arXiv subjects

Christian Ebenbauer

Publications and source records attributed to Christian Ebenbauer.

At least 37 records · Page 2Linked to original sources

arXiv2023

A structure exploiting SDP solver for robust controller synthesis

In this paper, we revisit structure exploiting SDP solvers dedicated to the solution of Kalman-Yakubovic-Popov semi-definite programs (KYP-SDPs). These SDPs inherit their name from the KYP Lemma and they play a crucial role in e.g. robustness analysis, robust state feedback synthesis, and robust estimator synthesis for uncertain dynamical systems. Off-the-shelve SDP solvers require $O(n^6)$ arithmetic operations per Newton step to solve this class of problems, where $n$ is the state dimension of the dynamical system under consideration. Specialized solvers reduce this complexity to $O(n^3)$. However, existing specialized solvers do not include semi-definite constraints on the Lyapunov matrix, which is necessary for controller synthesis. In this paper, we show how to include such constraints in structure exploiting KYP-SDP solvers.

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arXiv2022

Fast identification and stabilization of unknown linear systems

In the present work, a simple algorithm for stabilizing an unknown linear time-invariant system is proposed, assuming only that this system is stabilizable. The suggested algorithm is based on first performing a partial identification of the system and then stabilizing the controllable subsystem. It should be emphasized that our approach does not depend on any prior model knowledge and requires only a minimal number of samples.

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arXiv2022

Robust Differential Dynamic Programming

Differential Dynamic Programming is an optimal control technique often used for trajectory generation. Many variations of this algorithm have been developed in the literature, including algorithms for stochastic dynamics or state and input constraints. In this contribution, we develop a robust version of Differential Dynamic Programming that uses generalized plants and multiplier relaxations for uncertainties. To this end, we study a version of the Bellman principle and use convex relaxations to account for uncertainties in the dynamic program. The resulting algorithm can be seen as a robust trajectory generation tool for nonlinear systems.

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arXiv2022

Extremum Seeking with Intermittent Measurements: A Lie-brackets Approach

Extremum seeking systems are powerful methods able to steer the input of a (dynamical) cost function towards an optimizer, without any prior knowledge of the cost function. To achieve their objective, they typically combine time-periodic signals with the on-line measurement of the cost. However, in some practical applications, the cost can only be measured during some regular time-intervals, and not continuously, contravening the classical extremum seeking framework. In this paper, we first analyze how existing Lie-bracket based extremum seeking systems behave when being fed with intermittent measurements, instead of continuous ones. We then propose two modifications of those schemes to improve both the convergence time and the steady-state accuracy in presence of intermittent measurements. The performances of the different schemes are compared on a case study.

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arXiv2021

Gradient Approximation and Multi-Variable Derivative-Free Optimization based on Non-Commutative Maps

In this work, multi-variable derivative-free optimization algorithms for unconstrained optimization problems are developed. A novel procedure for approximating the gradient of multi-variable objective functions based on non-commutative maps is introduced. The procedure is based on the construction of an exploration sequence to specify where the objective function is evaluated and the definition of so-called gradient generating functions which are composed with the objective function such that the procedure mimics a gradient descent algorithm. Various theoretical properties of the proposed class of algorithms are investigated and numerical examples are presented.

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arXiv2021

Anytime Proximity Moving Horizon Estimation: Stability and Regret

In this paper, we address the efficient implementation of moving horizon state estimation of constrained discrete-time linear systems. We propose a novel iteration scheme which employs a proximity-based formulation of the underlying optimization algorithm and reduces computational effort by performing only a limited number of optimization iterations each time a new measurement is received. We outline conditions under which global exponential stability of the underlying estimation errors is ensured. Performance guarantees of the iteration scheme in terms of regret upper bounds are also established. A combined result shows that both exponential stability and a sublinear regret which can be rendered smaller by increasing the number of optimization iterations can be guaranteed. The stability and regret results of the proposed estimator are showcased through numerical simulations.

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arXiv2021

A Note on Nussbaum-type Control and Lie-bracket Approximation

In this paper, we propose an adaptive control law for completely unknown scalar linear systems based on Lie-bracket approximation methods. We investigate stability and convergence properties for the resulting Lie-bracket system, compare our proposal with existing Nussbaum-type solutions and demonstrate our results with an example. Even though we prove global stability properties of the Lie-bracket system, the stability properties of the proposed dynamics remain open, making the proposed control law an object of further studies. We elaborate the difficulties of establishing stability results by investigating connections to partial stability as well as studying the corresponding Chen-Fliess expansion.

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arXiv2021

Convex Synthesis of Accelerated Gradient Algorithms

We present a convex solution for the design of generalized accelerated gradient algorithms for strongly convex objective functions with Lipschitz continuous gradients. We utilize integral quadratic constraints and the Youla parameterization from robust control theory to formulate a solution of the algorithm design problem as a convex semi-definite program. We establish explicit formulas for the optimal convergence rates and extend the proposed synthesis solution to extremum control problems.

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arXiv2020

Control of Unknown (Linear) Systems with Receding Horizon Learning

A receding horizon learning scheme is proposed to transfer the state of a discrete-time dynamical control system to zero without the need of a system model. Global state convergence to zero is proved for the class of stabilizable and detectable linear time-invariant systems, assuming that only input and output data is available and an upper bound of the state dimension is known. The proposed scheme consists of a receding horizon control scheme and a proximity-based estimation scheme to estimate and control the closed-loop trajectory. Simulations are presented for linear and nonlinear systems.

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arXiv2020

Convex Synthesis of Accelerated Gradient Algorithms for Optimization and Saddle Point Problems using Lyapunov functions

This paper considers the problem of designing accelerated gradient-based algorithms for optimization and saddle-point problems. The class of objective functions is defined by a generalized sector condition. This class of functions contains strongly convex functions with Lipschitz gradients but also non-convex functions, which allows not only to address optimization problems but also saddle-point problems. The proposed design procedure relies on a suitable class of Lyapunov functions and on convex semi-definite programming. The proposed synthesis allows the design of algorithms that reach the performance of state-of-the-art accelerated gradient methods and beyond.

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arXiv2020

Online learning with stability guarantees: A memory-based real-time model predictive controller

We propose and analyze a real-time model predictive control (MPC) scheme that utilizes stored data to improve its performance by learning the value function online with stability guarantees. For linear and nonlinear systems, a learning method is presented that makes use of basic analytic properties of the cost function and is proven to learn the MPC control law and the value function on the limit set of the closed-loop state trajectory. The main idea is to generate a smart warm start based on historical data that improves future data points and thus future warm starts. We show that these warm starts are asymptotically exact and converge to the solution of the MPC optimization problem. Thereby, the suboptimality of the applied control input resulting from the real-time requirements vanishes over time. Simulative examples show that existing real-time MPC schemes can be improved by storing data and the proposed learning scheme.

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arXiv2019

Robust and structure exploiting optimization algorithms: An integral quadratic constraint approach

We consider the problem of analyzing and designing gradient-based discrete-time optimization algorithms for a class of unconstrained optimization problems having strongly convex objective functions with Lipschitz continuous gradient. By formulating the problem as a robustness analysis problem and making use of a suitable adaptation of the theory of integral quadratic constraints, we establish a framework that allows to analyze convergence rates and robustness properties of existing algorithms and enables the design of novel robust optimization algorithms with prespecified guarantees capable of exploiting additional structure in the objective function.

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arXiv2019

A Lie bracket approximation approach to distributed optimization over directed graphs

We consider a group of computation units trying to cooperatively solve a distributed optimization problem with shared linear equality and inequality constraints. Assuming that the computation units are communicating over a network whose topology is described by a time-invariant directed graph, by combining saddle-point dynamics with Lie bracket approximation techniques we derive a methodology that allows to design distributed continuous-time optimization algorithms that solve this problem under minimal assumptions on the graph topology as well as on the structure of the constraints. We discuss several extensions as well as special cases in which the proposed procedure becomes particularly simple.

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arXiv2018

Characterizing the learning dynamics in extremum seeking

We consider perturbation-based extremum seeking, which recovers an approximate gradient of an analytically unknown objective function through measurements. Using classical needle variation analysis, we are able to explicitly quantify the recovered gradient in the scalar case. We reveal that it corresponds to an averaged gradient of the objective function, even for very general extremum seeking systems. From this, we create a recursion which represents the learning dynamics along the recovered gradient. These results give rise to the interpretation that extremum seeking actually optimizes a function other than the original one. From this insight, a new perspective on global optimization of functions with local extrema emerges: because the gradient is averaged over a certain time period, local extrema might be evened out in the learning dynamics. Moreover, a multidimensional extension of the scalar results is given.

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arXiv2018

Derivative-Free Optimization Algorithms based on Non-Commutative Maps

A novel class of derivative-free optimization algorithms is developed. The main idea is to utilize certain non-commutative maps in order to approximate the gradient of the objective function. Convergence properties of the novel algorithms are established and simulation examples are presented.

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arXiv2018

On the Lie bracket approximation approach to distributed optimization: Extensions and limitations

We consider the problem of solving a smooth convex optimization problem with equality and inequality constraints in a distributed fashion. Assuming that we have a group of agents available capable of communicating over a communication network described by a time-invariant directed graph, we derive distributed continuous-time agent dynamics that ensure convergence to a neighborhood of the optimal solution of the optimization problem. Following the ideas introduced in our previous work, we combine saddle-point dynamics with Lie bracket approximation techniques. While the methodology was previously limited to linear constraints and objective functions given by a sum of strictly convex separable functions, we extend these result here and show that it applies to a very general class of optimization problems under mild assumptions on the communication topology.

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arXiv2018

A family of extremum seeking laws for a unicycle model with a moving target: theoretical and experimental studies

In this paper, we propose and practically evaluate a class of gradient-free control functions ensuring the motion of a unicycle-type system towards the extremum point of a time-varying cost function. We prove that the unicycle is able to track the extremum point, and illustrate our results by numerical simulations and experiments that show that the proposed control functions exhibit an improved tracking performance in comparison to standard extremum seeking laws based on Lie bracket approximations.

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arXiv2017

Iterative Learning and Extremum Seeking for Repetitive Time-Varying Mappings

In this paper, we develop an extremum seeking control method integrated with iterative learning control to track a time-varying optimizer within finite time. The behavior of the extremum seeking system is analyzed via an approximating system - the modified Lie bracket system. The modified Lie bracket system is essentially an online integral-type iterative learning control law. The paper contributes to two fields, namely, iterative learning control and extremum seeking. First, an online integral type iterative learning control with a forgetting factor is proposed. Its convergence is analyzed via $k$-dependent (iteration- dependent) contraction mapping in a Banach space equipped with $λ$-norm. Second, the iterative learning extremum seeking system can be regarded as an iterative learning control with "control input disturbance." The tracking error of its modified Lie bracket system can be shown uniformly bounded in terms of iterations by selecting a sufficiently large frequency. Furthermore, it is shown that the tracking error will finally converge to a set, which is a $λ$-norm ball. Its center is the same with the limit solution of its corresponding "disturbance-free" system (the iterative learning control law); and its radius can be controlled by the frequency.

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