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

3.1 Absolute Stability Problem
3.2 Popov Criterion

Properties of Non-linear systems

They do not follow the principle of superposition (linearity and homogenity).
They may have multiple equilibrium points.
They exhibit properties like limit-cycle, bifurcation, chaos.
For a sinusoidal input, the output signal may contain many harmonics and sub-harmonics with various amplitudes and phase differences. While for a linear system, we know that for u= A sin(ωt), output y = B sin(ωt+ φ).

Analysis and control of Non-linear Systems

Describing function method

Phase plane method

Lyapunov based methods
Input-output stability
Feedback linearization
Back-stepping
Sliding mode control

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

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

A is Hurwitz
(A,b) is controllable
(A,c) is observable
d>0 and
Φ ∈ (0,&infin)

then the above system is globally asymptotically stable if there exists a number r>0 such that
inf_{ω ∈ R} Re[(1+jωr)h(j&omega)] > 0