Nonlinear control 3359147 225399488 2008-07-13T14:01:19Z Jiuguang Wang 2430726 link [[Circle criterion]] using [[User:Edward/Find link|Find link]] '''Nonlinear control''' is a sub-division of [[control engineering]] which deals with the control of nonlinear systems. The behavior of a [[Nonlinearity|nonlinear system]] cannot be described as a linear function of the state of that system or the input variables to that system. For [[linear systems]], there are many well-established control techniques, for example [[root-locus]], [[Bode plot]], [[Nyquist criterion]], [[control theory|state-feedback]], [[control theory|pole-placement]] etc. == Properties of nonlinear systems == Some properties of nonlinear dynamic systems are * They do not follow the principle of [[superposition]] (linearity and homogeneity). * They may have multiple isolated equilibrium points. * They may exhibit properties such as [[limit-cycle]], [[bifurcation]], [[chaos]]. * Finite escape time: The state of an [[unstable]] nonlinear system can go to infinity in finite time. * nonlinear systems cannot be described in terms of their [[eigenvectors]], unlike, for instance, [[time invariant|translation-invariant]] linear systems that can be completely described by the input/output relationship of sinusoidal inputs. == Analysis and control of nonlinear systems == There are several well-developed techniques for analyzing nonlinear feedback systems: * [[Describing function]] method * [[Phase plane method|Phase plane]] method * [[Lyapunov stability]] analysis * [[Singular perturbation]] method * [[Vasile M. Popov|Popov]] criterion (described in ''The Lur'e Problem'' below) * [[Center manifold theorem]] * [[Small-gain theorem]] * [[Passivity analysis]] 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: * [[Gain scheduling]] 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: * [[Feedback linearization]] And [[Aleksandr Lyapunov|Lyapunov]] based methods: * [[Lyapunov redesign]] * Nonlinear Damping * [[Backstepping]] * [[Sliding mode control]] == Nonlinear feedback analysis - The Lur'e problem == An early nonlinear feedback system analysis problem was formulated by [[A. I. Lur'e]]. 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. :[[Image:lure.png|Lur'e Problem Block Diagram]] The linear part can be characterized by four matrices (A,B,C,D), while the nonlinear part is &Phi;(y) with <math>\frac{\Phi(y)}{y} \in [a,b],\quad a<b \quad \forall y </math> (a sector nonlinearity). === Absolute stability problem === Consider: # (A,B) is controllable and (C,A) is observable # two real numbers a, b with a<b, defining a sector for function &Phi; The 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. This is known as the Lur'e problem. There are two main theorems concerning the problem: * The [[Circle criterion]] * The [[Vasile M Popov|Popov]] criterion. ==== Popov criterion ==== The sub-class of Lur'e systems studied by [[Vasile M Popov|Popov]] is described by: <math> \begin{matrix} \dot{x}&=&Ax+bu \\ \dot{\xi}&=&u \\ y&=&cx+d\xi \quad (1) \end{matrix} </math> <math> \begin{matrix} u = -\phi (y) \quad (2) \end{matrix} </math> where x &isin; R<sup>n</sup>, &xi;,u,y are scalars and A,b,c,d have commensurate dimensions. The nonlinear element &Phi;: R &rarr; R is a time-invariant nonlinearity belonging to ''open sector'' (0, &infin;). This means that :&Phi;(0) = 0, y &Phi;(y) > 0, &forall; y &ne; 0; The transfer function from u to y is given by <br> :<math> H(s) = \frac{d}{s} + c(sI-A)^{-1}b \quad \quad</math> '''Theorem:''' Consider the system (1)-(2) and suppose # A is [[Routh_Hurwitz_stability_criterion|Hurwitz]] #(A,b) is controllable #(A,c) is observable #d>0 and # &Phi; &isin; (0,&infin;) then the system is [[globally asymptotically stable]] if there exists a number r>0 such that <br> inf<sub>&omega; &isin; R</sub> Re[(1+j&omega;r)h(j&omega;)] > 0 . Things to be noted: * The Popov criterion is applicable only to autonomous systems * The system studied by Popov has a pole at the origin and there is no direct pass-through from input to output * The nonlinearity &Phi; must satisfy an open sector condition == References == * A. I. Lur'e and V. N. Postnikov, "On the theory of stability of control systems," ''Applied mathematics and mechanics'', 8(3), 1944, (in Russian). * M. Vidyasagar, ''Nonlinear Systems Analysis'', second edition, Prentice Hall, Englewood Cliffs, New Jersey 07632. * A. Isidori, ''Nonlinear Control Systems,'' third edition, Springer Verlag, London, 1995. * H. K. Khalil, ''Nonlinear Systems,'' third edition, Prentice Hall, Upper Saddle River, New Jersey, 2002. ISBN 0130673897 * B. Brogliato, R. Lozano, B. Maschke, O. Egeland, "Dissipative Systems Analysis and Control", Springer Verlag, London, 2nd Editon, 2007. ==See also== * [[Phase-locked loop]] * http://www.mne.ksu.edu/research/laboratories/nonlinear-controls-lab [[ru:Нелинейное управление]] [[es:Control no lineal]] [[Category:Nonlinear control]]