Control of spatio-temporal pattern formation governed by geometrical models of interface evolution
2014 European Control Conference (ECC)
Numerous natural phenomena are characterized by spatio-temporal dynamics which give rise to time evolving spatial patterns. Although studies that address the problem of modelling these complex dynamics exist, a model based control approach for such systems is still a challenging task. The work in this thesis is concerned with the development of control methods for such spatio-temporal systems, where interface growth is represented using a geometric evolution law. In particular, the focus is set
... on the control of dendritic crystal growth and wind aided wildfire spread. In the first section of the thesis several modeling approaches used to describe spatio-temporal systems are discussed and partial differential equation models are selected as a suitable class of models for describing the system dynamics. Following this, the complex nonlinear dynamics found in dendritic crystal growth and wind aided wildfire evolution are analyzed through the scope of geometrical partial differential equation models of interface evolution. In the next part of the thesis, a new method for controlling the shape of evolving interfaces, such as the boundaries of dendritic crystals, described by a temperature-dependent geometrical evolution law, where the normal velocity of the growing interface depends on both local curvature and temperature, is presented. A study of the control method in both two and three dimensions is provided, using both linear and non-linear interface dynamics, to demonstrate the effectiveness of the proposed method. Following this, a novel model-based approach for controlling the evolving complex-shaped fire front using a level-set formulation of the model is presented. The work shows how key influencing parameters such as terrain topography and fuel loading, extracted directly from satellite imaging data, can be incorporated in the model. Simulation results show the numerical method is computationally efficient and examples are given with both synthetic and real data to illustrate the robustness of the containment method against changing wind speed, fuel loading and terrain topography. Based on the developments made in the previous chapters, a new optimization method for controlling wildfire evolution is presented. Using a level set formulation of the evolving interface, a cost function is derived that allows the selection of a spatial location, such that the firefighting action has the best results with respect to a predefined target (minimum fire area increase etc.). Several tuning parameters are incorporated within the optimization algorithm, enabling a wide range of fire combat scenarios to be implemented. Numerical simulations provided demonstrate the effectiveness of the method.