In the summary of the linked paper I only can read about "externally positive control loop." I think, this is not identical to "positive feedback".

In the text, I cannot find the term "positive feedback".

However, we always have positive feedback for oscillatory systems.

Dear Lutz von Wangenheim

As I hinted in my question, I had NO access to download and then read the paper I cited, although I am interested to have and read that paper.

However, I cited it to emphasize that, positive feedback loops are an open problem in control theory, hence are challenging. And I want to know what are the main challenges in this regard?

In Ogata control engineering text-book, he used the term positive feed-back for positive control-loops, page 373, Fourth Edition, reference W-4. And he cited the paper:

Article Analytical Representation of the Root-Locus

This paper was not also accessible through Sci-Hub as it seems there are some persistent errors downloading papers from ASME, and De Gruyter by Sci-Hub, both are problematic and are transformed to the same paper host-site when trying to be downloaded by Sci-Hub.

Moreover, please notice the image I attached to my answer, by which I am trying to sketch some examples for probably positive feedback in human gait control. You can experiment it yourself, by getting artificially an unstable backward movement and see you would automatically step backward to get your walking stabilized. The same happens in forward walking instability, while the direction of control-law and state x (foot position), are the same; u=kx. Whereas, in classical control problems we usually set u=-kx, as in control of an inverted pendulum moving on a fulcrum.

I think as Ogata mentioned in his text-book, many of biological systems are positive-feedback control systems, hence, it seems positive feedback control is more intelligent than negative feedback control. But it seems it is an open control problem, so it would be helpful if I had some references on this topic.

Dear Lutz von Wangenheim

Please notice the other image attached to my present comment, and please let me know your idea. Thank you.

" However, I cited it to emphasize that, positive feedback loops are an open problem in control theory, hence are challenging. And I want to know what are the main challenges in this regard?"

Really an open problem never discussed?

At first, we should not overlook the fact that positive fedback is used - for example - in active filters (bandwidth reduction). Positive feedback leads to stable control loops if the stability criterion is not violated.

And, yes - we can find many positive feedback effects in nature (biology, sociology, chemistry,...)

Dear Lutz von Wangenheim

Thank you. Please also comment about the recent image attached to my recent comment, before your last comment. With regard to that image, have you ever seen green-paths probably residing only in first or third quadrants of the phase-plane? I have not seen such phase plots. I have seen control text-books always emphasizing that, the phase-plane trajectory should reside in the second and fourth quadrants. Please let me know your viewpoints on this speculation.

In the previous image, xdot should be replaced with u(x) on the vertical axis, where x is the state variable and u(x) is the control-law.

I re-attached the image, here.

I have commenced a project on stabilization of positive feedback systems:

https://www.researchgate.net/project/stabilization-of-outer-loop-positive-feedback-systems

Two comments:

1.) I cannot understand how you arrive at the last line of the enclosed paper.

2.) The stabiliy limit of the most left feedback system (with pos. feedback) is GH=1. However, the same limit can be derived from the combination of both loops (G²H²=1). So - where is the advantage of this combination?

Dear Lutz von Wangenheim

To understand the last line of the enclosed paper, you may disintegrate the byproduct positive system into the couple of conflicting positive feedback system (left)+negative feedback system (right). I mean instead of assembling the two conflicting positive feedback system (left) and negative feedback system (right), to arrive at the byproduct positive system, you may move backward and dissolve the resultant (byproduct) positive system into the two conflicting feedback block systems. Since the single byproduct system (which is an externally positive system) is decomposed into the two conflicting left+right feedback systems, hence they are simply identical (left feedback system+right feedback system=byproduct positive system), and the two block diagram representations are actually the same, it is like disintegrating or factorization in simple math. Hence to nullify the out-put of the right system (the negative feedback system) is analogous to nullifying the output of the byproduct externally positive system (resultant system), no matter what is the out-put of the left (positive feedback system), which enters as an input to the right (negative feedback system).

Conclusion: to stabilize the externally positive feedback system (the byproduct of the left+right systems), we only need to stabilize the right negative system, no matter what is the out-put of the left positive system (which is the input to right negative system). The closed-loop transfer function of the right system is C/R=G/(1+GH), which should be stabilized, and this results in stabilization of the byproduct (resultant) positive system whose closed-loop transfer function is C/R=G^2/(1-G^2H^2).

This is simply to say that to stabilize the closed-loop transfer function of the byproduct positive system: C/R=G^2/(1-G^2H^2), we need to only stabilize the disintegrated negative feedback system (right) whose transfer function is:

C/R=G/(1+GH).

The last line of the enclosed paper is only a statement of this conclusion in other notation, nothing else differs. For concreteness, if you denote G^2 by F, and H^2 by M, then you simply arrive the same conclusion, reformulated as:

to stabilize the closed-loop transfer function of the byproduct positive system: C/R=G^2/(1-G^2H^2)=F/(1-FM), we need to only stabilize the disintegrated negative feedback system (right) whose transfer function is:

C/R=G/(1+GH)=sqrt(F)/[1+sqrt(F)*sqrt(M)]=sqrt(F)/[1+sqrt(FM)].

Feel free to share with me if you had any other query or idea on.

Thanks.

I must admit that I have some problems to follow your conclusion.

* At first, it is somewhat confusing to use the same symbols (C, R) for (a) the initial loop with pos. feedback as well as (b) for the series combination of the two loops.

So - from the beginning you are equalizing two different systems (S1 and S2). As a consequence - the signal output "C" appears BETWEEN both systems as well as at the OUTPUT of both systems.

* Independent from these formal arguments, I cannot follow the logic behind your assertion that stabilizing the loop with negative feedback (S1) would stabilize also the combination of both loops (S1 in series with S2). Does this statement result from your calculation?

* independent from all the calculations: If the first loop represents an unstable system S1, how can a second system S2 - connected to the ouput of S1 cause a stabilization of S1 (because there is no interaction between S1 and S2) ?

Dear Lutz von Wangenheim

Your best critical statement is here repeated as:

* independent from all the calculations: If the first loop represents an unstable system S1, how can a second system S2 - connected to the output of S1 cause a stabilization of S1 (because there is no interaction between S1 and S2) ?

My answer:

This is the main case which has been adroitly spotted and raised by you, and is well-pondered.

Although S1 is connected to S2 and both are feedback systems, however, I have shown them as purely open-loop systems with a simple output/input transfer function C/R. Whatever time signal output of the S2 (positive feedback system), constant, ramp, acceleration, pulse, impulse, then we can design S1 to be stable for such inputs. If S1 is stable for some of these typical inputs, then the out-put of S1 (negative feedback system) will always nullify in steady-state (stabilized). So do not consider the S2 as interfering coupled with S1. The only role of S2 is to shape the externally positive by-product (resultant) system S. Through another view-point S2 only generates a time signal and enters it as an input into the S1 which is to be stabilized, but S2 is only for the auxiliary abstract analysis and imagination, not for any other design purpose of the control system.

Your other well-pondered critical comment is repeated as:

* At first, it is somewhat confusing to use the same symbols (C, R) for (a) the initial loop with pos. feedback as well as (b) for the series combination of the two loops.

My answer:

I have denoted them to illustrate my idea and you are correct about the confusion to use the same symbols. They should be tagged by a subscript. But you may see the block diagram strategy with a closed eye to symbols as C/R. I mean you may only notice the signal flow and the transfer functions, and not paying much attention to the symbols for signal flow and I/O relations.

Feel free to share with me, any other comment, query, or idea on.

Thanks.

You wrote: "...we can design S1 to be stable for such inputs. If S1 is stable for some of these typical inputs, then the out-put of S1 (negative feedback system) will always nullify in steady-state (stabilized). "

My answer: When a system is unstable - it is unstable without any consideration of input signals. It does not need any input signals to show such an unstable behaviour.

Coming back to the general point:

Independent on your example with the two loops - the general question is:

When a system is unstable - how can it bestabilized connecting another system to its (idealized) signal output? This system acts as a load to the first one - without any influence on the behaviour of the first one (stable or unstable).

Dear Lutz von Wangenheim

The concept needs more insight. We consider the negative feedback system S1 with a gain K, even if it is unstable. If S1 is stable for some specific inputs, then the by-product system S, will be also stable for much of those inputs (that are assumed to be similar but not exactly to inputs of S2, as S2 translates the inputs of S into other types of input for S1, but not widely different when we suppose them as a generic time signal). If the system S1 is not stable, then we should try to stabilize it by a good selection of K in feedforward path of S1.

But the by-product system has an input and output. The input of the byproduct system (S) is the input of S2, while the output of S2 becomes the input of S1, and since S1 is stable the output of S1 will be damped, but this output of S1, is also the output of the byproduct system S. We only should switch in our imagination between the byproduct system S and the disintegrated systems (S1+S2), as we trace the input and output signals. It needs only an explicit illustration of the block diagram, but it could be also imagined with mind and sight together.

But the equivalence and analogies are based on the final conclusion of the paper, which is a simple mathematical statement of these abstract analysis and imagination as: G=sqrt(F), and M=sqrt(H). This analysis is based on the illustrated analogy between the by-product and the disintegrated systems (S=S1+S2). It only needs a careful look at the block diagram representation.

With regard to the inputs and outputs flown in this block diagram, one may conclude, the main duty, is to stabilize the negative feedback system S1, for a class of input time signals. But the mathematical connection of the negative feedback system S1 and the byproduct system S should be respected, as: G=sqrt(F), and M=sqrt(H).

Update: Notice that, if S2 has sine wave, cosine wave or any specific input, then the output of S2 would be similarly cosine or sine, which are entered as an input into negative feedback system S1. Hence, if we stabilize S1 for a range of specific time signal inputs, then we do not worry what happens to S2 input (which is also the input of S) after it enters S2 and leaves it in a different shape enroute to enter S1 (the stabilized negative feedback system), and leave S1 as a nullified output (which is also the output of the byproduct system S).

Feel free to share with me, any other comment, query, or idea on.

Thanks.

Dear Saeb AmirAhmadi Chomachar ,

As your arguments, examples and conclusions again are based on input and output signals of the systems S1, S2 and the combination of both - I only can repeat what I have written already:

When a system is unstable - it is unstable without any consideration of input signals. It does not need any input signals to show an unstable behaviour.

More than that, I kindly ask you to comment on the last part of my former answer, which was not related to your specific problem. Instead, it was a general question:

How can an unstable system be stabilized without changing some properties of the system? I think, it does not matter at all if the unknown output of this (unstable) system will be used as an input for any other system.

Dear Lutz von Wangenheim

That's going to be an interesting discussion. Now, something suddenly popped up in my mind, as a complementary statement, which could possibly settle the discussion.

I) We have stabilized S1 (the negative feedback system), hence we are sure about its stability. For stability of a negative feedback system (S1) we should have:

|G(s)H(s)|=100 (very large),

or

|G(s)H(s)|=.01 (very small).

II) If the positive feedback system S2 and the byproduct positive feedback system S are simultaneously unstable, we can assume:

|G(s)H(s)|≈1≈|G^2*H^2|,

We already knew from stability of S1 (the negative feedback system) that:

|G(s)H(s)|=100 (very large),

or

|G(s)H(s)|=.01 (very small).

For the both positive feedback systems to get simultaneously unstable, the total loop gain should be unity for both:

|G(s)H(s)|≈1≈|G^2*H^2|,

But S1 (negative feedback system) was stabilized. Hence:

|G(s)H(s)|=100 (very large),

or

|G(s)H(s)|=.01 (very small).

Case I:

|G(s)H(s)|=100 (very large), then:

|G(s)H(s)|= |100|≈1≈|10000|=|G^2*H^2|.

Hence, we encounter the contradiction: 100≈10000.

Case II:

|G(s)H(s)|=0.01 (very small).

|G(s)H(s)|= 0.01≈1≈|0.0001|=|G^2*H^2|,

Hence, again we encounter the contradiction: .01≈.0001.

Therefore logically, not to face a contradiction, only one of the positive systems, either S or S2, should be unstable, not both simultaneously. Since we stabilized S1 (the negative feedback system), we are sure that S2 (the positive feedback system) which is conjugate to S1, is certainly unstable. But only one of the positive feedback systems; S2, or, S, could be unstable. Since we revealed that S2 (conjugate of S1) is unstable, hence we are left with the non-discretionary conclusion that S (the byproduct positive feedback system) is stable. The proof is hopefully accomplished.

The reference URL for my proof:

https://www.quora.com/Can-a-control-system-be-stable-with-negative-gain-margin-and-positive-phase-margin

Also read: https://www.researchgate.net/post/What_feedback_convention_do_you_prefer

Dear Josef Punčochář

Your idea:

What "feedback convention" do you prefer?

which are followed up by some rather lengthy discussions below that RGate page, seem interesting and worthy, after I had a glance over. Whenever I find an opportunity, I would study them carefully, to figure out further ideas on a positive feedback controller and as well probably design of the electronic circuitry.

Thanks

Dear professors Lutz von Wangenheim and Josef Punčochář

I have managed to derive another stability condition for externally positive feedback systems, added as a new update on my ResearchGate project:

https://www.researchgate.net/project/stabilization-of-externally-positive-feedback-systems

Once again I had to think about this question. Let's take a look at the world of active RC filters - "resonance properties" (in passive circuits guaranteed by inductance) are there always realized by "handy" positive feedback (subcritical, of course).

So what about the predominance of negative feedback now :-)

Dear professor Josef Punčochář

Thank you for the comment.

@ Saeb AmirAhmadi Chomachar , I think there is a logical error in your derivation for the "new" stabilization method.

After connecting both simple feedback loops (positive and negative internal feedback) you have arrived at C/R=G²/(1-G²H²).

As a next step, you have allocated a new system with feedback to this new transfer function with G² in the forward path and H² in the positive feedback path.

However, such a step is not allowed because it is ambiguous (not unique).

For example, another representation for this function could be

C/R=- (1/H²)/[1-(1/G²H²)].

Dear professor Lutz von Wangenheim

Without loss of generality, assume unity feedback H=1, to resettle the challenge.

Beware that, any control system has infinitely many representations in Transfer Function or State Space model. But we mind only the logic-based representation of these available models.

Dear professor Josef Punčochář

https://www.quora.com/Can-a-control-system-be-stable-with-negative-gain-margin-and-positive-phase-margin

Through the URL above: It must be noted that for very large and very small loop gains, the transfer function behaves almost identically for both positive and negative feedback.

Therefore, for negative and analogously positive feedback control, we should look for very large or very small gains for stabilization. But based on my latest proof of positive feedback stability, the stable gain for positive feedback control is the stable gain for negative feedback control with 4th power, i,e, Gamma=Alpha^4. Hence, we may conclude, the positive feedback system would be stabilized by a very very large gain, or a very very small gain. For example, if the negative feedback system is luckily stabilized by the gains K=100, or K=.01, then the conjugate positive feedback system would be stabilized by gains K*=(10^2)^4=10^8,=100000000, or K*=(10^-2)^4=.00000001. Since positive feedback is critically unstable, then we can only turn for the very very small gains as K*=.00000001, but it seems, designing an electronic circuit for such a diminutively small gain controller would be so challenging.

"Without loss of generality, assume unity feedback H=1, to resettle the challenge.

Beware that, any control system has infinitely many representations in Transfer Function or State Space model. But we mind only the logic-based representation of these available models. "

I must admit that I cannot see how this answers my doubts.

Dear professor Lutz von Wangenheim

I suggest you, to see the proof only through algebraic viewpoint. I mean, do not prepossess with the block-diagram representation. Only consider signal flow and its correspondence to algebraic variables without any respect for subscripts, while you freeze the model in your mind, and interchange and switch your imagination, from block-diagram representation to algebraic variables, through a purely abstract mathematical logic, when you deal with the variables.

Moreover, I say that: I must admit that, I cannot see and understand what are your doubts.

In my opinion, it depends on the structure of the system itself and the conditions under which it operates. For example, positive feedback together with negative feedback can be applied to amplifier circuits, but the possibility of nondeterministic self-oscillations increases as a result of even random external noises.

"Moreover, I say that: I must admit that, I cannot see and understand what are your doubts."

I think I have expressed my doubts very clearly:

I think there is a logical error in your derivation for the "new" stabilization method.

...................

However, such a step is not allowed because it is ambiguous (not unique).

Regards

Dear professor Lutz von Wangenheim

Thanks for your remarks. Anyway, I still firmly stand by my claim. Regards.

Dear friends,

For simplicity, if the input signal is u and the output is y, let us assume that the feedback is unity.

I am a Control guy and negative feedback tells me that we measure the error, namely, the difference between the input command and the output, and the system is such that the error tends to zero.

For an extreme situation, when both G=1 and the Negative Feedback is H=-1, we get y=G/(1+GH)u=1/2 u.

If we use Positive Feedback, H=1, then we get

y=G(1-GH)u=1/0 u = infinite.

It would not help to connect the two systems in series.

I did not write these things to start any argument. It is only a first response to the question: Why did people think of negative feedback and why we usually implement negative feedback.

In more complex systems, as for the simple one G=-1, then we would use positive feedback because the negativity is already there in the forward path. (As a control guy, I would rather use the negative sign in the forward path, so the output moves in the same direction as the input.)

Regards,

Itzhak

Quote: It would not help to connect the two systems in series.

Yes - agreed.

That was the contents of my answer from Oct. 8th.

(... independent from all the calculations: If the first loop represents an unstable system S1, how can a second system S2 - connected to the ouput of S1 cause a stabilization of S1 (because there is no interaction between S1 and S2) ?

Of course, Lutz.

Some of my publications change what people (myself first) had got used to believe, and they met some strong opposition. Still, some careful rewriting and also adding some examples did happen to change even very strong opposite opinions.

Here, too, as I wrote, I only thought that some simple examples might help.

Regards,

Itzhak

Dear professor Itzhak Barkana

Thank you for your remarks.

Dear professor Lutz von Wangenheim

I should notify you that, my initial proof for positive feedback stability (screen-shot update to the project), was for non-unitary feedback systems. Whereas, my secondary (new) proof for positive feedback stability (attached as PDF to my project), is only for unitary feedback systems. If we assume the unity feedback-path gain as H=1, then the two proofs coincide and there might be no discrepancy. Worthy of note that, for null-stabilization, we may see the feedback-path gain (or analogously transfer function) in the feed-forward path, without any discrepancy. However, for command-following (control), the two systems (unitary feedback and non-unitary feedback) are not identical in terms of my ongoing analysis (I guess this is what you pointed out).

For linear control or Lurey systems, there is no difference between positive and negative feedback, just replace C in negative feedback with -C in positive feedback:

https://personalpages.manchester.ac.uk/staff/joaquin.carrasco/NIS_rev4.pdf

For nonlinear control, a similar result must be true just by playing with the sign of the error signal, but never thought carefully about this.

From the reading of the abstract, I would say that the authors don't mean positive feedback but positive systems, see Prof A Rantzer's work in this area.

Best regards,

Joaquin

Dear doctor Joaquin Carrasco

I had read the chapter in positive feedback loops in Ogata control engineering book in Farsi. I translated outer-loop in Farsi verbatim to external-loop in English. And then I wrongly cited the paper for externally positive feedback loops, which I meant positive outer feedback loops. It has been mentioned in Ogata: positive feedback loops, are usually stabilized by negative outer-loop feedback. But my research is on totally positive feedback outer-loop. I have not read the paper text, since as I had already mentioned in my discussions here in this thread, I have no access to the full-text of the paper. I wanted to only cite a paper for positive outer-loop (positive feedback control), NOT positive systems. I have attached a screen-shot from Ogata book, to my comment, here, for your notice. Thanks.

It is just a change in the sign. So basically, your critical point in the Nyquist plot becomes +1 instead of -1. But you can keep your critical point as -1 and then multiply by -1 your open loop transfer function. Similarly, imagine we only haver rlocus in MATLAB, if I want to see the rlocus with positive feedback, just change the sign of your plant. Some essential references in control theory use positive feedback only, e.g. Zhou-Doyle-Glover. Evidently applying these results to negative feedback is trivial.

Dear doctor Joaquin Carrasco

Thank you for the reference.

We use negative feedback because negative feedback reduces noise, makes the plant be stable and has some impacts on the gain. I use the book "Automatic control systems" by the authors: B.C. Kuo and F. Golnaraghi.

Dear Minh Ngoc Huynh

Thank you.

Quote: "We use negative feedback because negative feedback .... makes the plant be stable "

No, the opposite is true: Negative feedback has many advantages, HOWEVER: Stability will be degraded!! There is only one situation where negative feedback can improve the stability margin: A system with POSITIVE feedback can be stabilized using some negative feedback at the same time. (in parallel).

Dear professors Lutz von Wangenheim and Josef Punčochář

I have added a recent update to my project:

https://www.researchgate.net/project/stabilization-of-outer-loop-positive-feedback-systems

where I have addressed and solved the problem of controller design for double-integrator system in Laplace domain, using purely positive feedback. Now, it is your turn, you the electronic circuitry professors and experts, to discuss and investigate the possibility and practicality of building the PDN controllers (with the gains reported) in practice, and the challenges that might be faced through positive feedback.

Feedback also affects system performance in terms of bandwidth, input and output impedances, etc.

Dear Saeb AmirAhmadi Chomachar

Direct use of Math formulas can lead to very "interesting" conclusions. However, your scheme contains an unstable system followed by a stable one. Even for a simple example such as G=1/s (a simple integrator) and H=1, you get the first TF to be

TF1=C/R=1/(s-1) while the second is TF2=C/R=1/(s+1).

The impulse response (the same as the response to unity initial conditions) of the first system is exp(t), while the seconds is exp(-t). Therefore, the output of the unstable first system goes to infinity for any nonzero input, while the stable system follows it nicely on its way to infinity.

Have Fun,

Itzhak

Dear professor Itzhak Barkana

Thank you for the remark. You may be notified that, you have connected two SISO systems together, without any outer feed-back loop. When you draw a feed-back line from the out-put of the second system, to the in-put of the first system, then it would be judged better. The two systems in the feed-forward path, are interchangeable in their cascaded serial position. Therefore, if you imagine the stable system as the first system from left, then the first out-put is exponentially decaying as it enters the second unstable system which would also oust another exponentially decaying signal as an stabilizable system. The main mistake in your analysis, is to consider the systems to interact as decoupled and SISO in the feed-forward path, without the critically important outer complementary feed-back loop which connect the out-put of the second system to the input of the first system. Your counter-example is not well-posed.

Dear SaebSaeb AmirAhmadi Chomachar

(and I am Itzhak for all, no need for Prof. Dr., Mr., or Sir, except for strictly official occasions)

I was misled by your sentence about the product system C/R=F/(1-FM)

“which looks good.”

So, I thought that you consider this product result “good” and I did not see that you then go to use another feedback, which is meant to stabilize the combined system.

Still, for the simple example G=1/s (a simple integrator) and H=1, you get the first TF to be

TF1=C/R=1/(s-1) while the second is TF2=C/R=1/(s+1).

The total transfer function is TF=C/R=1/(s^2 -1)

Better try to see what feedback would stabilize the combined system for this specific example.

Despite other titles, I am an Engineer, busy solving real-world problems. So, before thinking of positive or negative feedback, the basic control problem is that you have an input Command u(t) that gives you the desired behavior of the Plant output y(t). To control this, you measure the error u(t)-y(t) and use it such that the error moves towards the desired zero value. This is what led people to talk about negative feedback.

Have Fun,

Itzhak

The concept of loop gain is not correct in physics. Therefore, Nyquist stability criterion is only used in classical Lab simulations.

And, all of physical signal generation networks are positive feedback systems.

Why don't we clarify the difference between the negative feedback and adaptive feedback network?

Saeb AmirAhmadi Chomachar check this research

Article A Family of Controllers based on Sector Nonlinear Functions:...

as it proposes control laws like the ones you put in your graph (the green and red one)

Dear María del Carmen Rodríguez Liñán

As I see, it seems the control-law in the paper is not what you addressed, as the out-put and control signal in the paper you have cited, vacillate both through negative and positive values. This is in contrast to Fig. 1 of the cited paper which looks like the green lines of my graph. Moreover, in Fig. 2 and Fig. 3 of the paper which you have cited, for the initial time-interval, the control-input and state-output are of conflicting sign (one is positive and the other is negative), and this violates the illustration in Fig. 1, and the positivity of the feed-back law.

Saeb AmirAhmadi Chomachar the function is a control input in terms of the error (error in x-axis, u in y-axis) residing in quadrants I and III. Therefore, both the argument (the error) and the control signal are the same sign all over.

María del Carmen Rodríguez Liñán

It is true, through the illustration and graph in Fig.1 . But getting deeper in computer simulation results of the paper, it seems there is a sign controversy in contrast to the claim in graph illustration of Fig. 1. I mean, in some time intervals, the state (out-put) and control signal are of conflicting mathematical sign, which violates the hypothesis of graph in Fig. 1. Anyway, thank you for citing the paper. I should study it further carefully:

I am talking with regard to the paper:

Family of controllers based on sector non‐linear functions: an application for first‐order dynamical systems,

Authored by:

Marlen Meza‐Sánchez

Maria del Carmen Rodríguez‐Liñán

Eddie Clemente

Dear Itzhak Barkana

I didn't claim that, the combined by-product system is absolutely stable. I claimed it is stabilizable. I derived a formula for the stabilizing gain of the positive-feedback system based on the stabilizing gain of the symmetrically similar negative-feedback system. The gains are related mathematically as; alpha^4=Gamma, whilst alpha is the feed-forward path gain of the stable negative-feedback system, and Gamma is the feed-forward path gain of the stabilizable positive-feedback system which are symmetric to one another with conflicting feedback signs.

Please see my ResearchGate project:

https://www.researchgate.net/project/stabilization-of-outer-loop-positive-feedback-systems

for the PDF of the proof.

I am afraid for the delay in my response. I missed to spot and I saw your comment just right now. Thanks.

Dear Saeb,

OK, we see things from different angles.

I tried to write a small explanation of why people, trying to make the output of a system follow an input command, saw the need of using the error signal and why this, in turn, led to negative feedback.

For quite a good number of decades, I had to put real things to work, machines to work better and better, faster and faster, or things to fly, and I never ever met or saw the need for positive feedback.

Still, various systems are various and may need various solutions, so just have fun and keep up our good work.

Thanks,

Itzhak

the only difference between positive and negative feedback, that positive yield inverse response

The "only difference"? Interesting finding....

Musaab Ali, could you care to explain?

Musaab Ali, an inverse response generally happens when the initial response of a dynamic system is in a direction opposite to the final outcome. I don't think the above simulation false in the category of an "inverse response' in the real sense of the term. Refer to the following link for clarification

http://facstaff.cbu.edu/rprice/lectures/inverse.html#:~:text=When%20the%20initial%20response%20of,with%20different%20directions%20and%20dynamics.

regardless of initial response controller with positive feedback can make inverse response and with negative feedback make direct response.

question here how to choose right controller

https://www.linkedin.com/in/dr-musaab-ali-05236761/detail/recent-activity/shares/

Musaab Ali, I think the error you have made is that you (blindly ?) trust the simulation resilts.

The simulation is not wrong, but very often the user and his interpretation of the results is wrong.

Note that the small-signal ac analysis is not able to reveal instabilities because during the analysis a signal and voltage sources are assumed to be ideal (no switching transients at t=0). Therefore, the program finds a closed-loop function, which is not valid in reality (only in theory).

The same happens for hand calculation - you can calculate a gain value for 100% positive feedback - this is similar to the case where a small ball is riding upon a larger one.

In theory, the result applies only if if there are absolutely no dirsturbing effects.

The same applies to transient analysis where ideal parts and macros are used without any parasitic delays.

Only if you switch-on the supply voltages at t=0, this analysis will show the real behaviour.

please see below for the difference:

https://www.albert.io/blog/positive-negative-feedback-loops-biology/