Eigenvalue placement for variable structure control systems.

Variable Structure Control is a well-known solution to the
problem of deterministic control of uncertain systems, since it is
invariant to a class of parameter variations. A central feature
of vsc is that of sliding motion, which occurs when the system
state repeatedly crosses certain subspaces in the state space.
These subspaces are known as sliding hyperplanes, and it is the
design of these hyperplanes which is considered in this thesis.
A popular method of hyperplane design is to specify
eigenvalues in the left-hand half-plane for the reduced order
equivalent system, and to design the control matrix to yield these
eigenvalues. A more general design approach is to specify some
region in the left-hand half-plane within which these eigenvalues
must lie. Four regions are considered in this thesis, namely a
disc, an infinite vertical strip, a sector and a region bounded by
two intersecting sectors.
The methods for placing the closed-loop eigenvalues within
these regions all require the solution of a matrix Riccati
equation : discrete or continuous, real or complex. The choice of
the positive definite symmetric matrices in these Riccati
equations affects the positioning of the eigenvalues within the
region. suitable selection of these matrices will therefore lead
to real or complex eigenvalues, as required, and will influence
their position within the chosen region.
The solution of the hyperplane design problem by a more
general choice of the closed-loop eigenvalues lends itself to the
minimization of the linear part of the control. A suitable choice
of the position of the eigenvalues within the required region
enables either the 2-norm of the linear part of the control, or
the condition number of the linear feedback to be minimized. The
choice of the range space eigenvalues may also be used, more
effectively, in this minimization.