arXiv subjects

Andrew R. Teel

Publications and source records attributed to Andrew R. Teel.

At least 19 recordsLinked to original sources

arXiv2026

On robustness, input-to-state stability and backstepping for stochastic differential equations

We study conditions under which stability of the origin of stochastic differential equations is robust to small perturbations. We express robustness in two ways, firstly in the sense that stochastic stability is maintained under small parametric perturbations not exceeding a state-dependent bound vanishing at the origin but positive elsewhere, and secondly via stochastic input-to-state stability (ISS) which allows non-zero perturbations everywhere. We prove the former property assuming the existence of a Lyapunov function certifying stochastic stability of the nominal system. Under the same assumption, stochastic ISS holds under a suitable state-dependent perturbation scaling. Stochastic exponential stability is maintained under proportionally bounded perturbations and implies exponential ISS even without perturbation scaling. Finally, we propose a novel approach to stochastic integrator backstepping in pure-feedback form that uses the tools from our robustness analysis.

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arXiv2026

Chain transitivity in generalized hybrid dynamics with application to simulation and stochastic approximation of hybrid systems

Asymptotic properties of discrete, stochastic approximations to hybrid systems, modeled as hybrid inclusions, are studied. First, the internal chain transitivity of omega-limits of solutions is concluded, along with other properties related to chain recurrence and transitivity. A concept of an asymptotic solution is proposed to describe any mapping that, asymptotically, resembles a solution, and for which the chain transitivity properties also turn out to hold. The mentioned developments are carried out in an abstract setting of a generalized hybrid system defined by a set of hybrid curves, each defined on a hybrid time domain, and possibly consisting of all solutions to a given hybrid inclusion. Then, more specific kinds of perturbed solutions to a hybrid inclusion are proposed and shown to include the solutions of a discretization and of a stochastic approximation to the hybrid inclusion. Consequently, appropriate discretizations and stochastic approximations of a hybrid inclusion produce mappings whose omega limits are internally chain transitive for the underlying hybrid inclusion.

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arXiv2026

Two-Timescale Asymptotic Simulations of Hybrid Inclusions with Applications to Stochastic Hybrid Optimization

Convergence properties of model-free two-timescale asymptotic simulations of singularly perturbed hybrid inclusions are developed. A hybrid inclusion combines constrained differential and difference inclusions to capture continuous (flow) and discrete (jump) dynamics, respectively. Sufficient conditions are established under which sequences of iterates and step sizes constitute a two-timescale asymptotic simulation of such a system, with limiting behavior characterized via weakly invariant and internally chain-transitive sets of an associated boundary layer and reduced system. To illustrate the applicability of these results, conditions are given under which a two-timescale stochastic approximation of a hybrid optimization algorithm asymptotically recovers the behavior of its deterministic counterpart.

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arXiv2025

Hybrid Set-Seeking Systems: Model-Free Feedback Optimization via Hybrid Inclusions

This article aims to provide an accessible, tutorial-style introduction to hybrid extremum-seeking systems, which are model-free, feedback-optimization controllers that incorporate hybrid dynamics, meaning both continuous-time and discrete-time behaviors. Such systems arise when advanced control and optimization tools are needed to overcome the limitations of smooth feedback methods and to satisfy demanding transient and steady-state requirements in high-performance applications. They also appear when controllers must operate on plants that inherently exhibit hybrid behaviors, as is common in cyber-physical and autonomous systems that rely on digital sensing, computation, and actuation. To study hybrid extremum-seeking dynamics through control-theoretic methods, we first review the key concepts that support the development of perturbation theory for hybrid inclusions, forming the basis for averaging and singular perturbation analyses. We then show how these ideas apply to the design and evaluation of hybrid extremum-seeking algorithms for static and dynamic plants. Several examples are presented, including set-valued and switching algorithms under different switching regimes such as arbitrarily fast switching, dwell-time and average dwell-time constraints, and average activation time conditions. We also discuss state-based switching extremum seeking for obstacle-avoidance problems and gradient-Newton switching schemes. Additional topics include momentum-based and reset-type extremum seeking, intermittent updates, slowly varying parameters, hybrid filters, and safety-aware schemes that incorporate constraints. Across all these settings, we illustrate how perturbation-based methods traditionally used for extremum-seeking control naturally extend to hybrid systems when mild regularity assumptions are satisfied, and solutions are modeled on hybrid time domains.

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arXiv2025

Data-Driven Control of Continuous-Time LTI Systems via Non-Minimal Realizations

This article proposes an approach to design output-feedback controllers for unknown continuous-time linear time-invariant systems using only input-output data from a single experiment. To address the lack of state and derivative measurements, we introduce non-minimal realizations whose states can be observed by filtering the available data. We first apply this concept to the disturbance-free case, formulating linear matrix inequalities (LMIs) from batches of sampled signals to design a dynamic, filter-based stabilizing controller. The framework is then extended to the problem of asymptotic tracking and disturbance rejection - in short, output regulation - by incorporating an internal model based on prior knowledge of the disturbance/reference frequencies. Finally, we discuss tuning strategies for a class of multi-input multi-output systems and illustrate the method via numerical examples.

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arXiv2025

Robust Recurrence of Discrete-Time Infinite-Horizon Stochastic Optimal Control with Discounted Cost

We analyze the stability of general nonlinear discrete-time stochastic systems controlled by optimal inputs that minimize an infinite-horizon discounted cost. Under a novel stochastic formulation of cost-controllability and detectability assumptions inspired by the related literature on deterministic systems, we prove that uniform semi-global practical recurrence holds for the closed-loop system, where the adjustable parameter is the discount factor. Under additional continuity assumptions, we further prove that this property is robust.

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arXiv2024

Derivative-Free Data-Driven Control of Continuous-Time Linear Time-Invariant Systems

This paper develops a data-driven stabilization method for continuous-time linear time-invariant systems with theoretical guarantees and no need for signal derivatives. The framework, based on linear matrix inequalities (LMIs), is illustrated in the state-feedback and single-input single-output output-feedback scenarios. Similar to discrete-time approaches, we rely solely on input and state/output measurements. To avoid differentiation, we employ low-pass filters of the available signals that, rather than approximating the derivatives, reconstruct a non-minimal realization of the plant. With access to the filter states and their derivatives, we can solve LMIs derived from sample batches of the available signals to compute a dynamic controller that stabilizes the plant. The effectiveness of the framework is showcased through numerical examples.

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arXiv2024

Emulation-based Stabilization for Networked Control Systems with Stochastic Channels

This paper studies the stabilization problem of networked control systems (NCSs) with random packet dropouts caused by stochastic channels. To describe the effects of stochastic channels on the information transmission, the transmission times are assumed to be deterministic, whereas the packet transmission is assumed to be random. We first propose a stochastic scheduling protocol to model random packet dropouts, and address the properties of the proposed stochastic scheduling protocol. The proposed scheduling protocol provides a unified modelling framework for a general class of random packet dropouts due to different stochastic channels. Next, the proposed scheduling protocol is embedded into the closed-loop system, which leads to a stochastic hybrid model for NCSs with random packet dropouts. Based on this stochastic hybrid model, we follow the emulation approach to establish sufficient conditions to guarantee uniform global asymptotical stability in probability. In particular, an upper bound on the maximally allowable transmission interval is derived explicitly for all stochastic protocols satisfying Lyapunov conditions that guarantee uniform global asymptotic stability in probability. Finally, two numerical examples are presented to demonstrate the derived results.

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arXiv2022

Concurrent learning in high-order tuners for parameter identification

High-order tuners are algorithms that show promise in achieving greater efficiency than classic gradient-based algorithms in identifying the parameters of parametric models and/or in facilitating the progress of a control or optimization algorithm whose adaptive behavior relies on such models. For high-order tuners, robust stability properties, namely uniform global asymptotic (and exponential) stability, currently rely on a persistent excitation (PE) condition. In this work, we establish such stability properties with a novel analysis based on a Matrosov theorem and then show that the PE requirement can be relaxed via a concurrent learning technique driven by sampled data points that are sufficiently rich. We show numerically that concurrent learning may greatly improve efficiency. We incorporate reset methods that preserve the stability guarantees while providing additional improvements that may be relevant in applications that demand highly accurate parameter estimates at relatively low additional cost in computation.

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arXiv2021

Passive soft-reset controllers for nonlinear systems

Soft-reset controllers are introduced as a way to approximate hard-reset controllers. The focus is on implementing reset controllers that are (strictly) passive and on analyzing their interconnection with passive plants. A passive hard-reset controller that has a strongly convex energy function can be approximated as a soft-reset controller. A hard-reset controller is a hybrid system whereas a soft-reset controller corresponds to a differential inclusion, living entirely in the continuous-time domain. This feature may make soft-reset controllers easier to understand and implement. A soft-reset controller contains a parameter that can be adjusted to better approximate the action of the hard-reset controller. Closed-loop asymptotic stability is established for the interconnection of a passive soft-reset controller with a passive plant, under appropriate detectability assumptions. Several examples are used to illustrate the efficacy of soft-reset controllers.

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arXiv2021

On Moment Matching for Stochastic Systems

In this paper we study the problem of model reduction by moment matching for stochastic systems. We characterize the mathematical object which generalizes the notion of moment to stochastic differential equations and we find a class of models which achieve moment matching. However, differently from the deterministic case, these reduced-order models cannot be considered "simpler" because of the high computational cost paid to determine the moment. To overcome this difficulty, we relax the moment matching problem in two different ways and we present two classes of reduced-order models which, approximately matching the stochastic moment, are computationally tractable.

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arXiv2021

Analyzing the Effect of Persistent Asset Switches on a Class of Hybrid-Inspired Optimization Algorithms

Convex optimization challenges are currently pervasive in many science and engineering domains. In many applications of convex optimization, such as those involving multi-agent systems and resource allocation, the objective function can persistently switch during the execution of an optimization algorithm. Motivated by such applications, we analyze the effect of persistently switching objectives in continuous-time optimization algorithms. In particular, we take advantage of existing robust stability results for switched systems with distinct equilibria and extend these results to systems described by differential inclusions, making the results applicable to recent optimization algorithms that employ differential inclusions for improving efficiency and/or robustness. Within the framework of hybrid systems theory, we provide an accurate characterization, in terms of Omega-limit sets, of the set to which the optimization dynamics converge. Finally, by considering the switching signal to be constrained in its average dwell time, we establish semi-global practical asymptotic stability of these sets with respect to the dwell-time parameter.

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arXiv2021

Hybrid Heavy-Ball Systems: Reset Methods for Optimization with Uncertainty

Momentum methods for convex optimization often rely on precise choices of algorithmic parameters, based on knowledge of problem parameters, in order to achieve fast convergence, as well as to prevent oscillations that could severely restrict applications of these algorithms to cyber-physical systems. To address these issues, we propose two dynamical systems, named the Hybrid Heavy-Ball System and Hybrid-inspired Heavy-Ball System, which employ a feedback mechanism for driving the momentum state toward zero whenever it points in undesired directions. We describe the relationship between the proposed systems and their discrete-time counterparts, deriving conditions based on linear matrix inequalities for ensuring exponential rates in both continuous time and discrete time. We provide numerical LMI results to illustrate the effects of our reset mechanisms on convergence rates in a setting that simulates uncertainty of problem parameters. Finally, we numerically demonstrate the efficiency and avoidance of oscillations of the proposed systems when solving both strongly convex and non-strongly convex problems.

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arXiv2020

Further results on synergistic Lyapunov functions and hybrid feedback design through backstepping

We extend results on backstepping hybrid feedbacks by exploiting synergistic Lyapunov function and feedback (SLFF) pairs in a generalized form. Compared to existing results, we delineate SLFF pairs that are ready-made and do not require extra dynamic variables for backstepping. From an (weak) SLFF pair for an affine control system, we construct an SLFF pair for an extended system where the control input is produced through an integrator. The resulting hybrid feedback asymptotically stabilizes the extended system when the synergy gap for the original system is strictly positive. To highlight the versatility of SLFF pairs, we provide a result on the existence of a SLFF pair whenever a hybrid feedback stabilizer exists. The results are illustrated on the 3D pendulum.

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arXiv2019

On reduction of differential inclusions and Lyapunov stability

In this paper, locally Lipschitz, regular functions are utilized to identify and remove infeasible directions from set-valued maps that define differential inclusions. The resulting reduced set-valued map is point-wise smaller (in the sense of set containment) than the original set-valued map. The corresponding reduced differential inclusion, defined by the reduced set-valued map, is utilized to develop a generalized notion of a derivative for locally Lipschitz candidate Lyapunov functions in the direction(s) of a set-valued map. The developed generalized derivative yields less conservative statements of Lyapunov stability theorems, invariance theorems, invariance-like results, and Matrosov theorems for differential inclusions. Included illustrative examples demonstrate the utility of the developed theory.

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arXiv2017

Invariance-like results for Nonautonomous Switched Systems

This paper generalizes the Lasalle-Yoshizawa Theorem to switched nonsmooth systems. Filippov and Krasovskii regularizations of a switched system are shown to be contained within the convex hull of the Filippov and Krasovskii regularizations of the subsystems, respectively. A candidate common Lyapunov function that has a negative semidefinite derivative along the trajectories of the subsystems is shown to be sufficient to establish LaSalle-Yoshizawa results for the switched system. Results for regular and non-regular candidate Lyapunov functions are presented using an appropriate generalization of the time derivative. The developed generalization is motivated by adaptive control of switched systems where the derivative of the candidate Lyapunov function is typically negative semidefinite.

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arXiv2017

Hybrid systems with memory: Existence and well-posedness of generalized solutions

Hybrid systems with memory refer to dynamical systems exhibiting both hybrid and delay phenomena. While systems of this type are frequently encountered in many physical and engineering systems, particularly in control applications, various issues centered around the robustness of hybrid delay systems have not been adequately dealt with. In this paper, we establish some basic results on a framework that allows to study hybrid systems with memory through generalized concepts of solutions. In particular, we develop the basic existence of generalized solutions using regularity conditions on the hybrid data, which are formulated in a phase space of hybrid trajectories equipped with the graphical convergence topology. In contrast with the uniform convergence topology that has been often used, adopting the graphical convergence topology allows us to establish well-posedness of hybrid systems with memory. We then show that, as a consequence of well-posedness, pre-asymptotic stability of well-posed hybrid systems with memory is robust.

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arXiv2016

Global asymptotic stability of a PID control system with Coulomb friction

We propose a model for representing a point mass subject to Coulomb friction in feedback with a PID controller, based on a differential inclusion comprising all the possible magnitudes of static friction during the stick phase. For this model we study the set of all equilibria and we establish its global asymptotic stability using a discontinuous Lyapunov-like function, and a suitable LaSalle's invariance principle. We finally use well-posedness of the proposed model to establish useful robustness results, including an ISS property from a suitable input in a perturbed context. Simulation results are also given to illustrate our statements.

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