# Nonlinear control

Nonlinear control

Nonlinear control is the area of control engineering specifically involved with systems that are nonlinear, time-variant, or both. Many well-established analysis and design techniques exist for LTI systems (e.g., root-locus, Bode plot, Nyquist criterion, state-feedback, pole placement); however, one or both of the controller and the system under control in a general control system may not be an LTI system, and so these methods cannot necessarily be applied directly. Nonlinear control theory studies how to apply existing linear methods to these more general control systems. Additionally, it provides novel control methods that cannot be analyzed using LTI system theory. Even when LTI system theory can be used for the analysis and design of a controller, a nonlinear controller can have attractive characteristics (e.g., simpler implementation, increased speed, or decreased control energy); however, nonlinear control theory usually requires more rigorous mathematical analysis to justify its conclusions.

## 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: Solutions of nonlinear systems may not exist for all times.

## Analysis and control of nonlinear systems

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

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:

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:

And Lyapunov based methods:

## Nonlinear feedback analysis – The Lur'e problem  Lur'e problem block diagram

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.

The linear part can be characterized by four matrices (A,B,C,D), while the nonlinear part is Φ(y) with $\frac{\Phi(y)}{y} \in [a,b],\quad a (a sector nonlinearity).

### Absolute stability problem

Consider:

1. (A,B) is controllable and (C,A) is observable
2. 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:

#### Popov criterion

The sub-class of Lur'e systems studied by Popov is described by: $\begin{matrix} \dot{x}&=&Ax+bu \\ \dot{\xi}&=&u \\ y&=&cx+d\xi \quad (1) \end{matrix}$ $\begin{matrix} u = -\phi (y) \quad (2) \end{matrix}$

where x ∈ Rn, 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 $H(s) = \frac{d}{s} + c(sI-A)^{-1}b \quad \quad$

Theorem: Consider the system (1)-(2) and suppose

1. A is Hurwitz
2. (A,b) is controllable
3. (A,c) is observable
4. d>0 and
5. Φ ∈ (0,∞)

then the system is globally asymptotically stable if there exists a number r>0 such that
infω ∈ R 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

## Theoretical results in nonlinear control

### Frobenius Theorem

The Frobenius theorem is a deep result in Differential Geometry. When applied to Nonlinear Control, it says the following: Given a system of the form $\dot x = \sum_{i=1}^k f_i(x) u_i(t) \,$

where $x \in R^n$, $f_1, \dots, f_k$ are vector fields belonging to a distribution Δ and ui(t) are control functions, the integral curves of x are restricted to a manifold of dimension m if span(Δ) = m and Δ is an involutive distribution.

• 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, 2nd edition, Prentice Hall, Englewood Cliffs, New Jersey 07632.
• A. Isidori, Nonlinear Control Systems, 3rd edition, Springer Verlag, London, 1995.
• H. K. Khalil, Nonlinear Systems, 3rd 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 edition, 2007.

Wikimedia Foundation. 2010.

### См. также в других словарях:

• nonlinear control — netiesinis reguliavimas statusas T sritis automatika atitikmenys: angl. nonlinear control vok. nichtlineare Regelung, f rus. нелинейное регулирование, n pranc. régulation non linéaire, f …   Automatikos terminų žodynas

• nonlinear control — netiesinis valdymas statusas T sritis automatika atitikmenys: angl. nonlinear control vok. nichtlineare Steuerung, f rus. нелинейное управление, n pranc. régulation non linéaire, f …   Automatikos terminų žodynas

• Control engineering — Control systems play a critical role in space flight Control engineering or Control systems engineering is the engineering discipline that applies control theory to design systems with predictable behaviors. The practice uses sensors to measure… …   Wikipedia

• Control theory — For control theory in psychology and sociology, see control theory (sociology) and Perceptual Control Theory. The concept of the feedback loop to control the dynamic behavior of the system: this is negative feedback, because the sensed value is… …   Wikipedia

• Control-Lyapunov function — In control theory, a control Lyapunov function V(x,u) is a generalization of the notion of Lyapunov function V(x) used in stability analysis. The ordinary Lyapunov function is used to test whether a dynamical system is stable (more… …   Wikipedia

• Nonlinear photonic crystal — Nonlinear photonic crystals are usually used as quasi phase matching materials. They can be either one dimensional or two dimensional. Nonlinear Photonic Crystals Broadly speaking, nonlinear photonic crystals (PC) are periodic structures… …   Wikipedia

• Nonlinear Dynamics (journal) — Nonlinear Dynamics, An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems   …   Wikipedia

• Control chart — One of the Seven Basic Tools of Quality First described by Walter A. Shewhart …   Wikipedia

• Nonlinear system — Not to be confused with Non linear editing system. This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity (disambiguation). In mathematics, a nonlinear system is one that does not satisfy the… …   Wikipedia

• Nonlinear filter — A nonlinear filter is a signal processing device whose output is not a linear function of its input. Terminology concerning the filtering problem may refer to the time domain (state space) showing of the signal or to the frequency domain… …   Wikipedia

### Поделиться ссылкой на выделенное

##### Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»