Accurate Linearization and Control of Non-linear Physical Systems using Increased Variables

先进的控制系统，如自动驾驶汽车、自主机器人和生物反应堆，表现出复杂的行为，不遵循简单的比例规则和线性关系。对许多工业部门的工程师来说，这些非线性系统的控制仍然是一个挑战。这项研究旨在将复杂的非线性系统简化为线性系统，同时保持原有的非线性行为。一旦转化为线性系统，控制设计就变得更容易，计算复杂性也大大降低。这一有利的结果是通过用比我们通常使用的更多的变量来表示系统来实现的。传统的使用有限变量的描述会导致非线性，而这种使用更多变量的新方法允许人们在线性域中处理非线性。这种新的方法可以使具有挑战性的非线性控制问题变得容易处理和简单，并可以为广泛的控制系统和产品创建实用的解决方案。该理论保证了精确线性化，但它不适用于具有主动控制输入的系统，而且为了实际应用，必须将变量的个数截断为有限维系统。此外，原始理论没有说明如何找到额外的变量，称为可观测，以表示提升空间中的非线性系统，在那里系统行为是线性的。在这里，我们将建立一种基于物理建模理论的寻找有效可观测量的系统方法，即键合图建模。在给定系统元件的物理连通性的情况下，定义了一类特殊的可观测对象，称为辅助变量。两个线性状态方程，一个用于状态变量，另一个用于辅助变量，表示提升空间中的非线性动力学。这称为双面(DF)线性化。这些辅助变量具有明确的物理意义，使用这些辅助变量进行线性反馈可以更好地告知控制器，并且性能优于相应的控制器。如果辅助变量或部分辅助变量是物理可测量的，则可以通过数据驱动的子空间方法识别线性模型。DF线性化对于非线性模型预测控制(MPC)特别有用，其中提升空间中的线性模型在有限时间范围内提供了精确的逼近，并将原始的非线性预测控制简化为线性预测控制。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Advanced control systems, such as self-driving cars, autonomous robots, and bio-reactors, exhibit complex behaviors, which do not follow simple proportional rules and linear relationships. Control of those non-linear systems remains a challenge for engineers in many industrial sectors. This research aims to reduce a complex, nonlinear system to a linear system while retaining the original nonlinear behaviors. Once converted to a linear system, control design becomes easier and computational complexity significantly reduces. This advantageous result is made possible by representing the system with more variables than we usually use. While a traditional description using a limited number of variables leads to non-linearity, this new method using more variables allows one to deal with non-linearity in a linear domain. With this new method, challenging non-linear control problems can be made tractable, and simple, and practical solutions may be created for a broad class of control systems and products.According to Koopman, an autonomous, nonlinear dynamical system can be represented as a linear system in an infinite dimensional space. The theory guarantees the exact linearization, but it is not applicable to systems with active control inputs, and the number of variables must be truncated to a finite dimensional system for practical use. Furthermore, the original theory does not state how to find additional variables, called observables, to represent a nonlinear system in the lifted space, where the system behaves linearly. Here, we will establish a systematic way of finding effective observables based on physical modeling theory, namely, Bond Graphs modeling. Given the physical connectivity of system elements, a special class of observables, called auxiliary variables, are defined. Two linear state equations, one for state variables and the other for auxiliary variables, represent the nonlinear dynamics in the lifted space. This is known as Dual-Faceted (DF) Linearization. These auxiliary variables possess clear physical meanings, and use of these auxiliary variables for linear feedback can better inform the controller and outperforms its counterpart. If the auxiliary variables, or part of them, are physically measurable, the linear model can be identified through a data-driven sub-space method. The DF Linearization is particularly useful for nonlinear Model Predictive Control (MPC), where the linear model in the lifted space provides an accurate approximation over a finite time horizon and reduces the original nonlinear MPC to a linear MPC. The optimization problem becomes convex and the computation time drastically reduces.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.