 0.0000 0.0000
0.0377 0.0377 0.0003 0.0003
0.0249 0.0249 0.0030 0.0030
EigenVector:
1.5124 0 0 0
0 1.5124 0 0
0 0 0.0966 0
0 0 0 0.0966
1.0e+05 *
0.0000 0.0012 0.0001 0.0009
0.0854 5.1051 0.3385 3.1935
t1 =
-0.0020 -0.9987 -0.0272 0.0483
-0.4652 -0.0065 0.1233 -0.1354
-0.6677 -0.0498 0.9859 -0.9820
-0.5812 -0.0059 0.1095 -0.1227
t2 =
1.1490 0 0 0
0 8.5000 0 0
2

0 0 9.0000 0
0 0 0 8.0000
ans =
1.1490
8.5000
9.0000
8.0000
L =
0.0987 1.7092
0.1680 0.0166
1.1157 0.0595
0.0196 0.0022
ans =
-1.0500 + 1.1000i
-1.0500 - 1.1000i
-0.0500 + 0.0900i
-0.0500 - 0.0900i
ans =
From input 1 to output...
32.018 (s+7.534) (s-6.785) (s-0.03526)
1: ---------------------------------------
(s-44.8) (s+13.63) (s-6.789) (s-0.1621)
1.3067e05 (s+8.767) (s-7.038) (s-2.049)
2: ---------------------------------------
(s-44.8) (s+13.63) (s-6.789) (s-0.1621)
From input 2 to output...
2.7555 (s-5.061) (s+4.681) (s+0.08024)
1: ---------------------------------------
(s-44.8) (s+13.63) (s-6.789) (s-0.1621)
25803 (s-30.06) (s+12.13) (s-0.2843)
2: ---------------------------------------
(s-44.8) (s+13.63) (s-6.789) (s-0.1621)
3

Continuous-time zero/pole/gain model.
For system stability
%To determine the equivalent compensator for the closed loop system
...transfer function.
figure(1)
rlocus(Gc(1,1))
grid on
for magnitude and phase plots
figure(2)
bode(Gc(1,1))
grid on
4

For System Controllability
disp('Testing for system controllability:')
sysCont=ctrbf(Ac,Bc,Cc);
ranQx=rank(sysCont);
if(ranQx==rank(Ac))
disp('System is controllable')
else
disp('System is NOT controllable')
end
Testing for system controllability:
System is controllable
5