Demonstrate eigenvector assignment using an STOVL Model for transition.
The example is taken from:
Lee, H. P., Jr., Yousseff, H.M. and R.P. Habek, "Application of Eigenstructure Assignment to the Design of STOVL Flight Control Systems," AIAA 88-4140-CP.
------------------------------------------------------------------------ See also STOVL, ESAssign, IC ------------------------------------------------------------------------
Contents
%--------------------------------------------------------------------------- % Copyright (c) 2003 Princeton Satellite Systems, Inc. % All rights reserved. %---------------------------------------------------------------------------
System
%------- g = STOVL('longitudinal transition'); disp(' ') disp('---------------------') disp('Open loop eigenvalues') disp('---------------------') eig(g)
---------------------
Open loop eigenvalues
---------------------
ans =
-2.5144 + 0i
1.8883 + 0i
-0.024921 + 0.1597i
-0.024921 - 0.1597i
-50 + 0i
-50 + 0i
-4 + 0i
Desired eigenvalues
%-------------------- j = sqrt(-1); lambda = [ -1.25 + j;... -1.25 - j;... -7.5;... -1.4];
Desired eigenvectors
%--------------------- vD = [ 1 nan 0 0;... nan 1 0 0;... 0 0 nan 1;... 0 0 1 nan;... nan*ones(3,4)]; disp(vD); fC = [1 1 0 0;... 0 0 1 0;... 0 0 0 1];
1 NaN 0 0
NaN 1 0 0
0 0 NaN 1
0 0 1 NaN
NaN NaN NaN NaN
NaN NaN NaN NaN
NaN NaN NaN NaN
Compute the gain and the achieved eigenvectors
%----------------------------------------------- disp(' ') disp('----') disp('Gain') disp('----') k = ESAssign( g, lambda, vD, fC ); disp(k);
----
Gain
----
Warning: The selected eigenvector for eigenvalue 2 leads to an ill-conditioned inverse.
Warning: The selected eigenvector for eigenvalue 4 leads to an ill-conditioned inverse.
1.0955 2.0764 0 0
0 0 -1.8914 0
0 0 0 0.53071
Create the closed loop system
%------------------------------ [a, b, c] = getabcd( g ); aCL = a - b*k*c; disp(' ') disp('-----------------------') disp('Closed loop eigenvalues') disp('-----------------------') eig(aCL)
-----------------------
Closed loop eigenvalues
-----------------------
ans =
-45.069 + 0i
-49.471 + 0i
-4.0474 + 0.29163i
-4.0474 - 0.29163i
-0.77002 + 1.0853i
-0.77002 - 1.0853i
-0.50182 + 0i
Digitize the closed loop system using a zero order hold
%-------------------------------------------------------- % Simulate %--------- x = [0;0;0;0;0;pi/180;0]; IC( g, x, 0.01, 1000 ); g = set( g, aCL, 'a' ); x = [0;0;0;0;0;pi/180;0]; IC( g, x, 0.01, 1000 ); %--------------------------------------
