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

自动驾驶汽车、自主机器人、生物反应器等先进的控制系统表现出复杂的行为，不遵循简单的比例规则和线性关系。在许多工业领域，控制这些非线性系统仍然是工程师面临的一个挑战。本研究旨在将复杂的非线性系统简化为线性系统，同时保留其原有的非线性行为。一旦转换为线性系统，控制设计变得更加容易，计算复杂度大大降低。这种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而实现的。传统的使用有限数量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内处理非线性。通过这种新方法，具有挑战性的非线性控制问题可以变得易于处理，并且可以为广泛的控制系统和产品创建简单实用的解决方案。根据库普曼的理论，一个自治的非线性动力系统可以表示为一个无限维空间中的线性系统。该理论保证了精确的线性化，但不适用于具有主动控制输入的系统，并且为了实际使用，必须将变量的数量截断为有限维系统。此外，原始理论并没有说明如何找到额外的变量，称为可观测值，以表示提升空间中的非线性系统，其中系统表现为线性。在这里，我们将建立一种基于物理建模理论的系统的寻找有效可观测物的方法，即键图建模。给定系统元素的物理连通性，定义了一类特殊的可观察对象，称为辅助变量。两个线性状态方程，一个表示状态变量，另一个表示辅助变量，表示提升空间中的非线性动力学。这就是所谓的双面（DF）线性化。这些辅助变量具有明确的物理意义，并且将这些辅助变量用于线性反馈可以更好地告知控制器并优于其对应变量。如果辅助变量或辅助变量的一部分在物理上是可测量的，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提供了有限时间范围内的精确近似值，并将原始非线性MPC减少为线性MPC。优化问题变得凸化，计算时间大大减少。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.