skip to main content


Search for: All records

Award ID contains: 1932189

Note: When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external site maintained by the publisher. Some full text articles may not yet be available without a charge during the embargo (administrative interval).
What is a DOI Number?

Some links on this page may take you to non-federal websites. Their policies may differ from this site.

  1. Abstract

    We study the problem of fitting a piecewise affine (PWA) function to input–output data. Our algorithm divides the input domain into finitely many regions whose shapes are specified by a user-provided template and such that the input–output data in each region are fit by an affine function within a user-provided error tolerance. We first prove that this problem is NP-hard. Then, we present a top-down algorithmic approach for solving the problem. The algorithm considers subsets of the data points in a systematic manner, trying to fit an affine function for each subset using linear regression. If regression fails on a subset, the algorithm extracts a minimal set of points from the subset (an unsatisfiable core) that is responsible for the failure. The identified core is then used to split the current subset into smaller ones. By combining this top-down scheme with a set-covering algorithm, we derive an overall approach that provides optimal PWA models for a given error tolerance, where optimality refers to minimizing the number of pieces of the PWA model. We demonstrate our approach on three numerical examples that include PWA approximations of a widely used nonlinear insulin–glucose regulation model and a double inverted pendulum with soft contacts.

     
    more » « less
  2. Free, publicly-accessible full text available May 14, 2025
  3. Free, publicly-accessible full text available May 14, 2025
  4. This paper presents a counterexample-guided iterative algorithm to compute convex, piecewise linear (polyhedral) Lyapunov functions for continuous-time piecewise linear systems. Polyhedral Lyapunov functions provide an alternative to commonly used polynomial Lyapunov functions. Our approach first characterizes intrinsic properties of a polyhedral Lyapunov function including its “eccentricity” and “robustness” to perturbations. We then derive an algorithm that either computes a polyhedral Lyapunov function proving that the system is asymptotically stable, or concludes that no polyhedral Lyapunov function exists whose eccentricity and robustness parameters satisfy some user-provided limits. Significantly, our approach places no a-priori bound on the number of linear pieces that make up the desired polyhedral Lyapunov function. The algorithm alternates between a learning step and a verification step, always maintaining a finite set of witness states. The learning step solves a linear program to compute a candidate Lyapunov function compatible with a finite set of witness states. In the verification step, our approach verifies whether the candidate Lyapunov function is a valid Lyapunov function for the system. If verification fails, we obtain a new witness. We prove a theoretical bound on the maximum number of iterations needed by our algorithm. We demonstrate the applicability of the algorithm on numerical examples. 
    more » « less
  5. Assistive robotic devices are a particularly promising field of application for neural networks (NN) due to the need for personalization and hard-to-model human-machine interaction dynamics. However, NN based estimators and controllers may produce potentially unsafe outputs over previously unseen data points. In this paper, we introduce an algorithm for updating NN control policies to satisfy a given set of formal safety constraints, while also optimizing the original loss function. Given a set of mixed-integer linear constraints, we define the NN repair problem as a Mixed Integer Quadratic Program (MIQP). In extensive experiments, we demonstrate the efficacy of our repair method in generating safe policies for a lower-leg prosthesis. 
    more » « less
  6. We present a framework that uses control Lyapunov functions (CLFs) to implement provably stable path-following controllers for autonomous mobile platforms. Our approach is based on learning a guaranteed CLF for path following by using recent approaches — combining machine learning with automated theorem proving — to train a neural network feedback law along with a CLF that guarantees stabilization for driving along low-curvature reference paths. We discuss how key properties of the CLF can be exploited to extend the range of the curvatures for which the stability guarantees remain valid. We then demonstrate that our approach yields a controller that obeys theoretical guarantees in simulation, but also performs well in practice. We show our method is both a verified method of control and better than a common MPC implementation in computation time. Additionally, we implement the controller on-board on a 18 -scale autonomous vehicle testing platform and present results for various robust path following scenarios. 
    more » « less
  7. Koyejo, S ; Mohamed, S ; Agarwal, A ; Belgrave, D ; Cho, K ; Oh, A. (Ed.)
  8. In this paper, we study the “decoding” problem for discrete-time, stochastic hybrid systems with linear dynamics in each mode. Given an output trace of the system, the decoding problem seeks to construct a sequence of modes and states that yield a trace “as close as possible” to the original output trace. The decoding problem generalizes the state estimation problem, and is applicable to hybrid systems with non-determinism. The decoding problem is NP-complete, and can be reduced to solving a mixed-integer linear program (MILP). In this paper, we decompose the decoding problem into two parts: (a) finding a sequence of discrete modes and transitions; and (b) finding corresponding continuous states for the mode/transition sequence. In particular, once a sequence of modes/transitions is fixed, the problem of “filling in” the continuous states is performed by a linear programming problem. In order to support the decomposition, we “cover” the set of all possible mode/transition sequences by a finite subset. We use well-known probabilistic arguments to justify a choice of cover with high confidence and design randomized algorithms for finding such covers. Our approach is demonstrated on a series of benchmarks, wherein we observe that relatively tiny fraction of the possible mode/transition sequences can be used as a cover. Furthermore, we show that the resulting linear programs can be solved rapidly by exploiting the tree structure of the set cover. 
    more » « less