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

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

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.