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.TH gfrancis 1 "April 1993" "Scilab Group" "Scilab Function"
.so ../sci.an
.SH NAME
gfrancis - Francis equations for tracking
.SH CALLING SEQUENCE
.nf
[L,M,T]=gfrancis(Plant,Model)
.fi
.SH PARAMETERS
.TP 15
Plant
: \fVsyslin\fR list
.TP 15
Model
: \fVsyslin\fR list
.TP 15
L,M,T
: real matrices
.SH DESCRIPTION
Given the the linear plant:
.nf
x'= F*x + G*u
y = H*x + J*u
.fi
and the linear model
.nf
xm'= A*xm + B*um
ym = C*xm + D*um
.fi
the goal is for the plant to track the model i.e. \fVe = y - ym ---> 0\fR
while keeping stable the state x(t) of the plant.
\fVu\fR is given by feedforward and feedback
.nf
u = L*xm + M*um + K*(x-T*xm) = [K , L-K*T] *(x,xm) + M*um
.fi
The matrices T,L,M satisfy generalized Francis equations
.nf
F*T + G*L = T*A
H*T + J*L = C
G*M = T*B
J*M = D
.fi
The matrix \fVK\fR must be chosen as stabilizing the pair \fV(F,G)\fR
See example of use in directory \fVdemos/tracking\fR.
.SH EXAMPLE
.nf
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)
.fi
.SH SEE ALSO
lqg, ppol
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