Nonlinear control

 

Zobacz też:

Nonlinear control is a sub-division of control engineering which deals with the control of nonlinear systems. The behavior of a 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, state-feedback, pole-placement etc.

Contents

Properties of nonlinear systems

Some properties of nonlinear dynamic systems are

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

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.

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 \frac{\Phi(y)}{y} \in [a,b],\quad a<b \quad \forall y (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:

References

See also