Dear Kranthi Kumar Deveerasetty, The bode plot of the open loop system provides the gain, gain margin and phase margin. Based on this data, we can design the feedback system in order to meet the system requirements like gain, stability, and so on.

Prof Ahmad,

my question is H(s) is the feedback, so how is G(s)H(s) is an open loop TF? Shouldn't G(s) be an open loop TF?

Dear Kranthi Kumar Deveerasetty, This is your question "In a Bode plot realization we will consider open loop tansfer function as G(s)H(s) why?"

i think we are taking 1+G(s)*H(s) as characteristic equation for root locus analysis(pole -zero position). and for bode plot we only take the TF of the system not entire loop.

For all plots the main idea of stability is that total loop gain protect should be less than 1 and total phase shift should be less than -180. such that the system will have damped oscillation or no oscillation.

There for even in Bod plot if there is a feedback H(s) it should be taken into account and not only G(S). So either you plot G(s) H(s) and compare with 1 magnitude or you plot G(s) and compare with the inverse of H(.s) plot for stability.

1.) A BODE plot consists of separate magnitude and phase drawings versus frequency (in logarithmic fashion). Thus, EACH transfer function of a four-pole can be visualized this way.

2.) In most cases, such a plot is used for designing and/or analyzing systems with feedback .

3.) In this case, it is important to discriminate (that means: to know the differences) between (a) "open-loop gain G(s)" (gain of the amplifier WITHOUT feedback) and

(b) "loop gain G(s)*H(s)" (gain of the complete loop).

4.) For stability investigations only the loop gain G*H matters (stability criterion).

5.) However, for finding the loop hain response G*H the Bode plot for G(s) is - in most cases - constructed first. Then, the magnitude response is multiplied with H(s). (Because of the log scale: Summation of dB values).

6.) Most simple: Draw G(s) as well as 1/H(s) in a common diagram. The "distance" between both curves (given in dB) is identical to the loop gain response G*H.

With other words: Considering the 1/H line as a new frequency axis, the G(s) magnitude response now gives you the loop gain magnitude G*H,

_________

Does this answer the question? If not -do not hesitate to ask again.

Dear Professor Farid Kadir,

I am not hear to confuse any one. In many text books they mentioned GH(s) for Bode plot problems.

Quote: Farid kadri: "I think you make a confusion, there is no open loop, we consider G(s)H(s) for stability investigations of the feedback loop."

I am afraid that your contribution can cause some confusion.

Of course, there is an open loop! The product G(s)*H(s) can be measured/simulated under open loop conditions only. Otherwise, you have no input and no output.

I have a very basic interpretation for this issue. let's have a look at this maybe you like it.

basically in control problems, a feedback system is stable if the product of the plant transfer function gain and feedback gain is less than zero plus the total phase of them is at least-180 deg. what does it mean? it means that supposing the feedback system with an arbitrary initial nonzero state in zero input meeting these criteria should not lead to an unbounded output. then if we look at this very basic definition we find out the necessity of the product of H*G.

I hope my words are clear and correct :)

I am very sorry - but, for my opinion, your words are neither clear nor correct.

1.) Question: What means "at least -180 deg"? The problem is in the negative sign. Therefore, do you mean that -185 deg are allowed or -175 deg?

2.) You did not mention the most important pre-condition of your "criterion": A negative sign at the summing junction where the input signal meets the feedback signal.

However, such a phase inversion at THAT point is not required - and in some cases not possible or even not allowed.. Instead, for stability we require (for low frequency signals, incl. DC) an odd number of phase inversions (1, 3, 5) which may be realized ANYWHERE in the loop.

3.) As a consequence, a minus sign must be incorporated into the criterion. Therefore, a correct and generally applicable stability criterion is NOT based on a phase shift of -180 deg for the transfer function of the simple product G*H.

4.) Instead, we require that the total gain around the loop (called "loop gain") - including the mentioned phase inversion - must be below unity at a frequency where the total loop phase is -360 deg. This is equivalent to the requirement that the phase magnitude must be below 360 deg for a loop gain of unity.

Dear All, Again I would like to draw our attention that the question is:

"In a Bode plot realization we will consider open loop tansfer function as G(s)H(s) why?"

Dear A. Ahmad,

Didn`t you read all posts?

The answer was already given in one of my my former post (post#6, point 4.).

For you I repeat: To check the stability of a system with feedback.

Dear Dr. Lutz von Wangenheim, Thanks. I want to clarify my understanding as below.

"The bode plot of the open loop system provides the gain, gain margin and phase margin. Based on this data, we can design the feedback system in order to meet the system requirements like gain, stability, and so on."

Dear Dr. Afaq Ahmad,

thanks for clarifying the point.

In order to avoid misunderstandings - may I repeat that in your reply the expression "open loop system" means in fact: LOOP GAIN.

This is a well known and established term for systems with feedback.

That means: We have to open the loop at one suitable point and inject (in reality or in simulation) a sinusoidal test signal for measuring the gain around the COMPLETE loop (including sign inversion at the summing junction).

The main purpose of such a test is to find the stability margin for the closed-loop system - applying one of the available stability criteria (which all are based on the general Nyquist stability criterion).

Dear all, thank you for your discussion.

@ Lutz von Wangenheim, Thank you very much for your valuable suggestion and above point is very helpful for understanding.

I would add that relating it to Bode Plot is only secondary.

In some control cases,indeed you just have an ope- loop transfer function G(s) that you close with (negative) unit feedback.

In general, however, you may have some feedback configuration H(s) and you close the loop after H(s). Therefore, when you "open" the loop, you make a cut exactly where it was closed, namely right before the diference joint. This way, you are left with the "Open Loop" G(s)*H(s) or, in short, GH(s).. Now, you plot the frequency response (Biode, Niquist, Nicholds, Root Locus) of the Open-Loop and draw conclusions about the closed loop.

You must be careful, though.

In the simple case, when you only have G(s), the regular diagram lines also tell you something about the expected behavior of the closed Loop T(s)=G(s)/(1+G(s)) (overshoot yes or no, etc.)

However, when you have GH(s), the same information is only valid wrt stability, because the "closed-loop" lines on the diagram relate to the the corresponding closed loop T(s)=GH(s)/(!+GHs)), while your real closed-loop is only T(s)=G(s)/(1+GH(s)). and one must take this into account, otherwise may have surprises.

This is one reason while many control designs prefer using a feedforwrd cointroller C(s) instead of feedback. So the new open loop plant is C(s)G(s) for which both the stability analysis and performance expectations hold. This is in particular true when you have a tracking system, where the output is suposed to track the input and this is what the unit feedback gives.

I think it is just for clarification to those who are studying it.

Quote: "As Itzhak mentioned , Bode is secondary !!"

I think, THIS remark is secondary because the questioner did ask, in particular, for the meaning of the expression G*H in the BODE diagram.

(1) Bode plot can be regarded as the extended version of Nyquist plot (or Nyquist stability criterion). Bode plot and Nyquist plot were proposed to determine the closed-loop stability by examining G(s)H(s) instead of 1+G(s)H(s). Sometimes G(s)H(s) can be regarded as "pseudo" open-loop transfer function.

In Nyquist plot, frequency is a parameter, and it is difficult to determine the frequency of a point on a Nyquist plot.

In Bode plot, magnitude (in db) and phase are plotted against frequency and that problem does not arise.

Bode plot is a powerful tool in electronics for determining the stability of op-amps and transistors, and it is essentially a plot of phase against frequency compared with gain against frequency.

Nyquist stability criterion is more useful when we analyze the system using system equations.

(2) The following example utilizes Nyquist stability criterion to illustrate Kranthi Kumar Deveerasetty's question "Why in a Bode plot realization we consider open loop transfer function as G(s)H(s)?"

Given a system with open-loop feedforward transfer function (OLTF) as G(s), if we place it in a closed-loop with negative-feedback transfer function H(s) (i.e., H(s) is applied in the feedback loop), then we can obtain the closed-loop transfer function (CLTF) as G(s)/[1+G(s)H(s)].

(2a) OLTF G(s) can be expressed as the ratio of two polynomials N(s) and D(s), i.e., G(s)=N(s)/D(s), where the roots of N(s) are called the zeros of G(s), and the roots of D(s) are the poles of G(s). The poles of G(s) are also said to be the roots of the polynomial D(s) = 0.

The open-loop stability of OLTF G(s) is determined by the values of its poles -- According to Routh–Hurwitz stability criterion, all the roots of the characteristic polynomial D(s) must have negative real parts to achieve open-loop stability.

In parallel with Routh–Hurwitz stability criterion, in complex analysis, Cauchy's argument principle demonstrates that if D(s) has d roots with positive real parts (i.e., G(s) has d poles in the right half plane), G(s) must encircle the origin (0, 0) anti-clockwise d times in order to achieve the open-loop stability of G(s). In other words, the number of encirclements of the origin (0,0) in anti-clockwise direction in the G(s) plane should be equal to the number of poles of G(s) in the right half s-plane.

(2b) For CLTF G(s)/[1+G(s)H(s)], 1+G(s)H(s) can be expressed as the ratio of two polynomials N'(s) and D'(s), i.e., 1+G(s)H(s)=N'(s)/D'(s).

Note that the roots of the polynomial N'(s) becomes the poles of CLTF G(s)/[1+G(s)H(s)].

It is tedious and complex work to apply Routh–Hurwitz stability criterion to directly determine the location of all the roots of high-order (>3) polynomial N'(s) especially in the case that we use Pade-approximation to approximate a time delay.

Instead, we can use Cauchy's argument principle to examine the number of encirclements of the point (0,0) in anti-clockwise direction in the 1+G(s)H(s) plane. However, it is still a complex and non-trivial task to examine the number of encirclements of the point (0,0) in anti-clockwise direction in the G(s)H(s) plane.

In order to make the solution more practical, Nyquist proposed to examine the number of encirclements of the point (-1,0) ( instead of the origin (0, 0) ) in anti-clockwise direction in the G(s)H(s) plane ( instead of 1+G(s)H(s) plane ) by converting 1+G(s)H(s)=0 to G(s)H(s)=-1.

Abduladhem Ali's comment, Lutz von Wangenheim's comment and Itzhak Barkana's comment give the similar presentation.

(2c) In summary, closed-loop stability is determined by the poles of CLTF G(s)/[1+G(s)H(s)]. Mathematical calculation indicates that the poles and zeros of G(s)H(s) contribute to the poles of CLTF G(s)/[1+G(s)H(s)].

Bode plot and Nyquist plot were proposed to determine the closed-loop stability by examining G(s)H(s) instead of 1+G(s)H(s). Sometimes G(s)H(s) can be regarded as "pseudo" open-loop transfer function.

For Nyquist plot, the number Z of unstable poles of the closed-loop system G(s)/[1+G(s)H(s)] in the right half s-plane is equal to the number P of unstable poles of the "pseudo" open-loop system G(s)H(s) in the right half s-plane plus the number N of encirclements of the the point (-1,0) in the G(s)H(s) plane -- clockwise encirclements of the point (-1,0) count as positive encirclements (>0) -- anti-clockwise encirclements of the point (-1,0) count as negative encirclements ( G(s)H(s)=[(s+6)(s+4)]/[(s-5)(s-3)] --> G(s)H(s) has P=2 unstable poles (s=5 and s=3) in the right-half s-plane

--> draw a Nyquist plot in the G(s)H(s) plane --> there are exactly 2 anti-clockwise encirclements of the point (-1, 0) in the G(s)H(s) plane (i.e., N=-2) --> Z=P+N=2+(-2)=0, i.e., G(s)/[1+G(s)H(s)] has no unstable poles --> the closed-loop system G(s)/[1+G(s)H(s)] is stable.

Example #2: G(s)=500/[(s+2)(s+7)] and H(s)=(s-2)/(s+50), --> G(s)H(s)=500(s-2)/[(s+2)(s+7)(s+50)] --> G(s)H(s) has no unstable poles in the right-half s-plane, s=-2, s=-7, s=-50 are stable poles, P=0

--> draw a Nyquist plot in the G(s)H(s) plane --> there are exactly one clockwise encirclements of the point (-1, 0) in the G(s)H(s) plane (i.e., N=1) --> Z=P+N=0+1=1, i.e., G(s)/[1+G(s)H(s)] has one unstable poles --> the closed-loop system G(s)/[1+G(s)H(s)] is unstable

Note that if we ignore H(s) --> G(s)=500/[(s+2)(s+7)] --> G(s) has no unstable poles in the right-half s-plane, s=-2, s=-7 are stable poles, P=0

--> draw a Nyquist plot in the G(s) plane --> there are no encirclements of the point (-1, 0) in the G(s) plane (i.e., N=0) --> Z=P+N=0+0=0, this only means that G(s)/[1+G(s)] has no unstable poles

--> This may generate a wrong image that the closed-loop system is stable --> However, the closed-loop system is unstable, because the closed-loop system is G(s)/[1+G(s)H(s)], NOT G(s)/[1+G(s)]

The above two examples are provided to answer

Kranthi Kumar Deveerasetty's question "why are we considering G(s)H(s) instead of G(s) as open loop system in Bode plot"

and Afaq Ahmad's question "In a Bode plot realization we will consider open loop tansfer function as G(s)H(s) why?".

(2d) More details can be found in the following links. http://en.wikipedia.org/wiki/Nyquist_stability_criterion http://en.wikipedia.org/wiki/Nyquist_plot http://en.wikipedia.org/wiki/Bode_plot http://ctms.engin.umich.edu/CTMS/index.php?example=Introduction§ion=ControlFrequency#33

More references are suggested by Aparna Murthy, "books written by Ogata, Kuo and Nagrath and Gopal will clearly define open loop vs closed loop systems." (Quote Aparna Murthy's comments)·

(2e) Both Bode plot and Nyquist plot (or Nyquist stability criterion) were selected by IEEE Control Society in 2000 as 25 Seminal Papers (1932-1981) for Control Theory developed in the twentieth century.   

H. Nyquist, "Regeneration theory," Bell System Technical Journal, vol. 11, 1932, pp. 126-147.  

H.W. Bode, "Relations between attenuation and phase in feedback amplifier design," Bell System Technical Journal, vol. 19, 1940, pp. 421-454.

http://www.ebay.com/ctg/Control-Theory-Twenty-Five-Seminal-Papers-1932-1981-by-Inc-Staff-IEEE-2000-Hardcover-/1776728

https://www.amazon.ca/Control-Theory-Twenty-Five-Seminal-1932-1981/dp/0780360214

4 co-authors of 25 Seminal Papers (1932-1981) awarded IEEE Medal of Honor (the highest IEEE award): Nyquist(1960), Kalman(1974), Bellman(1979), Astrom(1993), i.e., NBA (Sweden) and Kalman

http://www.ieee.org/about/awards/medals/medalofhonor.html http://en.wikipedia.org/wiki/IEEE_Medal_of_Honor

(3) Quote Afaq Ahmad's comment "The bode plot of the open loop system provides the gain margin and phase margin. Based on this data, we can design the feedback system in order to meet the system requirements like gain, stability, and so on."

According to Bode plot and Nyquist stability criterion, positive phase margin indicates a stable closed-loop feedback control system. A larger phase margin indicates a more stable control system but more sluggish response.

(3a) The following Automatica 2003 paper recommended that the choice of phase margin between 30 degree and 60 degree can achieve a good trade-off between system stability and responsiveness.

[HHHHD03] W.K. Ho, Y. Hong, A. Hansson, H. Hjalmarsson, and J.W. Deng, "Relay auto-tuning of PID controllers using iterative feedback tuning," Automatica 39 (1), January 2003, pp. 149-157. Available in the following RG link.

https://www.researchgate.net/publication/223504459_Relay_auto-tuning_of_PID_controllers_using_iterative_feedback_tuning

Co-author H. Hjalmarsson was elected to the Class of 2013 IEEE fellow due to his fundamental contribution to iterative feedback tuning.

 The paper [SLY17] extends iterative feedback tuning PID controller proposed by [HHHHD03] to enhance the handling and stability performance of an in-wheel motor-driven electric vehicle.

Note that a driveless car can be classified as an in-wheel motor-driven vehicle.

[SLY17] Y. Shi, Q. Liu, and F. Yu, "Design of an Adaptive FO-PID Controller for an In-Wheel-Motor Driven Electric Vehicle," SAE  International Journal of Commercial Vehicles, 10(1), March 2017, pp. 265-274.

http://papers.sae.org/2017-01-0427/

(3b) The following IEEE TCST 2001 paper uses both Bode plot and Nyquist stability criterion to analyze the closed-loop stability of a control system.

W.K. Ho, T.H. Lee, H.P. Han, and Y. Hong, "Self-Tuning IMC-PID Control with Interval Gain and Phase Margin Assignment," IEEE Transactions on Control Systems Technology, 9(3), May 2001, pp. 535-541. Available in the following RG Link.

H. Nyquist (Sweden) --> K.J. Astrom (Sweden) --> W.K. Ho (Sweden)

     |  Nyquist plot (published in 1932)

     |  Bell Labs

    V  Bode plot (published in 1940)

H.W. Bode (Harvard) --> K.S. Narendra (Harvard, Yale) --> T.H. Lee (Yale)

https://www.researchgate.net/publication/3332273_Self-tuning_IMC-PID_control_with_interval_gain_and_phase_marginsassignment

(3c) MapleSim Control Design Toolbox has implemented the classic PID controller tuning methods (co-invented by K.J. Astrom or his student W.K. Ho) based on gain-phase margin specifications (from Nyquist plot and Bode plot) 

C.C. Hang, K.J. Astrom, and W.K. Ho, "Refinements of the Ziegler-Nichols Tuning Formula," IEE Proceedings D, Control Theory and Applications, 138(2), 1991, pp. 111-118.

http://www.maplesoft.com/support/help/MapleSim/view.aspx?path=ControlDesign/GainPhaseMargin

(3d) Only PID Control and Smith Predictor were listed in the “Leaders of the Pack” InTech’s 50 most influential industry innovators since the year 1774. Available from the following link.

http://archive.today/2RoSK

PID Control was listed twice (the dominant control method in the industrial applications) -- (1) John G. Ziegler and Nathaniel B. Nichols and classical PID Control; (2) Karl Johan Astrom and modern PID Control (IEEE Medal of Honor, 1993)

http://en.wikipedia.org/wiki/IEEE_Medal_of_Honor

The following IEEE Globecom 2010 paper uses Nyquist stability criterion to design Redundant Retransmission Ratio Control (RRRC, implicit SIP overload control) algorithm for Voice over IP.

Y. Hong, C. Huang, and J. Yan, "Mitigating SIP Overload Using a Control-Theoretic Approach," Proceedings of IEEE Globecom, Miami, FL, U.S.A, December 2010.

https://www.researchgate.net/publication/221284946_Miigating_SIP_Overload_Using_a_Control-Theoretic_Approach

Journal version: Y. Hong, C. Huang, and J. Yan, "Applying control theoretic approach to mitigate SIP overload", Telecommunication Systems, 54(4), December 2013, pp. 387-404.

https://www.researchgate.net/publication/257667871_Applying_control_theoretic_approach_to_mitigate_SIP_overload

http://link.springer.com/article/10.1007/s11235-013-9744-8 

RRRC (implicit SIP overload control) algorithm has been quickly adopted by The Central Weather Bureau of Taiwan for their early earthquake warning system.

T.Y. Chi, C.H. Chen, H.C. Chao, and S.Y. Kuo, "An Efficient Earthquake Early Warning Message Delivery Algorithm Using an in Time Control-Theoretic Approach", 2011.

http://link.springer.com/chapter/10.1007%2F978-3-642-23641-9_15#

http://www.ipv6.org.tw/docu/elearning8_2011/1010004798p_3-7.pdf

Short review and comments on RRRC (implicit SIP overload control) algorithm by the former IEEE TAC Associate Editor S. Mascolo (Google Faculty Research Award 2014):

L. De Cicco, G. Cofano, and S. Mascolo,"Local SIP Overload Control: Controller Design and Optimization by Extremum Seeking", IEEE Transactions on Control of Network Systems, Vol. 2, Issue 3, September 2015, pp. 267-277.

http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=7035079

http://c3lab.poliba.it/images/f/f4/Dcm-tcns15.pdf

Google Faculty Research Award 2014 List

http://services.google.com/fh/files/blogs/googlefras-aug2014.pdf

(3e) The following IEEE TPDS 2007 paper uses both Bode plot and Nyquist stability criterion to design adaptive PI rate control protocol (API-RCP, an explicit congestion control protocol) for the Internet.

Y. Hong and O.W.W. Yang, "Design of Adaptive PI Rate Controller for Best-Effort Traffic in the Internet Based on Phase Margin," IEEE Transactions on Parallel and Distributed Systems, 18(4), April 2007, pp. 550-561.

http://www.researchgate.net/publication/3301176_Design_of_Adaptive_PI_Rate_Controller_for_Best-Effort_Traffic_in_the_Internet_Based_on_Phase_Margin

2 double-column pages Review, Comments, and Extensive Evaluation on API-RCP: H. Zhou, C. Hu, and L. He, "Improving the Efficiency and Fairness of eXplicit Control Protocol in Multi-Bottleneck Networks", Elsevier Computer Communications, 36(10-11), June 2013, pp. 1193-1208.

http://www.sciencedirect.com/science/article/pii/S0140366413001059

Real-world Linux implementation of API-RCP: Z. Liu, H. Wang, W. Hu, Y. Hong, O.W.W. Yang, and Y. Shu, "An Implementation and Experimental Study of the Adaptive PI Rate Control Protocol," Proceedings of IEEE HPSR, Paris, France, June 2009. The leading author Zhanliang Liu is Principle Engineer at Baidu, former Principle Engineer at Tencent, former software engineer at Microsoft.

https://www.researchgate.net/publication/224611517_An_implementation_and_experimental_study_of_the_Adaptive_PI_Rate_Control_Protocol

API-RCP (adaptive PI rate control algorithm) has been extended to vehicle traffic congestion management in vehicular ad-hoc networks (or a driveless car network)

B.K. Mohandas, R. Liscano, and O.W.W. Yang, "Vehicle traffic congestion management in vehicular ad-hoc networks", IEEE 34th Conference on Local Computer Networks (LCN 2009), 2009. http://ieeexplore.ieee.org/document/5355052/

API-RCP has been reviewed by the milestone paper [HCLMMWY10] which highligts AUTO21 research projects.

[HCLMMWY10] E. Hossain, G. Chow, V.C.M. Leung, R.D. McLeoda, J. Mišićd, V.W.S.Wong, and O. Yang, "Vehicular telematics over heterogeneous wireless networks: A survey", Computer Communications, 33(7), May 2010, pp. 775-793. http://www.sciencedirect.com/science/article/pii/S0140366410000022

Industry Canada has announced a five-year $145-million Automotive R&D Partnership Initiative to help the automotive sector address its innovation, productivity and competitiveness challenges. NRC is one of the main players in this initiative, investing an additional $30 million over the next five years in automotive R&D projects. https://www.nrc-cnrc.gc.ca/eng/achievements/highlights/2008/canada_automotive_industry.html

(4) Quote Jagadish KUMAR Bokam's follow-up question "Can we draw Bode plot and Nyquist plot for Non Unity feedback systems?"

A simple answer is Yes. We can Bode plot and Nyquist plot for Non Unity feedback systems where the open-loop transfer function is G(s)H(s) instead of G(s). Note that the feed-forward loop function G(s) = C(s)P(s) where C(s) is the controller and P(s) is the plant, H(s) is the feedback loop function.

Nyquist stability criterion was used to design API-RCP (adaptive PI rate control protocol) for next generation Internet traffic control in [HY07]. Both Bode plot and Nyquist plot were drawn in the original version and Nyquist plot was removed in the final version due to page limit. Both Bode plot and Nyquist stability criterion were used to provide the proof of Lemma 2 in Appendix A.

Y. Hong and O.W.W. Yang, "Design of Adaptive PI Rate Controller for Best-Effort Traffic in the Internet Based on Phase Margin," IEEE Transactions on Parallel and Distributed Systems, 18(4), April 2007, pp. 550-561.

https://www.researchgate.net/publication/4213069_Adaptive_multiloop_PI_rate-based_controller_design_for_a_MIMO_IP_router_based_on_phase_margin

Nyquist stability criterion was used to design IMC-PID controller for industrial process in [HLHH01]. Both Bode plot and Nyquist plot were drawn in the original version and Nyquist plot was removed in the final version to make presentation concise.

[HLHH01] W.K. Ho, T.H. Lee, H.P. Han, and Y. Hong, "Self-Tuning IMC-PID Control with Interval Gain and Phase Margin Assignment," IEEE Transactions on Control Systems Technology, 9(3), May 2001, pp. 535-541.

https://www.researchgate.net/publication/3332273_Self-tuning_IMC-PID_control_with_interval_gain_and_phase_marginsassignment

(5) Bode and Nyquist awarded American Society of Mechanical Engineers (ASME) Rufus Oldenburger Medal together in 1975.

https://www.asme.org/about-asme/get-involved/honors-awards/achievement-awards/rufus-oldenburger-medal

Evaluation on Nyquist plot by Hendrik W. Bode, Harvard University, IEEE Transactions on Automatic Control,  Hendrik W. Bode, USA, 1977.

"To control theorists, Nyquist is no doubt best known as the inventor of the Nyquist diagram, defining the conditions for stability of negative feedback systems. This has become a foundation stone for control theory the world over, applicable in a much wider range of situations than that for which it was orignally enunciated."

http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=1101666

(5a) Biography of Harry Nyquist, University of Cambridge, UK, 2003

"Between 1920 and 1940 he published a series of papers on research in telecommunications which are arguably the most outstanding set of scientific contributions since Newton (apart from Einstein!)."  http://babylon.acad.cai.cam.ac.uk/students/study/engineering/engineer03l/cenyquist.htm http://babylon.acad.cai.cam.ac.uk/students/study/engineering/engineer03l/ceframes.htm

(5b) K.J. Astrom, "Harry Nyquist (1889-1976): A Tribute to the Memory of an Outstanding Scientist," Royal Swedish Academy of Engineering Sciences, 2003. 

American Society of Mechanical Engineers (ASME) Nyquist Lecturer Award

https://www.asme.org/about-asme/get-involved/honors-awards/unit-awards/nyquist-lecturer

IEEE Control Systems Society Hendrik W. Bode Lecture Prize

http://www.ieeecss.org/awards/hendrik-w-bode-lecture-prize

(6) Discussions on control system design

"What are trends in control theory and its applications in physical systems (from a research point of view)?"

https://www.researchgate.net/post/What_are_trends_in_control_theory_and_its_applications_in_physical_systems_from_a_research_point_of_view2

Article Design of Adaptive PI Rate Controller for Best-Effort Traffi...

Article Self-tuning IMC-PID control with interval gain and phase mar...

Conference Paper Mitigating SIP Overload Using a Control-Theoretic Approach.

Article Relay auto-tuning of PID controllers using iterative feedback tuning

Conference Paper An implementation and experimental study of the Adaptive PI ...

Article Applying control theoretic approach to mitigate SIP overload

> Quote: "As Itzhak mentioned , Bode is secondary !!"

> I think, THIS remark is secondary because the questioner did ask, in particular,

> for the meaning of the expression G*H in the BODE diagram.

I could not guess any one would be offended by a response, in particular when the questioner isn't.

To my understanding, the question was about the use of CG(s) vs G(s) in Control Analysis. The excellent response of Yang Hong gives pretty much information and references.

As far as I am concerned and as I've always been trying to tell my students or colleagues, Bode plot is like another pencil or computer, namely, a tool. Yes, Bode plot is the "best" tool, as long as the lines are close enough to straight lines, when stability analysis, gain margin, phase margin, etc. are easy to establish. This convenience vanishes if one has to deal with any flexible mode, when the wild up-and-down runs of the gain and phase make analysis pretty dificult, if not impossible, This is without even mentioning when the Plant is open-loop unstable or non-minimum phase.

The "secret" in understanding this analysis lies in Nyquist plots, that teaches us about the number of encilrclement, etc, that is able to take into account open-loop "unstable" poles and/or zeros. The problem with Nyquist plot as an analysis tool is that it is difficult to "get the picture" from one plot. One needs the plotys at diffrent scales, one scale to see its general behavior and then some very enlarged plots of the region around the critical {-1, 0} and around the origin.

This is what brought many people to use Nichols plots for practical design. By using logaritmic scale for amplitude and by ploting gain vs phase on same diagram, it somehow combines the advantages of the two. It is also the only plot analysis where one can deal with 100 "identical" yet different machines and check that the one Controller that was designed for all. One can draw them all on one plot and can indeed deal with them all, because all that is neded is to make sure that, no matter how wild each individual plot may look, they all must leave the region around the critical center (-180, 0dB) free.

Quote Itzhak Barkana: "I could not guess any one would be offended by a response, in particular when the questioner isn't."

Dear Dr. I. Barkana, in case this sentence was adressed to me, I am sorry if I didn`t find the right words. I was not "offened" at all - it was my only intention to emphasize the fact that the questioner did ask for some clarification regarding the LOOP GAIN and - in this context - the role of the Bode diagram.

For my opinion, the question was not about the Bode diagram in general and/or in comparison with other graphical methods serving a similar purposes.

OK? Kind regards

Lutz vW

I thought Smiley was clear in my "quote of quotes," in particular because the apparent "offense" was addressed to a third party's opinion.

Second, even though sometimes people may want to call me Prof. or Dr., I am just Itzhak.

If everyone would only want to be sure that every word was perfect, I am afraid a very depressing silence would rule supreme, Facebook would go bankrupt, which does not seem to be the case.

Even though I myself can’t see why LOOP GAIN would necessarily ONLY relate to Bode diagram, best thing here is that there are different opinions and that each one is entitled to her/his own opinion.

Best regards,

Itzhak

Quote A. Murthy: "...graphs/plots are only secondary , it is the characteristic equation (as the name indicates) describes a system"

I think, it is not appropriate to generally qualify graph and plots as secondary. In many cases graphs are nothing else than a visualization of an equation.

Please remember: The well-known Nyquist plot is a graphical representation of the characteristic polynominal of the system.

And the BODE plot principle is nothing else than another graphical representation of the Nyquist curve for the loop gain (separation of magnitude and phase).

If anyone has an opinion, one should better express her/his own opinion, rather than seconding others.

My only definite opinion was and is "best thing here is that there are different opinions and that each one is entitled to her/his own opinion."

Only from my particular point of view and only in strict relation with some very specific question on CG(s) vs. G(s), my opinion was that the emphasis on immediately relating it to Bode plot was secondary. At no point did I claim that plots are secondary. On the contrary, they are useful tools and I think I tried to give a brief view of all useful plots.

In some domains, Bode plot could be the only useful and used tool, while others may only swear by Nichols that, agin, other designers might have hardly heard of, if at all.

this is my answer ( in brief ):

Closed-Loop Performance from Bode Plots

In order to predict closed-loop performance from open-loop frequency response, we need to have several concepts clear:

* The system must be stable in open-loop if we are going to design via Bode plots.

* If the gain crossover frequency is less than the phase crossover frequency (i.e.W gc < Wpc ), then the closed-loop system will be stable.

* For second-order systems, the closed-loop damping ratio is approximately equal to the phase margin divided by 100 if the phase margin is between 0 and 60 degrees. We can use this concept with caution if the phase margin is greater than 60 degrees.

* For second-order systems, a relationship between damping ratio, bandwidth frequency, and settling time is given by an equation described on the Extras: Bandwidth page.

* A very rough estimate that you can use is that the bandwidth is approximately equal to the natural frequency.

The 'open loop transfer function' of  a 'closed loop system' is defined as its 'loop gain'. For designing purpose, we make the feedback loop open from the comparator and then apply the input. now the output is taken after H(s)...thus the open loop transfer function is G(s)H(s).

A. Goyal - I am afraid your answer could cause some confusion (youare mixing "loop gain" and "open-loop gain"). Therefore, I like to comment on this. According to my knowledge and understanding the following definitions do exist:

1.) Open-loop gain Aol is the gain function of the active unit without any signal feedback,

2.) Loop gain Aloop is the gain of the whole feedback loop (active block and feedback path in series) under open-loop conditions. (Otherwise, no input-output relation can be defined and/or simulated).

3.) Closed-loop gain Acl is, of course, the gain of the complete circuit with feedback.

4.) All three symbols are combined in Blacks formula: Acl=Aol/(1-Aloop)

L. v. Wangenheim:  but if it is true then why we take the open loop transfer function as G(s)H(s)......if there is no feedback working, then H(s) should not come into the picture.

It depends on what you call H(s) and G(s), respectively. In most cases, H(s) is the forward transfer function (without feedback) and G(s) is the feedback function. Hence, G(s)*H(s) is the gain of  the (open) loop: Loop gain=G*H.

The crux is in the terminology:

Loop gain=Gain of the complete (open) loop

Open-loop gain = Gain of the active unit only - but under the condition of an open feedback loop (i.e. without feedback).

Let us take it easy.

Assume that you have a simple Plant with open loop transfer function G(s) and just unity feedback.

Then, the closed-loop is T(s)=G(s)/(1+G(s)).

Here, you are interested in the roots of 1+G(s). The “trick” is in Nyquist plot, which allows you to plot the open loop G(s) and think of the closed-loop eigenvalues, namely, how far G(s) is from the point -1.

Because, in STANDARD case, Bode plot is easier to plot and read, we plot amplitude and phase and compare with same point -1,which in Bode plot means magnitude 1 (0 dB) and phase -180⁰.

I prefer Nichols plots, which give you amplitude and phase on same plot and manage to give results in more complex case, when the ups-and downs in amplitude and phase become unreadable in Bode, and even if the open loop is unstable and non-minimum-phase.

There are helping lines in Nyquist and Nichols plots, which give you the closed-loop amplitude and phase for each value of the open-loop.

Now, I am used to call the feedforward G(s) and so, if you add some feedback H(s), you now have the sequence G1=G(s)H(s) and you close it after H(s). So now, indeed you have the open loop G(s)H(s) and “closed-loop” T1=G(s)H(s)/(1+G(s)H(s)). Like before, you plot G(s)H(s) and “think” of 1+G(s)H(s).

All helping plots in Nyquist and Nichols plots give you the “closed-loop” amplitude and phase of T1 for each value of the open-loop.

I wrote “closed-loop” within quotation marks, because this would be right only if you took the output after H(s). However, because the output is taken after G(s), the real closed-loop is T(s)=G(s)/(1+G(s)H(s))

Itzhak Barkana: Yes, perhaps this is the correct explanation. thanks for your clarification. 

@Itzhak Barkana: May I ask you two questions?

1.) What is the reason for introducing the "false" closed-loop (with quotation marks") in contrast to the "real closed-loop" at the end of your contribution? For my feeling, this can crate further confusion.

2.) I miss the term "loop gain" in your contribution (because of the confusion between the terms "loop gain" and "open-loop gain"). Instead, you are using again the term "open loop G(s)H(s)", which in reality is nothing else than the "loop gain".  This is an important difference (see my former contribution).

@Lutz von Wangenheim

I already defended your position when someone thought to divide us into parties.

I am not fanatic about my notations, my definitions or my paradigm.

For an experienced guy like you, whio msks the questions yet knows the answer, my "explanations" are redundant.

For somemone who asks the questions to get ans aswer, I thought it is easier to start with AZg(s0 and know rthat all diagrams relate to the corrsponding closed-loop G(s)/(+G(s)) and, if you then move to G(s)H(s) the diagrams arte corrdspondiongly. 

As far as stabiltiy of closed-loop is concerned, this closed-loop  is fine. However, I I used quation marksm, because you may actually take outptut from  G(s) or even decide on some partial G1(s) and this will give you the closed loop. 

@Itzhak Barkana, thank you for replying.

Perhaps I was not clear enough in explaining my doubts.

Quotes: "Now, I am used to call the feedforward G(s) and so, if you add some feedback H(s)",

and "....and “closed-loop” T1=G(s)H(s)/(1+G(s)H(s))"

If you define G(s) as the feedforward block and H(s) as feedback, I was surprised to read the closed-loop function T1 (as quoted above), because this assumes an output signal after the block named as "feedback". Of course, this is allowed and it is possible - however, in this case BOTH blocks are "feedforward blocks" with 100% feedback. Do you know what I mean? 

@Lutz von Wangenheim

Sorry, but somehow, it seems that I say Potato and you would like Poh-tah-toh.

Someone asked a question and my answer was for that that particular question, not to teach someone with your experience.

If you have the system G(s) and just unity feedback, then the loop is closed after that feedback, so when you want to open, you  open at same place were you closed it.

Same with feedback H(s). Now, you close the loop after H(s) so, when you open, you open at same place, so you are left with the open-loop TF G(s)H(s). What is clear, is that this is a TF. If you want to a call it gain, loop-gain, etc., is  another story and there is no argument here.

However, I also wanted people to be aware that standard plots of the open-loop G(s) also supply the diagrams for the closed-loop G(s)/(1+G(s)). Therefore, when you  deal with G(s)H(s), be aware that the diagrams give you the correspondingly "natural" closed-loop G(s)H(s)/(1+G(s)H(s)). This must be known and actually contains good information to caracterize the stability properties of the plant and the behavior of the closed-loop.  What you then call the input-output transfer function depends on  specific needs. You may supply position feedback or position plus velocity from G(s) to H(s) to guarantee stability, and yet, you may only measure velocity or only position as the desired output signal, so the closed-loop input-output transfer function can be some G3(s)/(1+G(s)H(s)).

We must be aware and make sure things are explained. In Control, we call non-minimum phase systems with zeros in the right half plane, while in Physics same name is for either poles or zeros in RHP. Shall I understand that at least one of them must be terrible wrong? :-)

Dear Sir, thank you for this long answer - and, certainly I agree with you in saying (quote) "We must be aware and make sure things are explained".

And exactly this was the reason for my comments.

Again In short: If a system with feedback has a feedforward block called G(s) (your example) and a feedback block called H(s) it is clear for me that the closed loop function is G/(1+GH) and NOT GH/(1+GH). Otherwise, it would be confusing (not correct?) to say that G(s) is the feedforward block. Thats my only point .

If I am wrong, please correct me, because only one formula can be correct - and I can see absolutely no relation to the question "potato" or  "Poh-tah-toh".

Regards

LvW

Sorry, but I have the impression that the one who asked the question does not have complains about the answer and so, I think I rest my case, in particular as I was not writing to you and as my explanations do not seem to add anything.

I only wanted people to know that the plots give you the NATURAL closed-loop of the natural open-loop G(s)H(s). If you use it or ignore it, is secondary. 

If  maybe writing that you are right is better, if this help. What you call closed-loop is your right, your own usage and your own business. Actually, Modern Control (in particular MIMO) found that the closed-loop system TF that you define could be "perfectly stable" when looking from one direction and may still oscillate like crazy or even diverge, because from that direction a zero may (apparently) cancel a bad pole, while some negligible (for any other practical purpose) input or disturbance from  another input may still blow things up. That's what led to checking 'internal stabilty' from  any possible source of signal to any possible output. So, one may have to check the "closed-loops" where H is FF and G is FB or GH is FB, or...

Best regards,

Itzhak

Perhaps it helps to recall the original question:

"For a open loop transfer function it is sufficient to write G(s) but why are we considering G(s)H(s) as open loop system in Bode plot? "

From this question I deduce that the questioner had some problems to understand the difference between "open loop transfer function" and "open loop system" (and he means: Loop gain). And I gave him an answer two days ago. 

I like to add that - according to my experience with students - a correct and unambiguous terminology is of great importance for a good understanding (example: "loop gain " vs. "open-loop gain").  And I am not sure if MIMO systems have a direct relation to the question as discussed in this thread.

Regards

LvW

>And I gave him an answer two days ago. And I gave him an answer two days ago. 

Let's STOP IT!

Suppose we have a Closed Loop Transfer Function as T = CGH/(1+CGH), where, C is controller, G is plant and H is feedback sensor transfer function. My question is: Can we design controller C for some specifications such as, Gain Margin, Phase Margin, phase cross-over frequency, etc. ?  

Yes, as far as stabilty is concerned, CGH is your Open-Loop Plant and its frequency plots and analysis tell you the Gain Margin, etc. for the closed loop T=CGH/(1+CGH)

However, at the end, as far as performance is concerned, the controlled output is the output of C, the forward path, so your transfer function  for step-response (or for response to any input command) is T=CG/(1+GCH).

Can we draw Bode plot and Nyquist plot for Non Unity feedback systems?

A simple answer is Yes. We can Bode plot and Nyquist plot for Non Unity feedback systems where the open-loop transfer function is G(s)H(s) instead of G(s). Note that the feed-forward loop function G(s) = C(s)P(s) where C(s) is the controller and P(s) is the plant, H(s) is the feedback loop function.

Nyquist stability criterion was used to design API-RCP (adaptive PI rate control protocol) for next generatio Internet traffic control in [HY07]. Both Bode plot and Nyquist plot were drawn in the original version and Nyquist plot was removed in the final version due to page limit. Both Bode plot and Nyquist stability criterion were used to provide the proof of Lemma 2 in Appendix A.

Y. Hong and O.W.W. Yang, "Design of Adaptive PI Rate Controller for Best-Effort Traffic in the Internet Based on Phase Margin," IEEE Transactions on Parallel and Distributed Systems, 18(4), April 2007, pp. 550-561.

https://www.researchgate.net/publication/4213069_Adaptive_multiloop_PI_rate-based_controller_design_for_a_MIMO_IP_router_based_on_phase_margin

Nyquist stability criterion was used to design IMC-PID controller for industrial process in [HLHH01]. Both Bode plot and Nyquist plot were drawn in the original version and Nyquist plot was removed in the final version to make presentation concise.

[HLHH01] W.K. Ho, T.H. Lee, H.P. Han, and Y. Hong, "Self-Tuning IMC-PID Control with Interval Gain and Phase Margin Assignment," IEEE Transactions on Control Systems Technology, 9(3), May 2001, pp. 535-541.

https://www.researchgate.net/publication/3332273_Self-tuning_IMC-PID_control_with_interval_gain_and_phase_marginsassignment

I have merged this answer to my original answer to the general question so that peer researchers can gain a big picture of Nyquist stability criterion and Bode plot.

Thank you very much for the information sir, I will go through the references.

T is the transfer function or overall gain of negative feedback control system. G is the open loop gain, which is function of frequency. H is the gain of feedback path, which is function of frequency.

Dear Kranthi Kumar Deveerasetty ;

Bode plot can give you a lot of information regarding the stability of the system, more precisely the phase and gain margins. That you can identify them from the Bode plot or directly from the Matlab code:

[Gm,Pm,Wgm,Wpm] = margin(sys)

Best regards

Open Loop Transfer Function -G(S)

Feedback Gain- H(S)

Closed Loop Transfer function --- G(S)/ (1+G(S)H(S))

Loop Transfer Function, Without Input is ---- G(S) H(S)

Characteristic Equation ----- 1+G(S)H(S)