Nonlinear control

<h2 id="properties-of-nonlinear-systems">Properties of nonlinear systems</h2>

Some properties of nonlinear dynamic systems are

<ul><li>They do not follow the principle of <a href="/facts/Superposition_principle/GKGs9C92">superposition</a> (linearity and homogeneity).</li>
<li>They may have multiple isolated equilibrium points.</li>
<li>They may exhibit properties such as <a href="/facts/Limit_cycle/LmM9EVoA">limit cycle</a>, <a href="/facts/Bifurcation_theory/sLsN54Rn">bifurcation</a>, <a href="/facts/Chaos_theory/LTC17vsF">chaos</a>.</li>
<li>Finite escape time: Solutions of nonlinear systems may not exist for all times.</li></ul>
<h2 id="analysis-and-control-of-nonlinear-systems">Analysis and control of nonlinear systems</h2>

There are several well-developed techniques for analyzing nonlinear feedback systems:

<ul><li><a href="/facts/Describing_function/SpPnV9wD">Describing function</a> method</li>
<li><a href="/facts/Phase_plane_method/MmvKENnS">Phase plane method</a></li>
<li><a href="/facts/Lyapunov_stability/r3je5zaQ">Lyapunov stability</a> analysis</li>
<li><a href="/facts/Singular_perturbation/YcNwvmC8">Singular perturbation</a> method</li>
<li>The <a href="/facts/Popov_criterion/yhqACsMk">Popov criterion</a> and the <a href="/facts/Circle_criterion/EScnVdJ5">circle criterion</a> for absolute stability</li>
<li><a href="/facts/Center_manifold_theorem/fHJPBTrA">Center manifold theorem</a></li>
<li><a href="/facts/Small-gain_theorem/QTmGnHwr">Small-gain theorem</a></li>
<li>Passivity analysis</li></ul>
Control design techniques for nonlinear systems also exist. These can be subdivided into techniques which attempt to treat the system as a linear system in a limited range of operation and use (well-known) linear design techniques for each region:

<ul><li><a href="/facts/Gain_scheduling/JPl9XEom">Gain scheduling</a></li></ul>
Those that attempt to introduce auxiliary nonlinear feedback in such a way that the system can be treated as linear for purposes of control design:

<ul><li><a href="/facts/Feedback_linearization/Dgn739RS">Feedback linearization</a></li></ul>
And <a href="/facts/Aleksandr_Lyapunov/1b0TlYBz">Lyapunov</a> based methods:

<ul><li><a href="/facts/Lyapunov_redesign/hQTCbam3">Lyapunov redesign</a></li>
<li><a href="/facts/Control-Lyapunov_function/scjr1L5Y">Control-Lyapunov function</a></li>
<li>Nonlinear damping</li>
<li><a href="/facts/Backstepping/8ENMxxUK">Backstepping</a></li>
<li><a href="/facts/Sliding_mode_control/218PP4uw">Sliding mode control</a></li></ul>
<h2 id="nonlinear-feedback-analysis--the-lure-problem">Nonlinear feedback analysis – The Lur'e problem</h2>

An early nonlinear feedback system analysis problem was formulated by <a href="/facts/Anatoliy_Lure/hVlUzeEG">A. I. Lur'e</a>.
Control systems described by the Lur'e problem have a forward path that is linear and time-invariant, and a feedback path that contains a memory-less, possibly time-varying, static nonlinearity.
The linear part can be characterized by four matrices (A,B,C,D), while the nonlinear part is Φ(y) with 
 
 
 
 
 
 
 Φ
 (
 y
 )
 
 y
 
 
 ∈
 [
 a
 ,
 b
 ]
 ,
 
 a
 
<h3>Absolute stability problem</h3>
Consider:

<ol><li>(A,B) is controllable and (C,A) is observable</li>
<li>two real numbers a, b with a < b, defining a sector for function Φ</li></ol>
The Lur'e problem (also known as the absolute stability problem) is to derive conditions involving only the transfer matrix H(s) and {a,b} such that x = 0 is a globally uniformly asymptotically stable equilibrium of the system. 
There are two well-known wrong conjectures on the absolute stability problem:

<ul><li>The <a href="/facts/Aizerman%2527s_conjecture/0pAynbK1">Aizerman's conjecture</a></li>
<li>The <a href="/facts/Kalman%2527s_conjecture/Nw7cF7XC">Kalman's conjecture</a>.</li></ul>
Graphically, these conjectures can be interpreted in terms of graphical restrictions on the graph of Φ(y) x y or also on the graph of dΦ/dy x Φ/y.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> There are counterexamples to Aizerman's and Kalman's conjectures such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists with a stable periodic solution—<a href="/facts/Hidden_oscillation/Tw9Fp243">hidden oscillation</a>.
There are two main theorems concerning the Lur'e problem which give sufficient conditions for absolute stability:

<ul><li>The <a href="/facts/Circle_criterion/EScnVdJ5">circle criterion</a> (an extension of the <a href="/facts/Nyquist_stability_criterion/J8rxK4yA">Nyquist stability criterion</a> for linear systems)</li>
<li>The <a href="/facts/Popov_criterion/yhqACsMk">Popov criterion</a>.</li></ul>
<h2 id="theoretical-results-in-nonlinear-control">Theoretical results in nonlinear control</h2>

<h3>Frobenius theorem</h3>
The <a href="/facts/Frobenius_theorem_(differential_topology)/A8gu0fjt">Frobenius theorem</a> is a <a href="/facts/Deep_result/fTRoAtn2">deep result</a> in differential geometry. When applied to nonlinear control, it says the following: Given a system of the form

x
              ˙
            
          
        
        =
        
          ∑
          
            i
            =
            1
          
          
            k
          
        
        
          f
          
            i
          
        
        (
        x
        )
        
          u
          
            i
          
        
        (
        t
        )
        
      
    
    {\displaystyle {\dot {x}}=\sum _{i=1}^{k}f_{i}(x)u_{i}(t)\,}

where 
 
 
 
 x
 ∈
 
 R
 
 n
 
 
 
 
 {\displaystyle x\in R^{n}}
 
, 
 
 
 
 
 f
 
 1
 
 
 ,
 …
 ,
 
 f
 
 k
 
 
 
 
 {\displaystyle f_{1},\dots ,f_{k}}
 
 are vector fields belonging to a distribution 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 and 
 
 
 
 
 u
 
 i
 
 
 (
 t
 )
 
 
 {\displaystyle u_{i}(t)}
 
 are control functions, the integral curves of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 are restricted to a manifold of dimension 
 
 
 
 m
 
 
 {\displaystyle m}
 
 if 
 
 
 
 span
 ⁡
 (
 Δ
 )
 =
 m
 
 
 {\displaystyle \operatorname {span} (\Delta )=m}
 
 and 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 is an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involutive</a> distribution.

<h2 id="see-also">See also</h2>
<ul><li>Feedback passivation</li>
<li><a href="/facts/Phase-locked_loop/sgmca8aQ">Phase-locked loop</a></li>
<li><a href="/facts/Small_control_property/wkbXCjD8">Small control property</a></li></ul>

<h2 id="further-reading">Further reading</h2>

<ul><li>Lur'e, A. I.; Postnikov, V. N. (1944). "К теории устойчивости регулируемых систем" [On the Theory of Stability of Control Systems]. Prikladnaya Matematika I Mekhanika (in Russian). 8 (3): 246–248.</li>
<li>Vidyasagar, M. (1993). Nonlinear Systems Analysis (2nd ed.). Englewood Cliffs: Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-623463-0.</li>
<li>Isidori, A. (1995). Nonlinear Control Systems (3rd ed.). Berlin: Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-19916-8.</li>
<li>Khalil, H. K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River: Prentice Hall. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-13-067389-3.</li>
<li>Brogliato, B.; Lozano, R.; Maschke, B.; Egeland, O. (2020). Dissipative Systems Analysis and Control (3rd ed.). London: Springer.</li>
<li>Leonov G.A.; Kuznetsov N.V. (2011). <a href="http://www.math.spbu.ru/user/nk/PDF/2011-DAN-Absolute-stability-Aizerman-problem-Kalman-conjecture.pdf">"Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems"</a> (PDF). Doklady Mathematics. 84 (1): 475–481. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1134%2FS1064562411040120">10.1134/S1064562411040120</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120692391">120692391</a>.</li>
<li>Bragin V.O.; Vagaitsev V.I.; Kuznetsov N.V.; Leonov G.A. (2011). <a href="http://www.math.spbu.ru/user/nk/PDF/2011-TiSU-Hidden-oscillations-attractors-Aizerman-Kalman-conjectures.pdf">"Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits"</a> (PDF). Journal of Computer and Systems Sciences International. 50 (5): 511–543. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1134%2FS106423071104006X">10.1134/S106423071104006X</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:21657305">21657305</a>.</li>
<li>Leonov G.A., Kuznetsov N.V. (2011). Sergio, Bittanti (ed.). <a href="http://www.math.spbu.ru/user/nk/PDF/2011-IFAC-Hidden-oscillations-control-systems-Aizerman-problem-Kalman.pdf">"Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems"</a> (PDF). IFAC Proceedings Volumes (IFAC-PapersOnline). Proceedings of the 18th IFAC World Congress. 18 (1): 2494–2505. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.3182%2F20110828-6-IT-1002.03315">10.3182/20110828-6-IT-1002.03315</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9783902661937.</li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://reference.wolfram.com/language/guide/NonlinearControlSystems.html">Wolfram language functions for nonlinear control systems</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">trim point <a href="http://www.mathworks.com/help/toolbox/simulink/slref/trim.html" target="_blank">http://www.mathworks.com/help/toolbox/simulink/slref/trim.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Naderi, T.; Materassi, D.; Innocenti, G.; Genesio, R. (2019). "Revisiting Kalman and Aizerman Conjectures via a Graphical Interpretation". IEEE Transactions on Automatic Control. 64 (2): 670–682. doi:10.1109/TAC.2018.2849597. ISSN 0018-9286. S2CID 59553748. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Nonlinear control open-in-new

Nonlinear control