Advanced Cell Mapping Algorithms For Dynamics,Control And Optimization

The cell mapping methods were originated by Hsu in 1980 s for global analysis of nonlinear dynamical systems that can have multiple steady-state responses including equilibrium states,periodic motions,and chaotic attractors.The cell mapping methods have been applied to deterministic,stochastic and fuzzy dynamical systems.Two important extensions of the cell mapping method have been developed to improve the accuracy of the solutions obtained in the cell state space.The first is the interpolated cell mapping which uses the cell mappings as a foundation to calculate point wise solutions without further numerical integrations of differential equations.The second is the sub-division technique of the set oriented method for improving the accuracy of the discrete solutions obtained with the cell mapping method.For a long time,the cell mapping methods have been applied to dynamical systems with low dimension until now.With the advent of cheap dynamic memory and massively parallel computing technologies such as the graphical processing units(GPUs),global analysis of moderate-to high-dimensional nonlinear dynamical systems becomes feasible.This thesis presents a unified parallel cell mapping framework with sub-division for multi-objective optimization,global analysis of nonlinear dynamical systems,zero finding and stability boundary searching.The parallel cell mapping method is implemented in a GPU that offers massive parallel computing power in a desktop computer.Both the simple cell mapping(SCM)and generalized cell mapping(GCM)are implemented in a hybrid manner.For the applications that requires only the steady state solutions,such as multiobjective optimization,attractor finding and zero locating,the solution process starts with a coarse cell partition to obtain a covering set of the steady-state responses,followed by the sub-division technique of the set-oriented method to enhance the accuracy and resolution of the steady-state responses.When the cells are small enough,no further sub-division is necessary.We propose to treat the solutions obtained by the cell mapping method on a sufficiently fine grid as a database,which provides a basis for the interpolated cell mapping to generate the point-wise approximation of the solutions without additional numerical integrations of differential equations.On the other hand,when global analysis is required,transient states must be considered to acquire the information like attraction domain.A modified analysis flow of global analysis of nonlinear systems with transient states brought in is then developed by taking advantage of parallel computing without sub-division.The analysis of generalized cell mapping takes advantage of graph theory algorithms and the theory of Markov chain.Note the one step cell mapping can be stored effectively in a sparse matrix.Once the transition matrix is set up,all information of the dynamical system,include chaotic attractor,attraction domain,domicile boundary,can be readily acquired by utilizing the recursive graph algorithm which exploits the recursion capability of modern CPU architecture.Plenty of examples on multi-objective optimization,analysis of nonlinear dynamics,zero finding and stability boundary searching are presented in this thesis to show the power of cell mapping algorithms.Multi-objective optimal design of full state feedback controls and PID controls subject to feedback time delay are presented as the application of a simple cell mapping hybrid algorithm.The goal of the design is to minimize several conflicting performance objective functions at the same time.The simple cell mapping method with a hybrid algorithm is used to find the multi-objective optimal design solutions.The multi-objective optimal design comes in a set of gains representing various compromises of the control system.Examples of regulation and tracking controls are presented to validate the control design.For nonlinear dynamical studies,a low dimensional benchmark example is presented in the paper first to illustrate the proposed parallel cell mapping framework.Global analysis of a six-dimensional Lorenz system and a three-dimensional plasma model are then carried out to show the effectiveness of the method on invariant set finding and global analysis respectively.Open questions on high dimensional analysis are discussed such as sampling technique,data visualization,interpolation accuracy and computational configurations.The zero finding algorithm is validated with several examples range from low to high dimensions.It turns out for high dimensional cases,the proposed algorithm can locate the global solutions very fast while still keep the accuracy in cellular space.Followed by the zero finding task,a parallel simple cell mapping based algorithm is proposed to extract stability boundary of potential field.The algorithm is validated with two and three dimensional problems.For a three dimensional problem involves million level cells,the proposed algorithm can accurately extract the stability boundary that has complicated structure in just a few seconds.