This paper presents the Lie-Backlund symmetry method to give the equivalence between differential equations and describe the equivalent transformation procedure of nonlinear control systems of partial differential equations. The equivalent linear systems found by solving the infinitesimal generator of one-parameter Lie groups with prolongations and the infinitesimal generator are used to construct the parameters of invertible mapping u. And the equivalence linear form of the nonlinear system is constructed via u. Some necessary conditions for mapping a nonlinear control system of PDE¡¦s to a linear control system of PDE¡¦s are discussed, and application of Lie-Backlund symmetries and invertible mapping u constructed linear time-invariant control system of partial differential equations.

Thesis (Ph. D.)--Electrical and Computer Engineering, Georgia Institute of Technology, 2004. / Habetler, Thomas, Committee Member ; Sadegh Nader, Committee Member ; Taylor David, Committee Member ; Tsiotras Panagiotis, Committee Co-Chair ; Heck-Ferri Bonnie, Committee Co-Chair. Vita. Includes bibliographical references (leaves 319-326).

The Reaction Wheel Pendulum (RWP) is an interesting nonlinear system. A prototypical control problem for the RWP is to stabilize it around the upright position starting from the bottom, which is generally divided into at least 2 phases: (1) Swing-up phase: where the pendulum is swung up and moves toward the upright position. (2) Stabilization phase: here, the pendulum is controlled to be balanced around the upright position. Previous studies mainly focused on an energy method in swing-up phase and a linearization method in stabilization phase. However, several limitations exist. The energy method in swing-up mode usually takes a long time to approach the upright position. Moreover, its trajectory is not controlled which prevents further extensions. The linearization method in the stabilization phase, can only work for a very small range of angles around the equilibrium point, limiting its applicability. In this thesis, we took the 2nd order state space model and solved it for a constant torque input generating the family of phase-plane trajectories (see Appendix A). Therefore, we are able to plan the motion of the reaction wheel pendulum in the phase plane and a sliding mode controller may be implemented around these trajectories. The control strategy presented here is divided into three phases. (1) In the swing up phase a switching torque controller is designed to oscillate the pendulum until the system’s energy is enough to drive the system to the upright position. Our approach is more generic than previous approaches; (2) In the catching phase a sliding surface is designed in the phase plane based on the zero torque trajectories, and a 2nd order sliding mode controller is implemented to drive the pendulum moving along the sliding surface, which improves the robustness compared to the previous method in which the controller switches to stabilization mode when it reaches a pre-defined region. (3) In the stabilization phase a 2nd order sliding mode integral controller is used to solve the balancing problem, which has the potential to stabilize the pendulum in a larger angular region when compared to the previous linearization methods. At last we combine the 3 phases together in a combined strategy. Both simulation results and experimental results are shown. The control unit is National Instruments CompactRIO 9014 with NI 9505 module for module driving and NI 9411 module for encoding. The Reaction Wheel Pendulum is built by Quanser Consulting Inc. and placed in UT’s Advanced Mechatronics Lab. / text

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In the past decades there has been onsiderable commercial and academic interest in methods for monitoring control system performance for linear systems. Far less has been written on control system performance for nonlinear dynamic / stochastic systems. This thesis presents research results on three control performance monitoring topics for the nonlinear systems: i) Controller assessment of a class of nonlinear systems: The use of autoregressive moving average (ARMA) models to assess the control loop performance for linear systems is well known. Classes of nonlinear dynamic / stochastic systems for which a similar result can be obtained are established for SISO discrete systems. For these systems, the performance lower bounds can be estimated from closed-loop routine operating data using nonlinear autoregressive moving average with exogenous inputs (NARMAX) models. ii) Variance decomposition of nonlinear systems / time series: We develop a variance decomposition approach to quantify the effects of different sources of disturbances on the nonlinear dynamic / stochastic systems. A method, called ANOVA-like decomposition, is employed to achieve this variance decomposition. Modifications of ANOVA-like decomposition are proposed so that the NOVA-like decomposition can be used to deal with the time dependency and the initial condition. iii) Parameter uncertainty effects on the variance decomposition: For the variance decomposition in the second part, the model parameters are assumed to be exactly known. However, parameters of empirical or mechanistic models are uncertain. The uncertainties associated with parameters should be included when the model is used for variance analysis. General solutions of the parameter uncertainty effects on the variance decomposition for the general nonlinear systems are proposed. Analytical solutions of the parameter uncertainty effects on the variance decomposition are provided for models with linear parameters. / Thesis (Ph.D, Chemical Engineering) -- Queen's University, 2007-10-17 16:02:26.376 / This work was sponsored by NSERC Discovery, NSERC Equipment, Shell Global Solutions, OGSST and QGA

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GE-model and equations of state are used to describe and predict phase equilibria. Current models have varying capabilities and some display selectivity for certain special mixtures. While many models are superior to others in their performance, all models share a common deficiency, the inability to simultaneously describe vapour-liquid (VLE) and liquid-liquid equilibria (LLE). Current models require separate parameters to describe the two equilibria. This formed the motivation for a non-linear transformation which was formulated by Rarey (2005). The transformation was applied to the concentration space. The clear advantage of such a transformation was that it could be easily applied to any model. The flexibility of the model was drastically increased. The effects were investigated on the local composition models, in particular the UNIQUAC model resulting in the FlexQUAC model. The model was used to regress a host of VLE and LLE data sets contained in the Dortmund Data Bank (DDB). The transformation had the desired effect on the flexibility of the model and the model was now able to describe VLE and LLE. However a symmetric transformation applied to the concentration space might not be effective in the description of systems exhibiting large difference in molecular size. This is a clear disadvantage of the proposed FlexQUAC model. In order to allow the model to cater to asymmetric systems, the transformation is now applied to the surface fraction of the residual contribution of the UNIQUAC model. The Guggenheim-Staverman expression in the combinatorial part was not transformed. Both the original combinatorial term and the more suitable modification of Weidlich and Gmehling (1987) were used. The newly formed model was called the FlexQUAC-Q model. The development of the FlexQUAC-Q model, derivation of activity coefficient expressions, model implementation and its performance analysis form the basis for this research study. The activity coefficient of the new model had to be re-derived due to the application of the transformation to the residual contribution of the UNIQUAC equation. The computation of the activity coefficient was programmed in FORTRAN and integrated into the regression tool (RECVAL) of the Dortmund Data Bank (DDB). The RECVAL tool was used to regress data sets contained in the DDB. Results obtained were comparable to those obtained using the GEQUAC model. The regression was also performed in EXCEL for the three models (UNIQUAC, FlexQUAC, FlexQUAC-Q). The regression in EXCEL was more rigorous and was used for the comparison of the objective functions and to obtain a set of unique model parameters for each data set. The performance of the FlexQUAC-Q model was assessed utilizing the same data sets used to analyse the performance of the FlexQUAC model. The model's performance was assessed in the regression of 4741 binary VLE data sets, 13 ternary VLE data sets and carefully select ternary LLE cases. The minor mean relative reduction of about 3% of the objective function using FlexQUAC-Q compared to FlexQUAC was observed compared to a reduction by about 53% relative to the UNIQUAC-results. It was necessary to illustrate that the new model does not degenerate the model's existing capabilities (e.g. ability to predict multi-component mixtures from binary data) and that the model performs as well as or superior to the UNIQUAC model. FlexQUAC-Q performed similarly to FlexQUAC. However the improvement in the qualitative description of data sets exhibiting asymmetry is apparent. Herein lies the justification of such a modification and this illustrates the preference of such a model when asymmetric systems are being considered. In addition, the FLEXQUAC-Q model can be adapted to be implemented into a group contribution method, a distinct advantage over the previous model FlexQUAC. The equations for the application of a non-linear transformation to a functional group activity coefficient model, UNIFAC are also explored in this study. The resulting model is referred to as FlexFaC. / Thesis (M.Sc.)-University of KwaZulu-Natal, Durban, 2007.

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