**Introduction to Nonlinear Systems.**

Lecturer:

Rafael Wisniewski
(office **C4-202**)

OVERVIEW:

Virtually
all physical systems are nonlinear in nature. Sometimes it is possible to
describe the operation of a physical system by a linear model, such as set of
ordinary linear differential equations. This is the case, for example, if the
mode of operation of the physical system does not deviate
too much form the nominal set of operating conditions. Thus the analysis of
linear systems occupies an important place in system theory. But in analyzing
the behavior of any physical system, one often encounters situations where the linearized model is inadequate or inaccurate; that is the
time when concepts of this course may prove useful.

Here
is what **Rudolf E. Kalman** once said about
nonlinear control

*Whenever the controlled variable y(t) is allowed to have large deviations from the steady
state, the linear constant-coefficient model will cease to represent the plant
accurately because of intrinsic nonlinearities involved in the description of
most natural dynamical phenomena. Equally important are intentionally
introduced nonlinearities that result from reasons of economy, simplicity and
reliability of engineering or from ignorance of the fact that the savings
achieved by nonlinear control devices may be negated by factors resulting from
the greater intrinsic difficulty of control. A good example of unavoidable
nonlinearities in the object to be controlled is a space vehicle whose rockets, at the current state of technology, can be
controlled only by turning them on or off. Continuously variable control action
(which is desirable) in this case is technologically impossible; the high cost
of rocket engines, however, justifies extremely sophisticated computer-based
technology to achieve optimal control. The purpose of the computer is then
simply one of switching the rockets on or off; the difficulty of the control
lies in the extreme precision with which the time must be determined when this
takes place.*

*When the basic mathematical data can be stated
in the conventional optimization framework, effective methods are available for
solving the optimal control problem even in the nonlinear case. The methods for
doing this are an extension of the classical methods of the calculus of
variations. These methods, however, have yielded little theoretical insight,
and their straightforward application becomes prohibitively expensive for
large-scale systems.*

*Even less satisfactory is the status of the
nonlinear optimal filtering and state estimation problem, which, as has been
noted, is a critical part of the general solution of the control problem.
Because of nonlinearity, the Duality Principle no longer applies; even the
formulation of the problem is controversial.*

*Nonlinear systems represent not a special case
but simply everything that is not subjected to the special assumption of
linearity; in a sense “nonlinear” is synonymous with “unknown.” Scientific
progress will undoubtedly occur by singling out special classes of systems
subject to restrictive structural assumptions other than linearity.*

REFLECTIONS:

The nonlinear control is important,
fun but difficult. In order to tackle these difficulties with success we shall
need some mathematical background in the first part of the course (5 mini-modules). The second part in turn will pay off
with a number of beautiful and powerful nonlinear control techniques. So
be patient. We think the highlights of the course will be Lyapunov
stability and feedback linearization. The Lyapunov
stability theorems are used for analysis of a nonlinear system. We answer the
questions: Is it asymptotically stable, is it locally stable/unstable.
Later on, we look at synthesis problems in **Nonlinear Control Methods** (5 extra mini-modules). Feedback linearization is a pure
geometrical method which helps you to find a certain nice map which translates
a nonlinear system into a linear one. Remark this is not a linearization; we
merely will change the coordinates such that your system appears linear.

PRACTICAL REMARKS:

We shall use my notes
and **H.K. Khalil,
Nonlinear Systems, Prentice 2002**. It is available in the campus bookshop. The exercises are compulsory to be delivered on
written a week after the lecture. The examination will be oral. You will be
allowed to bring maximum one slide for each topic. I will pose the list of
topics on the web page in due time.

MM1: Introduction to Nonliner Control

Written material: pages 1-

MM5: Stability of Nonlinear Systems

MM7: Potpourri in Stability of Nonlinear Systems