** Linear geometric control theory** was
initiated in the beginning of the 1970's. A good summary of the subject is the
book by Wonham.

The
term ``geometric'' suggests several things. First it suggests that the setting
is linear state space and the mathematics behind is primarily linear algebra
(with a geometric flavor). Secondly it suggests that the underlying methodology
is geometric. It treats many important system concepts, for example
controllability, as geometric properties of the state space or its subspaces.
These are the properties that are preserved under coordinate changes, for
example, the so-called invariant or controlled invariant subspaces. On the
other hand, we know that things like distance and shape do depend on the
coordinate system one chooses. Using these concepts the geometric approach
captures the essence of many analysis and synthesis problems and treats them in
a coordinate-free fashion. By characterizing the solvability of a control
problem as a verifiable property of some constructible subspace, calculation of
the control law becomes much easier. In many cases, the geometric approach can
convert what is usually a difficult nonlinear problem into a straight-forward
linear one.

The
linear geometric control theory was extended to** nonlinear systems** in the 1970's and 1980's (see the book by
Isidori). The underlying fundamental concepts are
almost the same, but the mathematics is different. For nonlinear systems the
tools from differential geometry are primarily used.

The
course compendium is organized as follows.

Chapter
1 is introduction; In Chapter 2,*
invariant and controlled invariant subspaces* will be discussed; In Chapter
3, the *disturbance decoupling problem*
will be introduced; In Chapter 4, we will introduce* transmission zeros* and their geometric interpretations; In Chapter
5, *non-interacting control and tracking*
will be studied as applications of the *zero
dynamics normal form*; In Chapter 6, we will discuss some *input-output behaviors* from a geometric
point of view; In Chapter 7, we will discuss the *output regulator problem* in some detail; In Chapter 8, we will
extend some of the central concepts in the geometric control to nonlinear
systems. Finally, in Chapter 9 some applications to *mobile robots* will be given.

In
the rest of the introduction, we use some *typical problems and examples* to illustrate the
advantages and basic ideas of geometric approaches.