 
Summary: ELEC 302 Solutions: HW #8
Problem [1]. Aircraft AFTI16 lateral motion. Consider the model given in HW #4. Compute the (infinite) con
trollability and observability gramians and hence deduce the states which are easiest/hardest to reach/observe.
Reduce the system to dimension 2 by keeping (i) the states which are easiest to reach and (ii) the states which are
easiest to observe. Are these reduced systems stable? Explain. Also, plot the frequency response of the original
system as well as of the reduced systems on the same (loglog) plot.
Solution. The following matlab code suffices.
A=[.746 .006 .999 .0369;12.9 .746 .387 0;4.31 .024 .174 0;0 1 0 0];
B=[.0012 .0092;6.05 .952;.416 1.76;0 0];
C=[0 0 1 0;0 0 0 1];
%gramians
P=lyap(A,B*B')
Q=lyap(A',C'*C)
%EVD of each gramian
[V,Dp]=eig(P)
[W,Dq]=eig(Q)
%state easiest to reach
V(:,4)
%state hardest to reach
V(:,1)
