Control theory - few questions - Page 1
Forum section: electronics / beginners

author: nForce
author_rank: Frequent Contributor
timestamp: July 09, 2017, 08:16:15 am
content: Hello, I have a few questions about control theory. I will be happy if someone can answer these questions.First, What does it mean adding a zero to the system? What is the response, and why do we do it?I will post more questions soon. One at the time.Thank you.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 09, 2017, 10:02:29 am
content: A pole or zero is a factor in the rational transfer function. Poles in the denominator, zeroes in the numerator.Okay, that doesn't help. So...The transfer function is the frequency response: H = F(out) / F(in), where F is some process variable (usually voltage).When the transfer function is expressed (or, often, approximated) as a rational equation (that is, the ratio of two polynomials), the polynomials can be factored. The factors may be real (unique or repeated), or complex (in conjugate pairs only).Because frequencies follow the imaginary axis (that is, the polynomials are in terms of j*omega), a real zero will put a local minima or inflection point in the transfer function, while a complex zero will have a local minima or true zero.Examples of real systems with zeroes: the Cauer / Elliptic filter, all-pass filters (where the zeroes are complemented by nearby poles), the peak current mode SMPS controller, etc. Anything non-minimum-phase or feed-forward has zeroes.Note that real systems also exhibit true delay, or diffusion (H ~ 1/sqrt(f)), or other phenomena that cannot be easily expressed as poles and zeroes. For example, a true delay must be approximated by an infinite number of poles at infinity (which, because the poles are at infinite frequency, they have no effect on frequency response at finite frequencies, but they do contribute phase shift, and hence delay). This is one of the downsides to rational transfer function analysis.Conversely, the addition of zeroes tends to reduce the phase shift, or delay. One cannot defeat causality, but feedforward can be quite useful.Tim

author: nForce
author_rank: Frequent Contributor
timestamp: July 09, 2017, 04:28:08 pm
content: QuoteHere you mean: j*(Small omega). So this omega is actually angular frequencyanda coefficient of imaginary unit in control theory?What are the consequences of this statement?

author: exe
author_rank: Unknown
timestamp: July 09, 2017, 07:24:02 pm
content: Quote from: nForce on July 09, 2017, 08:16:15 amI don't really know, but, if we are talking about opamp compensation, then there is a exceptionally good article on negative feedback, bode plots and compensation:https://www.allaboutcircuits.com/technical-articles/negative-feedback-part-1-general-structure-and-essential-concepts/. I strongly recommend read it first, then move on to more advanced articles in the Internet. And, of course, spice simulator and/or real circuit and an oscilloscope are your friends.

author: nForce
author_rank: Frequent Contributor
timestamp: July 10, 2017, 06:37:52 am
content: Quote from: exe on July 09, 2017, 07:24:02 pmNo, I don't need an opamp. I am talking about control theory in general.

author: MrAl
author_rank: Super Contributor
timestamp: July 10, 2017, 05:17:22 pm
content: Quote from: nForce on July 09, 2017, 08:16:15 amHi,First, posting questions one at a time is very good idea. It's easier to talk about one thing at a time and that way the reply time is not distributed over many questions, but just one.Zeros are frequencies that cause the response to become zero or near zero.Poles are frequencies that cause the response to become infinite or very high.Because poles can cause very high amplitudes, sometimes one may become a problem in a control network so the idea is to try to get rid of it or at least minimize it's effects. One way to do this, or at least try to, is by pole-zero cancellation. IF we have an offending pole we try to minimize it's effects on the overall system by attempting to cancel it with a zero. By adding a zero at a similar frequency, we either eliminate or make the effects less and so may obtain stability or just a better overall response. It's not always easy though because this requires the addition of real components, and real parts change value over time and temperature so it requires great care in use.Similarly, if we had an offending zero we may be able to compensate by using a pole.

author: nForce
author_rank: Frequent Contributor
timestamp: July 11, 2017, 06:58:06 am
content: MrAI, can you answer on this question (or teslacoil):Quote from: nForce on July 09, 2017, 04:28:08 pmThanks.

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 11, 2017, 07:24:36 pm
content: I guess so, but I don't see the insight of the observation. The definition arises from the Fourier transform, and more generally, from the Euler identity:\[ e^{a + jb} = e^a \left( \cos b + j \sin b \right) \]One consequence of this is that the Fourier transform of real-valued signals must be Hermitian, i.e., anywhere a complex value appears in the result, there must be a complementary pair. This is why poles and zeroes are either real (single or multiple), or complex (in conjugate pairs).It is possible to generate signals which are analytic (violates the above) in a sense, but only at a frequency offset, so that the image at negative frequency is Hermitian. That's a very in-depth way of saying: a QAM radio signal can have asymmetrical sidebands, because it's composed of independent I and Q components. That is, if you look at just the signal itself (ignoring its necessary image at negative frequency), it appears analytic, i.e., can be decomposed into A cos w + B sin w.Or actually, that's probably a shitty explanation, and a sloppy use of mathematical definitions that I'm slowly forgetting; but there does exist another transform, the Hartley transform, which goes between a Fourier-like representation, and a real signal, without getting complex numbers involved. That the transform exists proves that I-Q modulation is a complete modulation method.Tim

author: nForce
author_rank: Frequent Contributor
timestamp: July 12, 2017, 08:08:53 am
content: Sorry, but is there a particular reason why are frequencies on the imaginary axis? Could they be on real axis? Or is this just part of agreement?

author: MrAl
author_rank: Super Contributor
timestamp: July 12, 2017, 03:17:12 pm
content: Quote from: nForce on July 12, 2017, 08:08:53 amHi,In order to represent signals in the frequency domain we need two dimensions. That's like vectors where we need two values like [2,3] instead of just one like 5. The vector could be written out as 2i+3k where the i and k show us which dimension is which.In the case of the imaginary operator, it represents a 90 degree rotation and so the other axis is rotated by 90 degrees which acts similar to the 'y' axis in regular cartesian coordinates.Also, a physical view would be the side view of a exponentially changing helix which i think we call a projection of the 3d space onto a 2d plane. The exponential part which appears on the real axis changes the amplitude of the sinusoidal on the other axis. Since the exponential part is the envelope and is a real factor, it must be on the real axis.So it's really just a representation that must be shown in two dimensions. The orientation of the axes are usually shown with the jw axis vertical and the real axis horizontal.There are other geometrical views which bring other ideas into play too. The pole zero constellation shows a 2d 'relief' map where the poles have very high amplitude like the top of a hill and the zeros have zero amplitude like the deepest point of the valley.

author: CatalinaWOW
author_rank: Super Contributor
timestamp: July 12, 2017, 03:49:38 pm
content: While the answers above are all correct they may or may not "click" with someone with lesser experience or math skills. A simpler expansion may help set the answer.As stated above zeros are terms in the numerator of the transfer function when the transfer function has been reduced to standard form (a product of terms of the form (A+iB)). Since B contains the frequency term it will always go to a high magnitude as frequency gets large enough, so zeros represent some form of high pass filter. For the same reason poles will always represent some form of low pass filter. The other simple result is that zeros generally cause leading phase and poles lagging phase. While both of these simplifications cover up important details of behavior they can convey some general truths.Why do you use them? Lots of reasons. A simple example - a closed loop with 180 phase shift is in danger of being an oscillator. A well placed zero can reduce the phase shift until gain is below unity. Another simple example - a capacitively coupled circuit is intrinsically a high pass - and could theoretically have infinite noise at high frequencies. Poles inherent in all devices prevent this from being a free energy source, but an intentionally placed pole or poses can keep the noise from interfering with desired operations.

author: RoGeorge
author_rank: Super Contributor
timestamp: July 12, 2017, 04:33:02 pm
content: Quote from: CatalinaWOW on July 12, 2017, 03:49:38 pmWhy would a control loop oscillate at 180 phase shift instead of oscillating at 0 or 360 degrees phase shift?

author: Fire Doger
author_rank: Regular Contributor
timestamp: July 12, 2017, 05:18:02 pm
content: I will speak about response a bit and then will go further. I think whatYou got a system, you can analyze it in time domain or frequency domain. We use frequency because its better.So you take each individual component and use Laplace and you end up with a TF that represents your model. Or you can go with rule of thumb trial and adjustments if you are experienced and get the TF by viewing output for known input.There are 2 kind of response, Time and Frequency. Time is simple, for frequency there are a few analysis methods Bode, Nyquist and Nichols.From these you can see Trise, Oscillations, Settling Time, steady state error, Kmax, stability, and many more (i don't remember most of frequency i used more time response) its just a graphical way to see how TF is behaving, which means how your actual hardware is behaving. More specific results from these analysis are answered above.About the poles-zero:The poles are moving, and you want them in a specific position but they got a specific way of movement. Each pole goes to a zero, each zero can take only 1 pole. If poles are more than zeros then the singles goes to infinity.So to make them move to the points that you want you have to add a controller.Examples:Lead, Lag, Notch, PIDs and othersPI: 1 pole in (0,0) and 1 zeroK(s+z1)----------sPD: 1 zeroK(s+z1)So your target is a specific Trise, Overshoot etc, you find the desired points in root locus and you change the way that poles moves to pass over desired points by adding poles or zeros and the final closed loop TF is different and complies with your needs.How you choose what controller to use, or how to place poles and zeros its an other story and depends on the base TF, what it represents etc.

author: ammjy
author_rank: Regular Contributor
timestamp: July 12, 2017, 05:30:07 pm
content: Response.. in Control theory, Response is very important.we use Response for analyze system. via Response system, we can understand this system work fine, this system is stabilize.we can see Response in Frequency, Times area via graph from Transfer function.(why Transfer function - T.f maked via Raplace from differentil equation. via Raplace, Fucntion exist in S area. via Raplace, we can controling system "Algebraically" in S area.)via time response, we can see how system reacting in time. if graph is Vibrate to infinity, we can understand this system is not stabilized.via requency response, we can see vector orbit. via it(=where is the pole&zero) we can understand this system is not stabilized or stabilized. (about it u can search about Nyquist method, lot of method exist in control systems) <--- oh i see. in English it call pole and zero. right ? (^^);; for frequency response, we should convert transfer function to transfer function complex type (S->iw) because for making graph pole and zero, need a complex math.just via transer function, simply we can understand via denominator with Louth method.Nyquist method example !https://www.google.com/search?q=%D0%BD%D0%B0%D0%B9%D0%BA%D0%B2%D0%B8%D1%81%D1%82+%D0%B3%D1%80%D0%B0%D1%84&source=lnms&tbm=isch&sa=X&ved=0ahUKEwj8vtqhpYTVAhVHV7wKHekmADsQ_AUIBigB&biw=1440&bih=831#tbm=isch&q=%D0%BD%D0%B0%D0%B9%D0%BA%D0%B2%D0%B8%D1%81%D1%82%D0%B0+%D0%B3%D1%80%D0%B0%D1%84

author: CatalinaWOW
author_rank: Super Contributor
timestamp: July 12, 2017, 06:34:49 pm
content: Quote from: RoGeorge on July 12, 2017, 04:33:02 pmClosed loop controls subtract a modified version of the output from the input. While in principal it could be added, the normal result would be an exponentially growing output which is not usually useful. Given subtraction 180 phase is where the feedback lines up with the input. If the feedback exceeds losses it will oscillate.

author: coppice
author_rank: Super Contributor
timestamp: July 12, 2017, 07:15:49 pm
content: Quote from: RoGeorge on July 12, 2017, 04:33:02 pmYou apply negative feedback by using an inverting amp, so that gives you a nice 180 degree phase shift. However, phase shifts due to the capacitances and inductances in the circuit tamper with that nice inversion. If they tamper by a multiple of 180 degrees you have full on positive feedback. If they tamper by anything between 90 and 270 degrees you will have some amount of positive feedback. The phase different between where you are and where oscillation would start is called the phase margin of the design.Bottom line: the 180 degrees being referred to does not include the 180 degrees from direct inversion in the amp.

author: RoGeorge
author_rank: Super Contributor
timestamp: July 12, 2017, 07:24:41 pm
content: Quote from: CatalinaWOW on July 12, 2017, 06:34:49 pmI was thinking about adding instead of subtraction. Now it makes sense, thank you.Brian Douglas's lectures about control theory are the best I found so far. There is also a free e-book, not only the video lectures:https://www.youtube.com/user/ControlLectures/playlists

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 12, 2017, 08:37:20 pm
content: Quote from: nForce on July 12, 2017, 08:08:53 amIt's the definition of the Fourier transform, and it arises from the formulation of that transform, using the complex exponential function which has an oscillating behavior.I'm afraid it literally cannot be made any more basic or simple than that.Would you need these terms defined, too? This is all within the domain of calculus after all, not the easiest subject for many people...There are other transforms, the Laplace transform for example uses a one-sided exponential kernel so frequency takes real values. Most transfer functions can be converted between Laplace domain (s) and Fourier domain (w) with the identity:\[ s = j \omega \]If you find complex numbers less appealing, you may find Laplace more handy.Tim

author: bson
author_rank: Unknown
timestamp: July 12, 2017, 10:52:55 pm
content: Say you want a bandstop filter; it will look something like this: ---\___/---, meaning flat to where the response drops, then levels out, then returns back up, and then levels out. Imagine you're bending a straight piece of wire to look like a Bode plot of your desired response. You start from the left and bend the wire downwards - bending it downwards is apole. It points everything to the right of it downwards. Then after enough length to get the desired dip in the stop band you bend it again to level it out - this is azero. A zero points everything to the right of it upwards. Then follows another zero, and another pole. You have two poles on the outside and two zeros on the inside. Basically the list of poles and zeros (and their powers if any!) is a way to express the response; in filter design this is used to factor the transfer function into a bilinear statement of poles and zeros multiplied by a topology-dependent gain for the amplifier (which is expressed in terms of the same poles and zeros, as well as component values, permitting calculating component values based on the bilinear PZ factor - but this is way, way beyond beginner stuff and would be in a graduate level course). These are then solved for various kinds of responses (low pass, band stop, etc) so all you need to calculate the actual component values for an ideal model is your specific p's and z's. Which kind of makes sense - the components depend on the actual response... and the p's and z's express this.

author: nForce
author_rank: Frequent Contributor
timestamp: July 13, 2017, 09:20:53 am
content: Ok, let me continue with questions:LTI system is a differential equation with constant coefficients and D.E (Differential equation) must be linear?

author: T3sl4co1l
author_rank: Super Contributor
timestamp: July 13, 2017, 10:01:46 am
content: LTI is a bit more general than that, but for the SISO pole-zero case, that is correct. And the order of the differential equation equals the order of the auxiliary polynomial which gives the system poles and zeroes. (The solution to the differential equation is a series of exponential terms, with coefficients determined by that polynomial.)A nonlinear or time-varying system may be able to be approximated under small-signal or slowly-varying conditions, but the general solution cannot be expressed with such simple functions (polynomials and exponentials; it may not even be expressible with a finite series of elementary functions), and other analysis methods are necessary.Tim

author: MrAl
author_rank: Super Contributor
timestamp: July 13, 2017, 04:08:21 pm
content: Hello again,Here are some graphics to illustrate a geometrical interpretation.Note time 't' is upward in all but the last diagram.