PI controller tuning without Ziegler-Nichols methods

01 January 2012 26 8K Report

Hello, I need help of how to do PI controller tuning for a first order system without resorting to the measurements of the plant as in Nichols-Ziegler methods or the use of MATLAB graphs. I am modelling a permanent magnet motor drive and have a problem with trial and error methods which some of my colleagues are suggesting to me. I want to compute the Ki and Kp from the motor parameters such as R and L. I saw one video about doing this using root locus, but I tried it several times and I get a poor/slow settling time. Any help please!!

Yang Hong Popular answer

(1) A simple PI controller tuning without Z-N method can be founded in the following classical INFOCOM 2001 paper that applied PI control method for Internet congestion control.

CV Hollot, V Misra, D Towsley, and WB Gong, "On designing improved controllers for AQM routers supporting TCP flows," In Proceedings of IEEE INFOCOM, 2011, pp. 1726-1734.

http://scholar.google.ca/scholar?oi=bibs&hl=en&cluster=8548240540248358189

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://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 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

PI Controller is special form of PID Controller where D Controller parameter is set to zero.

(2) 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 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 minimize a given quadratic cost function of the system output error and control effort, thus solving the controller tuning issues caused by plant uncertainty. The goal of IFT is similar to Quantitative Feedback Theory proposed by Isaac Horowitz and his co-workers, as mentioned by Simon's comments.

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

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

By cooperating with his peer researchers including Stanford University researcher, H. Hjalmarsson integrated iterative feedback tuning with PID controller to solve controller tuning issues caused by plant uncertainty.

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.

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

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 and selected classic PID tuning methods (co-invented by K.J. Astrom and his student W.K. Ho) have 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: 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

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

(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), i.e., NBA (Sweden) and Kalman.

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

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

To celebrate the golden anniversary, IFAC Automatica has published in the 50th volume throughout the year, special survey/overview papers, on selected topics, some tracing the history on some mature topics and areas, others focusing on newly emerging areas, with forecasts into the future. The survey paper published by the 1st issue provides an overview of the developments in the control field from its early days.

K.J. Astrom and P.R. Kumar, "Control: A Perspective," Automatica, 50(1), January 2014, pp. 3-43.

http://www.sciencedirect.com/science/article/pii/S0005109813005037

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 Design of a PI Rate Controller for Mitigating SIP Overload

Article A Comparative Study of SIP Overload Control Algorithms

Article Applying control theoretic approach to mitigate SIP overload

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

Data Mitigating SIP Overload Using a Control-Theoretic Approach

Aviti Mushi

Romero, I did just that and a trial and error in tuning the Kp and Ki. Now, I get my q-current signal to follow the reference value. Only problem is that at the start of simulation, there is a big spiking instead of smooth or near smooth increment to the desired reference value. Any idea here as to how can I play around with the two gains?

Aviti Mushi

Thank you. I tried that and got some Kp and Ki which of course had a bump at the begining (it seems I did a sloppy job then). Now, I will continue with this method until I can tune the PI to eliminate the bumps. Also, I was advised by my teacher to try use pole assignment and decoupling control. I will get busy with them all for now..

Aviti Mushi

Romero, it worked. The method you suggested and the other methods like Internal model control, and pole placement by designing the damping ratio. It was a well deserving contribution from you which got me started in getting it easily done for the first order systems. I think for higher orders, I can as well use the same with small modifications to handle the added poles. Thank you.

Peter A Vanrolleghem

An alternative is to use the Cohen-Coon tuning method. With a simple step response experiment you get the necessary information to tune the controller

Mohan Krishnan

I need to apply this solution too in my work ... I have a plant that is a black box.

Jesus, when you said set kp = tao / tao_cl / K, did you mean kp = tao /( tao_cl *K)? Thanks!

Aviti Mushi

The Ki acts to reduce the stead state error (O'Dwyer, Aidan : PI and PID controller tuning rules: an overview and personal perspective. Proceedings of the IET Irish Signals and Systems Conference, pp. 161-166, Dublin Institute of Technology, June, 2006.)

Aviti Mushi

I read one paper about robust control using DOB with delay compensation with inverse first order-model.... a very informative paper on PMSMs...

Simon Stoyanov

@Peter Vanrolleghem: with a step response you usually 'overlook' the high-frequency (HF) behaviour. Most of the time there are just not enough points sampled in the step reponse to give a representation of the HF behaviour.

@Somesh Bhattacharya: The term robust control typically referred to (closed-loop) systems that can tolerate uncertainty in the plant. In this sense, a closed-loop system is robust depending on its ability to deliver the requiired performance when the controlled plant changes its behaviour, rather than on the controller type. While there are methods claiming to give a robust controller, the best robust conrollers in practice/industry are PID-based. It depends how you tune the (PID) controller. Ziegler-Nichols methods use step responses; the P, I and D gains can be also calculated using plant's frequency response. In the latter case you can provide robustness, by means of phase and gain margins or by limiting the sensitivity function (peak) - a more general and safe than the phase and gain margin.

Aviti Mushi

I have found another paper by Chou and Liaw, IEEE Transactions, Vol. 58, No. 10, 10 October 2011. Titled Dyanamic control and diagnostic friction estimation. For me the useful part is where the authors develop the robust speed error cancellation control to preserve the defined response trajectory. This is done for a case of a SPMSM. For my works, I am concerned with IPMSM. I think anyone interested in PMSM and robust control can find this paper interesting.

Anup Kumar

You can use some optimization technique to optimize the value of Kp & Ki. These optimization techniques are GA, PSO, HPSO, NRM and quassi NRM.

Aviti Mushi

Anup Kumar...can you point me to some simple document or a tutorial of those techniques? Because I have tried the MATLAB automatic optimization and I got results which I did not like. Probably I need to learn more of how to do it, and maybe do it manually thereafter I could employ the MATLAB automatic optimization.

Anup Kumar

Using PSO technique, the function to be optimized has to be taken as an objective function. One has to randomly generate the value of Kp, Ki & Kd using MATLAB coding and then create a population of Kp, Ki & Kd. Then initialization of position vector and velocity vector. After each iteration particle position and velocity will be updated and particle will always search for a global best position in the entire population for best objective function.

Yang Hong