Systems Non-Proper Transfer Fns: Causality?

[Tasos51]

In control systems, the definition of causality is taken as (in words): "the output does not depend on future inputs". For this definition, there exists a test: a system is causal if its impulse response h(t) is 0 for t<0. This is well documented. In parallel, and with no documentation, it is stated that an improper transfer function represents a non causal system. This makes the differentiator non causal. In some texts, this may stated otherwise, but there is NO text that states that an improper transfer function may relate to a causal system.

There is an alternative route to this, equally obscure. An improper transfer function cannot be represented in state space form. This form is only valid for causal systems.

Further, to properly deal with impulse response, we need generalized functions. Frankly, it goes beyond my capabilities. One thing I can mention though is that these functions possesses derivatives and integrals of any order, thus they behave differently from "ordinary" functions like the step or the impulse.

One last thing, @jasonRF where did you see the support of the unit doublet ?

 

[jasonRF]

One last thing, @jasonRF where did you see the support of the unit doublet ?

I learned it from A Guide to Distribution Theory and Fourier Transforms, by Strichartz. It was one of the required texts for a math course I took my senior year, and is pretty accessible.

You should be able to find it in other texts as well. For example, Distributions, Complex Variables and Fourier Transforms by Bremmermann has it as well.

Basically, they show that a distribution with point support is a finite linear combination of the delta distribution and its derivatives.

jason

 

[Tasos51]

I learned it from A Guide to Distribution Theory and Fourier Transforms, by Strichartz. It was one of the required texts for a math course I took my senior year, and is pretty accessible.

You should be able to find it in other texts as well. For example, Distributions, Complex Variables and Fourier Transforms by Bremmermann has it as well.

Basically, they show that a distribution with point support is a finite linear combination of the delta distribution and its derivatives.

jason

So, what is the support of the unit doublet ?

 

[jasonRF]

So, what is the support of the unit doublet ?

The support of $\delta^\prime(t)$ is $\{0\}$. That is, it is just a single point. Higher derivatives keep the same point support.

 

[Tasos51]

The support of $\delta^\prime(t)$ is $\{0\}$. That is, it is just a single point. Higher derivatives keep the same point support.

Could that mean that the system is not at initial rest, since at 0 it has some non-zero value ?
I mean, if the solution to the differential equation contains singularity terms concentrated at zero, this can be viewed as non-zero initial conditions, thus the system is not at initial rest.

 

[jasonRF]

I’m not sure I understand what you mean, but it sounds interesting. Could you elaborate?

Of course, within lumped circuit theory the impulse response of an ideal wire is $h(t)= \delta(t)$ so is also ‘singular’ at zero. So it must be related to having the derivatives of deltas…

Jason

 

[Tasos51]

I’m not sure I understand what you mean, but it sounds interesting. Could you elaborate?

Of course, within lumped circuit theory the impulse response of an ideal wire is $h(t)= \delta(t)$ so is also ‘singular’ at zero. So it must be related to having the derivatives of deltas…

Jason

I refer to the solution, not the impulse response. If it contains singularities at 0 (deltas and its derivatives), it means y(0) is not 0, which contradicts the requirement that a causal LTI system should be initially at rest. It is like having an instantaneous input a t=0, when there should be none (u(0)=0).

 

[jasonRF]

I think you are an assuming a particular domain of input functions. To me it seems that a differentiator at rest can only produce a singular output at zero if the input is not differentiable.

also, I still don’t understand how the singularity in the solution corresponds to initial conditions. Could you illustrate?

Edit: should add that part of why I’m confused is that the convolution of the input and the impulse response is by definition the zero state response. Or at least that is what I learned.

 

[jasonRF]

In case I haven’t made this clear enough, I still believe a differentiator, and hence a general improper system, is unstable, impractical and unrealizable. Further, it either doesn’t exist or has singular output (depending on perspective) when we mathematically analyze useful input functions of the form $H(t) e^{a t}$ (where I am using $H$ for the unit step function). It has plenty of problems, even though I believe it is causal.

Jason

 

[Tasos51]

I agree with all of the above, but not the last statement. I cannot "believe" in a mathematical statement: it is either proved or disproved. I believe though that the answer is difficult and invloves distribution theory. Another fact, as I mentioned, is that improper systems cannot be put in state-space form which is causal.
Finally I am afraid I cannot eleborate on my previous posts and perhaps they are nonsensial.

 

[jasonRF]

I cannot "believe" in a mathematical statement: it is either proved or disproved.

Agreed - I used a poor choice of words. Given a domain where the system is well defined (n-times differentiable inputs in the context of classical analysis, or simply use distribution theory throughout), then the nth order differentiator can be defined so that it is causal for all definitions of causality I have found. So improper systems can be causal.

My opinion is that it is nonsense to use a domain for which the system isn't always well-defined. I believe systems should map well-defined functions to well-defined functions, or distributions to distributions, or perhaps well-defined functions to distributions. That is where the 'belief' comes in. I don't at all think less of people who disagree with me on this!

If you don't know any distribution theory already, I do recommend the text by Gasquet and Witomski, who discuss conditions for causality in the context of continuous filters. It is more mathematical than Strichartz, though, who makes the subject very accessible but doesn't include a number of topics engineers really need.

Anyway, I think I have nothing more to say. Thanks for starting this thread - and thanks to DaveE who came up with the good example to force us to sharpen our thinking. I learned a lot. If you ever come up with a proof that satisfies you, please share it with us!

jason