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Scilab Help >> Control Systems - CACSD > Control Design > Tracking > gfrancis

# gfrancis

Francis equations for tracking

### Syntax

`[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)```

• 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