arXiv subjects

Christian Ebenbauer

Publications and source records attributed to Christian Ebenbauer.

At least 19 recordsLinked to original sources

arXiv2026

Lie bracket approximations of RMSprop for Extremum Seeking Control

This paper presents novel extremum seeking algorithms inspired by RMSProp dynamics and based on Lie bracket approximation techniques. To inherit the behavior of the classical continuous-time RMSProp algorithm, we introduce an extremum seeking scheme that exploits second-order Lie brackets, allowing the excitation of squared gradient terms in the resulting Lie bracket system. Moreover, we propose two modified RMSProp-type algorithms that can be implemented using only first-order Lie bracket approximations, thereby reducing the complexity of the extremum seeking control design. The three resulting extremum seeking algorithms are compared through numerical simulations.

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arXiv2026

Stochastic Gradient Descent with Momentum: Analysis and Synthesis via Integral Quadratic Constraints

This article applies dynamic integral quadratic constraints (IQCs) to the analysis and synthesis of accelerated stochastic gradient algorithms. We consider composite objective functions whose gradient can be approximated via mini-batch sampling and we model the resulting stochastic gradient oracle as a feedback nonlinearity in the spirit of Lur'e systems from robust control. Our first main contribution is a family of IQCs that characterize the second-order statistics of mini-batch gradients, extending the classical Zames--Falb multipliers. Our second contribution is a semidefinite-program for certifying exponential convergence rates of stochastic gradient algorithms, and a complementary condition for bounding the asymptotic variance caused by non-vanishing gradient noise. Our third contribution is a convex synthesis procedure that identifies mini-batch gradient algorithms with the smallest certifiable convergence rate.

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arXiv2026

Averaging of Random Vibrations in Mechanical Systems in the Sense of Ito, Stratonovich, and Sussmann

In this paper, we investigate a stochastic averaging principle for a large class of mechanical systems in the presence of random vibrations. We show that a known deterministic averaging principle for affine connection control systems with large-amplitude high-frequency inputs also holds for non-periodic stochastic inputs. The randomly vibrating mechanical system is described by a stochastic differential equation whose solutions do not depend on its interpretation in the sense of Ito, Stratonovich, or Sussmann. We also show that solutions of this stochastic differential equation can be directly computed from a single ordinary differential equation. We illustrate our theoretical results by the example of stochastic source seeking with a nonholonomic vehicle.

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arXiv2026

Global exponential stabilization of a force- and torque-actuated unicycle by flexible-step MPC

We study the problem of global exponential stabilization of a force- and torque-controlled unicycle model in discrete time. To this end, we extend a recently introduced approach to model predictive control (MPC) in which a flexible number of inputs is implemented in every iteration. We present the first flexible-step MPC protocol with state-dependent weights for average descent. Notably, the proposed method relies neither on a suitable design of running or terminal cost functions nor on a suitable choice of terminal constraints. Instead, stability is guaranteed through a generalized discrete-time control Lyapunov function. We establish a new theoretical framework for global exponential stabilization of general nonlinear discrete-time control systems by flexible-step MPC. The obtained results go beyond the unicycle example. However, given the importance of the unicycle dynamics, we make that a focal point of our work. For the particular case of the dynamic (second-order) unicycle model, we show that global exponential stability cannot be attained in the classical sense, but in a slightly weaker sense. The proposed flexible-step MPC method is shown to induce the best possible notion of global exponential stability for this model. We provide explicit rules for the choice of parameters, which guarantee feasibility and global exponential stability. Our numerical simulations show that the discrete MPC method also works very well in applications to a continuous-time torque-actuated unicycle.

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arXiv2026

Global Stability and Step Size Robustness of RMSProp

In this paper, an input-to-state Lyapunov function for the RMSProp optimization algorithm is introduced. Global asymptotic stability of the RMSProp algorithm for constant step sizes and robustness properties with respect to arbitrary bounded time-varying step size rules are established.

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arXiv2026

Extremum Seeking with Double Integrators in the Presence of Local Extrema

We study the problem of global extremum seeking in the presence of local extrema. We investigate two different perturbation-based methods: 1) a well-known classical extremum seeking scheme for steady-state output optimization, and 2) a source seeking scheme for a two-dimensional point mass. In each of these two scenarios, the closed-loop system involves a damped double integrator subject to an oscillatory force. An averaging analysis reveals that the respective averaged system is again a damped double integrator, but now subject to a potential force. The potential force is given by the gradient of a locally averaged objective function. Such a function is less prone to have undesired local extrema and is therefore better suited for global optimization. We provide sufficient conditions for semi-global practical uniform asymptotic stability of the closed-loop systems. The sufficient conditions only involve assumptions on the averaged objective function but not the original one.

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arXiv2026

Non-Local Extremum Seeking Based on the Divergence Theorem

We propose a new design strategy for extremum seeking control for a multi-dimensional single-integrator system in the presence of local extrema. The proposed method employs suitably designed sinusoidal dither signals, which force the single-integrator to a spherical motion. Over time, this spherical motion gives approximate access to an integral of the objective function over a sphere. Using the divergence theorem, we identify the integral over the sphere as the gradient of an integral over the enclosed ball. This integral over the ball defines a locally averaged objective function. The proposed extremum seeking method drives the system state into the gradient direction of the averaged objective function. Such a local average of the objective function can eliminate undesired local extrema and is therefore beneficial for global optimization. Under the assumption that the averaged objective function has no undesired critical points, we prove practical asymptotic stability of the closed-loop system. Our theoretical analysis takes sufficiently small $L_\infty$-measurement errors of the objective function into account.

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arXiv2025

A Poisson Jump-driven SDE Approach to Distributed Gradient Descent with Sparse Communication

To bridge the gap between idealised communication models and the stochastic reality of networked systems, we introduce a framework for embedding asynchronous communication directly into algorithm dynamics using stochastic differential equations (SDE) driven by Poisson Jumps. We apply this communication-aware design to the continuous-time gradient flow, yielding a distributed algorithm where updates occur via sparse Poisson events. Our analysis establishes communication rate bounds for asymptotic stability and, crucially, a higher, yet sparse, rate that provably any desired exponential convergence performance slower than the nominal, centralized flow. These theoretical results, shown for unconstrained quadratic optimisation, are validated by a numerical simulation.

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arXiv2025

Flexible-step MPC for Unknown Linear Time-Invariant Systems

We propose a novel flexible-step model predictive control algorithm for unknown linear time-invariant discrete-time systems. The goal is to asymptotically stabilize the system without relying on a pre-collected dataset that describes its behavior in advance. In particular, we aim to avoid a potentially harmful initial open-loop exploration phase for identification, since full identification is often not necessary for stabilization. Instead, the proposed control scheme explores and learns the unknown system online through measurements of inputs and states. The measurement results are used to update the prediction model in the finite-horizon optimal control problem. If the current prediction model yields an infeasible optimal control problem, then persistently exciting inputs are applied until feasibility is reestablished. The proposed flexible-step approach allows for a flexible number of implemented optimal input values in each iteration, which is beneficial for simultaneous exploration and exploitation. A generalized control Lyapunov function is included into the constraints of the optimal control problem to enforce stability. This way, the problem of optimization is decoupled from the problem of stabilization. For an asymptotically stabilizable unknown control system, we prove that the proposed flexible-step algorithm can lead to global convergence of the system state to the origin.

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arXiv2025

Stochastic Gradient Descent for Constrained Optimization based on Adaptive Relaxed Barrier Functions

This paper presents a novel stochastic gradient descent algorithm for constrained optimization. The proposed algorithm randomly samples constraints and components of the finite sum objective function and relies on a relaxed logarithmic barrier function that is appropriately adapted in each optimization iteration. For a strongly convex objective function and affine inequality constraints, step-size rules and barrier adaptation rules are established that guarantee asymptotic convergence with probability one. The theoretical results in the paper are complemented by numerical studies which highlight potential advantages of the proposed algorithm for optimization problems with a large number of constraints.

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arXiv2024

Time-delay Induced Stochastic Optimization and Extremum Seeking

In this paper a novel stochastic optimization and extremum seeking algorithm is presented, one which is based on time-delayed random perturbations and step size adaptation. For the case of a one-dimensional quadratic unconstrained optimization problem, global exponential convergence in expectation and global exponential practical convergence of the variance of the trajectories are proven. The theoretical results are complemented by numerical simulations for one- and multi-dimensional quadratic and non-quadratic objective functions.

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arXiv2024

Inferring Global Exponential Stability Properties using Lie-bracket Approximations

In the present paper, a novel result for inferring uniform global, not semi-global, exponential stability in the sense of Lyapunov with respect to input-affine systems from global uniform exponential stability properties with respect to their associated Lie-bracket systems is shown. The result is applied to adapt dither frequencies to find a sufficiently high gain in adaptive control of linear unknown systems, and a simple numerical example is simulated to support the theoretical findings.

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arXiv2024

Flexible-step MPC for Switched Linear Systems with No Quadratic Common Lyapunov Function

In this paper, we develop a systematic method for constructing a generalized discrete-time control Lyapunov function for the flexible-step Model Predictive Control (MPC) scheme, recently introduced in [2], when restricted to the class of linear systems. Specifically, we show that a set of Linear Matrix Inequalities (LMIs) can be used for this purpose, demonstrating its tractability. The main consequence of this LMI formulation is that, when combined with flexible-step MPC, we can effectively stabilize switched control systems, for which no quadratic common Lyapunov function exists.

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arXiv2024

Flexible-step Model Predictive Control based on Generalized Lyapunov Functions

We present a novel nonlinear model predictive control (MPC) scheme with relaxed stability criteria, based on the idea of generalized discrete-time control Lyapunov functions. These functions need to satisfy an average descent over a finite window of time, rather than a descent at every time step. One feature of this scheme is that it allows for implementing a flexible number of control inputs in each iteration, in a computationally attractive manner, while guaranteeing recursive feasibility and stability. The benefits of our flexible-step implementation are also demonstrated in an application to nonholonomic systems, where the one-step standard implementation may suffer from lack of asymptotic convergence.

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arXiv2024

On moment relaxations for linear state feedback controller synthesis with non-convex quadratic costs and constraints

We present a simple and effective way to account for non-convex costs and constraints~in~state feedback synthesis, and an interpretation for the variables in which state feedback synthesis is typically convex. We achieve this by deriving the controller design using moment matrices of state and input. It turns out that this approach allows the consideration of non-convex constraints by relaxing them as expectation constraints, and that the variables in which state feedback synthesis is typically convexified can be identified with blocks of these moment matrices.

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arXiv2023

Synthesis of constrained robust feedback policies and model predictive control

In this work, we develop a method based on robust control techniques to synthesize robust time-varying state-feedback policies for finite, infinite, and receding horizon control problems subject to convex quadratic state and input constraints. To ensure constraint satisfaction of our policy, we employ (initial state)-to-peak gain techniques. Based on this idea, we formulate linear matrix inequality conditions, which are simultaneously convex in the parameters of an affine control policy, a Lyapunov function along the trajectory and multiplier variables for the uncertainties in a time-varying linear fractional transformation model. In our experiments this approach is less conservative than standard tube-based robust model predictive control methods.

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arXiv2023

Optimization Algorithm Synthesis based on Integral Quadratic Constraints: A Tutorial

We expose in a tutorial fashion the mechanisms which underlie the synthesis of optimization algorithms based on dynamic integral quadratic constraints. We reveal how these tools from robust control allow to design accelerated gradient descent algorithms with optimal guaranteed convergence rates by solving small-sized convex semi-definite programs. It is shown that this extends to the design of extremum controllers, with the goal to regulate the output of a general linear closed-loop system to the minimum of an objective function. Numerical experiments illustrate that we can not only recover gradient decent and the triple momentum variant of Nesterov's accelerated first order algorithm, but also automatically synthesize optimal algorithms even if the gradient information is passed through non-trivial dynamics, such as time-delays.

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arXiv2023

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.

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