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Iterative learning control

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Iterative Learning Control (ILC) is an open-loop control approach of <a href="Process_control" class="mw-redirect" title="Process control">tracking control</a> for systems that work in a repetitive mode.<a href="#cite_note-1">[1]</a> Examples of systems that operate in a repetitive manner include <a href="Robot" title="Robot">robot</a> arm manipulators, chemical batch processes and <a href="Reliability_engineering" title="Reliability engineering">reliability testing</a> rigs. In each of these tasks the system is required to perform the same action over and over again with high <a href="Accuracy_and_precision" title="Accuracy and precision">precision</a>. This action is represented by the objective of accurately tracking a chosen reference signal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(t)}">

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<annotation encoding="application/x-tex">{\displaystyle r(t)}</annotation>

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</math><img src="./ec653b463c709d42cb85133595bf0da29801f6e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.698ex; height:2.843ex;" alt="{\displaystyle r(t)}" loading="lazy"> on a finite time interval.

Repetition allows the system to sequentially improve tracking accuracy, in effect learning the required input needed to track the reference as closely as possible. The learning process uses information from previous repetitions to improve the control signal, ultimately enabling a suitable control action to be found <a href="Iteration" title="Iteration">iteratively</a>. The <a href="Internal_model_(motor_control)" title="Internal model (motor control)">internal model</a> principle yields conditions under which perfect tracking can be achieved but the design of the control algorithm still leaves many decisions to be made to suit the application. A typical, simple control law is of the form:

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<annotation encoding="application/x-tex">{\displaystyle u_{p+1}=u_{p}+K*e_{p}}</annotation>

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</math><img src="./8c1bd32d25dfaeb1e1ea6c17806a43d5b26111b8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.22ex; height:2.843ex;" alt="{\displaystyle u_{p+1}=u_{p}+K*e_{p}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{p}}">

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<annotation encoding="application/x-tex">{\displaystyle u_{p}}</annotation>

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</math><img src="./a113d37c1d0f165030579e70c73c7608eb9c1e4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.389ex; height:2.343ex;" alt="{\displaystyle u_{p}}" loading="lazy"> is the input to the system during the pth repetition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{p}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{p}}</annotation>

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</math><img src="./d2f55fae5dbe513ef99bd6c42a815d30865fbe0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.143ex; height:2.343ex;" alt="{\displaystyle e_{p}}" loading="lazy"> is the tracking error during the pth repetition and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is a design parameter representing operations on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{p}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{p}}</annotation>

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</math><img src="./d2f55fae5dbe513ef99bd6c42a815d30865fbe0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.143ex; height:2.343ex;" alt="{\displaystyle e_{p}}" loading="lazy">.<a href="#cite_note-2">[2]</a> Achieving perfect tracking through iteration is represented by the mathematical requirement of convergence of the input signals as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> becomes large, whilst the rate of this convergence represents the desirable practical need for the learning process to be rapid. There is also the need to ensure good algorithm performance even in the presence of uncertainty about the details of process dynamics. The operation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> is crucial to achieving design objectives (i.e. trading off fast convergence and robust performance) and ranges from simple scalar gains to sophisticated optimization computations.<a href="#cite_note-3">[3]</a>

In many cases a low-pass filter is added to the input to improve performance. The control law then takes the form

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<annotation encoding="application/x-tex">{\displaystyle u_{p+1}=Q(u_{p}+K*e_{p})}</annotation>

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</math><img src="./ca619757167e5ef94a3c563c30fb9ad474cac4bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.868ex; height:3.009ex;" alt="{\displaystyle u_{p+1}=Q(u_{p}+K*e_{p})}" loading="lazy">

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">

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<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>

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</math><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> is a low-pass filtering matrix. This removes high-frequency disturbances which may otherwise be amplified during the learning process.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFS.Arimoto,_S._KawamuraF._Miyazaki1984" class="citation journal cs1">S.Arimoto, S. Kawamura; F. Miyazaki (1984). "Bettering operation of robots by learning". Journal of Robotic Systems. 1 (2): 123–140. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Frob.4620010203">10.1002/rob.4620010203</a>.</cite>

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<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFMoore1993" class="citation book cs1">Moore, K.L. (1993). Iterative Learning Control for Deterministic Systems. London: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-19707-9</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFOwens_D.H.Feng_K.2003" class="citation journal cs1">Owens D.H.; Feng K. (20 July 2003). "Parameter optimization in iterative learning control". International Journal of Control. 76 (11): 1059–1069. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0020717031000121410">10.1080/0020717031000121410</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120288506">120288506</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite id="CITEREFBristow,_D._A.Tharayil,_M.Alleyne,_A._G.2006" class="citation magazine cs1">Bristow, D. A.; Tharayil, M.; <a href="Andrew_G._Alleyne" title="Andrew G. Alleyne">Alleyne, A. G.</a> (2006). "A Survey of Iterative Learning Control A learning-based method for high-performance tracking control". IEEE Control Systems Magazine. Vol. 26. pp. 96–114.</cite>

</li>

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<ul><li><cite id="CITEREFS.Arimoto,_S._KawamuraF._Miyazaki1984" class="citation journal cs1">S.Arimoto, S. Kawamura; F. Miyazaki (1984). "Bettering operation of robots by learning". Journal of Robotic Systems. 1 (2): 123–140. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Frob.4620010203">10.1002/rob.4620010203</a>.</cite></li>

<li><cite id="CITEREFMoore1993" class="citation book cs1">Moore, K.L. (1993). Iterative Learning Control for Deterministic Systems. London: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-19707-9</bdi>.</cite></li>

<li><cite id="CITEREFJian_Xin_XuYing_Tan.2003" class="citation book cs1"><a href="Jian_Xin_Xu" title="Jian Xin Xu">Jian Xin Xu</a>; Ying Tan. (2003). <a rel="nofollow" class="external text" href="https://archive.org/details/linearnonlineari00xuji">Linear and Nonlinear Iterative Learning Control</a>. Springer-Verlag. p. <a rel="nofollow" class="external text" href="https://archive.org/details/linearnonlineari00xuji/page/n178">177</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-40173-3</bdi>.</cite></li>

<li><cite id="CITEREFBristow,_D._A.Tharayil,_M.Alleyne,_A._G.2006" class="citation magazine cs1">Bristow, D. A.; Tharayil, M.; <a href="Andrew_G._Alleyne" title="Andrew G. Alleyne">Alleyne, A. G.</a> (2006). "A Survey of Iterative Learning Control A learning-based method for high-performance tracking control". IEEE Control Systems Magazine. Vol. 26. pp. 96–114.</cite></li>

<li><cite id="CITEREFOwens_D.H.Feng_K.2003" class="citation journal cs1">Owens D.H.; Feng K. (20 July 2003). "Parameter optimization in iterative learning control". International Journal of Control. 76 (11): 1059–1069. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F0020717031000121410">10.1080/0020717031000121410</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120288506">120288506</a>.</cite></li>

<li><cite id="CITEREFOwens_D.H.Hätönen_J.2005" class="citation journal cs1">Owens D.H.; Hätönen J. (2005). "Iterative learning control — An optimization paradigm". Annual Reviews in Control. 29 (1): 57–70. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.arcontrol.2005.01.003">10.1016/j.arcontrol.2005.01.003</a>.</cite></li>

<li><cite id="CITEREFDaley_S.Owens_D.H.2008" class="citation journal cs1">Daley S.; Owens D.H. (2008). <a rel="nofollow" class="external text" href="http://zbc.uz.zgora.pl/Content/78717/AMCS_2008_18_3_3.pdf">"Iterative Learning Control – Monotonicity and Optimization"</a> (PDF). International Journal of Applied Mathematics and Computer Science. 18 (3): 179–293. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2478%2Fv10006-008-0026-7">10.2478/v10006-008-0026-7</a>.</cite></li>

<li><cite id="CITEREFWang_Y.Gao_F.Doyle_III,_F.J.2009" class="citation journal cs1">Wang Y.; Gao F.; Doyle III, F.J. (2009). "Survey on iterative learning control, repetitive control, and run-to-run control". Journal of Process Control. 19 (10): 1589–1600. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.jprocont.2009.09.006">10.1016/j.jprocont.2009.09.006</a>.</cite></li></ul>

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