Data-based Control of Feedback Linearizable Systems

We present an extension of Willems' Fundamental Lemma to the class of multi-input multi-output discrete-time feedback linearizable nonlinear systems. We study the effect of approximating the unknown nonlinearities with a choice of basis functions that depend only on input and output data, then provide error bounds on the results of the data-based simulation and output matching control problems. Furthermore, we use this data-based approximation of the trajectories of the nonlinear system to design a multi-step robust data-based nonlinear predictive control scheme. We show that this control scheme is recursively feasible and renders the closed-loop system practically exponentially stable. Finally, we illustrate our results on a model of a fully-actuated double inverted pendulum.Comment: submitted to IEEE Transactions on Automatic Contro

Data-driven Nonlinear Predictive Control for Feedback Linearizable Systems

We present a data-driven nonlinear predictive control approach for the class of discrete-time multi-input multi-output feedback linearizable nonlinear systems. The scheme uses a non-parametric predictive model based only on input and noisy output data along with a set of basis functions that approximate the unknown nonlinearities. Despite the noisy output data as well as the mismatch caused by the use of basis functions, we show that the proposed multistep robust data-driven nonlinear predictive control scheme is recursively feasible and renders the closed-loop system practically exponentially stable. We illustrate our results on a model of a fully-actuated double inverted pendulum.Comment: accepted to IFAC World Congress 2023. arXiv admin note: substantial text overlap with arXiv:2204.0114

Model-Based Control Using Koopman Operators

This paper explores the application of Koopman operator theory to the control of robotic systems. The operator is introduced as a method to generate data-driven models that have utility for model-based control methods. We then motivate the use of the Koopman operator towards augmenting model-based control. Specifically, we illustrate how the operator can be used to obtain a linearizable data-driven model for an unknown dynamical process that is useful for model-based control synthesis. Simulated results show that with increasing complexity in the choice of the basis functions, a closed-loop controller is able to invert and stabilize a cart- and VTOL-pendulum systems. Furthermore, the specification of the basis function are shown to be of importance when generating a Koopman operator for specific robotic systems. Experimental results with the Sphero SPRK robot explore the utility of the Koopman operator in a reduced state representation setting where increased complexity in the basis function improve open- and closed-loop controller performance in various terrains, including sand.Comment: 8 page

Feedback and Partial Feedback Linearization of Nonlinear Systems: A Tribute to the Elders

Arthur Krener and Roger Brockett pioneered the feedback linearization problem for control systems, that is, the transforming of a nonlinear control system into linear dynamics via change of coordinates and feedback. While the former gave necessary and sufficient conditions to linearize a system under change of coordinates only, the latter introduced the concept of feedback and solved the problem for a particular case. Their work was soon extended in the earlier eighties by Jakubczyk and Responder, and Hunt and Su who gave the conditions for a control system to be linearizable by change of coordinates and feedback (full rank and involutivity of the associated distributions). It turned out that those conditions are very restrictive; however, it was showed later that systems that fail to be linearizable can still be transformed into two interconnected subsystems: one linear and the other nonlinear. This fact is known as partial feedback linearization. For input-output systems with well-defined relative degree, coordinates can be found by differentiating the outputs. For systems without outputs, necessary and sufficient geometric conditions for partial linearization have been obtained in terms of the Lie algebra of the system; however, both results of linearization and partial feedback linearization lack practicability. Until recently, none has provided a way to actually compute the linearizing coordinates and feedback. In this paper, we propose an algorithm allowing to find the linearizing coordinates and feedback if the system is linearizable, and in the contrary, to decompose a system (without outputs) while achieving the largest linear subsystem. Those algorithms are built upon successive applications of the Frobenius theorem. Examples are provided to illustrate