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Carsten W. Scherer

Publications and source records attributed to Carsten W. Scherer.

At least 19 recordsLinked to original sources
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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arXiv2025

Distributionally robust LMI synthesis for LTI systems

This article shows that distributionally robust controller synthesis as investigated in \cite{taskesen2024distributionally} can be formulated as a convex linear matrix inequality (LMI) synthesis problem. To this end, we rely on well-established convexification techniques from robust control. The LMI synthesis problem we propose has the advantage that it can be solved efficiently using off-the-shelf semi-definite programming (SDP) solvers. In addition, our formulation exposes the studied distributionally robust controller synthesis problem as an instance of robust $H_2$ synthesis.

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arXiv2025

Distributional Robustness in Output Feedback Regret-Optimal Control

This paper studies distributionally robust regret-optimal (DRRO) control with purified output feedback for linear systems subject to additive disturbances and measurement noise. These uncertainties (including the initial system state) are assumed to be stochastic and distributed according to an unknown joint probability distribution within a Wasserstein ambiguity set. We design affine controllers to minimise the worst-case expected regret over all distributions in this set. The expected regret is defined as the difference between an expected cost incurred by an affine causal controller and the expected cost incurred by the optimal noncausal controller with perfect knowledge of the disturbance trajectory at the outset. Leveraging the duality theory in distributionally robust optimisation, we derive strong duality results for worst-case expectation problems involving general quadratic objective functions, enabling exact reformulations of the DRRO control problem as semidefinite programs (SDPs). Focusing on one such reformulation, we eliminate certain decision variables. This technique also permits a further equivalent reformulation of the SDP as a distributed optimisation problem, with potential to enhance scalability.

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arXiv2025

Multi-objective robust controller synthesis with integral quadratic constraints in discrete-time

This article presents a novel framework for the robust controller synthesis problem in discrete-time systems using dynamic Integral Quadratic Constraints (IQCs). We present an algorithm to minimize closed-loop performance measures such as the $\mathcal H_\infty$-norm, the energy-to-peak gain, the peak-to-peak gain, or a multi-objective mix thereof. While IQCs provide a powerful tool for modeling structured uncertainties and nonlinearities, existing synthesis methods are limited to the $\mathcal H_\infty$-norm, continuous-time systems, or special system structures. By minimizing the energy-to-peak and peak-to-peak gain, the proposed synthesis can be utilized to bound the peak of the output, which is crucial in many applications requiring robust constraint satisfaction, input-to-state stability, reachability analysis, or other pointwise-in-time bounds. Numerical examples demonstrate the robustness and performance of the controllers synthesized with the proposed algorithm.

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arXiv2025

Integral Quadratic Constraints on Linear Infinite-dimensional Systems for Robust Stability Analysis

This paper proposes a framework to assess the stability of an ordinary differential equation which is coupled to a 1D-partial differential equation (PDE). The stability theorem is based on a new result on Integral Quadratic Constraints (IQCs) and expressed in terms of two linear matrix inequalities with a moderate computational burden. The IQCs are not generated using dissipation inequalities involving the whole state of an infinite-dimensional system, but by using projection coefficients of the infinite-dimensional state. This permits to generalize our robustness result to many other PDEs. The proposed methodology is applied to a time-delay system and numerical results comparable to those in the literature are obtained.

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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

Gain-Scheduling Controller Synthesis for Nested Systems with Full Block Scalings

This work presents a framework to synthesize structured gain-scheduled controllers for structured plants whose dynamics change according to time-varying scheduling parameters. Both the system and the controller are assumed to admit descriptions in terms of a linear time-invariant system in feedback with so-called scheduling blocks, which collect all scheduling parameters into a static system. We show that such linear fractional representations permit to exploit a so-called lifting technique in order to handle several structured gain-scheduling design problems. These could arise from a nested inner and outer loop control configuration with partial or full dependence on the scheduling variables. Our design conditions are formulated in terms of convex linear matrix inequalities and permit to handle multiple performance objectives.

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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

Practical and Rigorous Uncertainty Bounds for Gaussian Process Regression

Gaussian Process Regression is a popular nonparametric regression method based on Bayesian principles that provides uncertainty estimates for its predictions. However, these estimates are of a Bayesian nature, whereas for some important applications, like learning-based control with safety guarantees, frequentist uncertainty bounds are required. Although such rigorous bounds are available for Gaussian Processes, they are too conservative to be useful in applications. This often leads practitioners to replacing these bounds by heuristics, thus breaking all theoretical guarantees. To address this problem, we introduce new uncertainty bounds that are rigorous, yet practically useful at the same time. In particular, the bounds can be explicitly evaluated and are much less conservative than state of the art results. Furthermore, we show that certain model misspecifications lead to only graceful degradation. We demonstrate these advantages and the usefulness of our results for learning-based control with numerical examples.

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arXiv2023

Robust Exponential Stability and Invariance Guarantees with General Dynamic O'Shea-Zames-Falb Multipliers

We propose novel time-domain dynamic integral quadratic constraints with a terminal cost for exponentially weighted slope-restricted gradients of not necessarily convex functions. This extends recent results for subdifferentials of convex function and their link to so-called O'Shea-Zames-Falb multipliers. The benefit of merging time-domain and frequency-domain techniques is demonstrated for linear saturated systems.

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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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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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arXiv2023

Input-Output-Data-Enhanced Robust Analysis via Lifting

Starting from a linear fractional representation of a linear system affected by constant parametric uncertainties, we demonstrate how to enhance standard robust analysis tests by taking available (noisy) input-output data of the uncertain system into account. Our approach relies on lifting the system and the construction of data-dependent multipliers. It leads to a test in terms of linear matrix inequalities which guarantees stability and performance for all systems compatible with the observed data if it is in the affirmative. In contrast to many other data-based approaches, prior physical or structural knowledge about the system can be incorporated at the outset by exploiting the power of linear fractional representations.

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arXiv2022

IQC Based Analysis and Estimator Design for Discrete-Time Systems Affected by Impulsive Uncertainties

We propose novel quadratic performance tests for linear discrete-time impulsive systems based on viewing these systems as feedback interconnections of some non-impulsive linear system with an impulsive operator. In order to systematically analyze such interconnections, we employ the framework of integral quadratic constraints and propose novel constraints of this kind for capturing the behavior of the involved impulsive operator. As a major benefit, the modularity of this framework permits seamless extensions to interconnections affected by heterogeneous uncertainties in a straightforward fashion. This contrasts with alternative approaches which are based on capturing the system's impulsive behavior by means of a clock. Building upon the developed analysis criteria, we characterize the existence of non-impulsive estimators for such impulsive interconnections in a lossless fashion and in terms of linear matrix inequalities. Finally, our approach is illustrated by means of several numerical examples.

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arXiv2022

A Dynamic S-Procedure for Dynamic Uncertainties

We show how to compose robust stability tests for uncertain systems modeled as linear fractional representations and affected by various types of dynamic uncertainties. Our results are formulated in terms of linear matrix inequalities and rest on the recently established notion of finite-horizon integral quadratic constraints with a terminal cost. The construction of such constraints is motivated by an unconventional application of the S-procedure in the frequency domain with dynamic Lagrange multipliers. Our technical contribution reveals how this construction can be performed by dissipativity arguments in the time-domain and in a lossless fashion. This opens the way for generalizations to time-varying or hybrid systems.

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arXiv2022

Lifting to Passivity for $\mathcal{H}_2$-Gain-Scheduling Synthesis with Full Block Scalings

We focus on the $\mathcal{H}_2$-gain-scheduling synthesis problem for time-varying parametric scheduling blocks with scalings. Recently, we have presented a solution of this problem for $D$- and positive real scalings by guaranteeing finiteness of the $\mathcal{H}_2$-norm for the closed-loop system with suitable linear fractional plant and controller representations. In order to reduce conservatism, we extend these methods to full block scalings by designing a triangular scheduling function and by introducing a new lifting technique for gain-scheduled synthesis that enables convexification.

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