Derivative after integration issue.

Why in simulink if you integrate and then derive a signal in the end you don't get the initial signal?

Answer 1

[Azzi Abdelmalek]

If you use these two blocks derivative block and Integrator block you will get the same result, however you can get, for certain cases a difference caused by numerical calculation errors. When the error occur during integration, obviously you will get error after derivation

[Vadims]

Well in some cases it could be a serious problem when using simulink. If you look at my Example , even a simple sine function has this issue. And since I am currently modeling errors of an inertial navigation system, I would not want to have final results blow up exponential due to Simulink imposed errors. If noone will have a better answer, I will accept your even though I knew that this could be the case. My intention was to address this issue.

[Azzi Abdelmalek]

In model configuration parameters, change the max step size, for example to 0.001, you will get a very small difference (1e-4)

Answer 2

[Matt J]

The derivative only reverses the integral of a continuous function, see

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus#First_part

[Azzi Abdelmalek]

Simulink provides continuous derivation, obviously, it's an approximation based on numerical solution

[Matt J]

Not sure what "continuous derivation" means.

I'm not sure it matters, though. If the continuous integral of a function is not differentiable at a certain point, the discrete approximation of a derivative there will give arbitrary results.

[Matt J]

For example, the integral of a step function (i.e., a ramp) is not differentiable at t=0. Therefore, different numerical approximations of the derivative there will give different results at that point. A right handed derivative will give the original value back, assuming t=0 is exactly sampled, but centered derivatives or other types will give something else.

[Azzi Abdelmalek]

Check it with simulink: derivative block and Integrator block : you will get the original signal

[Matt J]

Don't have Simulink. But if it's true, it simply implies that the derivative block does right handed derivatives.

[Vadims]

=< I am well aware of limitations when using derivatives and obviously I am using a continuous function. I have tried using sine and afterwards subtract original signal from ending one. The result is a sine function that also has a constant shift.

[Azzi Abdelmalek]

You mean a slight shift

[Matt J]

obviously I am using a continuous function.

How was it obvious? You just told us that now.

Anyway, if it is a slight shift, as Azzi proposes, I suspect it's because the differentiation step is using diff() internally, without any extrapolation. That's why the '1' is not recovered when diff() is applied in the example below.

    >> x=1:5

    x =

         1     2     3     4     5

    >> y=cumsum(x) %numerical integration

    y =

         1     3     6    10    15

    >> z=diff(y) %numerical differentiation

    z =

         2     3     4     5

Extrapolating the beginning of the signal with zeros is a solution for this,

    >> z2=diff([0,y])

    z2 =

         1     2     3     4     5

[Matt J]

It could also be because of mismatch between the integral approximation and the derivative approximation. If trapezoidal integration is used instead of cumsum() above, you get a mismatch when using diff:

    >> y2=cumtrapz(x)

    y2 =

             0    1.5000    4.0000    7.5000   12.0000

    >> z3=diff(y2)

    z3 =

        1.5000    2.5000    3.5000    4.5000

[Vadims]

Omg, can you read? I told, I was using simulink. And who in his right mind would ask about differenciating step values. Here you can see, that even sine has this problem. Note that original amplitude was 1 therefore difference with an aplitude of 0.06 and constant shift of -0.06 in not acceptable.

[Matt J]

Omg, can you read? I told, I was using simulink.

Yes, I didn't fail to see that we were talking about simulink. The manipulations I did at the command line are examples to show the issue. They are likely similar to what simulink is doing internally, and with similar issues.

And who in his right mind would ask about differenciating step values.

I don't know you and I don't know if you're in your right mind... But my remarks and examples of differentiator/integrator mismatch would apply to sines as well.