Number of zeroes more than the number of poles will lead to instability

I read in some book that if the order of the numerator is greater than the order of the denominator of the transfer function then the system becomes unstable. Why is that so?

Thanks in advance.

iVenky said:

I read in some book that if the order of the numerator is greater than the order of the denominator of the transfer function then the system becomes unstable. Why is that so?
Thanks in advance.

Click to expand...

No, that's not correct. There is no real system with a numerator that is larger in order than the denumerator.
The reason: Such a system would go to infinity for rising frequencies - and that is not possible in reality.
But - as far as stability is concerned - there is no stability problem in the classical sense.

It does not exist generally but even if it is.. the reason is phase margin. left hand zero causes indefinite oscillations. Right hand zero acts more or less like a let hand pole.

SWINI, I think you got it a bit confused here.

If the order of the numerator is greater than that of the denominator the system is called non-causal.
Since its output depends not only from the present and past input/output but also depends from future inputs.
This cannot be realized as a real time system but can be built digitally, postprocessing a large enough amount of data. Sometimes non-causal filters are used in practice.
Stability must be investigated

albbg said:

If the order of the numerator is greater than that of the denominator the system is called non-causal.
Since its output depends not only from the present and past input/output but also depends from future inputs.
This cannot be realized as a real time system but can be built digitally, postprocessing a large enough amount of data. Sometimes non-causal filters are used in practice.
Stability must be investigated

Click to expand...

How do you say that if the order of numerator is greater than that of denominator then the system depends on the future input?

Non-causality has nothing to do with the question.

A system like the described one has one or more poles at infinity.

Let's take the simplest example of such a system: an ideal differentiator.

H(s) = k*s (k is a constant)

It has one zero, no finite poles (just a pole at infinity).

It is causal and stable.
It is true that a real-life differentiator must have a pole at some finite frequency. But if the frequency of the pole is so high that the whole system (circuit and signal in which the differentiator is used) does not have response at that frequency and beyond, then it is a good approximation to consider that the pole does not exist (i.e. it is at infinity).

Regards

Z

I completely agree with Zorro. The problem of causality has nothing to do with the original question.
LvW

Zorro, the system H(s) = k*s is not-causal

Please have a look at:
**broken link removed** (chap. 7.3)
**broken link removed** (chap. 2.3.2)
http://disi.unitn.it/~palopoli/courses/SS/SSlect7.pdf (pag. 8 and pag. 23)

A differentiator is not anti-causal, which is what matters. I don't think not-causal is even a coherent term (unless you mean it's just memoryless, which is something completely different).

Anyways, yes a differentiator is a perfectly valid example of a stable system with more zeros than poles.

Yes, I was not discussing about the stability, but the causality

albbg said:

Please have a look at:
**broken link removed** (chap. 7.3)
**broken link removed** (chap. 2.3.2)
http://disi.unitn.it/~palopoli/courses/SS/SSlect7.pdf (pag. 8 and pag. 23)

Click to expand...

Hi albbg,

Thank you for the references. Let me make some comments:

Reference 1 treats discrete-time systems, that is not the case under consideration.

Reference 2 treats discrete-time systems too. The only reference to continuous-time is eq. (2.15) and the phrase "The order of b(s) must not be greater than the order of a(s) or the system will be non-causal." Unfortunately, it does not give a definition of causality and does not justify the statement.

Reference 3 (page 8) also lacks a definition and a justification of the statement.
In page 23 it says that a differentiator is not causal and that "In order to make it causal we have to have one or more poles that reduce the growth of the transfer function."
I agree with the second part (reduce the growth of the transfer function, that is related with my post #7 above) but not with the fact related with causality.

I would agree with a definition of causality more or less like this:
A system with input x(t) and output y(t) is causal if for any t0, y(t0) does not depend of x(t) for any t>t0.

Regards

Z

Interesting discussion; let me try to explain:

we can consider the step function 1(t-to) and the ramp function R(t-to) both starting at time "to" and always zero before "to". We know that:

R(t-to) = ∫1(t-to)*dt <==> dR(t-to)/dt = 1(t-to)

To integrate the function 1(t-to) to obtain the ramp we just need to know the value up to the actual one. We will have always zero until the time to has reached where the ramp will start to rise.

Considering instead the opposite operation, that is the time derivative of the ramp to obtain the step, if we consider only the values up to the present one we will have always zero until the time "to" is surpassed. But the discontinuity of the step 1(t-to) appears exactly at the istant "to": left limit (t -> to) of 1(t-to) = 0 but right limit (t -> to) of 1(t-to) = 1
The only way to correctly reconstruct it is to consider also the future input.

This means that the integrator (1/s) is a causal system while the differentiator (s) is a non-causal system.

By the way, the anti-causal systems (recalled by mtwieg) are those that consider only future inputs.

Hi albbg,

it was not a problem to follow your explanation as far as the mathematics is concerned.
However, I cannot agree that "the differentiator (s)" is a non-causal system.
For my opinion, there are two arguments against your position:
* In reality, neither a pure step with a rise time of zero seconds is possible nor a ramp that suddenly starts creating a response discontinuity;
* More than that, it is not possible to realize a clean differentiating circuit;
* The transfer function H(s) of any circuit (like a differentiating circuit) is valid for the steady-state only. This is not the case if we try to differentiate a ramp that starts at t=0.
Thus, it is not a surprise that the output does not jump immediately to the final step value.

Mostly because of the 3rd point mentioned above, I think your explanation cannot serve as an argument that supports a possible non-causality of an differentiating circuit.
Are there some other arguments pro or con?

---------- Post added at 12:11 ---------- Previous post was at 10:53 ----------

I summary, I tend to say:
A circuit designed to differentiate an input signal is able to work as a diffentiator only in steady state mode (after transients have disappeared).
Otherwise, it would be necessary to operate non-causal - and that's not possible in reality.

Yes, interesting discussion.


Related to albbg point of view:

There are several ways to define the step function, whose difference is the value it takes at the point of discontinuity. In order to avoid confusions, let me use the notation u(t) and take t0=0.
It is clear that u(t)=0 for t<0 and u(t)=1 for t>0. But what at t=0?
Consider these alternative forms:

(a) u(0)=0
(b) u(0)=1
(c) u(0)=1/2

Fourier theorem says that when a function has this type of discontinuity, at the disconinuity point its Fourier series or transform converges (but not uniformly) to the mean of the limits at left and right, so form (c) is the step we find in Fourier analysis.
Nevertheless, the difference between these 3 forms is a function of zero energy. So, from an engineering point of view, the 3 forms are equivalent (they are not distinguishable).

A problem with the ramp is that it is not differentiable at t=0, because the limits at left and right are not he same. If calculating its derivative you take limit at left (i.e. derivative at left) you find step in form (a), limit at right gives step in the form (b), and a "centered" lim(eps->0){[r(t0+eps/2)-r(t0-eps/2)]/2/eps} gives (c).

A "strictly mathematical" differentiator would not give any output at t=0 because "strictly mathematical derivative" does not exist at that point.
But a "physical world differentiator" must give an output at any time. All the three above mathematical models for representing it (derivative at left, at right or centered) are "practically" the same.
This subject can bring us to a different field, with other rather philisophical derivations: Up to which extent can mathematical models represent phisical or "real-world" systems? But let's leave this for another discussion.

If you find hard to conceive a transfer function V/V that is an ideal differentiator, consider a transfer function I/V (i.e. the output variable is current while the input variable is voltage). Such an ideal differentiator is an ideal capacitor. It is causal, stable and nice.
When a ramp voltage starting at t=0 is applied to an ideal capacitor, the current is a step.

A related V/V transfer function is H(S)=s/(s+1) . It is called sometimes "nonideal differentiator" and can be realizad as a highpass RC cell (the R converts I into V). When we solve this circuit, we treat without problem the C as as a differentiator I=C*dV/dt.


Related to LvW observations:

I would separate this discussion (causality of differentiators) from the transient and steady state problem. Transent behaviour characterizes systems that have memory (including "memory from the future" for non-causal systems!).
A differentiator has no memory. Its output does not depend of the past (nor of the future), but just of the rate of change of the input at the same instant.
One could consider that it "looks" at past (or future) a time eps (I would say that for this reason Albbg considered it non-causal), but eps->0, so eps is not really a memory but something needed in order to define the "rate of change".

Usually, the memory of a linear system is characterized by its impulse response. What is the impulse response of a differentiator? It is a monstruous thing called "doublet", that like Dirac Delta is 0 for any t different from 0. The impulse response of two cascaded differentiators? A triplet... and so on. [Recall that Dirac deltas, as well as its derivatives are not functions in standard sense, so we have to be cautious with them.]

Happy New Year for all!
Regards

Z

Zorro, thank you - a well written treatment of the questions related to the problem of electronically differentiating a signal.
I think you have explained the theoretical background why an electronic circuit designed to differentiate a signal is able to perform this task (of course, with limited accuracy) in a steady state mode only (as indicated already by the transfer function based on the the frequency variable s).

Wow I never thought non-causality problem is linked with a differentiator.

I want to know whether it is stable or not. What do you say on stability problem?

Thanks in advance.

It is true that for LTI systems if order of numerator>order of denominator then the system is not causal. It is also true that even though this appears in all textbooks, nowhere is a proof.
Regarding stability: assumming BIBO (bounded input-bounded output) definition, the system is unstable since it has a pole on +infinity. BIBO stability requires all poles strictly in left plane (real part<0).