Let's say that I got the transfer function matrix. I can use a state space model if I want it, but in this case, I will use a transfer function matrix:
$$ G(s) = \begin{bmatrix} \frac{5}{s^2 + 3s +6}\\ \frac{1}{s^2 + 0.1s +4} \end{bmatrix}$$
And I set the weighting matrices to:
$$W_1 = \frac{5}{s+1}$$ $$W_2 = \frac{1}{s+1}$$ $$W_3 = \frac{1}{s^2+s + 1}$$
Then I use MATLAB's command augw.m to create the agumented plant of transfer function matrix $G(s)$. https://se.mathworks.com/help/robust/ref/augw.html
The diagram for the augmented plant is:
>> G
Transfer function 'G' from input 'u1' to output ...
5
y1: -------------
s^2 + 3 s + 6
1
y2: ---------------
s^2 + 0.1 s + 4
Continuous-time model.
>> W1
Transfer function 'W1' from input 'u1' to output ...
5
y1: -----
s + 1
Continuous-time model.
>> W2
Transfer function 'W2' from input 'u1' to output ...
1
y1: -----
s + 1
Continuous-time model.
>> W3
Transfer function 'W3' from input 'u1' to output ...
1
y1: -----------
s^2 + s + 1
Continuous-time model.
>> P = augw(G, W1, W2, W3)
P.a =
x1 x2 x3 x4
x1 -1 0 0 0
x2 0 -1 0 0
x3 0 0 -1 0
x4 0 0 0 0
x5 0 0 0 1
x6 0 0 0 0
x7 0 0 0 0
x8 0 0 0 0
x9 0 0 0 0
x10 0 0 0 0
x11 0 0 0 0
x5 x6 x7 x8
x1 0 0 0 0
x2 0 0 0 0
x3 0 0 0 0
x4 -1 0 0 0
x5 -1 0 0 0
x6 0 0 -1 0
x7 0 1 -1 0
x8 0 0 0 4.768e-16
x9 0 0 0 4.487e-16
x10 0 0 0 -1
x11 0 0 0 0
x9 x10 x11
x1 0 0 5
x2 0 -5 1.908e-16
x3 0 0 0
x4 0 0 1
x5 0 0 0
x6 0 -1 3.816e-17
x7 0 0 0
x8 1.476e-15 4 -2.776e-16
x9 7.167e-16 -1.776e-15 -0.6
x10 -2.22e-16 -0.1 2.776e-16
x11 10 -3.553e-15 -3
P.b =
w1 w2 u1
x1 5 0 0
x2 0 5 0
x3 0 0 1
x4 0 0 0
x5 0 0 0
x6 0 0 0
x7 0 0 0
x8 0 0 -1
x9 0 0 -0.5
x10 0 0 0
x11 0 0 0
P.c =
x1 x2 x3 x4
z1 1 0 0 0
z2 0 1 0 0
z3 0 0 1 0
z4 0 0 0 0
z5 0 0 0 0
v1 0 0 0 0
v2 0 0 0 0
x5 x6 x7 x8
z1 0 0 0 0
z2 0 0 0 0
z3 0 0 0 0
z4 -1 0 0 0
z5 0 0 -1 0
v1 0 0 0 0
v2 0 0 0 0
x9 x10 x11
z1 0 0 0
z2 0 0 0
z3 0 0 0
z4 0 0 0
z5 0 0 0
v1 0 0 1
v2 0 -1 3.816e-17
P.d =
w1 w2 u1
z1 0 0 0
z2 0 0 0
z3 0 0 0
z4 0 0 0
z5 0 0 0
v1 1 0 0
v2 0 1 0
Input group 'W' = [1 2]
Input group 'U' = 3
Output group 'Z' = [1 2 3 4 5]
Output group 'V' = [6 7]
Continuous-time model.
>>
My question is:
The augmented plant in state space form, is displayed as:
$$\begin{bmatrix} \dot{x}\\ z\\ y \end{bmatrix}= P = \begin{bmatrix} A & B_1 & B_2 \\ C_1 & D_{11} & D_{12}\\ C_2 & D_{21} & D_{22} \end{bmatrix}\begin{bmatrix} x\\ u_1\\ u_2 \end{bmatrix}$$
So how should I interpret this from the terminal:
Input group 'W' = [1 2]
Input group 'U' = 3
Output group 'Z' = [1 2 3 4 5]
Output group 'V' = [6 7]
Does this explain which matrix represent, from P.a, P.b. P.c, P.d in the terminal, $A, B_1, B_2, C_1 , C_2, D_{11}, D_{12}, D_{21}, D_{22}$ ?
For example:
$C_2$ will be: $$C_2 = \begin{bmatrix} 0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 3.816e-17 \end{bmatrix}$$
$D_{21}$ will be:
w1 w2
z1 0 0
z2 0 0
z3 0 0
z4 0 0
z5 0 0
Right?
The w1 and w2 are reference vectors. In many cases, they are named as r1 and r2.