Bandwidth analysis of a system with multidimensional input

I am playing around with the built-in bandwidth function.

I can intuitively understand what is going on under the hood for a transfer function with scalar input and scalar output. However, how is the band width computed when the input is a vector of length > 1?

For instance, as in this example from the documentation:

A   = [-2 -1; 1 0];
B   = [1; 0];
C   = [1 2];
D   = 1;
sys = ss(A,B,C,D);

bandwidth(sys)

Answer 1

[Star Strider]

The length of the input vector is likely irrelevant, and is not even an argument to the bandwidth function. Note however that the bandwidth function only applies to SISO systems, or arrays of SISO systems (Find Bandwidth of Model Array), so it would not apply to MISO or MIMO systems.

[HaMo]

Apparently my ignorance is deeper than I thought. In my example above, the matrix eq is

x' = A x + B u

y = C x + D

If A is an 2x2 matrix, x and u must be 2x1. How is the gain and phase response determined when x and u has two values each at each sample time point?

[Star Strider]

Note that:

B   = [1; 0];

so the input is only to the first row of ‘A’. This is a SISO system. It doesn’t matter what ‘u’ is here.

[HaMo]

Thanks, that clarified things... a bit :)

Honestly I will need to look deeper into the underlying theory. I just thought that reverse-engineering the bandwidth function was a good start. Maybe if I can find the corresponding function in an open source implementation, like Python or octave.

Thanks for help

[Star Strider]

As always, my pleasure!

If you are taking a course in modern control, or linear control systems (it goes by several names), this will all be cleared up for you as the course proceeds. If you have not taken such a course, I definitely recommend it. An excellent text (in my opinion, and the one we used when I took a linear control systems course) is Chi-Tsong Chen Linear system Theory and Design (ISBN 0-19-511777-8). We used an earlier edition, and I still use it (this is the third edition) as my primary reference.

[HaMo]

After testing out a few different ideas in Matlab, I suddenly had an Eureka moment. This is the answer I was looking for:

A = [-2,-1;1,0];
B = [1;0];
C = [1,2];
D = 1;

% Laplace transformation:
% sX = AX + BU
% Y  = CX + D
%
% (sI - A)X = BU
% X = inv(sI - A) B U
% Y = C inv(sI - A) B U + D
% H(s) = C inv(sI - A) B 

sys = ss(A,B,C,D);
bandwidth(sys)

%% Built-in bode plot
figure(1)
bode(sys);

%% Home-made bode plot
s = (i*10.^(-2:.1:2))/(2*pi);    % Input frequencies
freq_response = nan(size(s));    % Frequency response vector
for i=1:length(s)
    freq_response(i) = C / (s(i) * eye(length(A)) - A) * B + D;
end

figure(2)
% Magnitude plot
subplot(2,1,1)
semilogx(abs(s), 10*log(abs(freq_response)))
% Phase plot
subplot(2,1,2)
semilogx(abs(s), 180/pi*phase(freq_response));

[Star Strider]

Great!

That example (from the documentation) is a SISO system.