# 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
- ppol

## History

5.4.0
Sl
is now checked for continuous time linear dynamical system. This modification has been introduced by this commit