Non-linear control

Non-linear control is a sub-division of control engineering which deals with the control of non-linear systems. The behaviour of a non-linear 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, state-feedback, pole-placement etc.

Properties of non-linear systems

Some properties of non-linear 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
For a sinusoidal input, the output signal may contain many harmonics and sub-harmonics with various amplitudes and phase differences (a linear system's output will only contain the sinusoid at the input)

Analysis and control of non-linear systems

The Lur'e problem

An early non-linear control 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, non-linearity.

The linear part can be characterized by four matrices (A,B,C,D), while the non-linear part is Φ(y) ∈ ^*, a

Absolute stability problem

Consider:
(A,B) is controllable and (C,A) is observable
two real numbers a, b with a
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 Popov criterion.
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 non-linear 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
A is 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
inf_{ω ∈ R} Re^* > 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
Non-linearity Φ belongs to an open sector
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.
See also

Phase-locked loop
Non-linear systems | Control theory
Vikipedio:Projekto matematiko/Ne-lineara rego

Gourt Home::Gourt :: Non-linear control

Properties of non-linear systems

Analysis and control of non-linear systems

The Lur'e problem

Absolute stability problem

Popov criterion

References

See also