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Robust reduced order control for nonlinear distributed systems of Burgers class.

机译:Burgers类非线性分布式系统的鲁棒降阶控制。

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In the presented application of control engineering to fluid flow dynamics, the basal and primal motivation arises from laminar flow control in aerodynamics. The governing physical model is identified as the Navier-Stokes equations. These equations are reviewed and derived in a general approach so that the presented techniques and results can be related to a variety of continuity problems. The key issues associated with this class of problems, distributed systems governed by nonlinear partial differential equations, are identified, and a mathematical benchmark problem reflecting those properties (the Burgers equation with periodic boundary conditions and a non-homogeneous forcing term is created. Thereby, Burgers' equation is linked by several means to distributed nonlinear systems: as an approximation of a two-dimensional channel flow problem; as the decisive factor in the creation of turbulence (original motivation); and as a modeling equation for traffic flow.;The benchmark problem is stated and classified as a continuous partial differential equation subject to periodic boundary conditions (of first and second order) and a non-homogeneous distributed forcing term (the control input). This setting has not yet been adequately addressed in previous research. A viscosity parameter kappa being an analogue to the inverse Reynolds number is incorporated. In the following, the viscous problem is solved analytically for a certain class of initial conditions while the general solution to the inviscid case, as well as the steady-state solution, are derived without limitations on the initial condition.;In order to provide a suitable formulation for control engineering, a semi-discretization (in space) is performed using a Galerkin finite element method. This results in a large-scale ODE system which is tested for consistency with the previously derived analytical solution. The resulting state-space formulation is expanded to an unprecedented 'real world' control loop design, including process disturbance, measurement noise, model-error, and model-reduction. Also, the benchmark problem is analyzed in control terms for stability and controllability. Thereby, a Lyaponuv based proof of exponential stability of the origin, for certain initial conditions, can be established. The exponential rate of decay is identified as being only dependent on the viscosity parameter kappa. It can be shown that a feedback law with positive semi-definite gain even improves exponential stability. An argumentation toward the controllability of constant equilibria, previously identified as being the only steady-state solutions, is made. For the nominal control, as well as for the required estimator, the linear quadratic regulator and the extended Kalman filter are briefly reviewed and derived for the benchmark problem. Additionally, model-error control synthesis is introduced in its one-step ahead prediction formulation for nonlinear distributed systems. This provides a computationally fast correction to cope with model-error and process disturbances. The implementation in previous research is briefly presented while a modification for application in this work is suggested. The derived and introduced techniques are subject to extensive numerical evaluation, and detailed results are given. Thereby, the combination of the linear quadratic regulator with model-error control synthesis reveals itself to be a powerful control tool, resulting in a fast attenuation of an initial distribution as well as a robust correction of process disturbance. Results hold in face of noisy measurements (additive white Gaussian noise) if the extended Kalman filter is added to the system. Due to the differentiating character of the predictive filter, the model-error correction has to be reprocessed when a reduced-order model is applied. Then, the results prove to be as powerful as in the full-order model case. A comprehensive Matlab code is provided.;The problem is approached from a 'worst case' point of view, where the applied disturbance and noise by far exceeds 'real world' dimensions. The reduced-order model is based on a coarse truncation of a linear global Galerkin finite element method; even better results are expected if refined techniques are applied. A brief review of previous research on PDE problems in control engineering, as well as a detailed reference to publications on Burgers' equation, is presented. Especially, results are compared to previous work at the Virginia Polytechnic Institute and State University. An outlook on future research and topics to be addressed concludes this thesis.
机译:在控制工程对流体流动动力学的提出的应用中,基础和主要动机来自空气动力学中的层流控制。控制物理模型被标识为Navier-Stokes方程。这些方程式可以通过一般方法进行审查和推导,从而使所提出的技术和结果可以与各种连续性问题相关。确定了与此类问题相关的关键问题,由非线性偏微分方程控制的分布式系统,并反映了反映这些特性的数学基准问题(创建了具有周期性边界条件和非齐次强迫项的Burgers方程。), Burgers方程通过多种方式与分布式非线性系统相关联:作为二维通道流动问题的近似;作为产生湍流(原始动力)的决定性因素;作为交通流的建模方程。基准问题被陈述并归类为服从周期性边界条件(一阶和二阶)和非均匀分布强迫项(控制输入)的连续偏微分方程,此设置在先前的研究中尚未得到充分解决。包含一个与雷诺数倒数类似的粘度参数kappa。通过分析解决一类初始条件的粘性问题,同时导出无粘性情况的一般解以及稳态解,而不受初始条件的限制。;为了提供合适的控制公式在工程中,使用Galerkin有限元方法进行半离散化(在空间中)。这导致了大规模ODE系统的测试,该系统与先前导出的分析解决方案的一致性。由此产生的状态空间公式扩展到了前所未有的“现实世界”控制回路设计,包括过程干扰,测量噪声,模型误差和模型简化。此外,还从控制方面分析了基准问题的稳定性和可控性。因此,对于某些初始条件,可以建立基于Lyaponuv的原点指数稳定性证明。指数衰减率被确定为仅取决于粘度参数κ。可以证明,具有正半定增益的反馈定律甚至可以提高指数稳定性。对先前被确定为唯一的稳态解的恒定平衡的可控性提出了争论。对于标称控制以及所需的估计器,对线性二次调节器和扩展卡尔曼滤波器进行了简要回顾,并得出了基准问题。此外,模型误差控制综合在非线性分布式系统的一步式预测公式中得到了介绍。这提供了计算上的快速校正以应对模型误差和过程干扰。简要介绍了以前的研究中的实现,同时建议对本工作中的应用进行修改。派生和引入的技术都经过广泛的数值评估,并给出了详细的结果。因此,线性二次调节器与模型误差控制综合的结合显示出自己是一个强大的控制工具,从而导致初始分布的快速衰减以及过程干扰的鲁棒校正。如果将扩展的卡尔曼滤波器添加到系统中,结果将面临嘈杂的测量结果(加性高斯白噪声)。由于预测滤波器的微分特性,当应用降阶模型时,必须重新处理模型误差校正。然后,结果证明像在全阶模型中一样强大。提供了全面的Matlab代码。;从“最坏情况”的角度解决了该问题,其中所施加的干扰和噪声远远超过了“现实世界”的维度。降阶模型基于线性全局Galerkin有限元方法的粗略截断;如果采用改进的技术,则有望获得更好的结果。简要回顾了先前对控制工程中PDE问题的研究,并提供了有关Burgers方程的出版物的详细参考。尤其是,将结果与弗吉尼亚理工学院和州立大学的先前工作进行了比较。本文对未来的研究前景和主题进行了展望。

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