Sliding mode control is a well-known approach to the problem of the control of uncertain
systems, since it is invariant to a class of parameter variations. Well-established investigations
have shown that the sliding mode controller/ observer is a good approach from
the point of view of robustness, implementation, numerical stability, applicability, ease of
design tuning and overall evaluation.
In the sliding mode control approach, the controller and/ or observer is designed so
that the state trajectory converges to a surface named the sliding surface. It is desired to
design the sliding surface so that the system stability is achieved.
Many new methods and design techniques for the sliding controller/ observer are
presented in this thesis.
LQ frequency shaping sliding mode is a way of designing the sliding surface and
control. By using this method, corresponding to the weighting functions in conventional
quadratic performance, a compensator can be designed.
The intention of observer design is to find an estimate for the state and, if the input
is unknown, estimate a suitable input. Using the sliding control input form, a suitable
estimated input can be obtained. The significance of the observer design method in this
thesis is that a discontinuous observer for full order systems, including disturbance inputs,
is designed. The system may not be ideally in the sliding mode and the uncertainty may
not satisfy the matching condition.
In discrete-time systems instead of having a hyperplane as in the continuous case,
there is a countable set of points comprising a so-called lattice; and the surface on which
these sliding points lie is named the latticewise hyperplane. Control and observer design
using the discrete-time sliding mode, the robust stability of the sliding mode dynamics
and the problem of stabilization of discrete-time systems are also studied.
The sliding mode control of time-delay systems is also considered. Time-delay sliding
system stability is studied for the cases of full information about the delay and also lack of
information. The sliding surface is delay-independent as for the traditional sliding surface,
and the reaching condition is achieved by applying conventional discontinuous control.
A well-known method of control design is to specify eigenvalues in a region in the
left-hand half-plane, and to design the gain feedback matrix to yield these eigenvalues.
This method can also be used to design the sliding gain matrix. The regions considered in
this thesis are, a sector, an infinite vertical strip, a disc, a hyperbola and the intersection
ii
of two sectors. Previous erroneous results are rectified and new theory developed.
The complex Riccati equation, positivity of a complex matrix and the control of
complex systems are significant problems which arise in many control theory problems
and are discussed in this thesis.