Demonstrate eigenvector assignment using an L1011 STOVL Model.
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 L1011, ESAssign, IC, PrintEigenvalues ------------------------------------------------------------------------
Contents
------------------------------------------------------------------------- Copyright (c) 2003 Princeton Satellite Systems, Inc. All rights reserved. -------------------------------------------------------------------------
System
%------- g = L1011; disp(' ') disp('---------------------') disp('Open loop eigenvalues') disp('---------------------') eig(g)
---------------------
Open loop eigenvalues
---------------------
ans =
-0.5 + 0i
-0.08819 + 1.2695i
-0.08819 - 1.2695i
-1.0855 + 0i
-0.0091648 + 0i
-20 + 0i
-25 + 0i
Desired eigenvalues
%-------------------- j = sqrt(-1); lambda = [ -1.5 + j*1.5;... -1.5 - j*1.5;... -2.0 + j*1.5;... -2.0 - j*1.5]; vD = [nan*ones(2,4);... 0 0 1 nan;... 1 nan 0 0;... 0 0 nan 1;... nan 1 0 0;... nan nan 0 0]; fC = [1 1 0 0;0 0 1 1]; r = [0 0 0 0;... 0 0 0 0;... 1 1 1 0;... 1 0 1 1;... 1 1 0 1;... 0 1 1 1;... 0 0 1 1];
Compute the gain and the achieved eigenvectors
%----------------------------------------------- f = {}; f{1} = ESAssign( g, lambda, vD ); f{2} = ESAssign( g, lambda, vD, [1 0 1 0;1 1 1 1] ); f{3} = ESAssign( g, lambda, vD, [1 0 1 0;0 1 1 1] ); f{4} = ESAssign( g, lambda, vD, [1 0 1 0;0 1 0 1] ); [a, b, c] = getabcd( g ); x = [.1;0;0;0;0;0;0]; for k = 1:4 fprintf('------------------\nCase %i\n------------------',k); gCL = g; aCL = a - b*f{k}*c; gainMatrix = f{k}; disp(gainMatrix); PrintEigenvalues( aCL, '%7.3f', 'Closed Loop Eigenstructure' ) disp(' ') PrintEigenvalues( struct('lambda',lambda,'vD',vD), '%7.3f', 'Desired Eigenstructure' ); gCL = set( gCL, aCL, 'a' ); IC( gCL, x, 0.01, 1000 ); end %--------------------------------------
------------------
Case 1
------------------ -3.3463 0.14974 4.8827 0.37556
-1.416 -2.4265 6.3571 -4.7824
Closed Loop Eigenstructure
-22.012 -17.060 -0.699 -1.500 + j 1.500 -1.500 - j 1.500 -2.000 + j 1.500 -2.000 - j 1.500
-------------------------------------------------------------------------------------------------------------------------
0.011 -0.997 0.538 0.762 + j 0.000 0.762 + j 0.000 -0.043 - j 0.000 -0.043 + j 0.000
0.999 -0.058 -0.743 0.129 - j 0.551 0.129 + j 0.551 0.829 + j 0.000 0.829 + j 0.000
-0.002 -0.001 -0.026 -0.000 - j 0.000 -0.000 + j 0.000 0.014 - j 0.207 0.014 + j 0.207
0.002 -0.044 0.127 0.249 + j 0.122 0.249 - j 0.122 -0.000 - j 0.001 -0.000 + j 0.001
0.053 0.017 0.018 0.000 - j 0.000 0.000 + j 0.000 0.282 + j 0.435 0.282 - j 0.435
0.000 -0.001 0.200 0.034 + j 0.125 0.034 - j 0.125 -0.002 + j 0.002 -0.002 - j 0.002
-0.000 0.001 -0.318 -0.010 - j 0.076 -0.010 + j 0.076 -0.000 + j 0.000 -0.000 - j 0.000
Desired Eigenstructure
-1.500 + j 1.500 -1.500 - j 1.500 -2.000 + j 1.500 -2.000 - j 1.500
----------------------------------------------------------------------------------------
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
0.000 + j 0.000 0.000 + j 0.000 1.000 + j 0.000 NaN + j 0.000
1.000 + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
0.000 + j 0.000 0.000 + j 0.000 NaN + j 0.000 1.000 + j 0.000
NaN + j 0.000 1.000 + j 0.000 0.000 + j 0.000 0.000 + j 0.000
NaN + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
------------------
Case 2
------------------ -3.3414 0 4.8713 0
-1.416 -2.4265 6.3571 -4.7824
Closed Loop Eigenstructure
-22.020 -17.118 -0.697 -1.497 + j 1.500 -1.497 - j 1.500 -1.971 + j 1.485 -1.971 - j 1.485
-------------------------------------------------------------------------------------------------------------------------
-0.022 -0.997 0.537 0.761 + j 0.000 0.761 + j 0.000 0.236 + j 0.039 0.236 - j 0.039
0.998 -0.058 -0.743 0.128 - j 0.551 0.128 + j 0.551 0.813 + j 0.000 0.813 + j 0.000
-0.002 -0.001 -0.027 0.000 + j 0.000 0.000 - j 0.000 0.047 - j 0.191 0.047 + j 0.191
0.001 -0.044 0.126 0.250 + j 0.122 0.250 - j 0.122 0.069 + j 0.045 0.069 - j 0.045
0.054 0.017 0.019 -0.001 - j 0.000 -0.001 + j 0.000 0.192 + j 0.446 0.192 - j 0.446
0.000 -0.001 0.200 0.034 + j 0.125 0.034 - j 0.125 0.007 + j 0.033 0.007 - j 0.033
-0.000 0.001 -0.320 -0.010 - j 0.077 -0.010 + j 0.077 -0.004 - j 0.019 -0.004 + j 0.019
Desired Eigenstructure
-1.500 + j 1.500 -1.500 - j 1.500 -2.000 + j 1.500 -2.000 - j 1.500
----------------------------------------------------------------------------------------
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
0.000 + j 0.000 0.000 + j 0.000 1.000 + j 0.000 NaN + j 0.000
1.000 + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
0.000 + j 0.000 0.000 + j 0.000 NaN + j 0.000 1.000 + j 0.000
NaN + j 0.000 1.000 + j 0.000 0.000 + j 0.000 0.000 + j 0.000
NaN + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
------------------
Case 3
------------------ -3.3414 0 4.8713 0
0 -2.4407 3.5084 -4.8536
Closed Loop Eigenstructure
-22.029 -17.169 -1.628 + j 1.675 -1.628 - j 1.675 -1.809 + j 1.334 -1.809 - j 1.334 -0.700
-------------------------------------------------------------------------------------------------------------------------
-0.022 -0.955 0.157 + j 0.342 0.157 - j 0.342 0.339 - j 0.054 0.339 + j 0.054 -0.541
0.998 0.292 0.828 + j 0.000 0.828 + j 0.000 0.718 + j 0.000 0.718 + j 0.000 0.741
-0.002 -0.002 0.073 - j 0.134 0.073 + j 0.134 0.090 - j 0.222 0.090 + j 0.222 0.005
0.001 -0.041 -0.004 + j 0.132 -0.004 - j 0.132 0.117 + j 0.030 0.117 - j 0.030 -0.128
0.054 0.040 0.107 + j 0.341 0.107 - j 0.341 0.133 + j 0.522 0.133 - j 0.522 -0.004
0.000 -0.001 -0.046 + j 0.035 -0.046 - j 0.035 0.027 + j 0.045 0.027 - j 0.045 -0.200
-0.000 0.001 0.028 - j 0.017 0.028 + j 0.017 -0.016 - j 0.028 -0.016 + j 0.028 0.320
Desired Eigenstructure
-1.500 + j 1.500 -1.500 - j 1.500 -2.000 + j 1.500 -2.000 - j 1.500
----------------------------------------------------------------------------------------
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
0.000 + j 0.000 0.000 + j 0.000 1.000 + j 0.000 NaN + j 0.000
1.000 + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
0.000 + j 0.000 0.000 + j 0.000 NaN + j 0.000 1.000 + j 0.000
NaN + j 0.000 1.000 + j 0.000 0.000 + j 0.000 0.000 + j 0.000
NaN + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
------------------
Case 4
------------------ -3.3414 0 4.8713 0
0 -2.4582 0 -4.9307
Closed Loop Eigenstructure
-22.001 -17.165 -1.432 + j 1.738 -1.432 - j 1.738 -2.035 + j 1.280 -2.035 - j 1.280 -0.670
-------------------------------------------------------------------------------------------------------------------------
0.022 -0.961 -0.086 + j 0.341 -0.086 - j 0.341 0.200 - j 0.060 0.200 + j 0.060 -0.491
-0.998 0.270 0.822 + j 0.000 0.822 + j 0.000 0.775 + j 0.000 0.775 + j 0.000 0.740
0.002 -0.002 0.066 - j 0.160 0.066 + j 0.160 0.007 - j 0.227 0.007 + j 0.227 0.219
-0.001 -0.041 -0.085 + j 0.091 -0.085 - j 0.091 0.072 + j 0.008 0.072 - j 0.008 -0.105
-0.054 0.039 0.183 + j 0.343 0.183 - j 0.343 0.275 + j 0.471 0.275 - j 0.471 -0.147
-0.000 -0.001 -0.057 - j 0.006 -0.057 + j 0.006 0.020 + j 0.022 0.020 - j 0.022 -0.187
0.000 0.001 0.031 + j 0.008 0.031 - j 0.008 -0.013 - j 0.013 -0.013 + j 0.013 0.309
Desired Eigenstructure
-1.500 + j 1.500 -1.500 - j 1.500 -2.000 + j 1.500 -2.000 - j 1.500
----------------------------------------------------------------------------------------
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
NaN + j 0.000 NaN + j 0.000 NaN + j 0.000 NaN + j 0.000
0.000 + j 0.000 0.000 + j 0.000 1.000 + j 0.000 NaN + j 0.000
1.000 + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
0.000 + j 0.000 0.000 + j 0.000 NaN + j 0.000 1.000 + j 0.000
NaN + j 0.000 1.000 + j 0.000 0.000 + j 0.000 0.000 + j 0.000
NaN + j 0.000 NaN + j 0.000 0.000 + j 0.000 0.000 + j 0.000
