linearization of non-linear systems

[steveB]

What you say above is critical here. Could you please elaborate on it a little using some signal?

What physical system can the function f(x)=ax+b represent? The function f(x)=ax could represent an ohmic resistor circuit.

I've attached a picture of a noninverting amplifier. The figure shows the formulas when the voltage reference is nonzero and when the reference voltage is zero. If you look at these formulas closely, you will realize that one of them is a y=mx function and the other is a y=mx+b function. And, that's all there is to it. If your input and output voltages are referenced to ground, and your opamp is referenced to Vref with Vref not equal to zero, then it's a nonlinear circuit, even when the signals are all in the linear range of the opamp itself.

You are quite correct in saying that it's a trap and I have already fallen into this trap and hit my face on a box full of rotten tomatoes!

Now, wipe the rotten tomatoes off your face, pick yourself up and pat yourself on the back for learning yet another important lesson. You're on the road to becoming an expert because the person you perceive to be the expert is always the one that keeps getting up and has wiped more tomatoes off his face than you have.

 

[PG1995]

Thank you, Steve.

Let's talk about op-amp in general. An op-amp always amplifies the differential voltage of its inputs. I believe it functions as a linear device when it has a negative feedback because then it adjusts its output in such a way that the differential voltage is almost zero, and its input is also almost zero. In short, the negative feedback is what makes it a linear device. It has been quite a while since I took op-amp course so please correct me I'm wrong or missing some other important detail here.

Let's take a look at reactive elements which are mostly considered linear devices. Please have a look here and kindly help me. Thanks.

Regards
PG

Helpful link(s):
1: https://www.electro-tech-online.com/threads/op-amp-comparator-floyd13-3.126224/

 

[steveB]

For the cap and inductor, derivatives and integrals are linear operations. Check it mathematically if you don't see this. These operators are not line functions y=mx+b.

For the OPAMP, I just showed you an OPAMP circuit with feedback that was not strictly linear. But, generally opamps are considered to operate linearly within the power supply range when there is good feedback. However, even when there is good feedback, you must remember that the linearity is only good at low frequency. At high frequency the feedback gain is reduces and you will see the nonlinearity of the opamp circuitry, and OPAMPs also have slew rate limits which is clearly a nonlinearity.

 

[MrAl]

Hi again,

There's a really good book out there on non linear systems and linearization of those systems and if and when it is possible from a state space point of view. Unfortunately i cant remember the name of the book now because it's been maybe 20 years since i covered it. It's a pretty big, thick book too with lots of information in it, but what i gathered from it after reading almost the whole thing from cover to cover and trying examples was that there are too many times when the system can not be linearized or handled in a linear fashion with regard to stability, so i abandoned it in favor of numerical techniques. That wasnt an entirely good idea, but it wasnt too bad because i didnt need it except for maybe when someone asked a question about that kind of thing.

I guess that doesnt help much, except to let you know that there is reading material out there that takes this stuff very seriously and you can find it if you have an interest and take the time to look for it. I'll try to remember the name of the book, but it wasnt cheap anyway somewhere around $80 USD probably more now.

 

[PG1995]

Thank you, Steve, MrAl.

It really looks like that I'm back to square one! Anyway, let's try again.

...derivatives and integrals are linear operations.

Yes, they are linear **broken link removed**. But I take linearity of systems and linearity of the operators as entirely different things. Perhaps, I'm missing something.

Moreover, now I think that I came up with bad examples here. Do you think a capacitor or inductor can be looked as systems in themselves? To me, they are devices and not systems. A good example of a system would be an RC circuit made up of two linear devices, a resistor and a capacitor.

Let's talk about op-amp in general. An op-amp always amplifies the differential voltage of its inputs. I believe it functions as a linear device when it has a negative feedback because then it adjusts its output in such a way that the differential voltage is almost zero, and its input is also almost zero. In short, the negative feedback is what makes it a linear device. It has been quite a while since I took op-amp course so please correct me I'm wrong or missing some other important detail here.

Do you think what I said above is correct? I'm asking this because I would like to know if I have the fundamental concept correct. It has been quite a while since I studied op-amp.

An RC circuit in zero-state is a good example of a linear system (likewise, I believe an LC circuit is also a linear system). But I was missing an important term from the definition of a linear system, i.e. "zero-state response".

But an RC circuit which is not in zero-state (i.e. with capacitor charged) is not an example of a linear system.

It has been said that a system represented by a function of type f(x)=mx+b is not a linear system where "m" is slope and "b" is constant. I believe it can also be written as output=m(input)+{output even when the input is zero} where "m" is slope. Please correct me if I'm wrong. By the way, in view of the definitions such a function, f(x)=mx+b, cannot represent a linear system because it contradicts the requirement the system being in zero-state.

Let's discuss your op-amp example in a little detail. By the way, if you could come up with a simpler example then it might help. Anyway, please have a look here.

How would you define a linear system in general? Forget the definitions such as homogeneity and additivity for a moment. Above all, one shouldn't confuse a linear system with a mathematical linear function, i.e. line. For example, you can see here that an RC circuit in zero-state represents a linear system but still you don't get a linear or line graph. Is this correct?

Does a linear system always consist of linear components such as resistors, capacitors, etc? Can a diode (non-linear component) be a part of a linear system?

Regards
PG

Helpful links:
1: http://www.thefreedictionary.com/linear operator
2: **broken link removed**
3: **broken link removed**
4: http://mathworld.wolfram.com/LinearOperator.html

 

[steveB]

Yes, they are linear **broken link removed**. But I take linearity of systems and linearity of the operators as entirely different things. Perhaps, I'm missing something.

OK, you can take them differently if you want. But, you asked about caps and inductors and showed the relations. Those devices are defined by the linear operators (i said operators, not operations).

Moreover, now I think that I came up with bad examples here. Do you think a capacitor or inductor can be looked as systems in themselves? To me, they are devices and not systems. A good example of a system would be an RC circuit made up of two linear devices, a resistor and a capacitor.

Do you think what I said above is correct? I'm asking this because I would like to know if I have the fundamental concept correct. It has been quite a while since I studied op-amp.

Sounds fine to me.

An RC circuit in zero-state is a good example of a linear system (likewise, I believe an LC circuit is also a linear system). But I was missing an important term from the definition of a linear system, i.e. "zero-state response".

But an RC circuit which is not in zero-state (i.e. with capacitor charged) is not an example of a linear system.

Theses are considered to be linear systems. If something causes an offset nonlinearity, or other behavior that appears nonlinear, in these systems, you can easily identify it, in your context, by applying the definition of linearity. Rarely do you need to worry about it in linear circuit theory.

Zero state response is specified to eliminate the natural response to initial conditions. These natural responses are not responses to input signals.

It has been said that a system represented by a function of type f(x)=mx+b is not a linear system where "m" is slope and "b" is constant. I believe it can also be written as output=m(input)+{output even when the input is zero} where "m" is slope. Please correct me if I'm wrong. By the way, in view of the definitions such a function, f(x)=mx+b, cannot represent a linear system because it contradicts the requirement the system being in zero-state.

Yes.

Let's discuss your op-amp example in a little detail. By the way, if you could come up with a simpler example then it might help. Anyway, please have a look here.

That's about as simple as it gets, from my point of view. If you can think of something simpler, I will consider it. I'm basically confused by what you are saying here. We have a simple definition of what linearity is, and the circuit fails the definition. Why do we need to talk about black and white cats?

How would you define a linear system in general? Forget the definitions such as homogeneity and additivity for a moment. Above all, one shouldn't confuse a linear system with a mathematical linear function, i.e. line. For example, you can see here that an RC circuit in zero-state represents a linear system but still you don't get a linear or line graph. Is this correct?

Yes, but why forget about definitions? They provide the basis for proving what you said.

Does a linear system always consist of linear components such as resistors, capacitors, etc? Can a diode (non-linear component) be a part of a linear system?

Generally, linear systems are made up of linear components. Generally, nonlinear components will result in a nonlinear system. Generally, all rules have exceptions. Generally, the rule that "all rules have exceptions" is a rule that also has exceptions. Generally, thinking more is better than thinking less, but sometimes thinking too much can cause even more confusion.

 

[MrAl]

Hi again PG,

To add a little here...

You're asking a question about the general overview of a circuit in terms of it's components alone. This is not always possible to do, if it ever is. That's because the linear or non linear nature of the circuit usually also depends on the viewpoint of the circuit, or how the circuit components are being used.

For example, we can define a typical circuit behavior with resistor and capacitor as:
1/(s+A)

which is considered linear in the frequency domain, but then we know we can also define an RC network with:
1-e^(-t/RC)

which is a non linear equation in time.

So which is it then? In both cases we have R and C (and a voltage source). But in one case we have a linear circuit and the other is non linear. In the linear case we used a sine source, in the non linear case we used a step source. So we see that depending on how we USE the components and how we view the behavior also plays a role in whether we consider it linear or non linear.

It also depends on bias point, and again how we use the circuit. In the simple example of:
y=M*x

this is completely linear. But in the example of:
y=M*x+B

as you pointed out, this is strictly speaking non linear because it has an offset 'B'. But in this second example we may consider it still linear when we operate such that the operation does not go beyond some extremes and the output depends mostly on M*x and not much on 'B'.

So the best definition of linear and non linear comes from the describing equation for what it is we are looking at and how it is used. There will be certain components that will usually be used as non linear, but we still have to make sure they are really used that way to call them non linear.

The ordinary silicon diode is a non linear device, but it can be used in a linear circuit in a linear manner when it has a constant bias current.

The diode can also be used in a circuit that is both linear and non linear at the same time, maybe depending on the type of excitation, but realizing both excitations types are valid in the same circuit and may be used in the same application. For a good example, consider a gain control circuit using a diode as the gain control element. The diode's internal impedance decreases with current, so the more current we apply the lower the impedance, and this is used to vary the AC gain of a circuit in a simple AC voltage divider.
The circuit is basically non linear, but because we apply a constant DC current to the diode the response of the AC signal looks very much linear for small signals. But for large signals it would still appear non linear and distort the signal. So we have to know the difference there. And also, when we vary the current we vary the gain, and that's non linear too.
So there we have two different forms of non linearity, one with amplitude change and one with forward bias change, and they are both used in the same circuit within the same application (gain control circuit). But with proper design the output AC is perfectly linear even though we can control it in a non linear fashion.

We can view this application wide to see how this can vary the way we define it based on such an extraordinary factor as "who is operating the circuit".
Strange as this sounds, imagine we have the gain control circuit in an amplifier that gets sold to the public. In the first application, the user is given a pot to vary that varies the current through the diode. In that application the user has non linear control. In the second application, the pot is only available to the factory worker who adjusts the gain of the circuit to some factory preset level, so the user never gets to adjust the pot. To the factory worker, it's non linear, but to the user, it's a linear circuit because the gain never changes.

How to rationalize all this.
Well, the response of the AC signal is M*x, so it's linear, and the response with the bias current is m*x, so it's non linear (m now a variable too not a constant). So it depends on the input/output equation, not the components themselves. If you have a non linear output with input then you have a non linear circuit, and if you have a linear output with input then you have a linear circuit.

Very often however when we have a non linear circuit behavior anywhere within the circuit we call the whole thing non linear. So we would probably deem the gain control circuit a non linear circuit overall.

You might also note that often we linearize a circuit in theory. But we never actually change the circuit itself, we just change the equations a little. So we start with non linear equations and end with linear equations. What changed? Not the circuit components themselves or the circuit connections, just the way we viewed the circuit.

 

[PG1995]

Thank you very much, Steve, MrAl.

I still need to confirm few details from my previous post because I'm not sure if those have been answered directly. But first I will focus on the main issue.

Please have a look here and kindly help me with these four queries. Thanks.

Q5: A resistor is a linear device because the relationship between current and voltage is linear - when voltage is increased, the current also increases proportionally. How would you define linearity of capacitor and inductor along the same lines in DC analysis?

Q6: Almost every circuit has two kinds of responses, transient response and steady state response, where transient response dies away as the time tends to infinity. If it is said that a circuit or electrical system is a linear system, it means that the relationship between input and output is of linear nature. Let's focus on an RC circuit in zero-state with a DC power supply. You can see here that it is a linear system even in view of its transient response in addition to being a linear system in steady state. If an electrical system is a linear one considering its steady state response, then does this mean that it is also linear system when its transient response is considered?

Q7: For an RC circuit the transient response is given as Vout(t)=Vs{1-e^(-t/RC)} where Vs is a DC supply. Let's take R=0, then Vout(t)=Vs{1-1}=0. Could you please tell me that why this is so?

Thank you for very much for your time and guidance.

Regards
PG

 

[steveB]

OK, we must be going too fast here. I'm not going to try to answer all these questions until we deal with question #1.

Why do you think my answer to question 1 would be yes? I've already giving that circuit as an example of nonlinearity. I was clear that the nonlinearity is not due to the supply rails but due to the offset from the voltage reference. So, strictly that circuit is nonlinear.

However, if you re-reference the input and output signals to the voltage reference, it is linear again. Also, that circuit is linear to AC signals that operate around a DC operating point.

Until we are clear on this, we can't go any further.

 

[PG1995]

Thank you, Steve.

I was clear that the nonlinearity is not due to the supply rails but due to the offset from the voltage reference. So, strictly that circuit is nonlinear.

If for the sake of argument, we can omit the qualifier "strictly" then would you agree that the circuit in Fig. 1 exhibits linearity at least over a short range? It's fine to me even if we don't omit the qualifier because then I would say that the circuit in Fig. 2 is also a non-linear one in strict sense of the word.

I'm not disputing what you have said but rather I'm being a kind of naughty skeptic! Thanks.

Regards
PG

 

[steveB]

Your wording is very confusing. When you say 'over short range', what does this mean? Are you talking about AC operating around a DC bias point?

Also, Fig. 2 is non-linear only because of the supply rail limits. Within those limits, it is strictly linear.

So, we are seriously miscommunicating here.I'm not sure that you are getting it yet.

 

[PG1995]

Thanks.

Actually I was just thinking of Vin as a variable DC source in both figures. For Fig. 1, if you vary the voltage between 0V<Vin<6V, the op-amp system functions as a linear system, and for Fig. 2 the range is 0V<Vin<4.5.

I'm sorry if it's still unclear. Please let me know if that's the case so that I can try to ask it differently after giving it some time and more thought.

Regards
PG

 

[steveB]

I still don't understand your basis to say that Fig. 1 is operating as a linear system. I've stated that it isn't linear for the reason that it does not meet the definition of a linear system. I pointed out the two cases (in/out referenced to Vref and AC around a DC operating point) where you can say it's linear. If you are talking about one of the two exceptions I mentioned, then we agree. If you are saying it's linear for another reason, then I don't understand why you think it's linear.

 

[MrAl]

Hi,

Are we talking about the Figure 1 op amp circuit with Vin and Vref=3v?

That circuit is not linear in the strictest sense, but we often refer to it as linear because it responds with a straight line. It does not meet the mathematical definition but it's still referred to as linear in many cases. It depends how you look at it. If the next stage is also biased for example we might deem the whole circuit linear.
It's good to look at these specific examples i think.

Also, in another post i noticed the question about Vc=1-e^-t/RC and letting R=0, but that's not really possible because that equation assumes a voltage excitation and with R=0 that would mean we would be applying a step change in DC voltage right across a lone capacitor, which gives rise to infinite current so is not allowed really.
If we did allow it in theory though, we would find e^(-t/RC) as R goes to zero to be equal to 0, so we would have:
Vc=1-0=1
not:
Vc=1-1=0
anyway.

Some other examples:
The time response of a capacitor to a DC current could be called linear, and the time response of an inductor to a DC voltage could also be called linear, however there is also usually storage involved (past history) so it gets more complicated.
The time response of a capacitor to an AC voltage is linear (if 2vac and 0.2amps, then if 4vac then 0.4 amps).
The time response of an RC to a voltage is non linear.

 

[PG1995]

Thank you, Steve, MrAl.

That circuit is not linear in the strictest sense, but we often refer to it as linear because it responds with a straight line. It does not meet the mathematical definition but it's still referred to as linear in many cases.

Thanks, MrAl. Yes, this is exactly what I had in mind and was trying to convey. Mathematically or according to the system theory, it is not a linear circuit or system because it doesn't comply with the definitions of homogeneity and additivity.

Let's see what Steve has to say about this.

Regards
PG

 

[steveB]

Yes, this is exactly what I had in mind and was trying to convey. Mathematically or according to the system theory, it is not a linear circuit or system because it doesn't comply with the definitions of homogeneity and additivity.

Let's see what Steve has to say about this.

What I have to say is that you shouldn't call a system that responds with a linear function (y=mx+b) a "linear system" unless b=0.

If Q1 is meant to ask if Fig. 1 and Fig. 2 behave as linear functions (and not linear systems), then I will say yes, they are linear functions within the range of the power supply rails. However, Fig. 1 is not a linear system, while Fig. 2 is a linear system (assuming ideal components) provided that the output is within the range of the power supply rails.

Does this opinion match your expectations and understanding?

 

[PG1995]

Thanks a lot for the help and patience, Steve.

What I have to say is that you shouldn't call a system that responds with a linear function (y=mx+b) a "linear system" unless b=0.

If Q1 is meant to ask if Fig. 1 and Fig. 2 behave as linear functions (and not linear systems), then I will say yes, they are linear functions within the range of the power supply rails. However, Fig. 1 is not a linear system, while Fig. 2 is a linear system (assuming ideal components) provided that the output is within the range of the power supply rails.

Does this opinion match your expectations and understanding?

Actually what I was trying to say was that the circuit from Fig. 1 is also, loosely speaking, a linear system. I don't see why we can't call it a linear system when the relationship between the input and output is of linear nature. I believe you don't like to agree with what I say because it goes against the definitions. But then I would say that remember your opinion about those "math guys" who love to look at every math detail in strictest sense of math definitions just to frustrate engineers like you.

I think it's okay because at least now you understand my confusion and what I was thinking. Thank you.

Regards
PG

 

[steveB]

PG,

Anyone is free to create their own definitions and use them as they like. The main problem with this is you can then run into communications problems by not using standard terminology. Having clear definitions is just as useful in engineering as in mathematics. So, when you break tradition and modify definitions, you have a responsibility to clearly define your terms so that people know what you mean.

In system theory, linearity is a critical property that allows great simplifications for analysis because superposition can be applied and Laplace/Fourier theory is much more fruitful in application. So, "speaking loosely" is a very dangerous thing to do for engineering reasons. Why? Because if I do a stability analysis on a system that I think is linear, but really is not, and use linear theory to make conclusions, I may predict adequate stability margins are in place, when in reality, the system may not have good margins. Hence, people could be killed and/or money could be lost. Who will be "frustrated" then?

I like to say that the mathematicians like to "throw spears" or "shoot arrows" at us. This is just my way of saying that they will often be critical of the non-rigorous and intuitive approach we sometimes take. The conflict between math and science/engineering is that the mathematician requires complete rigor, but this is often in conflict with being productive in science and engineering. So, the mathematicians do not "frustrate" me, or engineers in general, but perhaps the reverse is sometimes true.

So, there is a very fine balance we must use here. We, as engineers and scientists, can't be so rigorous that we limit our progress by not using theory that we believe is correct because the mathematicians have not placed that theory on firm rigorous ground yet. Also, we can't be so careless as to blatantly disregard known rigorous rules that must be applied to get correct answers. When Fourier developed his transform theory, many were critical because there was not a rigorous footing. Likewise, when Dirac developed the delta-function, mathematicians cringed. But, the scientists kept using the theory and making progress. Only later did the mathematicians place the theory on solid footing.

 

[MrAl]

Hi Steve and PG,

I think the reason Steve is disagreeing with you PG is because he wants to stick to the most strictest definition of linear, while i think you and i are willing to relax the definition when applied to electrical circuits.
Yes, it's true that some things may not work the same with these pseudo linear networks, but we still think of them as linear in some cases realizing the limits. The limits play more of a role in those kinds of networks.

Another good example of where we relax the definition, maybe even bypass it altogether, is in the AC averaged model. Take a buck circuit, which is a switching power supply that takes a higher voltage and steps it down to a lower voltage. There's nothing linear about that, or is there? The transfer of energy requires a switch that opens and closes so how could that ever be called linear.
Yet in the averaged AC model we get exactly that, linear operation. It's almost as if we removed the switch and replaced it with a linear equivalent circuit. And what we get is a linear response, exactly what we get from a linear voltage regulator. So we reduce the entire switching circuit to a linear one and that makes it much easier to study some aspects of the circuit that would be much harder to do otherwise.

But when we do this we dont just forget about the differences, and we impost limits usually on the operating range and also on what it is we can actually look at (study) using this method. We dont just call it 'linear' and then just forget about everything else. We of course remember that we linearized this circuit and that brought with it various assumptions and limits. However, system wide it can often be used in place of the real non linear circuit and simplify the whole scheme of things greatly.

So calling something linear that is not strictly linear comes with a price: we have to remember the differences and take the necessary precautions. But we often still call it linear because it still stands a large class away from the very non linear which is no where near linear.

If you were asked this question on a test you'd have to know what the professor's context was.

Come to think of it, is a linear voltage regulator really linear? It takes in voltages from maybe 10 to 20 volts and outputs +5v for example. That doesnt sound linear to me, yet we call it a "linear" voltage regulator.

I believe we have to take a look at slightly more complicated circuits to understand these concepts better.

 

[steveB]

Hi Mr Al,

Those are good points. However, I find your wordings in this thread much less objectionable than what PG is trying to say. You can note that in all of your discussion here, you use the term "linear" and not "linear system". To me there is enough leeway when you don't add the word "system". "Linear" can mean different things in different contexts, whereas "linear system" usually has special meaning to electrical engineers. Only you know if your word choice was conscious or unconscious, but either way, your experience and knowledge leads you to not say "linear system" for a system that does not meet the definition we all use and accept.

But, as I said, we can use any terms we want to, but we should be clear about meanings in serious work. Obviously casual conversation is less critical. In your buck converter example, you are quite clear on what you are saying and doing. You have a nonlinear circuit, and you describe a process of developing an average model. Averaging switch system models do not always yield a linear system (it depends on the exact converter type), but often they do , or when they don't, they can often be approximated as a linear system, particularly for the common converter topologies. For some topologies, that does not work well and you have to linearize the nonlinear averaged model and consider different operating points in your analysis.

The buck converter example really highlights the importance of "linear system" analysis. We are able to take a circuit that would be very hard to model and develop control algorithms for, and we convert it into a linear system model that is "good enough" to allow all the linear analysis and linear control theory to be applied to the problem.