Data-driven nonparametric models gain importance as control systems are increasingly applied in domains where classical system identification is difficult, e.g., because of the system's complexity, sparse training data or its probabilistic nature. Gaussian process state space models (GP-SSM) are a data-driven approach which requires only high-level prior knowledge like smoothness characteristics. Prior known properties like stability are also often available but rarely exploited during modeling. The enforcement of stability using control Lyapunov functions allows to incorporate this prior knowledge, but requires a data-driven Lyapunov function search. Therefore, we propose the use of Sum of Squares to enforce convergence of GP-SSMs and compare the performance to other approaches on a real-world handwriting motion dataset.
When first principle models cannot be derived due to the complexity of the real system, data-driven methods allow us to build models from system observations. As these models are employed in learning-based control, the quality of the data plays a crucial role for the performance of the resulting control law. Nevertheless, there hardly exist measures for assessing training data sets, and the impact of the distribution of the data on the closed-loop system properties is largely unknown. This paper derives -based on Gaussian process models -an analytical relationship between the density of the training data and the control performance. We formulate a quality measure for the data set, which we refer to as ρ-gap, and derive the ultimate bound for the tracking error under consideration of the model uncertainty. We show how the ρ-gap can be applied to a feedback linearizing control law and provide numerical illustrations for our approach.
We propose a novel framework for constructing linear time-invariant (LTI) models for datadriven representations of the Koopman operator for a class of stable nonlinear dynamics. The Koopman operator (generator) lifts a finite-dimensional nonlinear system to a possibly infinite-dimensional linear feature space. To utilize it for modeling, one needs to discover finite-dimensional representations of the Koopman operator. Learning suitable features is challenging, as one needs to learn LTI features that are both Koopman-invariant (evolve linearly under the dynamics) as well as relevant (spanning the original state) -a generally unsupervised learning task. For a theoretically well-founded solution to this problem, we propose learning Koopman-invariant coordinates by composing a diffeomorphic learner with a lifted aggregate system of a latent linear model. Using an unconstrained parameterization of stable matrices along with the aforementioned feature construction, we learn the Koopman operator features without assuming a predefined library of functions or knowing the spectrum, while ensuring stability regardless of the operator approximation accuracy. We demonstrate the superior efficacy of the proposed method in comparison to a state-ofthe-art method on the well-known LASA handwriting dataset.
Determining the region of attraction of nonlinear systems is a difficult problem, which is typically approached by means of Lyapunov theory. State of the art approaches either provide high flexibility regarding the Lyapunov function or parallelizability of computation. Aiming at both, flexibility and parallelizability, we propose a method to obtain a Lyapunov-like function for stability analysis by learning the infinite horizon cost function with a Gaussian process based on approximate dynamic programming. We develop a novel approach to characterize the region of attraction using a Lyapunov-like function, which is analyzed with a sampling-based interval analysis algorithm. Since each interval can be examined independently, the algorithm allows both parallelizable analysis and flexible construction of the Lyapunov-like function.R + /N + all real/integer positive numbers, In the n × n identity matrix, · the Euclidean norm and < ·, · > the scalar product.
Learning-based techniques are increasingly effective at controlling complex systems. However, most work done so far has focused on learning control laws for individual tasks. Simultaneously learning multiple tasks on the same system is still a largely unaddressed research question. In particular, no efficient state space exploration schemes have been designed for multi-task control settings. Using this research gap as our main motivation, we present an algorithm that approximates the smallest data set that needs to be collected in order to achieve high performance across multiple control tasks. By describing system uncertainty using a probabilistic Gaussian process model, we are able to quantify the impact of potentially collected data on each learning-based control law. We then determine the optimal measurement locations by solving a stochastic optimization problem approximately. We show that, under reasonable assumptions, the approximate solution converges towards the exact one. Additionally, we provide a numerical illustration of the proposed algorithm.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.