| | | Scilab Reference Manual | |
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augment — augmented plant
[P,r]=augment(G) [P,r]=augment(G,flag1) [P,r]=augment(G,flag1,flag2)
| 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 |
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)];
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))
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