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# Non-linear control

Non-linear systems are those systems whose input-output behaviour are very much unpredictable. For linear systems, we have a lot of well-established control techniques like root-locus, Bode plot, Nyquist criterion, state-feedback, pole-placement etc. Here we will explore the control techniques for non-linear systems.

 Table of contents 1 Properties of Non-linear systems 2 Analysis and control of Non-linear Systems 3 The Lur'e Problem

## 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.

### Absolute Stability Problem

Given the

• (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(.) 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.

We will discuss two main theorems concerning Lure's problem.

• The Circle Criterion
• The Popov Criterion.

### Popov Criterion

The class of systems studied by Popov is described by

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

Things to be noted
• Popov criterion is applicable only to autonomous systems.
• The system studied by Popov has a pole at the origin and there is no throughput from input to output.
• Non-linearity &Phi belongs to a open sector.

Theorem:
Consider the system (1) and (2) and suppose
1. A is Hurwitz
2. (A,b) is controllable
3. (A,c) is observable
4. d>0 and
5. Φ ∈ (0,&infin)

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