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Frank Allgöwer

Publications and source records attributed to Frank Allgöwer.

At least 91 records · Page 5Linked to original sources
arXiv2023

Lipschitz constant estimation for 1D convolutional neural networks

In this work, we propose a dissipativity-based method for Lipschitz constant estimation of 1D convolutional neural networks (CNNs). In particular, we analyze the dissipativity properties of convolutional, pooling, and fully connected layers making use of incremental quadratic constraints for nonlinear activation functions and pooling operations. The Lipschitz constant of the concatenation of these mappings is then estimated by solving a semidefinite program which we derive from dissipativity theory. To make our method as efficient as possible, we exploit the structure of convolutional layers by realizing these finite impulse response filters as causal dynamical systems in state space and carrying out the dissipativity analysis for the state space realizations. The examples we provide show that our Lipschitz bounds are advantageous in terms of accuracy and scalability.

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arXiv2023

Robust data-driven control for nonlinear systems using the Koopman operator

Data-driven analysis and control of dynamical systems have gained a lot of interest in recent years. While the class of linear systems is well studied, theoretical results for nonlinear systems are still rare. In this paper, we present a data-driven controller design method for discrete-time control-affine nonlinear systems. Our approach relies on the Koopman operator, which is a linear but infinite-dimensional operator lifting the nonlinear system to a higher-dimensional space. Particularly, we derive a linear fractional representation of a lifted bilinear system representation based on measured data. Further, we restrict the lifting to finite dimensions, but account for the truncation error using a finite-gain argument. We derive a linear matrix inequality based design procedure to guarantee robust local stability for the resulting bilinear system for all error terms satisfying the finite-gain bound and, thus, also for the underlying nonlinear system. Finally, we apply the developed design method to the nonlinear Van der Pol oscillator.

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arXiv2023

Stochastic Model Predictive Control using Initial State and Variance Interpolation

We present a Stochastic Model Predictive Control (SMPC) framework for linear systems subject to Gaussian disturbances. In order to avoid feasibility issues, we employ a recent initialization strategy, optimizing over an interpolation of the initial state between the current measurement and previous prediction. By also considering the variance in the interpolation, we can employ variable-size tubes, to ensure constraint satisfaction in closed-loop. We show that this novel method improves control performance and enables following the constraint closer, then previous methods. Using a DC-DC converter as numerical example we illustrated the improvement over previous methods.

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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

Distributed Model Predictive Control for Periodic Cooperation of Multi-Agent Systems

We consider multi-agent systems with heterogeneous, nonlinear agents subject to individual constraints that want to achieve a periodic, dynamic cooperative control goal which can be characterised by a set and a suitable cost. We propose a sequential distributed model predictive control (MPC) scheme in which agents sequentially solve an individual optimisation problem to track an artificial periodic output trajectory. The optimisation problems are coupled through these artificial periodic output trajectories, which are communicated and penalised using the cost that characterises the cooperative goal. The agents communicate only their artificial trajectories and only once per time step. We show that under suitable assumptions, the agents can incrementally move their artificial output trajectories towards the cooperative goal, and, hence, their closed-loop output trajectories asymptotically achieve it. We illustrate the scheme with a simulation example.

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arXiv2023

Shared Network Effects in Time- versus Event-Triggered Consensus of a Single-Integrator Multi-Agent System

Event-triggered control has the potential to provide a similar performance level as time-triggered (periodic) control while triggering events less frequently. It therefore appears intuitive that it is also a viable approach for distributed systems to save scarce shared network resources used for inter-agent communication. While this motivation is commonly used also for multi-agent systems, a theoretical analysis of the impact of network effects on the performance of event- and time-triggered control for such distributed systems is currently missing. With this paper, we contrast event- and time-triggered control performance for a single-integrator consensus problem under consideration of a shared communication medium. We therefore incorporate transmission delays and packet loss in our analysis and compare the triggering scheme performance under two simple medium access control protocols. We find that network effects can degrade the performance of event-triggered control beyond the performance level of time-triggered control for the same average triggering rate if the network is used intensively. Moreover, the performance advantage of event-triggered control shrinks with an increasing number of agents and is even lost for sufficiently large networks in the considered setup.

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arXiv2023

Practical exponential stability of a robust data-driven nonlinear predictive control scheme

We provide theoretical guarantees for recursive feasibility and practical exponential stability of the closed-loop system of a feedback linearizable nonlinear system when controlled by a robust data-driven nonlinear predictive control scheme. This technical report serves as a supplementary material to our recent paper "Data-driven Nonlinear Predictive Control for Feedback Linearizable Systems". The arguments shown in this report follow similar steps to those for the LTI case, since feedback linearizable systems are linear in transformed coordinates. However, the proof was suitably adapted to match the nonlinear setting under consideration, and the differences are emphasized throughout the proof.

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arXiv2023

Data-driven Nonlinear Predictive Control for Feedback Linearizable Systems

We present a data-driven nonlinear predictive control approach for the class of discrete-time multi-input multi-output feedback linearizable nonlinear systems. The scheme uses a non-parametric predictive model based only on input and noisy output data along with a set of basis functions that approximate the unknown nonlinearities. Despite the noisy output data as well as the mismatch caused by the use of basis functions, we show that the proposed multistep robust data-driven nonlinear predictive control scheme is recursively feasible and renders the closed-loop system practically exponentially stable. We illustrate our results on a model of a fully-actuated double inverted pendulum.

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arXiv2023

Gaussian inference for data-driven state-feedback design of nonlinear systems

Data-driven control of nonlinear systems with rigorous guarantees is a challenging problem as it usually calls for nonconvex optimization and requires often knowledge of the true basis functions of the system dynamics. To tackle these drawbacks, this work is based on a data-driven polynomial representation of general nonlinear systems exploiting Taylor polynomials. Thereby, we design state-feedback laws that render a known equilibrium point globally asymptotically stable while operating with respect to a desired quadratic performance criterion. The calculation of the polynomial state feedback boils down to a single sum-ofsquares optimization problem, and hence to computationally tractable linear matrix inequalities. Moreover, we examine state-input data in presence of Gaussian noise by Bayesian inference to overcome the conservatism of deterministic noise characterizations from recent data-driven control approaches for Gaussian noise.

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arXiv2023

Self-triggered output feedback control for nonlinear networked control systems based on hybrid Lyapunov functions

Most approaches for self-triggered control (STC) of nonlinear networked control systems (NCS) require measurements of the full system state to determine transmission times. However, for most control systems only a lower dimensional output is available. To bridge this gap, we present in this paper an output-feedback STC approach for nonlinear NCS. An asymptotically stable observer is used to reconstruct the plant state and transmission times are determined based on the observer state. The approach employs hybrid Lyapunov functions and a dynamic variable to encode past state information and to maximize the time between transmissions. It is non-conservative in the sense that the assumptions on plant and controller are the same as for dynamic STC based on hybrid Lyapunov functions with full state measurements and any asymptotically stabilizing observer can be used. We conclude that the proposed STC approach guarantees asymptotic stability of the origin for the closed-loop system.

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arXiv2023

Data-based Control of Feedback Linearizable Systems

We present an extension of Willems' Fundamental Lemma to the class of multi-input multi-output discrete-time feedback linearizable nonlinear systems, thus providing a data-based representation of their input-output trajectories. Two sources of uncertainty are considered. First, the unknown linearizing input is inexactly approximated by a set of basis functions. Second, the measured output data is contaminated by additive noise. Further, we propose an approach to approximate the solution of the data-based simulation and output matching problems, and show that the difference from the true solution is bounded. Finally, the results are illustrated on an example of a fully-actuated double inverted pendulum.

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arXiv2023

Data-Driven Reachability Analysis from Noisy Data

We consider the problem of computing reachable sets directly from noisy data without a given system model. Several reachability algorithms are presented for different types of systems generating the data. First, an algorithm for computing over-approximated reachable sets based on matrix zonotopes is proposed for linear systems. Constrained matrix zonotopes are introduced to provide less conservative reachable sets at the cost of increased computational expenses and utilized to incorporate prior knowledge about the unknown system model. Then we extend the approach to polynomial systems and, under the assumption of Lipschitz continuity, to nonlinear systems. Theoretical guarantees are given for these algorithms in that they give a proper over-approximate reachable set containing the true reachable set. Multiple numerical examples and real experiments show the applicability of the introduced algorithms, and comparisons are made between algorithms.

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arXiv2023

A quantitative and constructive proof of Willems' Fundamental Lemma and its implications

Willems' Fundamental Lemma provides a powerful data-driven parametrization of all trajectories of a controllable linear time-invariant system based on one trajectory with persistently exciting (PE) input. In this paper, we present a novel proof of this result which is inspired by the classical adaptive control literature and differs from existing proofs in multiple aspects. The proof involves a quantitative and directional PE notion, allowing to characterize robust PE properties via singular value bounds, as opposed to binary rank-based PE conditions. Further, the proof is constructive, i.e., we derive an explicit PE lower bound for the generated data. As a contribution of independent interest, we generalize existing PE results from the adaptive control literature and reveal a crucial role of the system's zeros.

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arXiv2022

Linear Data-Driven Economic MPC with Generalized Terminal Constraint

In this paper, we propose a data-driven economic model predictive control (EMPC) scheme with generalized terminal constraint to control an unknown linear time-invariant system. Our scheme is based on the Fundamental Lemma to predict future system trajectories using a persistently exciting input-output trajectory. The control objective is to minimize an economic cost objective. By employing a generalized terminal constraint with artificial equilibrium, the scheme does not require prior knowledge of the optimal equilibrium. We prove that the asymptotic average performance of the closed-loop system can be made arbitrarily close to that of the optimal equilibrium. Moreover, we extend our results to the case of an unknown linear stage cost function, where the Fundamental lemma is used to predict the stage cost directly. The effectiveness of the proposed scheme is shown by a numerical example.

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arXiv2022

Data-driven inference on optimal input-output properties of polynomial systems with focus on nonlinearity measures

In the context of dynamical systems, nonlinearity measures quantify the strength of nonlinearity by means of the distance of their input-output behaviour to a set of linear input-output mappings. In this paper, we establish a framework to determine nonlinearity measures and other optimal input-output properties for nonlinear polynomial systems without explicitly identifying a model but from a finite number of input-state measurements which are subject to noise. To this end, we deduce from data for the unidentified ground-truth system three possible set-membership representations, compare their accuracy, and prove that they are asymptotically consistent with respect to the amount of samples. Moreover, we leverage these representations to compute guaranteed upper bounds on nonlinearity measures and the corresponding optimal linear approximation model via semi-definite programming. Furthermore, we extend the established framework to determine optimal input-output properties described by time domain hard integral quadratic constraints.

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arXiv2022

Robust peak-to-peak gain analysis using integral quadratic constraints

This work provides a framework to compute an upper bound on the robust peak-to-peak gain of discrete-time uncertain linear systems using integral quadratic constraints (IQCs). Such bounds are of particular interest in the computation of reachable sets and the $\ell_1$-norm, as well as when safety-critical constraints need to be satisfied pointwise in time. The use of $ρ$-hard IQCs with a terminal cost enables us to deal with a wide variety of uncertainty classes, for example, we provide $ρ$-hard IQCs with a terminal cost for the class of parametric uncertainties. This approach unifies, generalizes, and significantly improves state-of-the-art methods, which is also demonstrated in a numerical example.

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arXiv2022

Model-Based and Data-Driven Control of Event- and Self-Triggered Discrete-Time LTI Systems

The present paper considers the model-based and data-driven control of unknown linear time-invariant discrete-time systems under event-triggering and self-triggering transmission schemes. To this end, we begin by presenting a dynamic event-triggering scheme (ETS) based on periodic sampling, and a discrete-time looped-functional approach, through which a model-based stability condition is derived. Combining the model-based condition with a recent data-based system representation, a data-driven stability criterion in the form of linear matrix inequalities (LMIs) is established, which also offers a way of co-designing the ETS matrix and the controller. To further alleviate the sampling burden of ETS due to its continuous/periodic detection, a self-triggering scheme (STS) is developed. Leveraging pre-collected input-state data, an algorithm for predicting the next transmission instant is given, while achieving system stability. Finally, numerical simulations showcase the efficacy of ETS and STS in reducing data transmissions as well as of the proposed co-design methods.

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arXiv2022

Neural network training under semidefinite constraints

This paper is concerned with the training of neural networks (NNs) under semidefinite constraints, which allows for NN training with robustness and stability guarantees. In particular, we focus on Lipschitz bounds for NNs. Exploiting the banded structure of the underlying matrix constraint, we set up an efficient and scalable training scheme for NN training problems of this kind based on interior point methods. Our implementation allows to enforce Lipschitz constraints in the training of large-scale deep NNs such as Wasserstein generative adversarial networks (WGANs) via semidefinite constraints. In numerical examples, we show the superiority of our method and its applicability to WGAN training.

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