Collision Avoidance Safety Filter for an Autonomous E-Scooter using Ultrasonic Sensors

In this paper, we propose a collision avoidance safety filter for autonomous electric scooters to enable safe operation of such vehicles in pedestrian areas. In particular, we employ multiple low-cost ultrasonic sensors to detect a wide range of possible obstacles in front of the e-scooter. Based on possibly faulty distance measurements, we design a filter to mitigate measurement noise and missing values as well as a gain-scheduled controller to limit the velocity commanded to the e-scooter when required due to imminent collisions. The proposed controller structure is able to prevent collisions with unknown obstacles by deploying a reduced safe velocity ensuring a sufficiently large safety distance. The collision avoidance approach is designed such that it may be easily deployed in similar applications of general micromobility vehicles. The effectiveness of our proposed safety filter is demonstrated in real-world experiments.

SafEDMD: A certified learning architecture tailored to data-driven control of nonlinear dynamical systems

The Koopman operator serves as the theoretical backbone for machine learning of dynamical control systems, where the operator is heuristically approximated by extended dynamic mode decomposition (EDMD). In this paper, we propose Stability- and certificate-oriented EDMD (SafEDMD): a novel EDMD-based learning architecture which comes along with rigorous certificates, resulting in a reliable surrogate model generated in a data-driven fashion. To ensure trustworthiness of SafEDMD, we derive proportional error bounds, which vanish at the origin and are tailored for control tasks, leading to certified controller design based on semi-definite programming. We illustrate the developed machinery by means of several benchmark examples and highlight the advantages over state-of-the-art methods.

Novel Quadratic Constraints for Extending LipSDP beyond Slope-Restricted Activations

Recently, semidefinite programming (SDP) techniques have shown great promise in providing accurate Lipschitz bounds for neural networks. Specifically, the LipSDP approach (Fazlyab et al., 2019) has received much attention and provides the least conservative Lipschitz upper bounds that can be computed with polynomial time guarantees. However, one main restriction of LipSDP is that its formulation requires the activation functions to be slope-restricted on $[0,1]$, preventing its further use for more general activation functions such as GroupSort, MaxMin, and Householder. One can rewrite MaxMin activations for example as residual ReLU networks. However, a direct application of LipSDP to the resultant residual ReLU networks is conservative and even fails in recovering the well-known fact that the MaxMin activation is 1-Lipschitz. Our paper bridges this gap and extends LipSDP beyond slope-restricted activation functions. To this end, we provide novel quadratic constraints for GroupSort, MaxMin, and Householder activations via leveraging their underlying properties such as sum preservation. Our proposed analysis is general and provides a unified approach for estimating $\ell_2$ and $\ell_\infty$ Lipschitz bounds for a rich class of neural network architectures, including non-residual and residual neural networks and implicit models, with GroupSort, MaxMin, and Householder activations. Finally, we illustrate the utility of our approach with a variety of experiments and show that our proposed SDPs generate less conservative Lipschitz bounds in comparison to existing approaches.

Lipschitz-bounded 1D convolutional neural networks using the Cayley transform and the controllability Gramian

We establish a layer-wise parameterization for 1D convolutional neural networks (CNNs) with built-in end-to-end robustness guarantees. Herein, we use the Lipschitz constant of the input-output mapping characterized by a CNN as a robustness measure. We base our parameterization on the Cayley transform that parameterizes orthogonal matrices and the controllability Gramian for the state space representation of the convolutional layers. The proposed parameterization by design fulfills linear matrix inequalities that are sufficient for Lipschitz continuity of the CNN, which further enables unconstrained training of Lipschitz-bounded 1D CNNs. Finally, we train Lipschitz-bounded 1D CNNs for the classification of heart arrythmia data and show their improved robustness.

Convolutional Neural Networks as 2-D systems

This paper introduces a novel representation of convolutional Neural Networks (CNNs) in terms of 2-D dynamical systems. To this end, the usual description of convolutional layers with convolution kernels, i.e., the impulse responses of linear filters, is realized in state space as a linear time-invariant 2-D system. The overall convolutional Neural Network composed of convolutional layers and nonlinear activation functions is then viewed as a 2-D version of a Lur'e system, i.e., a linear dynamical system interconnected with static nonlinear components. One benefit of this 2-D Lur'e system perspective on CNNs is that we can use robust control theory much more efficiently for Lipschitz constant estimation than previously possible.

Lipschitz constant estimation for 1D convolutional neural networks

In this work, we propose a dissipativity-based method for Lipschitz constant estimation of 1D convolutional neural networks (CNNs). In particular, we analyze the dissipativity properties of convolutional, pooling, and fully connected layers making use of incremental quadratic constraints for nonlinear activation functions and pooling operations. The Lipschitz constant of the concatenation of these mappings is then estimated by solving a semidefinite program which we derive from dissipativity theory. To make our method as efficient as possible, we take the structure of convolutional layers into account realizing these finite impulse response filters as causal dynamical systems in state space and carrying out the dissipativity analysis for the state space realizations. The examples we provide show that our Lipschitz bounds are advantageous in terms of accuracy and scalability.

Neural network training under semidefinite constraints

This paper is concerned with the training of neural networks (NNs) under semidefinite constraints. This type of training problems has recently gained popularity since semidefinite constraints can be used to verify interesting properties for NNs that include, e.g., the estimation of an upper bound on the Lipschitz constant, which relates to the robustness of an NN, or the stability of dynamic systems with NN controllers. The utilized semidefinite constraints are based on sector constraints satisfied by the underlying activation functions. Unfortunately, one of the biggest bottlenecks of these new results is the required computational effort for incorporating the semidefinite constraints into the training of NNs which is limiting their scalability to large NNs. We address this challenge by developing interior point methods for NN training that we implement using barrier functions for semidefinite constraints. In order to efficiently compute the gradients of the barrier terms, we exploit the structure of the semidefinite constraints. In experiments, we demonstrate the superior efficiency of our training method over previous approaches, which allows us, e.g., to use semidefinite constraints in the training of Wasserstein generative adversarial networks, where the discriminator must satisfy a Lipschitz condition.

Data-Driven Reachability Analysis from Noisy Data

We consider the problem of computing reachable sets directly from noisy data without a given system model. Several reachability algorithms are presented, and their accuracy is shown to depend on the underlying system generating the data. First, an algorithm for computing over-approximated reachable sets based on matrix zonotopes is proposed for linear systems. Constrained matrix zonotopes are introduced to provide less conservative reachable sets at the cost of increased computational expenses and utilized to incorporate prior knowledge about the unknown system model. Then we extend the approach to polynomial systems and under the assumption of Lipschitz continuity to nonlinear systems. Theoretical guarantees are given for these algorithms in that they give a proper over-approximative reachable set containing the true reachable set. Multiple numerical examples show the applicability of the introduced algorithms, and accuracy comparisons are made between algorithms.

Linear systems with neural network nonlinearities: Improved stability analysis via acausal Zames-Falb multipliers

In this paper, we analyze the stability of feedback interconnections of a linear time-invariant system with a neural network nonlinearity in discrete time. Our analysis is based on abstracting neural networks using integral quadratic constraints (IQCs), exploiting the sector-bounded and slope-restricted structure of the underlying activation functions. In contrast to existing approaches, we leverage the full potential of dynamic IQCs to describe the nonlinear activation functions in a less conservative fashion. To be precise, we consider multipliers based on the full-block Yakubovich / circle criterion in combination with acausal Zames-Falb multipliers, leading to linear matrix inequality based stability certificates. Our approach provides a flexible and versatile framework for stability analysis of feedback interconnections with neural network nonlinearities, allowing to trade off computational efficiency and conservatism. Finally, we provide numerical examples that demonstrate the applicability of the proposed framework and the achievable improvements over previous approaches.

Offset-free setpoint tracking using neural network controllers

In this paper, we present a method to analyze local and global stability in offset-free setpoint tracking using neural network controllers and we provide ellipsoidal inner approximations of the corresponding region of attraction. We consider a feedback interconnection using a neural network controller in connection with an integrator, which allows for offset-free tracking of a desired piecewise constant reference that enters the controller as an external input. The feedback interconnection considered in this paper allows for general configurations of the neural network controller that include the special cases of output error and state feedback. Exploiting the fact that activation functions used in neural networks are slope-restricted, we derive linear matrix inequalities to verify stability using Lyapunov theory. After stating a global stability result, we present less conservative local stability conditions (i) for a given reference and (ii) for any reference from a certain set. The latter result even enables guaranteed tracking under setpoint changes using a reference governor which can lead to a significant increase of the region of attraction. Finally, we demonstrate the applicability of our analysis by verifying stability and offset-free tracking of a neural network controller that was trained to stabilize an inverted pendulum.