arXiv subjects

Julian Berberich

Publications and source records attributed to Julian Berberich.

At least 19 recordsLinked to original sources

arXiv2026

Koopman meets input-output data: Data-driven output-feedback control of nonlinear systems with closed-loop guarantees

Data-driven control of nonlinear systems from input-output measurements remains a fundamental challenge, as existing approaches with rigorous closed-loop guarantees predominantly require access to full state measurements. In this paper, we address this gap by proposing a data-driven output-feedback controller design method for nonlinear systems that provides provable closed-loop guarantees while operating solely on measured input-output data. Our approach combines Koopman operator theory with an extended state representation of the nonlinear system constructed from input-output trajectories. This allows us to obtain a bilinear surrogate model directly from data, on which robust state-feedback design methods can be applied. By exploiting the observability of the underlying nonlinear system, we establish exponential stability of the extended state, which in turn implies exponential convergence of the original system state to the origin. Finally, we validate our theoretical findings in numerical simulations.

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arXiv2026

Data-Driven Min-Max MPC with Integral Quadratic Constraints

Data-driven control of nonlinear systems with rigorous guarantees is a challenging control problem. Integral quadratic constraints (IQCs) provide a powerful framework for modeling nonlinearities. This paper presents a data-driven min-max model predictive control (MPC) synthesis method for unknown systems subject to (nonlinear) uncertainties using the IQC framework. The unknown system matrices are characterized by a set-membership representation using the input-state data and the knowledge of the IQCs. We derive two semidefinite programs (SDPs) that minimize an upper bound on the worst-case cost over all possible system dynamics and uncertainties. By iteratively solving these SDPs, the proposed state-feedback control law is obtained. We further prove that the resulting closed-loop system is exponentially stable and satisfies the input and state constraints. A numerical example demonstrates the validity of the proposed method.

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arXiv2026

A Koopman Set-Membership Approach for Nonlinear Data-Driven Control with Stability Guarantees

This paper proposes a data-driven controller design method for unknown nonlinear systems based on a Koopman bilinear realization. Using Koopman operator theory, the nonlinear system can be represented as a bilinear discrete-time system with a residual error term. The residual error is proportionally bounded by the norm of the lifted state and input, while the system matrices of the bilinear model are unknown. Assuming that bounds on the residual error are available, the unknown system matrices are characterized via a set-membership representation using the collected input-state data pairs of the nonlinear system. A data-driven controller design method is proposed to ensure stability for all bilinear systems within this set-membership description and for all admissible residual errors. More specifically, we design a rational state-feedback controller that stabilizes the bilinear model with residual error and, consequently, the original nonlinear system, by solving a sum-of-squares (SOS) program. The effectiveness of the proposed approach is demonstrated through numerical examples.

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arXiv2026

Adaptive Data-Driven Min-Max MPC for Linear Time-Varying Systems

In this paper, we propose an adaptive data-driven min-max model predictive control (MPC) scheme for discrete-time linear time-varying (LTV) systems. We assume that prior knowledge of the system dynamics and bounds on the variations are known, and that the states are measured online. Starting from an initial state-feedback gain derived from prior knowledge, the algorithm updates the state-feedback gain using online input-state data. To this end, a semidefinite program (SDP) is solved to minimize an upper bound on the infinite-horizon optimal cost and to derive a corresponding state-feedback gain. We prove that the resulting closed-loop system is exponentially stabilized and satisfies the constraints. Further, we extend the proposed scheme to LTV systems with process noise. The resulting closed-loop system is shown to be robustly stabilized to a robust positive invariant (RPI) set. Finally, the proposed methods are demonstrated by numerical simulations.

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arXiv2026

Noise Resilience and Robust Convergence Guarantees for the Variational Quantum Eigensolver

Variational Quantum Algorithms (VQAs) are a class of hybrid quantum-classical algorithms that leverage on classical optimization tools to find the optimal parameters for a parameterized quantum circuit. One relevant application of VQAs is the Variational Quantum Eigensolver (VQE), which aims at steering the output of the quantum circuit to the ground state of a certain Hamiltonian. Recent works have provided global convergence guarantees for VQEs under suitable local surjectivity and smoothness hypotheses, but little has been done in characterizing convergence of these algorithms when the underlying quantum circuit is affected by noise. In this work, we derive an upper bound on the error on the optimal parameters of a VQE under the effect of different coherent and incoherent noise processes. We then procced to show robust convergence guarantees of the algorithm to the perturbed optimal parameters. Our work provides novel theoretical insight into the behavior of VQAs subject to noise. Furthermore, we accompany our results with numerical simulations implemented via Pennylane.

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arXiv2025

Robustness of quantum algorithms: Worst-case fidelity bounds and implications for design

Errors occurring on noisy hardware pose a key challenge to reliable quantum computing. Existing techniques such as error correction, mitigation, or suppression typically separate the error handling from the algorithm analysis and design. In this paper, we develop an alternative, algorithm-centered framework for understanding and improving the robustness against errors. For a given quantum algorithm and error model, we derive worst-case fidelity bounds which can be efficiently computed to certify the robustness. We consider general error models including coherent and (Markovian) incoherent errors and allowing for set-based error descriptions to address uncertainty or time-dependence in the errors. Our results give rise to guidelines for robust algorithm design and compilation by optimizing our theoretical robustness measure. We demonstrate the practicality of the framework with numerical results on algorithm analysis and robust optimization, including the robustness analysis of a 50-qubit modular adder circuit.

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arXiv2025

Model predictive quantum control: A modular approach for efficient and robust quantum optimal control

Model predictive control (MPC) is one of the most successful modern control methods. It relies on repeatedly solving a finite-horizon optimal control problem and applying the beginning piece of the optimal input. In this paper, we develop a modular framework for improving efficiency and robustness of quantum optimal control (QOC) via MPC. We first provide a tutorial introduction to basic concepts of MPC from a QOC perspective. We then present multiple MPC schemes, ranging from simple approaches to more sophisticated schemes which admit stability guarantees. This yields a modular framework which can be used 1) to improve efficiency of open-loop QOC and 2) to improve robustness of closed-loop quantum control by incorporating feedback. We demonstrate these benefits with numerical results, where we benchmark the proposed methods against competing approaches.

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arXiv2025

An overview of Koopman-based control: From error bounds to closed-loop guarantees

Controlling nonlinear dynamical systems remains a central challenge in a wide range of applications, particularly when accurate first-principle models are unavailable. Data-driven approaches offer a promising alternative by designing controllers directly from observed trajectories. A wide range of data-driven methods relies on the Koopman-operator framework that enables linear representations of nonlinear dynamics via lifting into higher-dimensional observable spaces. Finite-dimensional approximations, such as extended dynamic mode decomposition (EDMD) and its controlled variants, make prediction and feedback control tractable but introduce approximation errors that must be accounted for to provide rigorous closed-loop guarantees. This survey provides a systematic overview of Koopman-based control, emphasizing the connection between data-driven surrogate models, approximation errors, controller design, and closed-loop guarantees. We review theoretical foundations, error bounds, and both linear and bilinear EDMD-based control schemes, highlighting robust strategies that ensure stability and performance. Finally, we discuss open challenges and future directions at the interface of operator theory, approximation theory, and nonlinear control.

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arXiv2025

Robust feedback-based quantum optimization: analysis of coherent control errors

The Feedback-based Algorithm for Quantum Optimization (FALQON) is a Lyapunov inspired quantum algorithm proposed to tackle combinatorial optimization problems. In this paper, we examine the robustness of FALQON against coherent control errors, a class of multiplicative errors that affect the control input. We show that the algorithm is asymptotically robust with respect to systematic errors, and we derive robustness bounds for independent errors. Finally, we propose a robust version of FALQON which minimizes a regularized Lyapunov function. Our theoretical results are supported through simulations.

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arXiv2025

The interplay of robustness and generalization in quantum machine learning

While adversarial robustness and generalization have individually received substantial attention in the recent literature on quantum machine learning, their interplay is much less explored. In this chapter, we address this interplay for variational quantum models, which were recently proposed as function approximators in supervised learning. We discuss recent results quantifying both robustness and generalization via Lipschitz bounds, which explicitly depend on model parameters. Thus, they give rise to a regularization-based training approach for robust and generalizable quantum models, highlighting the importance of trainable data encoding strategies. The practical implications of the theoretical results are demonstrated with an application to time series analysis.

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arXiv2025

Bilinear Data-Driven Min-Max MPC: Designing Rational Controllers via Sum-of-squares Optimization

We propose a data-driven min-max model predictive control (MPC) scheme to control unknown discrete-time bilinear systems. Based on a sequence of noisy input-state data, we state a set-membership representation for the unknown system dynamics. Then, we derive a sum-of-squares (SOS) program that minimizes an upper bound on the worst-case cost over all bilinear systems consistent with the data. As a crucial technical ingredient, the SOS program involves a rational controller parameterization to improve feasibility and tractability. We prove that the resulting data-driven MPC scheme ensures closed-loop stability and constraint satisfaction for the unknown bilinear system. We demonstrate the practicality of the proposed scheme in a numerical example.

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arXiv2025

Kernel-based error bounds of bilinear Koopman surrogate models for nonlinear data-driven control

We derive novel deterministic bounds on the approximation error of data-based bilinear surrogate models for unknown nonlinear systems. The surrogate models are constructed using kernel-based extended dynamic mode decomposition to approximate the Koopman operator in a reproducing kernel Hilbert space. Unlike previous methods that require restrictive assumptions on the invariance of the dictionary, our approach leverages kernel-based dictionaries that allow us to control the projection error via pointwise error bounds, overcoming a significant limitation of existing theoretical guarantees. The derived state- and input-dependent error bounds allow for direct integration into Koopman-based robust controller designs with closed-loop guarantees for the unknown nonlinear system. Numerical examples illustrate the effectiveness of the proposed framework.

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arXiv2024

Bringing Quantum Systems under Control: A Tutorial Invitation to Quantum Computing and Its Relation to Bilinear Control Systems

Quantum computing comes with the potential to push computational boundaries in various domains including, e.g., cryptography, simulation, optimization, and machine learning. Exploiting the principles of quantum mechanics, new algorithms can be developed with capabilities that are unprecedented by classical computers. However, the experimental realization of quantum devices is an active field of research with enormous open challenges, including robustness against noise and scalability. While systems and control theory plays a crucial role in tackling these challenges, the principles of quantum physics lead to a (perceived) high entry barrier for entering the field of quantum computing. This tutorial paper aims at lowering the barrier by introducing basic concepts required to understand and solve research problems in quantum systems. First, we introduce fundamentals of quantum algorithms, ranging from basic ingredients such as qubits and quantum logic gates to prominent examples and more advanced concepts, e.g., variational quantum algorithms. Next, we formalize some engineering questions for building quantum devices in the real world, which requires the careful manipulation of microscopic quantities obeying quantum effects. To this end for N-level systems, we introduce basic concepts of (bilinear) quantum systems and control theory including controllability, observability, and optimal control in a unified frame. Finally, we address the problem of noise in real-world quantum systems via robust quantum control, which relies on a set-membership uncertainty description frequently employed in control.

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arXiv2024

Koopman-based control of nonlinear systems with closed-loop guarantees

In this paper, we provide a tutorial overview and an extension of a recently developed framework for data-driven control of unknown nonlinear systems with rigorous closed-loop guarantees. The proposed approach relies on the Koopman operator representation of the nonlinear system, for which a bilinear surrogate model is estimated based on data. In contrast to existing Koopman-based estimation procedures, we state guaranteed bounds on the approximation error using the stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD) framework. The resulting surrogate model and the uncertainty bounds allow us to design controllers via robust control theory and sum-of-squares optimization, guaranteeing desirable properties for the closed-loop system. We present results on stabilization both in discrete and continuous time, and we derive a method for controller design with performance objectives. The benefits of the presented framework over established approaches are demonstrated with a numerical example.

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arXiv2024

Data-Driven Min-Max MPC for LPV Systems with Unknown Scheduling Signal

This paper presents a data-driven min-max model predictive control (MPC) scheme for linear parameter-varying (LPV) systems. Contrary to existing data-driven LPV control approaches, we assume that the scheduling signal is unknown during offline data collection and online system operation. Assuming a quadratic matrix inequality (QMI) description for the scheduling signal, we develop a novel data-driven characterization of the consistent system matrices using only input-state data. The proposed data-driven min-max MPC minimizes a tractable upper bound on the worst-case cost over the consistent system matrices set and over all scheduling signals satisfying the QMI. The proposed approach guarantees recursive feasibility, closed-loop exponential stability and constraint satisfaction if it is feasible at the initial time. We demonstrate the effectiveness of the proposed method in simulation.

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arXiv2024

Koopman-based control using sum-of-squares optimization: Improved stability guarantees and data efficiency

In this paper, we propose a novel controller design approach for unknown nonlinear systems using the Koopman operator. In particular, we use the recently proposed stability- and feedback-oriented extended dynamic mode decomposition (SafEDMD) architecture to generate a data-driven bilinear surrogate model with certified error bounds. Then, by accounting for the obtained error bounds in a controller design based on the bilinear system, one can guarantee closed-loop stability for the true nonlinear system. While existing approaches over-approximate the bilinearity of the surrogate model, thus introducing conservatism and providing only local guarantees, we explicitly account for the bilinearity by using sum-of-squares (SOS) optimization in the controller design. More precisely, we parametrize a rational controller stabilizing the error-affected bilinear surrogate model and, consequently, the underlying nonlinear system. The resulting SOS optimization problem provides explicit data-driven controller design conditions for unknown nonlinear systems based on semidefinite programming. Our approach significantly reduces conservatism by establishing a larger region of attraction and improved data efficiency. The proposed method is evaluated using numerical examples, demonstrating its advantages over existing approaches.

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arXiv2024

Robustness and Generalization in Quantum Reinforcement Learning via Lipschitz Regularization

Quantum machine learning leverages quantum computing to enhance accuracy and reduce model complexity compared to classical approaches, promising significant advancements in various fields. Within this domain, quantum reinforcement learning has garnered attention, often realized using variational quantum circuits to approximate the policy function. This paper addresses the robustness and generalization of quantum reinforcement learning by combining principles from quantum computing and control theory. Leveraging recent results on robust quantum machine learning, we utilize Lipschitz bounds to propose a regularized version of a quantum policy gradient approach, named the RegQPG algorithm. We show that training with RegQPG improves the robustness and generalization of the resulting policies. Furthermore, we introduce an algorithmic variant that incorporates curriculum learning, which minimizes failures during training. Our findings are validated through numerical experiments, demonstrating the practical benefits of our approach.

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arXiv2024

Robustness of optimal quantum annealing protocols

Noise in quantum computing devices poses a key challenge in their realization. In this paper, we study the robustness of optimal quantum annealing protocols against coherent control errors, which are multiplicative Hamlitonian errors causing detrimental effects on current quantum devices. We show that the norm of the Hamiltonian quantifies the robustness against these errors, motivating the introduction of an additional regularization term in the cost function. We analyze the optimality conditions of the resulting robust quantum optimal control problem based on Pontryagin's maximum principle, showing that robust protocols admit larger smooth annealing sections. This suggests that quantum annealing admits improved robustness in comparison to bang-bang solutions such as the quantum approximate optimization algorithm. Finally, we perform numerical simulations to verify our analytical results and demonstrate the improved robustness of the proposed approach.

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