The following five research topics are living. All of them offer fantastic potentials for developing postgraduate research theses and dissertations. Perspective students are encouraged to contact me (xxia@postino.up.ac.za) for further information and discussion.

1) Steady States of Nonlinear Systems

2) $\Delta$-modulated Feedback Control

3) Modelling of HIV Infection

4) Modelling and Control of Heavy-haul Trains

5) Energy Optimisation

Topics 1) and 2) are of control theoretic in nature. To pick up these topics, you should be quite theoretically oriented.

Topic 3) is good for those who do not want to go beyond engineering and medical science. Extra-ordinary opportunities exist for pure as well as practical research.

Topic 4) is industrial-project motivated. A good platform has been laid. There is a need to re-connect with industry to further the project implementation.

Topic 5) is a relative new topic. Major funding is sought after, and great opportunities are offered to those who are keen on relating what is learnt and developed, in the framework of CNES, to meet the national needs.

Some detailed descriptions of each topic follow:

1) Steady States of Nonlinear Systems

We direct our attention to the existence of steady states of
nonlinear systems, with a particular interestin the
case when the exogenous system is unstable. We start with some easy
characterizations of the linear cases. Some sufficient conditions are given for
the existence of steady states of nonlinear systems. Applications of the steady
state theory could be made to nonlinear observer design and the output
regulation of nonlinear systems with unstable exogenous signals. The new
definition of nonlinear steady state is based on the following observation. For
neutrally stable exogenous signals, the center manifold theory can be applied
and a geometric description on the existence of steady state can be formulated
as the controlled invariance and attractiveness of the solution manifold of the
center manifold equation. Such a formulation tells more geometric information
about steady states than the analytic definition which uses the existence of a
suitable input function and an initial state. For unstable exogenous signals
where the center manifold theory is not applicable, the geometric description
is adopted as a definition for nonlinear steady states. As
the same as Isidori and Byrnes¡¯ approach, the
exogenous system has not been assigned any initial value so that a class of
exogenous signals are considered simultaneously. Furthermore, the
obtained definition gives both the local and global version of steady states in
a unified way which enables one to give the existence criteria for local and
global steady states in some unified way too. Examples from forced and unforced
systems are also included to show that the above geometric definition is reasonable
for the corresponding systems.

2) Periodicity in $\Delta$-modulated Feedback Control

$\Delta$-modulated feedback control of a linear system
introduces nonlinearity into the system through switchings
between two input values. It has been found that $\Delta$-modulation gives rise
to periodic orbits. The existence of periodic points of all orders of Sigma-Delta
modulation with ``leaky'' integration is completely characterized by some
interesting groups of polynomials with ``sign'' coefficients. The results are
naturally generalized to

Sigma-Delta modulations with multiple delays, $\Delta$-modulations in the ``downlink'', unbalanced $\Delta$-modulations and systems with two-level quantized feedback. Further extensions relate to the existence of periodic points arising from $\Delta$-modulated feedback control of a stable linear system in an arbitrary direction, for which some necessary and sufficient conditions are given.

3) Modelling of HIV Infection

The research of the pathogenesis of HIV has reached a point
where control system engineering can play a constructive role. The parameter
estimation schemes are also useful for immune prognosis. Anti-retroviral
therapies are usually effective in suppressing the viral load in a short period
of time. The response of immune systems takes a longer time (usually more than
a month) to manifest. In current clinical practice, immune responses are usually
monitored after six months after treatment. It is desirable to predict the
immune system response with less than a month's samples. Preliminary results in
this regard have been done for six naive patients at CHU of Nantes, France, and
further experiments are currently done for more difficult patients.

A large amount of raw data have been
accumulated around the world of HIV patients under HAART. The drug
effectiveness and therapeutical failure models have
to be populated with different HIV subtypes and with different geographic data.
For this purpose, an automated computer routine has the advantage, and is also
under current investigation. The system modelling
ideas can certainly be borrowed to HIV and TB co-infections and other
infectious disease.

One of the challenges from a parameter estimation's viewpoint is limited and poor data. Perhaps the biggest scientific open problem is whether a true cure for HIV/AIDS can eventually be found. If there is a cure, then one thing is certain that it comes as a treatment strategy which includes not only combination of drugs but also the manner in which these drugs are administered. From a practical point of view, affordable and effective means to prevent and stop infection and progression is the key to the success of the fight against AIDS. Applying these ideas to create suitable educational platforms for the high-risk groups of people is also deemed necessary.

4) Modelling and Control of Heavy-haul Trains

A model-based control of heavy haul trains is proposed in terms of optimal scheduling and cruising. A longitudinal dynamical model is set up for an ECP train, and validated against the real data. The model is then employed to reveal the heuristic scheduling procedure currently in use and to design an optimal scheduling scheme. A linearized model and the LQR technique are used to derive an optimal cruising control. With different optimization parameters, the simulation results of different controllers are given. It is shown that the optimal scheduling is meaningful, especially for fully independent control and for long heavy haul trains. Implementation issues of the proposed driving strategies should be pointed out.

5) Energy Optimisation

This is a very recent research interest initiated by the national needs of energy expansion. The following research themes are identified.

(i) Renewable energy;

(ii) Control system approach to energy management;

(iii) Pricing issues of electricity;

(iv) Demand side management;

(v) Unit commitment and deregulated electricity market;

(vi) Distributed generation.

The above research topics involve the subjects of energy systems, control theory, econometrics, optimization, computer science and financial mathematics. Some preliminary results have been obtained. The following are detailed information about some of the research themes, more information about the research problems and references can be accessed in one of the four modules: Advanced Research Topics for postgraduates, which we have developed in CNES. Interested parties can also contact me (xxia@postino.up.ac.za) for further details.

The Control of Hot Water Cylinders

The problems about the control of hot water cylinders(HWC) have attract great attention of ESKOM several
years ago, and some research problems
related to HWC have been financially supported by ESKOM. In fact, the 2006
International Conference on Domestic Use of Energy in

Pricing Problems in Electricity Market

The pricing problems in electricity market include how to find a pricing method which is more reasonable than existing methods, and how to compare existing pricing methods. The existing pricing scheme of ESKOM has been classified into several groups according to the different types of end users. It is worthy to investigate if the existing schemes are reasonable. For example, for residential home users, the price of electricity is fixed and the Time-of-Use tariff is not adopted. Intuitively, this is not reasonable. Therefore it is interesting and important to set up a system which is able to assess if a pricing scheme is reasonable, and in particular, what is reasonable. As for the problem of comparing existing pricing methods, it is well known that the debate of marginal pricing(MP) and pay-as-bid(PAB) has attracted a lot of research work. However most of the comparison of MP and MPB are only based on qualitative descriptions, and some existing results by quantitative descriptions are limited on various hypotheses on the market. It is a good starting point for us to begin our research on electricity market by considering the comparison of MP and PAB under released conditions of the market. Some preliminary result on the comparison of MP and PAB has been obtained.

Other Potential Problems in Electricity Market

Some other problems such as bidding and auction in electricity market, demand side management under capacity subscription, applications of game theory in electricity market, etc, have also the potential to attract financial support. We would identify the details after finishing some of the problems mentioned in the above two research topics.