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<title>Machine learning control</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Machine learning control</span></span>

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<p><b>Machine learning control</b> (<b>MLC</b>) is a subfield of <a href="Machine_learning" title="Machine learning">machine learning</a>, <a href="Intelligent_control" title="Intelligent control">intelligent control</a>, and <a href="Control_theory" title="Control theory">control theory</a> which aims to solve <a href="Optimal_control" title="Optimal control">optimal control</a> problems with machine learning methods. Key applications are complex nonlinear systems for which <a href="Linear_control_theory" class="mw-redirect" title="Linear control theory">linear control theory</a> methods are not applicable.

</p>

<div class="mw-heading mw-heading2"><h2 id="Types_of_problems_and_tasks">Types of problems and tasks</h2></div>

<p>Four types of problems are commonly encountered:

</p>

<ul><li>Control parameter identification: MLC translates to a parameter identification<sup id="cite_ref-Baeck1993_1-0" class="reference"><a href="#cite_note-Baeck1993-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> if the structure of the control law is given but the parameters are unknown. One example is the <a href="Genetic_algorithm" title="Genetic algorithm">genetic algorithm</a> for optimizing coefficients of a <a href="PID_controller" class="mw-redirect" title="PID controller">PID controller</a><sup id="cite_ref-Benard2015aiaa_2-0" class="reference"><a href="#cite_note-Benard2015aiaa-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> or discrete-time optimal control.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li>

<li>Control design as <a href="Regression_analysis" title="Regression analysis">regression</a> problem of the first kind: MLC approximates a general nonlinear mapping from sensor signals to actuation commands, if the sensor signals and the optimal actuation command are known for every state. One example is the computation of sensor feedback from a known <a href="Full_state_feedback" title="Full state feedback">full state feedback</a>. <a href="Neural_network_(machine_learning)" title="Neural network (machine learning)">Neural networks</a> are commonly used for such tasks.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li>

<li>Control design as regression problem of the second kind: MLC may also identify arbitrary nonlinear control laws which minimize the <a href="Loss_function" title="Loss function">cost function</a> of the plant. In this case, neither a model, the control law structure, nor the optimizing actuation command needs to be known. The optimization is only based on the control performance (cost function) as measured in the plant. <a href="Genetic_programming" title="Genetic programming">Genetic programming</a> is a powerful regression technique for this purpose.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li>

<li>Reinforcement learning control: The control law may be continually updated over measured performance changes (rewards) using <a href="Reinforcement_learning" title="Reinforcement learning">reinforcement learning</a>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:0_7-0" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Adaptive_Dynamic_Programming">Adaptive Dynamic Programming</h2></div>

<p>Adaptive Dynamic Programming (ADP), also known as approximate dynamic programming or neuro-dynamic programming, is a machine learning control method that combines reinforcement learning with dynamic programming to solve optimal control problems for complex systems. ADP addresses the "<a href="Curse_of_dimensionality" title="Curse of dimensionality">curse of dimensionality</a>" in traditional dynamic programming by approximating value functions or control policies using parametric structures such as neural networks. The core idea revolves around learning a control policy that minimizes a long-term cost function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J}">

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

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</math></span><img src="./359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"></span>, defined as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle J(x(t))=\int _{t}^{\infty }e^{-\gamma (\tau -t)}r(x(\tau ),u(\tau ))\,d\tau }">

</mrow>

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</msubsup>

<msup>

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</msup>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle J(x(t))=\int _{t}^{\infty }e^{-\gamma (\tau -t)}r(x(\tau ),u(\tau ))\,d\tau }</annotation>

</semantics>

</math></span><img src="./88f83b31bb6ead1f4a4821dc0574cadb3ffd20fb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.951ex; height:5.843ex;" alt="{\displaystyle J(x(t))=\int _{t}^{\infty }e^{-\gamma (\tau -t)}r(x(\tau ),u(\tau ))\,d\tau }" loading="lazy"></span> , where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

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

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> is the system state, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"></span> is the control input, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span> is the instantaneous reward, and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

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

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</math></span><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"></span> is a discount factor. ADP employs two interacting components: a critic that estimates the value function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(x)\approx J(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V(x)\approx J(x)}</annotation>

</semantics>

</math></span><img src="./5592b46bdf3cc5bc95fdabf83dbef81fa4765443.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.635ex; height:2.843ex;" alt="{\displaystyle V(x)\approx J(x)}" loading="lazy"></span>, and an actor that updates the control policy <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x)}">

</mstyle>

</mrow>

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

</semantics>

</math></span><img src="./b1e5ff65a28eed29d36ddae9c6ae3b596fd14370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.469ex; height:2.843ex;" alt="{\displaystyle u(x)}" loading="lazy"></span>. The critic and actor are trained iteratively using temporal difference learning or gradient descent to satisfy the <a href="Hamilton%E2%80%93Jacobi%E2%80%93Bellman_equation" title="Hamilton–Jacobi–Bellman equation">Hamilton-Jacobi-Bellman (HJB) equation</a>:  

</p><p> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min _{u}\left(r(x,u)+{\frac {\partial V}{\partial x}}f(x,u)\right)=0,}">

</mrow>

</munder>

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<mrow>

<mfrac>

<mrow>

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<mrow>

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</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \min _{u}\left(r(x,u)+{\frac {\partial V}{\partial x}}f(x,u)\right)=0,}</annotation>

</semantics>

</math></span><img src="./dba3c0f0708d9bcaafa3f2ffd4019ae005242d8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.706ex; height:6.176ex;" alt="{\displaystyle \min _{u}\left(r(x,u)+{\frac {\partial V}{\partial x}}f(x,u)\right)=0,}" loading="lazy"></span>

</p><p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,u)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x,u)}</annotation>

</semantics>

</math></span><img src="./08d97aa51b73aa6e397524e96619b8ca9c73163e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.781ex; height:2.843ex;" alt="{\displaystyle f(x,u)}" loading="lazy"></span> describes the system dynamics. Key variants include heuristic dynamic programming (HDP), dual heuristic programming (DHP), and globalized dual heuristic programming (GDHP).<sup id="cite_ref-:0_7-1" class="reference"><a href="#cite_note-:0-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>ADP has been applied to robotics, power systems, and autonomous vehicles, offering a data-driven framework for near-optimal control without requiring full system models. Challenges remain in ensuring stability guarantees and convergence for general nonlinear systems.  

</p>

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

<p>MLC has been successfully applied

to many nonlinear control problems,

exploring unknown and often unexpected actuation mechanisms. Example applications include:

</p>

<ul><li><a href="Spacecraft_attitude_control" class="mw-redirect" title="Spacecraft attitude control">spacecraft attitude control</a>,<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup></li>

<li>thermal control of buildings,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup></li>

<li>feedback control of <a href="Turbulence" title="Turbulence">turbulence</a>,<sup id="cite_ref-Benard2015aiaa_2-1" class="reference"><a href="#cite_note-Benard2015aiaa-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li>

<li>and <a href="Remotely_operated_underwater_vehicle" title="Remotely operated underwater vehicle">remotely operated underwater vehicles</a>.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup></li></ul>

<p>Many more engineering MLC application are summarized in the review article of PJ Fleming & RC Purshouse (2002).<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p><p>As is the case for all general nonlinear methods,

MLC does not guarantee convergence,

<a href="Algorithmic_efficiency" title="Algorithmic efficiency">optimality</a>, or <a href="Robustness_(computer_science)" title="Robustness (computer science)">robustness</a> for a range of operating conditions.

</p>

<ul><li><a href="Reinforcement_learning" title="Reinforcement learning">Reinforcement learning</a></li></ul>

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

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<li id="cite_note-Baeck1993-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Baeck1993_1-0">^</a></b></span> <span class="reference-text">Thomas Bäck & Hans-Paul Schwefel (Spring 1993) <a rel="nofollow" class="external text" href="https://doi.org/10.1162/evco.1993.1.1.1">"An overview of evolutionary algorithms for parameter optimization"</a>, <a href="Evolutionary_Computation_(journal)" title="Evolutionary Computation (journal)">Journal of Evolutionary Computation (MIT Press)</a>, vol. 1, no. 1, pp. 1-23</span>

</li>

<li id="cite_note-Benard2015aiaa-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Benard2015aiaa_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Benard2015aiaa_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">N. Benard, J. Pons-Prats, J. Periaux, G. Bugeda, J.-P. Bonnet & E. Moreau, (2015) <a rel="nofollow" class="external text" href="https://arc.aiaa.org/doi/abs/10.2514/6.2015-2957">"Multi-Input Genetic Algorithm for Experimental Optimization of the Reattachment Downstream of a Backward-Facing Step with Surface Plasma Actuator"</a>, Paper AIAA 2015-2957 at 46th AIAA Plasmadynamics and Lasers Conference, Dallas, TX, USA, pp. 1-23.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">Zbigniew Michalewicz, Cezary Z. Janikow & Jacek B. Krawczyk (July 1992) <a rel="nofollow" class="external text" href="https://doi.org/10.1016/0898-1221(92)90094-X">"A modified genetic algorithm for optimal control problems"</a>, [Computers & Mathematics with Applications], vol. 23, no 12, pp. 83-94.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">C. Lee, J. Kim, D. Babcock & R. Goodman (1997) <a rel="nofollow" class="external text" href="https://dx.doi.org/10.1063/1.869290">"Application of neural networks to turbulence control for drag reduction"</a>, <a href="Physics_of_Fluids" title="Physics of Fluids">Physics of Fluids</a>, vol. 6, no. 9, pp. 1740-1747</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">D. C. Dracopoulos & S. Kent (December 1997) <a rel="nofollow" class="external text" href="https://doi.org/10.1007/BF01501508">"Genetic programming for prediction and control"</a>, Neural Computing & Applications (Springer), vol. 6, no. 4, pp. 214-228.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Andrew G. Barto (December 1994) <a rel="nofollow" class="external text" href="https://doi.org/10.1016/0959-4388(94)90138-4">"Reinforcement learning control"</a>, <a href="Current_Opinion_in_Neurobiology" class="mw-redirect" title="Current Opinion in Neurobiology">Current Opinion in Neurobiology</a>, vol. 6, no. 4, pp. 888–893</span>

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</style><cite id="CITEREFJiangJiang2017" class="citation book cs1">Jiang, Yu; Jiang, Zhong-Ping (2017-05-30). <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/book/10.1002/9781119132677"><i>Robust Adaptive Dynamic Programming</i></a> (1 ed.). Wiley. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2F9781119132677">10.1002/9781119132677</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-119-13264-6</bdi>.</cite></span>

</li>

<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">Dimitris. C. Dracopoulos & <a href="Antonia_J._Jones" title="Antonia J. Jones">Antonia. J. Jones</a> (1994)

<a href="https://doi.org/10.1007/BF01414807" class="extiw external" title="doi:10.1007/BF01414807">Neuro-genetic adaptive attitude control</a>, Neural Computing & Applications (Springer), vol. 2, no. 4, pp. 183-204.</span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Jonathan A. Wright, Heather A. Loosemore & Raziyeh Farmani (2002) <a rel="nofollow" class="external text" href="https://doi.org/10.1016/S0378-7788(02)00071-3">"Optimization of building thermal design and control by multi-criterion genetic algorithm</a>, [Energy and Buildings], vol. 34, no. 9, pp. 959-972.</span>

</li>

<li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text">Steven J. Brunton & Bernd R. Noack (2015) <a rel="nofollow" class="external text" href="https://doi.org/10.1115/1.4031175">Closed-loop turbulence control: Progress and challenges</a>, <a href="Applied_Mechanics_Reviews" title="Applied Mechanics Reviews">Applied Mechanics Reviews</a>, vol. 67, no. 5, article 050801, pp. 1-48.</span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text">J. Javadi-Moghaddam, & A. Bagheri (2010 <a rel="nofollow" class="external text" href="https://doi.org/10.1016/j.eswa.2009.06.015">"An adaptive neuro-fuzzy sliding mode based genetic algorithm control system for under water remotely operated vehicle"</a>, <a rel="nofollow" class="external text" href="https://www.journals.elsevier.com/expert-systems-with-applications/">Expert Systems with Applications</a>, vol. 37 no. 1, pp. 647-660.</span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">Peter J. Fleming, R. C. Purshouse (2002 <a rel="nofollow" class="external text" href="https://doi.org/10.1016/S0967-0661(02)00081-3">"Evolutionary algorithms in control systems engineering: a survey"</a>

<a href="https://nl.wikipedia.org/wiki/Control_Engineering_Practice" class="extiw external" title="nl:Control Engineering Practice">Control Engineering Practice</a>, vol. 10, no. 11, pp. 1223-1241</span>

</li>

</ol></div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

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<ul><li>Dimitris C Dracopoulos (August 1997) <a rel="nofollow" class="external text" href="https://www.springer.com/fr/book/9783540761617">"Evolutionary Learning Algorithms for Neural Adaptive Control"</a>, Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-76161-7</bdi>.</li>

<li>Thomas Duriez, Steven L. Brunton & <a href="Bernd_Noack" title="Bernd Noack">Bernd R. Noack</a> (November 2016) <a rel="nofollow" class="external text" href="https://www.springer.com/fr/book/9783319406237">"Machine Learning Control - Taming Nonlinear Dynamics and Turbulence"</a>, Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-40624-4</bdi>.</li></ul>

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