Why is the transfer function called the characteristic equation of system?
Dear Chitta, characteristic equation is not the same as transfer function, but it is the denominator of transfer function! Stability and other issues are well described in following link:http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
Characteristic equation is not the transfer function, it is the denominator of transfer function whereas the transfer function gives the relation between input and output of the system.
Quote: "In case of voltage transfer function
Poles are the frequencies at which the output voltage is infinite and Zeroes are those at which the output voltage is zero when a unit impulse is applied."
I think, this contribution needs some clarification:
* The term "pole" is defined for transfer functions (better: system functions) that are written in terms of complex frequencies "s".
That means: You never will be able to measure an infinite amplitude at the pole frequency. It is just a kind of artificial variable that was introduced to simplify expressions (based on the LAPLACE transform).
In reality, the magnitude of a transfer function will drop in the vicinity of the pole frequency (eventually, in conjunction with a gain peaking).
* Regaring the "zero", something similar applies - however, it is important if you have a real or a complex zero. But this has nothing to do with a "unit impulse".
In case of a real zero, the transfer function magnitude can be measured to be really zero (example: notch filter, elliptical filter functions).
In analogy to the effects caused by complex poles, a complex zero causes a rising magnitude (or can compensate the falling of the amplitude as in all pass filter circuits)
Actually one can have different interptretation of them. one nice view point which i really like is as follow:
if you want to get some information about the characteristics of a system, the most significant things are thoes poles and zeros.
i suggest you to search for characteristic equation of a system, then you will see that this equation gives us eiugenvalues of a system and bay havin the eigenvalues and eigenn vectors of a system you can judge that system that how strongly it is oriented respect to its eigenvalues and eigen vectors.
if this point of veiw attracts you i can let you know about the some other literatures in which you can find nice information.
Chitta, the transfer function is telling you what kind of an OUTPUT the system produces to a given INPUT ...
*** Think of POLEs and ZEROs as INFINITY's and ZEROs.
*** At ZEROs, the system produces ZERO output ... At POLEs, the system produces INFINITE output ... Obviously, you cannot produce infinite voltage with any electronics :) So, it means that, the output will be unbounded (in theory) and SATURATED AT THE HIGHEST POSSIBLE VALUE (in practice).
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Now, let's talk about a specific case: The TRANSFER FUNCTION can be the IMPEDANCE of a filter, it will be zero (short circuit) at zeros, and INFINITY (open circuit) at poles ...
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EXAMPLE: Take an inductor and a capacitor, and connect them in parallel. Their impedances are Ls and 1/Cs ... So, the parallel inductor and capacitor will have an impedance of Ls/(1+s^2LC) ... Substitute s=j*2*PI*f. This means that, it has a ZERO at f=0 and a POLE at (2*PI*f)^2=LC (meaning, POLE at f=1/(2*PI*sqrt(LC)).
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It is clear what the ZERO means. It means that, if f=0 (i.e., NO oscillation activity is present or in other words, if you apply a DC voltage to the pins), since the capacitor is open circuit and the inductor is SHORT circuit, inductor will short circuit the capacitor, and the resulting impedance is ZERO. Since our transfer function is the impedance, we have ZERO impedance, and, thus, it corresponds to the ZERO of the transfer function.
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The POLE is a little less obvious. Let's assume that, C=1 microfarad, and L=1 microhenry. So, C=1E-6, L=1E-6. The POLE of the impedance is at f=159,236 Hz.
This means that, if you apply a sine wave of frequency 159 KHz to the pins of the parallel capacitor and resistor, since the impedance is INFINITY at that frequency, the oscillation will be forever sustained and never lost ...
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However, of course, these components will have a little bit of resistance which will make them non-ideal which will eventually kill the oscillation ...
Makes sense ?
Prof Tolga Soyata,
your contribution is very helpful to understand the physical significance of poles and zeros.
thank you very much
Quote Tolga Soyata: "At POLEs, the system produces INFINITE output ... Obviously, you cannot produce infinite voltage with any electronics :) So, it means that, the output will be unbounded (in theory) and SATURATED AT THE HIGHEST POSSIBLE VALUE (in practice). "
Tolga - I am sorry, but I strongly disagree!!!
Question: What does a simple first order lowpass if it is excited with a frequency which equals the pole frequency? Answer: It drops by 3 dB (and, of course, goes NOT to infinity).
Explanation (as I have given already in my last contribution):
Poles and zeros of a transfer function are defined for COMPLEX FREQUENCIES which, however, cannot be produced in reality.
Nobody can excite a system, for example, with a frequency that has a negative and real value "sigma". Remember: s=sigma+j*w.
Therefore, it is a pure imagination (mathematical abstraction) that the transfer function would go to infinite!
Something similar aplies to the zeros of a system:
You must discriminate between real and complex zeros. Real zeros can be implemented for realizing, for example, notch filters or elliptical lowpass filters.
But there are also complex zeros which do NOT force the transfer function to go to zero. Of course, they influence the magnitude and phase response.
As a simple example, take the well-known allpass function:
It has the same number of poles and zeros - and what is the result:
Constant ampltude independent on frequency.
Lutz, you seem to bring up quite a few questions in your questions usually. This time, let's approach it more systematically. I will put my questions up, we will try to converge to an answer, otherwise, we will hit a POLE of this question's transfer function :) and the system will go unstable :)
Every listening, please contribute ... If you say AGREED, I will ask the next question. If you say DISAGREED, state your reasoning ... We will discuss it before we move on ...
QUESTION 1: The concept of TRANSFER FUNCTION can be defined as anything you want. For example, if I define the IMPEDANCE as the transfer function, then, Z(s)=N(S)/D(S) ... It is a fancy way of saying "a function with a numerator and denominator which both depend on "s" ... So, clearly, if N(s) goes to zero, the transfer function Z(s) will go to zero. If the D(s) goes to zero, the transfer function Z(s) will go to infinity ... What does the "dependence on "s" mean ? " It means, dependence on FREQUENCY ... In other words, at different frequencies, the system has a different value for a transfer function ...
AGREED ? OR DISAGREED ?
QUESTION 2: I would never need the concept of POLEs and ZEROs if I was only using resistors. For example, imagine a resistor divider :
http://en.wikipedia.org/wiki/Resistor_divider#Resistive_divider_2
Imagine Z1 and Z2 being a only a resistor, R1, and R2. So, the transfer function is:
H=Vout/Vin=R2/(R1+R2).
The transfer function is a FIXED value and it doesn't depend on frequency, and there are no POLEs or ZEROs, since there is no way to make either R2 (numerator) or R1+R2 (denominator) zero ...
What is the physical meaning of this ? Simple ... resistors do not cause a PHASE CHANGE ... They only attenuate the signal ... And, this behavior doesn't depend on frequency (ignoring parasitic inductance and capacitances).
Our wonderful ancestors, Euler, Gauss, Laplace, figured out a genius way to model the PHASE CHANGE : Use a two dimensional value for voltage and current. In other words, use COMPLEX numbers instead of REAL numbers ... for VOLTAGE, CURRENT, and IMPEDANCE ... Notice, I can no longer call it RESISTANCE, since IMPEDANCE automatically implies that, it is a complex number ... If you use complex numbers, you can notate PHASE and AMPLITUDE under the same umbrella ...
So, If I say, Z=1 ohm , it is really R=1 Ohm, since it is a real number. So, VOLTAGE and CURRENT will always have the same phase. If I apply a sine wave voltage and measure current, since V=IR, V and I will be THE SAME PHASE. No phase difference ...
But, If I say, Z=Ls/(1+LCs^2) , then, it is a complex number, which implies that, the voltage and current will have different phases.
AGREED or DISAGREED ?
Hello Tolga - this is my answer to QUESTION 1: DISAGREED.
Reasoning: The initial question clearly was related to a four-pole system (mentioning of a characteristic equation). Thus, the commonly agreed definition for the term "transfer function" is to be used: Output-to-input ratio of a frequency-dependent system in the COMPLEX frequency domain (s=sigma+jw).
This is my understanding of a transfer function. Consequently, zeros and poles are located in the complex s-plane.
However, there is one special case with poles located directly on the jw axis.
In this case I agree with you that the output (theoretically) would go to infinite because the oscillation condition is fulfilled. But this is just a theoretical exercise because nobody would excite such a system with an input signal (it is an oscillator).
By the way - I never would say that the input impedance of a system or of a two-pole is a "transfer function". Of course, the impedance of an LC tank circuit is (theoretically) infinite at resonance. But - for my understanding - this has nothing to do with the pole of a transfer function as mentioned in the topic question.
More than that, in your contribution you spoke about an electronic system that will be "SATURATED AT THE HIGHEST POSSIBLE VALUE (in practice)".
To me., you were not referring to a passive two-pole but, instead, to an active four-pole. Am I wrong?
Hello Tolga - this is my answer to QUESTION 2: AGREED (with ???)
Tolga, please help me. I really don`t know how to comment.
You have mentioned various basic rules (resistive voltage divider) - and I am happy to agree. But so what? To what extent are these examples related to the question ?
I see no connection to the question of poles and zeros.
For clarification: My only point of disagreement was your claim that (Quote)
"At POLEs, the system produces INFINITE output" .
And my reasoning was: This is just a theoretical model based on complex frequencies which cannot produced in reality.
Thus, the system will not be (Quote) "SATURATED AT THE HIGHEST POSSIBLE VALUE (in practice)". In contrast, the pole frequency leads to a decrease of the magnitude response.
Am I in error?
Lutz, I am progressively getting to answering that statement. You are very detail oriented, and since you bring up a bunch of questions at the same time, I cannot tell what you are objecting to usually ... So, I am tracing the steps back ... It proved to be useful, since you are not agreeing with Question 1 ... So, we will go back to that ...
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In Question 1, you said, "The initial question clearly was related to a four-pole system " ... How do you gather that ???? I clearly see a question from a person that wanted to get an INTUITIVE feeling for the POLEs and ZEROs. There were no details like FOUR POLES ?? or something like that !!! There is absolutely nothing wrong with getting an INTUITIVE UNDERSTANDING of complicated concepts ... I wish somebody explained me the INTUITIVE meaning of a lot of things before they buried me in mathematics 25 years ago !!!! It would have saved me years in learning !!!
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As far as the TRANSFER FUNCTION, we will define it like this: In the parallel L, C I described, let the input voltage and current of the system be Vin and Iin, and the output voltage and current of the system be Iout and Vout. I will SHORT CIRCUIT the output pin to the ground, so, Vout=0. See attached Figure ... I am trying to find a transfer function for this system. I define it as the TRANSCONDUCTANCE, since I cannot define it as the voltage gain (Vout/Vin) ... So, my transfer function is H(s)=Iout(s)/Vin(s). Clearly, if I had only resistors inside, I could write H=Iout/Vin. But, I have other elements that change phase. So, I have to define it as H(s) in the complex domain ... Do you accept this transconductance as a transfer function ... AGREED or DISAGREED ... ?
Hello Tolga, I suppose my answer to all of your questions can be rather short.
1.) The questioner (Chitta Behera) did ask for the physical significance of poles and zeros in a transfer function - and, at the same time "Why is the transfer function called the characteristic equation of system?"
I must confess, I do not need any "intuitive feeling" to know that he/she speaks about a classical four-pole system with an input and an output signal that allows the definition of a transfer function in the classical sense. As you know, the characteristic polynominal appears as a denominator of a four-pole system function.
In this context, I like to remind you once again on your own words, which also are based on a four-pole system: (Quote)
"At POLEs, the system produces INFINITE output ... Obviously, you cannot produce infinite voltage with any electronics :) So, it means that, the output will be unbounded (in theory) and SATURATED AT THE HIGHEST POSSIBLE VALUE (in practice)." (End of quote).
2.) Starting with your passive example circuit you write "we will define like this...".
Of course, you are free to define for yourself - however, in communication with other people it is helpful to use commonly agreed definitions. Otherwise, misunderstandings cannot be avoided.
Therefore, regarding to your last question: I DISAGREE.
Reasoning: I think, the ratio Iout/Vin is neither a transfer function nor a TRANSCONDUCTANCE because Iout=Iin. There is no transfer operation from input to output. It is simply an input conductance Y(s).
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Finally, it would be very interesting for me to hear if there is any RG forum member who can give any reference to prove that the voltage-to-current ratio (or vice versa) of a two-pole is called "transfer function".
Sorry for being again "very detail oriented". However, perhaps I can learn something from this discussion.
Lutz, the TRANSFER FUNCTION does not have to be relating only input voltages, and output voltages. It can be used in control systems, where the INPUT has nothing to do with a voltage, neither does the output. Furthermore, TRANSFER FUNCTION is absolutely not limited to Electronics ... It could also be used in OPTICS ... Mechanical Systems ... anywhere, where you can relate output and input entities ... such as currents and voltages and vibration amount, material stress, etc ...
It is the relation between input and output entities ...
The above figure I am showing is a transconductance amplifier, so, it is perfectly acceptable to use input voltage as the input ENTITY, and the output CURRENT as the output ENTITY ... So, the transfer function is : H(s)=Iout/Vin=(1+s^2LC+1)/Ls.
Here is an example transconductance amplifier from ON Semiconductor: NE5517 ...
......... http://www.onsemi.com/pub_link/Collateral/NE5517-D.PDF
In this amplifier, the input is a VOLTAGE, and the output is a CURRENT. So, the value TRANSFER FUNCTION is given in units of Siemens. Page 8 mentions clearly mentions that, the transconductance of this amplifier is its TRANSFER FUNCTION.
That being the case, at f=0, the transfer function approaches infinity. This is very intuitive, since H(s)=(1+s^2LC)/Ls, and the system has a POLE at f=0. This means that, at DC (i.e., f=0), the system SHORT CIRCUITs the input to the output ... This is when the Iout wants to go to infinity ... Hence, my quote ...
"At POLEs, the system produces INFINITE output ... Obviously, you cannot produce infinite VOLTAGE with any electronics :) So, it means that, the output will be unbounded (in theory) and SATURATED AT THE HIGHEST POSSIBLE VALUE (in practice)."
In this specific case, I should have said infinite CURRENT ... But, I can find you another example, where, the voltage counterpart is true too ...
Tolga - it is really not necessary to explain to me that a four-pole transfer function is not necessarily restricted to voltages. If you would read my contributions again, you will see that I never have used the term "voltage". I only spoke about signals. So you didn`t really address the issue.
Since a couple of years I am designing OTA and CC circuits - and I even know what a transimpedance amplifier is. Thus, I know the TRUE meaning of the term TRANSconductance.
I am sure you will agree that the expression you call "transfer function H(s)" is nothing else than the input conductance Y(s)=1/sL + sC of a simple idealized LC tank circuit.
I repeat: According to my memory and my knowledge there is nobody who ever has used the term "transfer function" instead of "input impedance/conductance".
And it is more than trivial to state that the dc current through an ideal inductor approaches infinity. I am aware of this fact already since a couple of years.
Final Question: Do you really believe that this two-pole example meets the problem as described by the RG member Chitta Behera. I doubt.
CHITTA BEHERA - as you have initiated this discussion about transfer functions, poles. zeros and the characteristic equation it would be interesting to hear from you.
Has your knowledge improved a bit? Further clarification?
(I know, we didn`t stress up to now the characteristic equation - however, I am not sure if you are still interested).
@Lutz, I did gave @Chitta the answers 17 days ago regarding the characteristic equation and transfer function...,... etc. I do ,as You, expect her response.
@Ljubomir - ohh yes, correct. I forgot that you were the first who has answered the question regarding the characteristic equation.
To me, it is really phantastic to see that the solution of the time domain diff. equation appears also as denominator of a systems s-domain transfer function.
Thank you for giving the link to the MIT paper.
@Lutz, sorry if I missed something, but who did said next/citing : "To me, it is really phantastic to see that the solution of the time domain diff. equation appears also as denominator of a systems s-domain transfer function." !? Who does not make the difference between "s" and "t" domain, or in better words, between diff. equation and characteristic equation!?
Although circuit theory enjoys being able to calculate this formula for Z=Ls/(1+s^LC) I just displayed above, I am trying to get an accurate chronology of events that lead to the development of this. I will put the steps down here, and everybody, please if you know the answer, let me know, so, we can constructively develop this chronology. I might be wrong in some of them, in this case, we will freshen up the list ... Please kindly send your comments and we can revise the list.
***** EVENT 1 (1790-1800) ::: Pierre-Simon Laplace ::: LAPLACE TRANSFORM ::: which allowed conversion of ODE's to significantly simpler equations such as s+1 ... It wasn't invented to solve anything electronic, but will later be used for electronics since a network of circuit elements can be analyzed using ODEs ...
***** EVENT 2 (1831) ::: Michael Faraday ::: Faraday's law, Electromagnetic induction, the root of the V=LdI/dt formula. We got the inductor figured out !
***** EVENT 3 (18xx) ::: ??WHO??? ::: capacitor formula, root of I=CdV/dt
***** EVENT 4 (18xx) ::: ???WHO??? ::: CIRCUIT THEORY ::: connecting them together, by using Laplace transform and defining the impedance, which allows us to define the impedance of the LC tank circuit as Ls/(1+s^2LC) ... More importantly defining VOLTAGE and CURRENT as TWO DIMENSIONAL entities in the complex domain ...
***** EVENT 5 ::: (18xx) ::: ???WHO??? ::: Solving the equation, assigning s=j*2*PI*f, and the concept of POLEs and ZEROs ...
I am just trying to appreciate the formula which we can now quickly derive here, but, there is nearly a century of work behind it. I want to go down the memory lane and list these events chronologically. Help me please, so, we can get this list together ... I will reorganize any incorrect information ... Thanks. Tolga
Hey Tolga - thank you for starting this interesting tour through the history of network and system theory. However, I think this will be a good subject for starting a new discussion (and a new thread).
On the other hand - I am not sure if this tour helps clarifying things within THIS sequence of questions and answers (physical meaning of poles, zeros, characteristic equation).
Therefore, coming back to your last contribution (example: Input conductance of a simple LC tank at f=0 identical to dc), I think that for dc we can forget the capacitor and even the inductance - which means: Your example reduces to an idealized piece of wire (R=0), which indeed (theoretically) allows a current approaching infinity.
However, my question is: Does this model explain the physical meaning of a transfer function´s pole?
Ljubomir, thank you for correcting me. It seems I am getting older and older...
I simply forgot to write the middle part of the sentence (...the characteristic equation as...).
Thus, it must be:
....to see that the characteristic equation as solution of the time domain diff. equation appears also as denominator....
@Ljubomir, I have the feeling that I even could improve the exactness of my sentence above:
...to see that the characteristic equation as the source of the solution of the time domain diff. equation appears also as denominator....
"Conductance" vs "transconductance"... We talk about "conductance" when the input voltage is across and the output current through the same 2-terminal element (1-port), and about "transconductance" when the input voltage is across one and the output current through other port of a 4-terminal element. But if we draw the element as a 2-port "black box (see the attached picture below) and think of it as of a 4-terminal element, we can talk about its "transconductance"... but this is only an illusion...
Generally speaking, the relation (function) between the output and input quantity in both cases (1-port and 2-port element) can be considered as a kind of a transfer function Y = f(X). While in the fist case the two quantities are heterogenious (current and voltage), it is accepted it to be called "IV characteristic"; in the second case, the two quantities can be both heterogenious (current and voltage) and homogenious (current and current, voltage and voltage).
"But if we draw the element as a 2-port "black box (see the attached picture below) and think of it as of a 4-terminal element, we can talk about its "transconductance"... but this is only an illusion..."
Cyril - does this mean ("only an illusion") that it would be wrong to use for your example the term "transconductance"? By the way: I would say that it is wrong.
But I still wonder what kind of conductance we can observe in the active (inverting) version of this circuit - "conductance" or "transconductance"? Is there any difference with the passive version in regards to this classification? The confusing fact is that the input and output current are the same as in the passive version...
Cyril - regarding the "illusion":
I think, even a black box cannot change a two-pole into a four-pole (or three-pole) because in fact there are only 2 poles - and not 3 or 4 poles. The fictitous pole at the "output" of the series resistor R is identical with ground and cannot be regarded as a pole carrying any signal.
Cyril - as far as the inverting opamp configuration is concerned - I think, we cannot observe any transconductance.
1.) Of course, we can observe a conductance (or even two): There is an input conductance and an output conductance. This is in accordance with the classical 4-pole theory.
2.) For my opinion, the term "transconductance" (in analogy to the term "gain") is defined as the output current-to-input voltage ratio of an active 4-pole - with the aim to describe the internal voltage-to-current transfer properties of an active device.
This means, that the device to be described by this transconductance (gm) must provide an output current at its output node. This applies. for example, to transistors (of course including some non-idealities) and OTA`s.
However, it does NOT apply to the shown circuit (inverter) which represents a voltage source (low output resistance). Of course the ouput voltage causes a current Iout through the connected feedback resistor. And this current Iout does NOT depend on the value of that resistor.
I know, that - regarding this current (which is NOT ground referenced!) - some books call this configuration a (floating) constant current source. However, I don`t think that this configuration allows to define a "transconductance" because the current Iout is not a current that flows through an externally connected load. Instead, the feedback resistor itself is the main part of the configuration that enables this current to be constant (due to the feedback effects).
EDIT: Because of the above mentioned properties I think we should NOT call the shown circuit "voltage-to-current converter".
Lutz, thank you for the profound analysis! I need some time to realize it....
the significance of poles and zeros in a transfer function can be better understood by making a partial fraction realization of the transfer function.The poles determine the transient response of the system (rise time, peak time and overshoot), where as the zeros help in determine the residues of the partial fraction expansion along with the poles.
The poles of transfer function is a characteristics of the system (the response of the system) while the zero represent the force variable
In brief:
1-The poles describe the main dynamics ( they are the eigenvalues) of the matrix of dynamics. All of them have to be in the stable region in order to have a system stable. All or some of them have to be perhaps re-allocated via feedback in order to have a closed -loop prescribed stability degree in the open-loop system has not such a degree. All of them have to be able yo be fixed in prescribed positions if the system is controllable etc.
2-The zeros are important to several levels: they influence the relative stability (roughly speaking : how far or close the frequency response is from the critical stability point (-1, j0) in the Nyquist plot or how far is it from loosing prescribed stability margings in the Bode plot ) , they can then improve or worsen the transient responses. If some of them is (are) unstable, critically stable (or even stable with poor stability degree), it has to be transmitted to the reference model if a model-following controller is designed since it cannot be cancelled so as to avioiding internal instability. In this case, the reference model is not of fully free choice by the designer since some of its zeros have to be prefixed by the above reason.
From the multi-scale control scheme perspective, poles of a given plant P represent the basic modes of the plant, e.g., if the plant has two real poles then this implies that the plant has two basic modes, each is a first-order system with real coefficients. It should be noted that, the plant P can be decomposed into its basic modes (i.e., simple sub-systems) by using the partial fraction expansion. These modes might be multi-scale in nature, meanings that they have different speed of responses to a same input, i.e., a manipulated variable. While poles represent basic sub-systems of the plant, the combination of these sub-systems leads to zeros of the plant; physically, these zeros determine the overall shape of the plant response, e.g., a RHP zero means that the plant could exhibit an inverse-response type, if it has at least two poles - a plant P has two sub-systems (modes) having different speed of responses yields an inverse-response overall. It is interesting to note that, a plant shows more complex responses, the more number of sub-systems that it has even though each sub-system exhibit only simply monotonic response, i.e., a first-order sub-system is monotonic. A physical example, a dining room with several dozens people produces a more complex noise pattern than a dining room with only one or two people, where each people represent a sub-system.
Jobrun, nice analogy with dining room people....also, I find that zeros can be overly annoying, when they are unstable, i.e. RHP zeros. They put upper limit on the root locus of closed loop poles under influence of a controller. Several efforts, especially in the output tracking researches make estimate of these unstable zeros and turns them into stable zeros. Of course the procedure sometimes gets involved mathematically.
It is true. For model- following control design, the plant zeros have to be either cancelled by the controller or transmitted to the reference model . In this second case, the model is not of full free design since it has some plant zeros as its own zeros. In the first case, the cancelled plant zeros are allocated as controller poles and result to be closed-loop poles as well . So, if the plant has unstable zeros, they cannot be cancelled ( to avoid closed-loop instability) so that they are " transmitted" to the reference model. As a result, the reference model, which is an analytical description of the suited closed-loop behavior, is not of full free choice by the designer.
Concerning the use of transfer functions in circuit theory, it can be pointed out that there are typical standarized connections for the feedback-loop to the feed-forward -loop which facilitate calculations of the forms series-shunt, shunt-series , etc. ( the four possible connections) to give relevance either to either currents in nodes or voltages in branches, depending on the kind of connection. This emphasizes in addition that transfer functions can be adimensional ( both outputs and inputs either currents or voltages) or dimensional ( output voltage or current and input the contraries). The better right choice depends on each particular circuit but the relevant generated information, such as stability , relative stability, transients etc. ) is independent of the particular choice.
Wow last comment 4 years ago! No further progress, interest on this question? I was really surprised when I stumbled on it, and especially surprised when no one mentioned ENERGY anywhere in the discussion. The poles and zeros model (approximately) the topological structure of a system in terms of how energy is transmitted, stored and lost in a system. Encountering infinities at poles just abstract. Energy is always limited and nonlinearity always spoils the linear model when a resonance is encountered.
Dear @Kamel, you must cite the original resource which was used in your answer, just in order to avoid plagiarism.
Have you used this MIT resource?
Understanding Poles and Zeros
http://web.mit.edu/2.14/www/Handouts/PoleZero.pdf
You may visit my answer posted 5 years ago. You should read previous answers.
Mike, your comment is very insightful in the sense of physical significance. Poles and zeros are abstracted characteristics of Dynamic systems. Dynamic Systems are inclusively the systems that involve energy storage elements.
While energy is transferred from their input ports to their output ports the interaction between different storage elements shapes this transfer in a unique style based on :
- the initial amount of energy stored within each storing element
- the size of each storing element
- the relationship between the storing elements imposed by the physical structure and connections between these energy storage elements.
- The pace (rate or frequency) of applying the energy at the input port(s) of the system
This unique style (or profile ) of time response between the input(s) and output(s) of these dynamic systems is mainly portrayed using differential equations.
For well known inputs, the dynamic systems may afford bounded output or unbounded outputs. Our tools for investigating the output/input relationships is mathematics. Poles and zeros are the means for distinguishing two extreme responses of these systems to their inputs, Zeros highlight the zero response, this is the case in which the system can't convey any energy transfer from the input port to the output port, it can be inferred that energy perfectly trapped and oscillates internally between the energy storage elements of the system such that the system can neither accept more energy from input nor release any energy to output.
Poles highlights the unbounded response (favorably called unstable response in control literature), this is the case where energy storage in some elements provides additional leverage to input energy in times and amounts that continually boost the energy available at the output ports and ultimately leads to unbounded output.
After the mathematical surgery to quantify the zeros and poles using appropriate tools (e.g. Laplace transform), one can look back at the very structure of the system to gain more specific physical inference about the significance of each pole and zero in terms of the amounts and structural relationships between the energy storage elements of the system in one side and the frequencies of the exciting input in other side (e.g. in electrical circuits inductance and capacitance, in mechanical systems mass or inertia, transitional or torsional springs...etc)
Some fine resources regarding energy in this thread.
Linear Graph Modeling: One-Port Elements
" System dynamics provides a unified framework for characterizing the dynamic behavior of systems of interconnected one-port elements in the different energy domains, as well as in non-energetic systems. In this handout the one-port element descriptions are integrated into a common description by recognizing similarities between the elemental behavior in the energy domains, and by defining analogies between elements and variables in the various domains. The formulation of a unified framework for the description of elements in the energy domains provides a basis for development of unified methods of modeling systems which span several energy domains..."
http://web.mit.edu/2.14/www/Handouts/OnePorts.pdf
Passivity analysis of linear physical systems with internal energy sources modelled by bond graphs
"Integrated dynamic systems such as mechatronic or control systems generally contain passive elements and internal energy sources that are appropriately modulated to perform the desired dynamic actions. The overall passivity of such systems is a useful property that relates to the stability and the safety of the system, in the sense that the maximum net amount of energy that the system can impart to the environment is limited by its initial state. In this paper, conditions under which a physical system containing internal modulated sources is globally passive are investigated using bond graph modelling techniques. For the class of systems under consideration, bond graph models include power bonds and active (signals) bonds modulating embedded energy sources, so that the continuity of power (or energy conservation) in the junction structure is not satisfied. For the purpose of the analysis, a so-called bond graph pseudo-junction structure is proposed as an alternative representation for linear time-invariant (LTI) bond graph models with internal modulated sources ..."
Article Passivity analysis of linear physical systems with internal ...
Dr. Ljubomir Jacić , it is a great honor for me to get my answer recommended by your good self. Thank you. Thank you too for the valuable references you provided.
In few words, poles amplify the input, while zeroz reduce the input.
Follow This Link Please:
https://en.wikibooks.org/wiki/Control_Systems/Poles_and_Zeros
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