Suitably qualified people will be considered as candidates for PhDs
in the Theory and Applications of Nonlinear Dynamics and Control Theory.
If you have a degree in the areas of Mathematics, Physics or Engineering
and are interested in working in this field, please contact Mario
di Bernardo.
Funding could be available from grant awarding bodies, such as the
EPSRC and the European Union. Departmental bursaries may also be
available. Possible projects include the study of nonlinear phenomena
in systems of relevance in Engineering, the analysis of bifurcations and
chaos in piecewise smooth dynamical systems and more generally the modelling,
analysis and control of complex dynamical systems, i.e.:
This project is concerned with the analysis and control of the statistical properties of a widely used class of power converters, the so-called DC/DC converters [1]. In particular, we shall seek to analyse the stochastic properties of a set of discrete maps derived from these systems [4]. The analysis will be developed by focussing on the average behaviour of voltage and current waveforms associated with power converters operating in chaotic regime. In so doing we shall seek to characterise the probability density functions associated to different converter configurations and the related power spectral densities. The aim will be that of relating these stochastic properties with the parameters of the circuits [3]. This in turn could lead to the interesting possibility of designing the spectral properties of the system under investigation by appropriately controlling its chaotic dynamics. To this end, we shall seek to extend and apply the control approach based on the idea of stabilising an arbitrary invariant measure recently presented in [5]. We anticipate that this will result in the careful analysis of the inverse Frobenius-Perron Problem as outlined in [5].
References
[1] J.B. Deane & D.C. Hamill, "Instability, subharmonics and chaos in power electronics systems", IEEE Trans. on Power Electronics, 5:3, pp. 260-268, July 1990
[2] A. Baranovski, W. Schwarz, A. Moegel, “Statistical Analysis and
Design of Chaotic Switched Dynamical Systems”, Proceeding IEEE ISCAS99,
vol. 5,
pp. 467-470, May 1999
[3] A. Lasota, M.C. Mackey, “Probabilistic properties of determinstic systems”, Cambridge University Press, 1985
[4] M. di Bernardo, F. Vasca, “On Discrete Time Maps for the analysis of bifurcations and Chaos in DC/DC converters”, to appear IEEE Trans. on Circuits and Systems I, July 1999
[5] E. M. Bollt, “On the Inverse Frobenius-Perron Problem: Global Stabilisation
of Arbitrary Invariant Measures”, to appear on International Journal of
Bifurcations and Chaos, 1999
At the department of Engineering Mathematics, we have recently devised new method to control this class of dynamical systems joining the wider research-effort to understand and exploit the occurrence of nonlinear phenomena such as bifurcations and deterministic chaos [2]-[4].
This project is concerned with the analysis and investigation of novel control schemes aimed at exploiting the chaotic nature of the systems involved to achieve given control objectives. Preliminary results show that by using chaos, optimal control strategies for example can be used effectively to ame the dynamics of a fully nonlinear system. Surprisingly this seems to indicate the possibility of improved minimization algorithms, based on Pontryagin's Maximum Principle, to solve effectively the control problems [5].
The research student will be involved with both the analytical and numerical
investigation of these schemes and encouraged to develop new theoretical
control strategies that make explicit use of bifurcations and chaos
exhibited by the systems involved.
Applications to communications, flight dynamics and mechanical engineering
will be considered with the possibility of carrying out experimental investigations
in the Automatic Control Laboratory of the University of Bristol. The work
will be carried out in strict collaboration with coworkers at
the University of Texas at Houston, USA and the University of California
at Berkeley and visits to both these institution will be carried
References
----------
[1] M.di Bernardo and G. Chen in "Controlling Chaos and Bifurcations in Engineering Systems", CRC Press, 1999
[2] D.P. Stoten, M. di Bernardo, "An Application of the Minimal Control Synthesis Algorithm to the Control and Synchronization of Chaos", International Journal of Control, Vol. 6, pp. 925-938, December 1996.
[3] M. di Bernardo, "A purely adaptive controller to synchronize and control chaotic systems",Physics Letters A, May 13, vol. 214, 3/4, p. 139, 1996.
[4] M. di Bernardo, "An Adaptive Approach to the Control and Synchronization of Continuous- Time Chaotic Systems", International Journal of Bifurcation and Chaos, Vol.6, No. 3, pp. 557-568, 1996.
[5] S. Perry, "Analysis and Control of a double inverted pendulum",
M.Eng. Project, University of Bristol, 1999 (preliminary report)
Most of the theory of nonlinear dynamics has focussed on the analysis
of nonlinear phenomena for smooth dynamical systems. In so doing, efficient
algorithms have been proposed to continue their solutions, analyse their
Lyapunov exponents and in general characterise their nonlinear behaviour.
In many applications, though, the occurrence of discontinuous events (such as switchings in electronic circuits or impacts in mechanical devices) takes it necessary to consider nonsmooth or discontinous dynamical systems. Specifically, systems characterised by switching between different configurations whenever certain conditions are satisfied [2]-[4].
Understanding the complex behaviour often exhibited by this class of
dynamical systems is often a cumbersome task. It has been shown,
for example, that nonsmooth system can exhibit novel class of bifurcations
termed border-collisions or grazing which have been observed in real-world
systems [1]. Nevertheless, few algorithms have been proposed to investigate
numerically the behaviour of this type of systems and even fewer have
been actually implemented.
The aim of this project is to study and implement appropriate methods
to continue the solutions of a given piecewise smooth dynamical system,
compute its Lyapunov exponent, classify its bifurcations and
characterise its behaviour. The platform chosen is the powerful software
package MATLAB and the associated toolbox SIMULINK which offer an ideal
tool for numerical investigations of these systems. As part of the
project, specific routines in C or FORTRAN will also haveto be written.
This project is very well suited for students with a keen interest in
Mathematics and Computer Science. It involves both aspects of theory and
computer programming. By the end of the research period, the student
will be confident in an increasingly important area of modern Mathematics.
At the same time he will be an expert in the analysis and characterisation
of discontinuous dynamical systems which are extremely relevant in
applications.
References
[1] M. di Bernardo, M.I. Feigin, S.J. Hogan, M.E. Homer, "Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems", Chaos, Solitons and Fractals, vol. 10, no. 11, pp. 1881-1908, 1999.
[2] G.M. Maggio, M. di Bernardo, M.P. Kennedy, "Nonsmooth bifurcations in a piecewise linear model of the Colpitts Oscillator", IEEE Transactions on Circuits and Systems Part I (submitted) 1999.
[3] M. di Bernardo, F. Vasca, "On Discrete Time Maps for the analysis of bifurcations and Chaos in DC/DC converters", IEEE Transactions on Circuits and Systems Part I (to appear).
[4] M. di Bernardo, C.J. Budd, A.R. Champneys, "Grazing, Skipping and Sliding: analysis of the non-smooth dynamics of the DC/DC buck converter", Nonlinearity, vol. 11, no. 4, pp. 858-890, July 1998.
(see here for some other examples)