**Energy and Geometry in Nonlinear Control**
Workshop Prof. van der Schaft
n this course an
introduction is given to geometric nonlinear control theory. In the
first part of the course the basic concepts of nonlinear
controllability and observability are treated with tools from
coordinate-free analysis such as Lie brackets of vector fields. This
is illustrated by simple examples from mechanical systems. Next
issues around (partial) feedback linearization are addressed where we
wish to transform the nonlinear control system into a linear one, in
order to apply linear control techniques. Examples to tracking
control will be included. Second part of the talk is concerned with a
basic treatment of stability and stabilization of nonlinear control
systems. Focus is on linearization and the use of Lyapunov functions.
Dissipative systems are introduced, and the basic small-gain and
passivity theorems are given in time-domain. In the third part of the
talk we concentrate on nonlinear systems with physical structure, in
particular port-Hamiltonian systems as arising from network
modelling. The relation with dissipative systems are given, and
control strategies based on the physical properties of the system are
indicated.

References:

H. Nijmeijer, A.J. van der
Schaft,

Nonlinear Dynamical Control Systems,

Springer-Verlag,
New York, 1990 (4th printing 1998), p. xiii+467.

A.J. van der
Schaft,

L_{2}-Gain and Passivity Techniques in Nonlinear
Control,

Lect. Notes in Control and Inf. Sciences, Vol. 218,
Springer-Verlag, Berlin, 1996, p. 168, 2nd revised and enlarged
edition, Springer-Verlag, London, 2000 (Springer Communications and
Control Engineering series), p. xvi+249.

R. Ortega, A.J. van
der Schaft, I. Mareels & B.M. Maschke:

Putting energy back in
control, Control Systems Magazine, 21, pp. 18-33, 2001.