augw :: Functions (Robust Control Toolbox™)

augw

Purpose

State-space or transfer function plant augmentation for use in weighted mixed-sensitivity H and H₂ loopshaping design

Syntax

```
P = AUGW(G,W1,W2,W3) 
```

Description

P = AUGW(G,W1,W2,W3) computes a state-space model of an augmented LTI plant P(s) with weighting functions W₁(s), W₂(s), and W₃(s) penalizing the error signal, control signal and output signal respectively (see block diagram) so that the closed-loop transfer function matrix is the weighted mixed sensitivity

where S, R and T are given by

The LTI systems S and T are called the sensitivity and complementary sensitivity, respectively.

Figure 6-1: Plant Augmentation

For dimensional compatibility, each of the three weights W₁, W₂ and W₃must be either empty, a scalar (SISO) or have respective input dimensions N_Y, N_U, and N_Y where G is N_Y-by-N_U. If one of the weights is not needed, you may simply assign an empty matrix [ ]; e.g., P = AUGW(G,W1,[],W3) is P(s) as in the Algorithm section below, but without the second row (without the row containing W2).

Algorithm

The augmented plant P(s) produced by is

Partitioning is embedded via P=mktito(P,NY,NU), which sets the InputGroup and OutputGroup properties of P as follows

[r,c]=size(P);
P.InputGroup  = struct('U1',1:c-NU,'U2',c-NU+1:c);
P.OutputGroup = struct('Y1',1:r-NY,'Y2',r-NY+1:r);

Examples

s=zpk('s'); G=(s-1)/(s+1);
W1=0.1*(s+100)/(100*s+1); W2=0.1; W3=[];
P=augw(G,W1,W2,W3);
[K,CL,GAM]=hinfsyn(P); [K2,CL2,GAM2]=h2syn(P);
L=G*K; S=inv(1+L); T=1-S; 
sigma(S,'k',GAM/W1,'k-.',T,'r',GAM*G/W2,'r-.')
legend('S = 1/(1+L)','GAM/W1', 'T=L/(1+L)','GAM*G/W2',2)

Limitations

The transfer functions G, W₁, W₂ and W₃must be proper, i.e., bounded as or, in the discrete-time case, as . Additionally, W₁, W₂ and W₃ should be stable. The plant G should be stabilizable and detectable; else, P will not be stabilizable by any K.