gfrancis - Francis equations for tracking
Given the the linear plant:
x'= F*x + G*u
y = H*x + J*u
and the linear model
xm'= A*xm + B*um
ym = C*xm + D*um
the goal is for the plant to track the model i.e. e = y - ym ---> 0 while keeping stable the state x(t) of the plant. u is given by feedforward and feedback
u = L*xm + M*um + K*(x-T*xm) = [K , L-K*T] *(x,xm) + M*um
The matrices T,L,M satisfy generalized Francis equations
F*T + G*L = T*A
H*T + J*L = C
G*M = T*B
J*M = D
The matrix K must be chosen as stabilizing the pair (F,G) See example of use in directory demos/tracking .
Plant=ssrand(1,3,5);
[F,G,H,J]=abcd(Plant);
nw=4;nuu=2;A=rand(nw,nw);
st=maxi(real(spec(A)));A=A-st*eye(A);
B=rand(nw,nuu);C=2*rand(1,nw);D=0*rand(C*B);
Model=syslin('c',A,B,C,D);
[L,M,T]=gfrancis(Plant,Model);
norm(F*T+G*L-T*A,1)
norm(H*T+J*L-C,1)
norm(G*M-T*B,1)
norm(J*M-D,1)