Nonlinear control
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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 Φ(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 Φ
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 ∈ R<sup>n</sup>, ξ,u,y are scalars and A,b,c,d have commensurate dimensions. The nonlinear element Φ: R → R is a time-invariant nonlinearity belonging to ''open sector'' (0, ∞). This means that
:Φ(0) = 0, y Φ(y) > 0, ∀ y ≠ 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
# Φ ∈ (0,∞)
then the system is [[globally asymptotically stable]] if there exists a number r>0 such that <br>
inf<sub>ω ∈ R</sub> Re[(1+jωr)h(jω)] > 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 Φ 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
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[[Category:Nonlinear control]]