arXiv subjects

Mitchell Black

Publications and source records attributed to Mitchell Black.

34 records · Page 2Linked to original sources

arXiv2024

Risk-Aware Fixed-Time Stabilization of Stochastic Systems under Measurement Uncertainty

This paper addresses the problem of risk-aware fixed-time stabilization of a class of uncertain, output-feedback nonlinear systems modeled via stochastic differential equations. First, novel classes of certificate functions, namely risk-aware fixed-time- and risk-aware path-integral-control Lyapunov functions, are introduced. Then, it is shown how the use of either for control design certifies that a system is both stable in probability and probabilistically fixed-time convergent (for a given probability) to a goal set. That is, the system trajectories probabilistically reach the set within a finite time, independent of the initial condition, despite the additional presence of measurement noise. These methods represent an improvement over the state-of-the-art in stochastic fixed-time stabilization, which presently offers bounds on the settling-time function in expectation only. The theoretical results are verified by an empirical study on an illustrative, stochastic, nonlinear system and the proposed controllers are evaluated against an existing method. Finally, the methods are demonstrated via a simulated fixed-wing aerial robot on a reach-avoid scenario to highlight their ability to certify the probability that a system safely reaches its goal.

math.OC↗

arXiv2023

Advances in the Theory of Control Barrier Functions: Addressing Practical Challenges in Safe Control Synthesis for Autonomous and Robotic Systems

This tutorial paper presents recent work of the authors that extends the theory of Control Barrier Functions (CBFs) to address practical challenges in the synthesis of safe controllers for autonomous systems and robots. We present novel CBFs and methods that handle safety constraints (i) with time and input constraints under disturbances, (ii) with high-relative degree under disturbances and input constraints, and (iii) that are affected by adversarial inputs and sampled-data effects. We then present novel CBFs and adaptation methods that prevent loss of validity of the CBF, as well as methods to tune the parameters of the CBF online to reduce conservatism in the system response. We also address the pointwise-only optimal character of CBF-induced control inputs by introducing a CBF formulation that accounts for future trajectories, as well as implementation challenges such as how to preserve safety when using output feedback control and zero-order-hold control. Finally we consider how to synthesize non-smooth CBFs when discontinuous inputs and multiple constraints are present.

math.OC↗

arXiv2023

Safe Control Synthesis for Hybrid Systems through Local Control Barrier Functions

Control Barrier Functions (CBF) have provided a very versatile framework for the synthesis of safe control architectures for a wide class of nonlinear dynamical systems. Typically, CBF-based synthesis approaches apply to systems that exhibit nonlinear -- but smooth -- relationship in the state of the system and linear relationship in the control input. In contrast, the problem of safe control synthesis using CBF for hybrid dynamical systems, i.e., systems which have a discontinuous relationship in the system state, remains largely unexplored. In this work, we build upon the progress on CBF-based control to formulate a theory for safe control synthesis for hybrid dynamical systems. Under the assumption that local CBFs can be synthesized for each mode of operation of the hybrid system, we show how to construct CBF that can guarantee safe switching between modes. The end result is a switching CBF-based controller which provides global safety guarantees. The effectiveness of our proposed approach is demonstrated on two simulation studies.

eess.SY↗

arXiv2023

Fixed-Time Parameter Adaptation for Safe Control Synthesis

This paper introduces a parameter adaptation-based control law for a class of nonlinear, control-affine, safety-critical systems subject to additive, parameter-affine model uncertainty. It is shown that the uncertainty is learned in fixed-time, i.e., within a finite time independent of the initial parameter estimates, without requiring persistence of excitation or rank conditions on the associated regressor matrix. A monotonically decreasing bound on the error of the estimated uncertainty is derived and used in the proposed control barrier function-based control law for guaranteed safety of the system trajectories. It is then proven that the parameter adaptation law is robust to bounded measurement noise and non-parametric perturbations to the system in that the estimated uncertainty still converges to a known neighborhood of the true uncertainty in fixed-time. The advantages of the approach are highlighted in a comparative numerical study, and the method is validated on a quadrotor trajectory-tracking problem subject to safety requirements and an unknown wind field.

math.OC↗

arXiv2023

An Incremental Span-Program-Based Algorithm and the Fine Print of Quantum Topological Data Analysis

We introduce a new quantum algorithm for computing the Betti numbers of a simplicial complex. In contrast to previous quantum algorithms that work by estimating the eigenvalues of the combinatorial Laplacian, our algorithm is an instance of the generic Incremental Algorithm for computing Betti numbers that incrementally adds simplices to the simplicial complex and tests whether or not they create a cycle. In contrast to existing quantum algorithms for computing Betti numbers that work best when the complex has close to the maximal number of simplices, our algorithm works best for sparse complexes. To test whether a simplex creates a cycle, we introduce a quantum span-program algorithm. We show that the query complexity of our span program is parameterized by quantities called the effective resistance and effective capacitance of the boundary of the simplex. Unfortunately, we also prove upper and lower bounds on the effective resistance and capacitance, showing both quantities can be exponentially large with respect to the size of the complex, implying that our algorithm would have to run for exponential time to exactly compute Betti numbers. However, as a corollary to these bounds, we show that the spectral gap of the combinatorial Laplacian can be exponentially small. As the runtime of all previous quantum algorithms for computing Betti numbers are parameterized by the inverse of the spectral gap, our bounds show that all quantum algorithms for computing Betti numbers must run for exponentially long to exactly compute Betti numbers. Finally, we prove some novel formulas for effective resistance and effective capacitance to give intuition for these quantities.

quant-ph↗

arXiv2023

Understanding Oversquashing in GNNs through the Lens of Effective Resistance

Message passing graph neural networks (GNNs) are a popular learning architectures for graph-structured data. However, one problem GNNs experience is oversquashing, where a GNN has difficulty sending information between distant nodes. Understanding and mitigating oversquashing has recently received significant attention from the research community. In this paper, we continue this line of work by analyzing oversquashing through the lens of the effective resistance between nodes in the input graph. Effective resistance intuitively captures the ``strength'' of connection between two nodes by paths in the graph, and has a rich literature spanning many areas of graph theory. We propose to use total effective resistance as a bound of the total amount of oversquashing in a graph and provide theoretical justification for its use. We further develop an algorithm to identify edges to be added to an input graph to minimize the total effective resistance, thereby alleviating oversquashing. We provide empirical evidence of the effectiveness of our total effective resistance based rewiring strategies for improving the performance of GNNs.

cs.LG↗

arXiv2023

Consolidated Control Barrier Functions: Synthesis and Online Verification via Adaptation under Input Constraints

In this paper, we develop a novel adaptation-based approach to constrained control design under multiple state and input constraints. Specifically, we introduce a method for synthesizing any number of time-varying candidate control barrier functions (CBF) into one consolidated CBF (C-CBF) candidate, and propose a predictor-corrector optimization-based adaptation law for the weights of the constituent constraint functions that certifies the C-CBF as valid for a class of nonlinear, control-affine systems. We prove this result by showing that the adapted weights are guaranteed to confer sufficient control authority to meet the new, adaptive C-CBF condition in perpetuity despite input constraints, which thereby permits its use in a quadratic program based control law. We then illustrate the performance of our controller on an academic example, and further highlight that it is successful even for constraint functions with higher or mixed relative-degree by simulating a reach-avoid problem for bicycle robots, which we use to demonstrate how our approach out-performs two baseline approaches.

math.OC↗

arXiv2023

Safety Under Uncertainty: Tight Bounds with Risk-Aware Control Barrier Functions

We propose a novel class of risk-aware control barrier functions (RA-CBFs) for the control of stochastic safety-critical systems. Leveraging a result from the stochastic level-crossing literature, we deviate from the martingale theory that is currently used in stochastic CBF techniques and prove that a RA-CBF based control synthesis confers a tighter upper bound on the probability of the system becoming unsafe within a finite time interval than existing approaches. We highlight the advantages of our proposed approach over the state-of-the-art via a comparative study on an mobile-robot example, and further demonstrate its viability on an autonomous vehicle highway merging problem in dense traffic.

eess.SY↗

arXiv2022

Safe Control Design for Unknown Nonlinear Systems with Koopman-based Fixed-Time Identification

We consider the problem of safe control design for a class of nonlinear, control-affine systems subject to an unknown, additive, nonlinear disturbance. Leveraging recent advancements in the application of Koopman operator theory to the field of system identification and control, we introduce a novel fixed-time identification scheme for the infinitesimal generator of the infinite-dimensional, but notably linear, Koopman dynamical system analogous to the nonlinear system of interest. That is, we derive a parameter adaptation law that allows us to recover the unknown, residual nonlinear dynamics in the system within a finite-time independent of an initial estimate. We then use properties of fixed-time stability to derive an error bound on the residual vector field estimation error as an explicit function of time, which allows us to synthesize a provably safe controller using control barrier function based methods. We conduct a quadrotor-inspired case study in support of our proposed method, in which we show that safe trajectory tracking is achieved despite unknown, nonlinear dynamics.

math.OC↗

arXiv2022

Future-Focused Control Barrier Functions for Autonomous Vehicle Control

In this paper, we introduce a class of future-focused control barrier functions (ff-CBF) aimed at improving traditionally myopic CBF based control design and study their efficacy in the context of an unsignaled four-way intersection crossing problem for collections of both communicating and non-communicating autonomous vehicles. Our novel ff-CBF encodes that vehicles take control actions that avoid collisions predicted under a zero-acceleration policy over an arbitrarily long future time interval. In this sense the ff-CBF defines a virtual barrier, a loosening of which we propose in the form of a relaxed future-focused CBF (rff-CBF) that allows a relaxation of the virtual ff-CBF barrier far from the physical barrier between vehicles. We study the performance of ff-CBF and rff-CBF based controllers on communicating vehicles via a series of simulated trials of the intersection scenario, and in particular highlight how the rff-CBF based controller empirically outperforms a benchmark controller from the literature by improving intersection throughput while preserving safety and feasibility. Finally, we demonstrate our proposed ff-CBF control law on an intersection scenario in the laboratory environment with a collection of 5 non-communicating AION ground rovers.

math.OC↗

arXiv2022

Adaptation for Validation of a Consolidated Control Barrier Function based Control Synthesis

We develop a novel adaptation-based technique for safe control design in the presence of multiple control barrier function (CBF) constraints. Specifically, we introduce an approach for synthesizing any number of candidate CBFs into one consolidated CBF candidate, and propose a parameter adaptation law for the weights of its constituents such that the controllable dynamics of the consolidated CBF are non-vanishing. We then prove that the use of our adaptation law serves to certify the consolidated CBF candidate as valid for a class of nonlinear, control-affine, multi-agent systems, which permits its use in a quadratic program based control law. We highlight the success of our approach in simulation on a multi-robot goal-reaching problem in a crowded warehouse environment, and further demonstrate its efficacy experimentally in the laboratory via AION ground rovers operating amongst other vehicles behaving both aggressively and conservatively.

math.OC↗

arXiv2022

Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes

We describe a nearly-linear time algorithm to solve the linear system $L_1x = b$ parameterized by the first Betti number of the complex, where $L_1$ is the 1-Laplacian of a simplicial complex $K$ that is a subcomplex of a collapsible complex $X$ linearly embedded in $\mathbb{R}^{3}$. Our algorithm generalizes the work of Black et al.~[SODA2022] that solved the same problem but required that $K$ have trivial first homology. Our algorithm works for complexes $K$ with arbitrary first homology with running time that is nearly-linear with respect to the size of the complex and polynomial with respect to the first Betti number. The key to our solver is a new algorithm for computing the Hodge decomposition of 1-chains of $K$ in nearly-linear time. Additionally, our algorithm implies a nearly quadratic solver and nearly quadratic Hodge decomposition for the 1-Laplacian of any simplicial complex $K$ embedded in $\mathbb{R}^{3}$, as $K$ can always be expanded to a collapsible embedded complex of quadratic complexity.

math.NA↗

arXiv2022

Finding minimum bounded and homologous chains in simplicial complexes with bounded-treewidth 1-skeleton

We consider two problems on simplicial complexes: the Optimal Bounded Chain Problem and the Optimal Homologous Chain Problem. The Optimal Bounded Chain Problem asks to find the minimum weight $d$-chain in a simplicial complex $K$ bounded by a given $(d{-}1)$-chain, if such a $d$-chain exists. The Optimal Homologous Chain problem asks to find the minimum weight $(d{-}1)$-chain in $K$ homologous to a given $(d{-}1)$-chain. Both of these problems are NP-hard and hard to approximate within any constant factor assuming the Unique Games Conjecture. We prove that these problems are fixed-parameter tractable with respect to the treewidth of the 1-skeleton of $K$.

cs.CG↗

arXiv2022

ETH-tight algorithms for finding surfaces in simplicial complexes of bounded treewidth

Given a simplicial complex with $n$ simplices, we consider the Connected Subsurface Recognition (c-SR) problem of finding a subcomplex that is homeomorphic to a given connected surface with a fixed boundary. We also study the related Sum-of-Genus Subsurface Recognition (SoG) problem, where we instead search for a surface whose boundary, number of connected components, and total genus are given. For both of these problems, we give parameterized algorithms with respect to the treewidth $k$ of the Hasse diagram that run in $2^{O(k \log k)}n^{O(1)}$ time. For the SoG problem, we also prove that our algorithm is optimal assuming the exponential-time hypothesis. In fact, we prove the stronger result that our algorithm is ETH-tight even without restriction on the total genus.

cs.CG↗

arXiv2020

A Fixed-Time Stable Adaptation Law for Safety-Critical Control under Parametric Uncertainty

We present a novel technique for solving the problem of safe control for a general class of nonlinear, control-affine systems subject to parametric model uncertainty. Invoking Lyapunov analysis and the notion of fixed-time stability (FxTS), we introduce a parameter adaptation law which guarantees convergence of the estimates of unknown parameters in the system dynamics to their true values within a fixed-time independent of the initial parameter estimation error. We then synthesize the adaptation law with a robust, adaptive control barrier function (RaCBF) based quadratic program to compute safe control inputs despite the considered model uncertainty. To corroborate our results, we undertake a comparative case study on the efficacy of this result versus other recent approaches in the literature to safe control under uncertainty, and close by highlighting the value of our method in the context of an automobile overtake scenario.

math.OC↗

arXiv2020

A Quadratic Program based Control Synthesis under Spatiotemporal Constraints and Non-vanishing Disturbances

In this paper, we study the effect of non-vanishing disturbances on the stability of fixed-time stable (FxTS) systems. We present a new result on FxTS, which allows a positive term in the time derivative of the Lyapunov function with the aim to model bounded, non-vanishing disturbances in the system dynamics. We characterize the neighborhood to which the system trajectories converge, as well the time of convergence to this neighborhood, in terms of the positive and negative terms that appear in the time derivative of the Lyapunov function. Then, we use the new FxTS result and formulate a quadratic program (QP) that yields control inputs which drive the trajectories of a class of nonlinear, control-affine systems to a goal set in the presence of control input constraints and non-vanishing, bounded disturbances in the system dynamics. We consider an overtaking problem on a highway as a case study, and discuss how to setup the QP for the considered problem, and how to make a decision on when to start the overtake maneuver, in the presence of sensing errors.

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