unexact calculations when using delayss

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David T_
David T_ 2012-10-5
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
  2 个评论
Azzi Abdelmalek
Azzi Abdelmalek 2012-10-5
what do you want to do with your mimo system
David T_
David T_ 2012-10-8
编辑：David T_ 2012-10-8
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.

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回答（1 个）

Matt Kindig
Matt Kindig 2012-10-5
编辑：Matt Kindig 2012-10-5
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).
  1 个评论
David T_
David T_ 2012-10-8
编辑：David T_ 2012-10-8
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.)

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