Hello Mr. Silva,

at first - a small correction. I think the real pole in your function G(s) is located in the left half of the s-plane leading to an expression (s+0.004), right?

Now to your question:

Of course, the "simplification" of the transfer function is correct - however, only as long as you really are able to realize both singularities (pole and zero) at the same frequency (watch tolerances!).

You are loosing nothing - in contrary, this is a classical method to shape the gain function of an opamp in order to improve its stability properties (in case of feedback).

If both frequencies are not ideally matched - there are only minor consequences as far as the gain response (frequency domain) is concerned. However, the consequences in the time domain can be severe because the settling time of the step response can be enlarged drastically.

I hope this answers some parts of your question.

Lutz vW

Thank you Lutz,

but the pole is correct, is an unstable pole.

I've noticed when I project a PD or PID controller to the simplified transfer function and after test the controller at complete transfer function, I've noticed the long settling time, due to the pole next to origin and I noticed also an increase of the overshoot because the zero.

I'm starting to understand the problem, this dipole has a physical meaning I guess! This connect the translation and rotational movement at rigid body( when it is flying at atmosphere)!

Hi Adolfo,

in this case, I doubt if cancellation is possible.

Perhaps the following link can help

http://teal.gmu.edu/~gbeale/ece_720/internal-stability.pdf

If such simplifications are made, we get a stable system for whatever gain K, while the original system is stable only for a gain less than a specific value. This is a major difference.

Yes - I agree. However, I think this applies only if this "simplification" is introduced into the formula (as a pure mathematical effect). Does this mean that it will work also in practice (cancellation in a real system)? I think, the above mentioned link gives an answer.

Rami Maher,

do you have any text or article talk about this dipole cancellation?

I noticed that, but it's kind of weird to me, because I'm "ignoring" an unstable pole. And when I projected to system without this pole, and after put the controller at the complete transfer function, this pole become stable.

Is that possible because I'm projecting my system to have a bandwidth greater than the "dipole" frequency (low frequency)?

Hello Silva,

Lutz had answer your question. I think you are doing simulation. For real system, regulated to set point may works, but if the cancellation is not perfect, load disturbance and measurement noise will be amplified, and stability will be lost. In other aspect the robustness of controller is very bad.

Dear Tao Wang,

this cancellation are done only to project the controller, since this zero and poles are from my plant!

This cancellation is not perfect for sure, the pole is unstable and the zero is on left plane.

That model I put first is the same that are described by Greensite ( Analysis and Design Launch Vehicle) and he does the same simplification, but he doesn't explain. But it works!

Hello, Silva

Yes, the simplification is possible since they are very close. But in this situation, it is conditional stable, so the gain must take great care of. Maybe you can add disturbance on your actuator to check if the gain is enough to deal with the possible disturbance on the load.

As Tao Wang pointed out, robustness is keyword here. According to Folly (2007), there are at least two methods to prevent pole-zero cancellation. One of them is classical (Chiang & Safonov, “Synthesis using bilinear pole shifting transform”, 1992).

Hi Adolfo,

do you have MATLAB Version 7.10.0.499 (R2010a) or higher , go to help and write down: "Model Reduction and Approximation" , you will get alot of answer but look for " Model Order Reduction Commands" that will help you, I am very sure. if you need any help on that I can help if you like!! You can used for SISO & MIMO type of systems.

Hi again, you can check your model reduction result by preforming/drawing bode plots of the original and the reduce one at the same plot and the how accurate your reduction is!!!???

First of all, there are lots of mistakes in your equations. Particularly, it should be (s²+2wn*ksi*s + wn²) with wn=3 and ksi=0.01/3=0.033 instead of (s²+2wns + wn²)=(s+0.02+3²). This means that your plant is both unstable and almost purely oscillatory.

In open loop, your approximation is hardly acceptable. For example, the step response of the original plant and that of the simplified model will be very close to each other for time instants short enugh but will infinitely diverge with increasing time. This can be easily checked using Simulink.

The best insight into the closed loop phenomena can be gained using root locus. Since you have one zero and two complex poles, there is a branch starting at unstable pole and ending in stable zero, so that real pole will get stable even for relatively small values of the P controller's gain. The remaining branches will go to infinity approaching asymptotes that are parallel to imaginary axis and cross the real axis at roughly -0.02. The entire CL system becomes stable and will be still almost purely oscillatory but the frequency of oscilations will increase.

Large control progress can be reached applying a series connection of two phase lead compensators whose pole magnitude is 5-10 times greather than that of the zero. Then, for large gain values, the model of the plant will become completely irrelevant. However, to avoid excessive control action, the setpoint should then be passed through a reasonably slow low pass second order filter. The dynamics of the entire CL system will then approximately equal to the dynamics of that filter.

Actually, one phase lead compensator is sufficient to get the properties described in my previous answer.

A little out of track but I have a suggestion. Transform your transfer function model into state space model and then find reduced system matrix A will give you better results in your applications.

(a) For stable poles, I agree with Md. Abdus Samad's comment "I think transfer function may be simplified by eliminating those pole (and zero ) that are quite far from dominant pole (and zero ) in s-plane."

(b) For unstable poles (such as s=0.04 in Adolfo Silva's question "What am I losing by doing that, since my simplification is cancelling a dipole at low frequencies?")

Control tutorial (provided by the following Matlab link) titled with "Extras: Pole-Zero Cancellation" claims that "the pole-zero cancellation is theoretically okay, but practically unobtainable. Even if a perfect pole-zero cancellation were possible, the system would not be practically implementable because of internal stability issues."

As demonstrated by an example from the following Matlab link, "When an open-loop system has right-half-plane poles (in which case the system is unstable), one idea to alleviate the problem is to add zeros at the same locations as the unstable poles, to in effect cancel the unstable poles. Unfortunately, this method is unreliable. The problem is that when an added zero does not exactly cancel the corresponding unstable pole (which is always the case in real life), a part of the root locus will be trapped in the right-half plane. This causes the closed-loop response to be unstable."

http://ctms.engin.umich.edu/CTMS/index.php?aux=Extras_PZ

I also agree with Marian Blachuta's comments.

(1) For industrial control community,

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

http://www.isa.org/InTechTemplate.cfm?Section=Article_Index1&template=/ContentManagement/ContentDisplay.cfm&ContentID=28889

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

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

The next popular method is Smith Predictor: Otto J.M. Smith and Smith Predictor.

http://en.wikipedia.org/wiki/Otto_J._M._Smith

A typical PID tuning procedure: (1) Use relay control to estimate the control model (or control plant); (2) Use Z-N formula to initialize Kp and Ki; (3) use trial and error to adjust Kp and Ki or other method such as iterative feedback tuning (IFT), internal model control (IMC), etc.

H. Hjalmarsson was elected to the Class of 2013 IEEE fellow last year due to his fundamental contribution to iterative feedback tuning. The key contribution of IFT is tuning controller parameters for those control model (or control plant) whose parameters are difficult to be identified relatively accurately, in other words, iterative feedback tuning was proposed to solve controller tuning issues caused by plant uncertainty.

H. Hjalmarsson, M. Gevers, S. Gunnarsson, and O. Lequin, "Iterative feedback tuning: theory and applications," IEEE Control Systems Magazine, vol.18, no.4, Aug 1998, pp .26-41,

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=710876

WK Ho, Y Hong, A Hansson, H Hjalmarsson, and JW Deng, "Relay auto-tuning of PID controllers using iterative feedback tuning," Automatica 39 (1), January 2003, pp. 149-157. Available from the following RG Link.

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

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 from the following RG Link.

https://www.researchgate.net/publication/3332273_Self-tuning_IMC-PID_control_with_interval_gain_and_phase_marginsassignment?ev=prf_pub

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

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=67610

This paper has been implemented by Maplesoft Inc. for MapleSim Control Design Toolbox

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

K.J. Åström, T. Hägglund, C.C. Hang, and W.K. Ho, "Automatic tuning and adaptation for PID controllers - a survey," Control Engineering Practice, 1(4), August 1993, pp.699-714.

http://www.sciencedirect.com/science/article/pii/096706619391394C

K.J. Åström, C.C. Hang, P. Persson, and W.K. Ho, "Towards intelligent PID control," Automatica, 28(1), January 1992, pp.1-9.

http://www.sciencedirect.com/science/article/pii/000510989290002W

Control theorectic approaches have been applied to model the interactions between an overloaded SIP server and its upstream servers as a feedback control system in two different scenarios - round trip delay control (IEEE ICC 2011) and redundant retransmission ratio control (IEEE Globecom 2010).

Round-Trip Delay Control (RTDC, implicit SIP overload control) algorithm: Y. Hong, C. Huang, and J. Yan, "Design Of A PI Rate Controller For Mitigating SIP Overload," Proceedings of IEEE ICC, Kyoto, Japan, June 2011.

https://www.researchgate.net/publication/224249824_Design_of_a_PI_Rate_Controller_for_Mitigating_SIP_Overload

RTDC implicit SIP overload control algorithm has been recommended as White Paper by TechRepublic (CBS Interactive)

http://www.techrepublic.com/whitepapers/design-of-a-pi-rate-controller-for-mitigating-sip-overload/25142469

IEEE ICC 2011 presentation slides for RTDC implicit SIP overload control can be downloaded from the following RG link.

https://www.researchgate.net/publication/257945199_Round-Trip_Delay_Control_(RTDC)_For_Mitigating_SIP_Overload_(IEEE_ICC_2011_Slides)

Redundant Retransmission Ratio Control (RRRC, implicit SIP overload control) algorithm: 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

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 this implicit SIP overload control algorithm by former IEEE TAC Associate Editor S. Mascolo:

L. De Cicco, G. Cofano, and S. Mascolo,"Local SIP Overload Control", Proceedings of WWIC, June 2013.

http://link.springer.com/chapter/10.1007%2F978-3-642-38401-1_16#

http://c3lab.poliba.it/images/2/2a/SipOverload_WWIC13.pdf

IEEE GLOBECOM 2010 presentation slides for RRRC implicit SIP overload control can be downloaded from the following RG link.

https://www.researchgate.net/publication/258555827_Mitigating_SIP_Overload_Using_a_Control-Theoretic_Approach

Journal paper (SIP Overload Control) not only conducts more theoretical analysis of Round trip delay control (RTDC) and Redundant retransmission ratio control (RRRC), but also discusses how to apply RTDC algorithm to mitigate SIP overload for both SIP over UDP and SIP over TCP (with TLS).

Y. Hong, C. Huang, and J. Yan, "Applying control theoretic approach to mitigate SIP overload", Telecommunication Systems, 54(4), 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

Survey on SIP overload control algorithms: Y. Hong, C. Huang, and J. Yan, "A Comparative Study of SIP Overload Control Algorithms", IGI Global, 2012, pp. 1-20.

http://www.researchgate.net/publication/231609451_A_Comparative_Study_of_SIP_Overload_Control_Algorithms

http://www.igi-global.com/chapter/comparative-study-sip-overload-control/67496

Open-source SIP/VoIP project discussion archive

http://lists.sip-router.org/pipermail/sr-users/2013-April/077596.html

IETF-RFC "SIP Overload Control" discussion archive

http://www.ietf.org/mail-archive/web/sip-overload/current/msg00919.html

API-RCP(TCP Congestion Control):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

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: "An Implementation and Experimental Study of the Adaptive PI Rate Control Protocol," Proceedings of IEEE HPSR, Paris, France, June 2009.

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

(2) For academic control community,

ACC 2012 Plenary and Semi-Plenary Lectures

Plenary: Accomplishments and Prospects of Control

Abstract: There are many success stories and major achievements in control, something remarkable for a field as young as 50 years. In this talk we will sample these accomplishments and speculate about the future prospects of control. While there are examples of feedback from ancient times, extensive use of feedback paralleled industrialization: steam, electricity, communication, transportation etc. Control was established as a field in the period 1940-1960, when the similarities of control in widely different fields were recognized. Control constituted a paradigm shift in engineering that cut across the traditional engineering disciplines: mechanical, electrical, chemical, aerospace. A holistic view of control systems with a unified theory emerged in the 1950s, triggered by military efforts during the Second World War. The International Federation of Automatic Control (IFAC) was formed in 1956. Education in control spread rapidly to practically all engineering disciplines. Conferences and journals also appeared. A second phase, driven by the space race and the emergence of computers, started around 1960. Theory developed dramatically as did industrial applications. A large number of sub-specialties appeared and perhaps due to this, the holistic view of the field was lost. In my opinion we are now entering a third phase driven by the ubiquitous use of control and a strong interest in feedback and control among fellow scientists in physics and biology. There are also new areas driven by networked systems, autonomy and safe design of large complex systems. What will happen next depends largely on how we respond to the new challenges and on how we manage to recapture the holistic view of systems. Key issues will be education and interaction with other disciplines.

Semi-Plenary Lectures also discuss promising research topics.

http://a2c2.org/conferences/acc2012/plenary.php

8 wonderful presentations in control symposium "Paths Ahead in the Science of Information and Decision Systems", Massachusetts Institute of Technology (MIT), USA, November 12 – 14, 2009.

http://paths.lids.mit.edu/papers_mitter.html

Panel on Future Directions in Control, Dynamics, and Systems, Richard M. Murray (chair), California Institute of Technology, April 2002

http://www.cds.caltech.edu/~murray/cdspanel/

R. Murray, K. Astrom, S. Boyd, R. Brockett, and G. Stein, "Future Directions in Control in an Information-Rich World," IEEE Control Systems Magazine, 23(2), April 2003, pp.20-33.

http://www.stanford.edu/~boyd/papers/cds_panel.html

Currently US National Science Foundation (NSF) funding support emphasizes the real-life applications and/or industry needs instead of pure theoretical work in control.

"The Control Systems (CS) program supports fundamental research on control theory and control technology driven by real life applications. .... that are motivated and derived from real-life applications and/or industry needs."

http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=13575

(3) Control Theory developed in the twentieth century: 25 Seminal Papers (1932-1981) Selected by IEEE Control Society in 2000

http://www.wiley.com/WileyCDA/WileyTitle/productCd-0780360214,descCd-tableOfContents.html

http://ieeeexplore.com/xpl/bkabstractplus.jsp?bkn=5265919

IEEE Control Systems Award

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

IFAC Giorgio Quazza Medal

http://www.ifac-control.org/awards/major-medals

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).

http://www.ieee.org/about/awards/medals/medalofhonor.html

R.E. Bellman and K.J. Astrom, "On structural identifiability," Mathematical Biosciences, 7(3-4), 1970, pp. 329–339.

http://www.sciencedirect.com/science/article/pii/002555647090132X

Richard E. Bellman Control Heritage Award for US control systems engineers and scientists (US citizenship)

http://a2c2.org/awards/richard-e-bellman-control-heritage-award

Ragazzini's notable students are Rudolf Kalman (see Kalman filters), Eliahu Ibraham Jury (see Z-transform) and Lotfi Asker Zadeh (see Fuzzy sets and Fuzzy logic).

https://en.wikipedia.org/wiki/John_Ragazzini

K. Astrom, E.I. Jury, and R. Agniel, "A numerical method for the evaluation of complex integrals," IEEE Transactions on Automatic Control, vol.15, no.4, Aug 1970, pp.468-471.

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1099492

"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." Hendrik W. Bode, Harvard University, USA, 1977.

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

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

K.J. Åström, "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

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

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

Conference Paper Design of a PI Rate Controller for Mitigating SIP Overload

Data Round-Trip Delay Control (RTDC) For Mitigating SIP Overload ...

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

Data Mitigating SIP Overload Using a Control-Theoretic Approach

Article Applying control theoretic approach to mitigate SIP overload

Article A Comparative Study of SIP Overload Control Algorithms

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

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

You are referring to the issue of "manual model reduction" or "practical model reduction" usually followed by experienced designers in the particular application under consideration (of course there is the most appropriate Model reduction via minimization of norms, decompositions, or via projection, which comprises more analytical/mathematical techniques on particular metrics to minimize, e.g. ||G(s)-G_r(s) ||_infty < gamma).

Just very briefly on manual model reduction (mainly on TFs) (I will not go into details on other model reduction methods, there are many books you could have a look):

Unstable pole/zero cancellation must be avoided (see internal stability), stable zeros (esp. real) far from the origin (s-plane) could be neglected, stable poles (esp. real) far from the origin (and esp. far from the expected CL dominant poles) could be neglected. Of course, unstable poles must be retained always! In addition, you have to be very careful with unstable zeros (as these impose limitations in CL performance) must be retained. Once you get the "reduced" design model, you should design a robust controller (the price to pay... for robustness to neglected dynamics), many designers use Gain margins and Phase margins as first metrics for robustness assessment, although it is usually more appropriate to refer to Module Margin (i.e. ||S(s)||_infty, S(s) being the sensitivity function of the closed loop).

you may also want to have a look in the book Feedback Control Dynamics by Franklin et al. The section on effect of additional poles/zeros on the response of 2nd order systems, for the record.