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Ajuda Scilab >> CACSD > augment

# augment

augmented plant

### Calling Sequence

```[P,r]=augment(G)
[P,r]=augment(G,flag1)
[P,r]=augment(G,flag1,flag2)```

### Arguments

G

linear system (`syslin` list), the nominal plant

flag1

one of the following (upper case) character string: `'S'`, `'R'`, `'T'` `'SR'`, `'ST'`, `'RT'` `'SRT'`

flag2

one of the following character string: `'o'` (stands for 'output', this is the default value) or `'i'` (stands for 'input').

P

linear system (`syslin` list), the ``augmented'' plant

r

1x2 row vector, dimension of `P22 = G`

### Description

If `flag1='SRT'` (default value), returns the "full" augmented plant

```[ I | -G]   -->'S'
[ 0 |  I]   -->'R'
P = [ 0 |  G]   -->'T'
[-------]
[ I | -G]```

`'S'`, `'R'`, `'T'` refer to the first three (block) rows of `P` respectively.

If one of these letters is absent in `flag1`, the corresponding row in `P` is missing.

If `G` is given in state-space form, the returned `P` is minimal. `P` is calculated by: `[I,0,0;0,I,0;-I,0,I;I,0,0]*[I,-G;0,I;I,0]`.

The augmented plant associated with input sensitivity functions, namely

```[ I | -I]   -->'S'  (input sensitivity)
[ G | -G]   -->'R'  (G*input sensitivity)
P = [ 0 |  I]   -->'T'  (K*G*input sensitivity)
[-------]
[ G | -G]```

is obtained by the command `[P,r]=augment(G,flag,'i')`. For state-space `G`, this `P` is calculated by: `[I,-I;0,0;0,I;0,0]+[0;I;0;I]*G*[I,-I]` and is thus generically minimal.

Note that weighting functions can be introduced by left-multiplying `P` by a diagonal system of appropriate dimension, e.g., `P = sysdiag(W1,W2,W3,eye(G))*P`.

Sensitivity functions can be calculated by `lft`. One has:

For output sensitivity functions [P,r]=augment(P,'SRT'): lft(P,r,K)=[inv(eye()+G*K);K*inv(eye()+G*K);G*K*inv(eye()+G*K)];

For input sensitivity functions [P,r]=augment(P,'SRT','i'): lft(P,r,K)=[inv(eye()+K*G);G*inv(eye()+K*G);K*G*inv(eye()+G*K)];

### Examples

```G=ssrand(2,3,2); //Plant
K=ssrand(3,2,2); //Compensator
[P,r]=augment(G,'T');
T=lft(P,r,K);   //Complementary sensitivity function
Ktf=ss2tf(K);Gtf=ss2tf(G);
Ttf=ss2tf(T);T11=Ttf(1,1);
Oloop=Gtf*Ktf;
Tn=Oloop*inv(eye(Oloop)+Oloop);
clean(T11-Tn(1,1));
//
[Pi,r]=augment(G,'T','i');
T1=lft(Pi,r,K);T1tf=ss2tf(T1); //Input Complementary sensitivity function
Oloop=Ktf*Gtf;
T1n=Oloop*inv(eye(Oloop)+Oloop);
clean(T1tf(1,1)-T1n(1,1))```

• lft — linear fractional transformation
• sensi — sensitivity functions

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