gfrancis

Francis equations for tracking

Calling Sequence

[L,M,T]=gfrancis(Plant,Model)

Arguments

:Plant a continuous time dynamical system in state-space

representation.

: :Model a continuous time dynamical system in state-space

representation.

: :L,M,T real matrices :

Description

Given 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.

Examples

Plant=`ssrand`_(1,3,5);
[F,G,H,J]=`abcd`_(Plant);
nw=4;nuu=2;A=`rand`_(nw,nw);
st=`max`_(`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)

See Also

• `lqg`_ LQG compensator
• `ppol`_ pole placement

History

Version Description 5.4.0 Sl is now checked for continuous time linear dynamical system. This modification has been introduced by this `commit`_ .. _ppol: ppol.html .. _lqg: lqg.html .. _commit: http://gitweb.scilab.org/?p=scilab.git;a=commit;h=3d7083daae3339813ba747c8adcda1f9599bb80d