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# feedback

feedback operation

Sl=Sl1/.Sl2

### Arguments

Sl1,Sl2

linear systems (syslin list) in state-space or transfer form, or ordinary gain matrices.

Sl

linear system (syslin list) in state-space or transfer form

### Description

The feedback operation is denoted by /. (slashdot). This command returns Sl=Sl1*(I+Sl2*Sl1)^-1, i.e the (negative) feedback of Sl1 and Sl2. Sl is the transfer v -> y for y = Sl1 u, u = v - Sl2 y.

The result is the same as Sl=LFT([0,I;I,-Sl2],Sl1).

 Caution: do not use with decimal point (e.g. 1/.1 is ambiguous!)

### Examples

S1=ssrand(2,2,3);S2=ssrand(2,2,2);
W=S1/.S2;
ss2tf(S1/.S2)
//Same operation by LFT:
ss2tf(lft([zeros(2,2),eye(2,2);eye(2,2),-S2],S1))
//Other approach: with constant feedback
BigS=sysdiag(S1,S2); F=[zeros(2,2),eye(2,2);-eye(2,2),zeros(2,2)];
Bigclosed=BigS/.F;
W1=Bigclosed(1:2,1:2);   //W1=W (in state-space).
ss2tf(W1)
//Inverting
ss2tf(S1*inv(eye()+S2*S1))

• lft — linear fractional transformation
• sysdiag — соединение блочно-диагональной системы
• augment — augmented plant
• obscont — observer based controller

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