unexact calculations when using delayss

Hello community,

I'm using a state-space 2x3 MIMO model with inputDelays. When defining the model, however, I get some unwanted behaviour. Here is an example:

% set A
A=zeros(2);
% set B
B=[pi 2*pi 1;
    0   1  0];
% set C
C=eye(2);
% set D
D=zeros(2,3);
% set delaystruct
delay=struct('delay',{3},'a', [],'b', -B,'c', [],'d', []);
% set G
G=delayss(A, B, C, D, delay);
G.b

When computing the inputmatrix with delays set to zero we get

   4.4409e-016  8.8818e-016            0
  -1.1102e-016            0 -4.1633e-017

although it should be zeros.

This bothers me even more when looking at output 2, which should definitely not depend on input 1 nor input 3. You see this effect quite well when looking at G in ltiview and activating 'normalize'

ltiview(G)

I hope you have some ideas how to solve this problem.

Greetz and thank you very much,

David

[Azzi Abdelmalek]

what do you want to do with your mimo system

[David T_]

I will use it in simulations (Matlab/Simulink) as the process-model and later on want to include it in a MPC. I guess the unbounded error will fade in my MPC since I will use a state observer.

Answer 1

[Matt Kindig]

Hi David,

This is due to the fact that Matlab (like any computational program) stores numbers in a finite-precision length. In the case of Matlab doubles, it is a 64-bit binary number, according to the IEEE 754 floating-point specification. Since all values have a finite precision, numerical issues such as this are common (and in fact are expected). See, for example, http://matlab.wikia.com/wiki/FAQ#Why_is_0.3_-_0.2_-_0.1_.28or_similar.29_not_equal_to_zero.3F

Typically you would use some tolerance to detect a "zero" in your system. For example, if you have a matrix A,

A =   [4.4409e-016  8.8818e-016            0;
  -1.1102e-016            0 -4.1633e-017];

you can check whether is is within your tolerance as such

isZero = all(abs(A(:)) < 1e-15);   %arbitrarily chosen tolerance

You'll see that isZero is true (1).

[David T_]

Hi Matt,

thanks for the quick answer. I did consider problems due to finite precision. So I'll go with it, at least for the I/O pairs I did define a dependency.

In my opinion, however, this does not fully explain why delayss somewhat 'creates' unwanted dependencies. Furthermore, if you look at how output 2 reacts to a step-input on input 1, you see that it gets unbounded (due to the unstable pole of my model).

Since I want to use my MIMO in a quite lengthy Simulation, I want to make sure my errors keep small. In the above example the error does accumulate very slowly, I know, but the system I use has unstable poles (meaning Re(s_p)>0; contrary to the above integrator). So I fear that the errors will corrupt the model.

Furthermore I use delayss quite often. And if this is a usual case, then I will have to look into my subsystems for unwanted I/O pairings... which I rather not.

I guess I will have to separate my MIMO and not use delayss?! Or does anybody experience something similar when combining systems?

Greetings, David

PS: Thanks Matt for the link, btw. It led to some very interesting insights on how MATLAB treats roots. (Among other stuff.)