Causality in differential equations

[JTC]

Hello,

I am studying control theory. And I have encountered something I have never considered or thought about.

Consider a system with y as the output differential equation and u as the input.

a_ny⁽ⁿ⁾ + ... + a₁y⁽¹⁾ + a₀y = b_mu^(m) + ... + b₁u⁽¹⁾ + b₀u

Here, the subscripts indicate different constants (I am considering a Linear, Time Invariant system).
The superscripts indicate the order of the derivative.

I have now read that if m > n, the system is not causal.

Could someone explain:
Without reference to discrete time steps as encountered in a numerical system.
Without reference to the poles and zeros of the Laplace transform...

And with just looking at the equation above and the order of m and n, can someone explain why
m < n is the requirement for causality?

 

[andrewkirk]

Perhaps it helps to think about the very simple DE: $F = ma = mx^{(2)}$ where $x$ is position. This is Newton's third law. of motion. We think of the force $F$ as the input and $x$ as the output. The force affects what the value of $x$ will be in the future, because it affects a higher derivative of $x$. Generalising that, we can think of the inputs affecting the future of the outputs if they affect higher derivatives of the outputs than there are of the inputs, which corresponds to $n>m$ in your equation. In a sense the 'distance into the future' that the impact of one of the variables propagates is related to the highest order of the derivatives of the other variable in the equation. the variable with the lowest order propagates its influence furthest into the future.

This is rather hand-wavy. But that's consistent with the usual notion of the word 'causal', which is very vague and non-specific. Is there a definition of 'causal' in control theory that is precise enough to allow testing and reasoning about it?

 

[JTC]

I do not understand this

The force affects what the value of $x$ will be in the future, because it affects a higher derivative of $x$.

Can you explain this in and of itself?
AND
Can you explain this in the context of F, and its OWN derivatives: F, F', F'', F'''?

 

[andrewkirk]

Can you explain this in and of itself?

Well, given a record of F, the force applied over time, and the position x over time, if we make a small change to the force in time interval $(t,t+\delta t)$, that will make changes to the values of $x$ at times later than the end of that interval. If we instead make a small change to the value of $x$ in that interval, it enables us, by reverse engineering, to deduce changes to what $F$ must have been at times prior to the end of that interval, but the implied value of F after the end of that interval does not change.

Can you explain this in the context of F, and its OWN derivatives: F, F', F'', F'''?

All that's needed is to recognise that $F=ma$ is the same as $F = m \ddot x$ which is the same as $F = m x^{(2)}$, which is the same as
$$ mx^{(2)} = F^{(0)}$$
which is a differential equation of the type you describe above, with $x$ taking the role of $y$ and $F$ taking the role of $u$. We do not need to put higher derivatives of either $F$ or $x$ in the equation, although we could always include them with zero coefficients if we wanted to.

Also, have you been given a formal definition of 'causal' in your course? If not, the statement about the equation not being causal is vague and any explanation will necessarily be correspondingly vague.

 

[Tasos51]

I am sorry to dissapoint you, but this is an unanswered question. At least rigorously.
There are a lot of descriptive answers like the one above, but none to satisfy a strict mathematical approach.
For example in the answer above, one may define the derivative in past time i.e. in ( t, t-δx(t) ) and then the argument fails.
I have posted the exact same question elsewhere and have not receiced a satisfactory answer.

 

[mpresic3]

If I remember control theory correctly, The u's are the m+1 inputs, and the y's are the n + 1 outputs. It may be (I will need to look up the definition of causal in this context), that with the number of output states exceeding the number of input states, it may be impossible to control the inputs in such a way as to achieve the correct outputs. I will look at some texts to see if I can give a better answer.

I have to recall, does this have to do with controllability. I will look up soon

 

[mpresic3]

I looked up in my control theory book Modern Control Theory, by William L Brogan, Third Edition. I have not explored the issue in depth but the discussion after 3.4.4. State Equations from Transfer Functions on page 88 to address your question directly. It gives the transfer function Y(s) / U(s) as the ratio of two polynomials. Please see the discussion. It is probably in the other texts too. what text are you using?