Problem with using mussv

I have a state space representation of a system and I am interested in its frequency response. I calculate it using the function freqresp and the resulting array has dimensions: 3*(N-1)x3*(N-1)x100 (because the length of my frequency vector is 100). This works as expected, and the next thing I do is to rearrange the array that I got (by adding zeros) using the cat command to have dimensions: 3*(N-1)x9*(N-1)x100. This also works as expected. The reason for doing that is because I want to calculate bounds on the structured singular values of this rearranged array, and my uncertainty set is comprised of 3 blocks of dimensions: (N-1)x3*(N-1).

For clarification I provide a relevant part of my code:

N = 30;
R = 690;

w_vec = logspace(-5,5,100);

Blocks = [N-1,3*(N-1);N-1,3*(N-1);N-1,3*(N-1)];
state_space = ss(A,B,C_grad,D);
freq_data = freqresp(state_space,w_vec);
O = zeros(N-1,3*(N-1),length(w_vec));
freq_data_rearanged = cat(1,cat(2,freq_data(1:N-1,:,:),O,O), ...
                         cat(2,O,freq_data(N:2*(N-1),:,:),O), ...
                         cat(2,O,O,freq_data(2*N-1:3*(N-1),:,:)));

bounds = mussv(freq_data_rearanged,Blocks,'Ufs');
       

My problem is that I get the error: The dimensions of matrix are incompatible with the BLK dimensions.

I don't understand why this is happening, because as far as I understand my uncertainty matrix has dimensions: 3*(N-1)x9*(N-1) and the frequency response (named freq_data_rearanged in the code) has dimensions 3*(N-1)x9*(N-1)x100.

What am I doing wrong here?

Answer 1

[Paul]

Hi Ofek,

N = 30;
w_vec = logspace(-5,5,100);

"I calculate it using the function freqresp and the resulting array has dimensions: 3*(N-1)x3*(N-1)x100 (because the length of my frequency vector is 100)."

Based on that statement, the system state_space will have 3*(N-1) inputs and 3*(N-1) outputs. Since that system is not given, I'll create one with just a few states and compute its frequency response

rng('default');
state_space = rss(3,3*(N-1),3*(N-1));
size(state_space)

State-space model with 87 outputs, 87 inputs, and 3 states.

freq_data = freqresp(state_space,w_vec);
size(freq_data)

ans = 1×3

87 87 100

The block structure is

Blocks = [N-1,3*(N-1);N-1,3*(N-1);N-1,3*(N-1)]

Blocks = 3×2

29 87 29 87 29 87

which means the system has three full blocks of complex uncertainty, each with dimension of 29 x 87.

Create the augmented system

O = zeros(N-1,3*(N-1),length(w_vec));
freq_data_rearanged = cat(1,cat(2,freq_data(1:N-1,:,:),O,O), ...
                         cat(2,O,freq_data(N:2*(N-1),:,:),O), ...
                         cat(2,O,O,freq_data(2*N-1:3*(N-1),:,:)));
size(freq_data_rearanged)

ans = 1×3

87 261 100

Each page of freq_data_rearanged, calli it M, is 87 x 261, i.e., it represents a system with 87 outputs and 261 inputs. But the uncertainty is a block diagonal matrix, call it D, with dimensions that are also 87 x 261. But M and D have to be compatible for multiplication (I - M*D is the quantity of interest), which they are not. Either D should be 261 x 87 and M 87 x 261, or vice versa.

Here's the result assuming the uncertainty blocks need to be transposed

bounds1 = mussv(freq_data_rearanged,fliplr(Blocks),'Ufs');
figure
semilogx(w_vec,db(squeeze(bounds1)))

And here is the result assuming the system needs to be transposed

bounds2 = mussv(pagetranspose(freq_data_rearanged),Blocks,'Ufs');
figure
semilogx(w_vec,db(squeeze(bounds2)))

[Ofek Frank-Shapir]

Thank you!

this indeed solved the problem.

[Paul]

You're quite welcome.

If you don't mind me asking, I'm curious about this procedure. Why is the original system 87 x 87 which then needs to be block diagonalized into 87 x 261? That seems atypical for an uncertain, physical system.

Also, freq_data_rearanged could be constructed with one call to append if freq_data is an frd object and mussv shold work fine if freq_data_rearranged is an frd object.