How to convert state space to transfer function

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Ali Almakhmari
Ali Almakhmari 2023-7-26
评论： Ali Almakhmari 2023-8-1
I have the following state space:
Where x is 12 by 1, A is 12 by 12, B is 12 by 3, w is 12 by 1, y is 6 by 1, H is 6 by 12, and v is 6 by 1. How can I convert this to transfer functions in MATLAB? I wanted to use the ss2tf command but I realized that it takes the state space in the format of:
Which is not what I have.

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Sam Chak
Sam Chak 2023-7-26
Hi @Ali Almakhmari
I think you can rearrange the matrices manually because
can be expressed as
.
So, if , then in compact form, it looks like
.
Now you should be able to use ss2tf(A, B, C, D).
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Sam Chak
Sam Chak 2023-7-26
Ran in:
Hi @Ali Almakhmari
According to the theoretical mathematics presented, and based on the info you provided, It should work. Have you tried and show us the result? Here is an example:
A = [0 1; % state matrix
-1 -2];
G1 = [0; % input matrix of u
1]
G1 = 2×1
0 1
G2 = eye(2) % input matrix of w
G2 = 2×2
1 0 0 1
G3 = zeros(2, 3) % input matrix of v
G3 = 2×3
0 0 0 0 0 0
B = [G1 G2 G3]; % augmented input matrix B
C = [1 0]; % state matrix
D = [0, 0 0, 1 1 1]; % augmented direct feedforward matrix D
% State-Space Model
sys = ss(A, B, C, D)
sys = A = x1 x2 x1 0 1 x2 -1 -2 B = u1 u2 u3 u4 u5 u6 x1 0 1 0 0 0 0 x2 1 0 1 0 0 0 C = x1 x2 y1 1 0 D = u1 u2 u3 u4 u5 u6 y1 0 0 0 1 1 1 Continuous-time state-space model.
% Transfer functions
P = tf(sys) % predicted number of TFs: 1*6 = 6 Tfs
P = From input 1 to output: 1 ------------- s^2 + 2 s + 1 From input 2 to output: s + 2 ------------- s^2 + 2 s + 1 From input 3 to output: 1 ------------- s^2 + 2 s + 1 From input 4 to output: 1 From input 5 to output: 1 From input 6 to output: 1 Continuous-time transfer function.
Ali Almakhmari
Ali Almakhmari 2023-8-1
Thank you so much. You have been very helpful. One last question I have, what if in the new system you told me to use, I only care about the first three rows of transfer functions: I care about the transfer functions in terms of u only in the [u; v; w] vector. So a total of 6*3 transfer functions. How can I tell ss2tf to do that for me? Because I dont want it to waste so much time solving for all the other ones.

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