Table of contents |
2 Analysis and control of Non-linear Systems 3 The Lur'e Problem |
In this section, we will study the stability of an important class of control systems namely feedback systems whose forward path contains a linear time-invariant subsystem and whose feedback path contains a memory-less and possibly time-varying non-linearity. This class of problem is named for A. I. Lur'e.
The linear part is characterized by four matrices (A,B,C,D). The non-linear part is &Phi &isin [a,b], a
We will discuss two main theorems concerning Lure's problem.
where x ∈ Rn, &xi,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, &infin). This means that
&Phi(0) = 0, y &Phi(y) > 0, &forall y &ne 0; (3)
The transfer function from u to y is given by
Further reading:
Properties of Non-linear systems
Analysis and control of Non-linear Systems
The Lur'e Problem
Absolute Stability Problem
Given the
The problem is to derive conditions involving only the transfer matrix H(.) and the numbers a,b, such that x=0 is a globally uniformly asymptotically stable equilibrium of the system (1)-(3) for every function &Phi &isin [a,b]. This is also known as Lure's problem.Popov Criterion
The class of systems studied by Popov is described by
Things to be noted
Theorem:
Consider the system (1) and (2) and suppose
then the above system is globally asymptotically stable if there exists a number r>0 such that
infω ∈ R Re[(1+jωr)h(j&omega)] > 0