arXiv subjects

Frank Allgöwer

Publications and source records attributed to Frank Allgöwer.

At least 109 records · Page 6Linked to original sources

arXiv2022

Combining Prior Knowledge and Data for Robust Controller Design

We present a framework for systematically combining data of an unknown linear time-invariant system with prior knowledge on the system matrices or on the uncertainty for robust controller design. Our approach leads to linear matrix inequality (LMI) based feasibility criteria which guarantee stability and performance robustly for all closed-loop systems consistent with the prior knowledge and the available data. The design procedures rely on a combination of multipliers inferred via prior knowledge and learnt from measured data, where for the latter a novel and unifying disturbance description is employed. While large parts of the paper focus on linear systems and input-state measurements, we also provide extensions to robust output-feedback design based on noisy input-output data and against nonlinear uncertainties. We illustrate through numerical examples that our approach provides a flexible framework for simultaneously leveraging prior knowledge and data, thereby reducing conservatism and improving performance significantly if compared to black-box approaches to data-driven control.

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arXiv2022

Self-triggered MPC robust to bounded packet loss via a min-max approach: extended version

Networked Control Systems typically come with a limited communication bandwidth and thus require special care when designing the underlying control and triggering law. A method that allows to consider hard constraints on the communication traffic as well as on states and inputs is self-triggered model predictive control (MPC). In this scheme, the optimal length of the sampling interval is determined proactively using predictions of the system behavior. However, previous formulations of self-triggered MPC have neglected the widespread phenomenon of packet loss, such that these approaches might fail in practice. In this paper, we present a novel self-triggered MPC scheme which is robust to bounded packet loss by virtue of a min-max optimization problem. We prove recursive feasibility, constraint satisfaction and convergence to the origin for any possible packet loss realization consistent with the boundedness constraint, and demonstrate the advantages of the proposed scheme in a numerical example.

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arXiv2022

Analysis of Time- versus Event-Triggered Consensus for a Single-Integrator Multi-Agent System

It is well known that the employed triggering scheme has great impact on the control performance when control loops operate under scarce communication resources. Various practical and simulative works have demonstrated the potential of event-triggered control to reduce communication while providing a similar performance level when compared to time-triggered control. For non-cooperative networked control systems, analytical performance comparisons of time- and event-triggered control support this finding under certain assumptions. While being well-studied in the non-cooperative setting, it remains unclear if and how the performance relationship of the triggering schemes is altered in a multi-agent system setup. To close this gap, in this paper, we consider a homogeneous single-integrator multi-agent consensus problem for which we compare the performance of time- and event-triggered control schemes analytically. Under the assumption of equal average triggering rates, we use the long-term average of the quadratic deviation from consensus as a performance measure to contrast the triggering schemes. Contrary to the non-cooperative setting, we prove that event-triggered control performs worse than time-triggered control beyond a certain number of agents in this setup. In addition, we derive the asymptotic order of the performance measure as a function of the number of agents under both triggering schemes.

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arXiv2022

Stability in data-driven MPC: an inherent robustness perspective

Data-driven model predictive control (DD-MPC) based on Willems' Fundamental Lemma has received much attention in recent years, allowing to control systems directly based on an implicit data-dependent system description. The literature contains many successful practical applications as well as theoretical results on closed-loop stability and robustness. In this paper, we provide a tutorial introduction to DD-MPC for unknown linear time-invariant (LTI) systems with focus on (robust) closed-loop stability. We first address the scenario of noise-free data, for which we present a DD-MPC scheme with terminal equality constraints and derive closed-loop properties. In case of noisy data, we introduce a simple yet powerful approach to analyze robust stability of DD-MPC by combining continuity of DD-MPC w.r.t. noise with inherent robustness of model-based MPC, i.e., robustness of nominal MPC w.r.t. small disturbances. Moreover, we discuss how the presented proof technique allows to show closed-loop stability of a variety of DD-MPC schemes with noisy data, as long as the corresponding model-based MPC is inherently robust.

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arXiv2022

Data-Driven Control of Distributed Event-Triggered Network Systems

The present paper deals with data-driven event-triggered control of a class of unknown discrete-time interconnected systems (a.k.a. network systems). To this end, we start by putting forth a novel distributed event-triggering transmission strategy based on periodic sampling, under which a model-based stability criterion for the closed-loop network system is derived, by leveraging a discrete-time looped-functional approach. Marrying the model-based criterion with a data-driven system representation recently developed in the literature, a purely data-driven stability criterion expressed in the form of linear matrix inequalities (LMIs) is established. Meanwhile, the data-driven stability criterion suggests a means for co-designing the event-triggering coefficient matrix and the feedback control gain matrix using only some offline collected state-input data. Finally, numerical results corroborate the efficacy of the proposed distributed data-driven ETS in cutting off data transmissions and the co-design procedure.

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arXiv2022

Stochastic Model Predictive Control using Initial State Optimization

We propose a stochastic MPC scheme using an optimization over the initial state for the predicted trajectory. Considering linear discrete-time systems under unbounded additive stochastic disturbances subject to chance constraints, we use constraint tightening based on probabilistic reachable sets to design the MPC. The scheme avoids the infeasibility issues arising from unbounded disturbances by including the initial state as a decision variable. We show that the stabilizing control scheme can guarantee constraint satisfaction in closed loop, assuming unimodal disturbances. In addition to illustrating these guarantees, the numerical example indicates further advantages of optimizing over the initial state for the transient behavior.

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arXiv2022

Data-driven analysis and controller design for discrete-time systems under aperiodic sampling

This article is concerned with data-driven analysis of discrete-time systems under aperiodic sampling, and in particular with a data-driven estimation of the maximum sampling interval (MSI). The MSI is relevant for analysis of and controller design for cyber-physical, embedded and networked systems, since it gives a limit on the time span between sampling instants such that stability is guaranteed. We propose tools to compute the MSI for a given controller and to design a controller with a preferably large MSI, both directly from a finite-length, noise-corrupted state-input trajectory of the system. We follow two distinct approaches for stability analysis, one taking a robust control perspective and the other a switched systems perspective on the aperiodically sampled system. In a numerical example and a subsequent discussion, we demonstrate the efficacy of our developed tools and compare the two approaches.

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arXiv2022

Dynamic self-triggered control for nonlinear systems with delays

Self-triggered control (STC) is a resource efficient approach to determine sampling instants for Networked Control Systems (NCS). Recently, a dynamic STC strategy based on hybrid Lyapunov functions for nonlinear NCS has been proposed in Hertneck and Allgöwer (2021b), however with the limitation to NCS without transmission delays. In this paper, we extend this strategy for nonlinear NCS with transmission delays. The capability to handle systems with delays makes it possible to use the resulting dynamic STC mechanism in many practical scenarios where instant transmissions without delays cannot be guaranteed. The proposed dynamic STC mechanism guarantees stability despite bounded transmission delays. The effectiveness of the mechanism is illustrated with a numerical example and compared to state-of-the art literature.

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arXiv2022

Transient Performance of Tube-based Robust Economic Model Predictive Control

In this paper, we provide non-averaged and transient performance guarantees for recently developed, tube-based robust economic model predictive control (MPC) schemes. In particular, we consider both tube-based MPC schemes with and without terminal conditions. We show that the closed-loop performance obtained by applying such MPC schemes is approximately optimal when evaluated both on finite and infinite time horizons. These performance bounds are similar to those derived previously for nominal economic MPC. The theoretical results are discussed in a numerical example.

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arXiv2022

Safe Exploration and Escape Local Minima with Model Predictive Control under Partially Unknown Constraints

In this paper, we propose a novel model predictive control (MPC) framework for output tracking that deals with partially unknown constraints. The MPC scheme optimizes over a learning and a backup trajectory. The learning trajectory aims to explore unknown and potentially unsafe areas, if and only if this might lead to a potential performance improvement. On the contrary, the backup trajectory lies in the known space, and is intended to ensure safety and convergence. The cost function for the learning trajectory is divided into a tracking and an offset cost, while the cost function for the backup trajectory is only marginally considered and only penalizes the offset cost. We show that the proposed MPC scheme is not only able to safely explore the unknown constraints, but also escape from local minima that may arise from the presence of obstacles. Moreover, we provide formal guarantees for convergence and recursive feasibility of the MPC scheme, as well as closed-loop constraint satisfaction. Finally, the proposed MPC scheme is demonstrated in simulations using an example of autonomous vehicle driving in a partially unknown environment where unknown obstacles are present.

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arXiv2022

Model predictive control for linear uncertain systems using integral quadratic constraints

In this work, we propose a tube-based MPC scheme for state- and input-constrained linear systems subject to dynamic uncertainties characterized by dynamic integral quadratic constraints (IQCs). In particular, we extend the framework of $ρ$-hard IQCs for exponential stability analysis to external inputs. This result yields that the error between the true uncertain system and the nominal prediction model is bounded by an exponentially stable scalar system. In the proposed tube-based MPC scheme, the state of this error bounding system is predicted along with the nominal model and used as a scaling parameter for the tube size. We prove that this method achieves robust constraint satisfaction and input-to-state stability despite dynamic uncertainties and additive bounded disturbances. A numerical example demonstrates the reduced conservatism of this IQC approach compared to state-of-the-art robust MPC approaches for dynamic uncertainties.

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arXiv2022

Robust stability analysis of a simple data-driven model predictive control approach

In this paper, we provide a theoretical analysis of closed-loop properties of a simple data-driven model predictive control (MPC) scheme. The formulation does not involve any terminal ingredients, thus allowing for a simple implementation without (potential) feasibility issues. The proposed approach relies on an implicit description of linear time-invariant systems based on behavioral systems theory, which only requires one input-output trajectory of an unknown system. For the nominal case with noise-free data, we prove that the data-driven MPC scheme ensures exponential stability for the closed loop if the prediction horizon is sufficiently long. Moreover, we analyze the robust data-driven MPC scheme for noisy output measurements for which we prove closed-loop practical exponential stability. The advantages of the presented approach are illustrated with a numerical example.

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arXiv2022

Linear tracking MPC for nonlinear systems Part II: The data-driven case

We present a novel data-driven model predictive control (MPC) approach to control unknown nonlinear systems using only measured input-output data with closed-loop stability guarantees. Our scheme relies on the data-driven system parametrization provided by the Fundamental Lemma of Willems et al. We use new input-output measurements online to update the data, exploiting local linear approximations of the underlying system. We prove that our MPC scheme, which only requires solving strictly convex quadratic programs online, ensures that the closed loop (practically) converges to the (unknown) optimal reachable equilibrium that tracks a desired output reference while satisfying polytopic input constraints. As intermediate results of independent interest, we extend the Fundamental Lemma to affine systems and we derive novel robustness bounds w.r.t. noisy data for the open-loop optimal control problem, which are directly transferable to other data-driven MPC schemes in the literature. The applicability of our approach is illustrated with a numerical application to a continuous stirred tank reactor.

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arXiv2022

Linear tracking MPC for nonlinear systems Part I: The model-based case

We develop a tracking model predictive control (MPC) scheme for nonlinear systems using the linearized dynamics at the current state as a prediction model. Under reasonable assumptions on the linearized dynamics, we prove that the proposed MPC scheme exponentially stabilizes the optimal reachable equilibrium w.r.t. a desired target setpoint. Our theoretical results rely on the fact that, close to the steady-state manifold, the prediction error of the linearization is small and hence, we can slide along the steady-state manifold towards the optimal reachable equilibrium. The closed-loop stability properties mainly depend on a cost matrix which allows us to trade off performance, robustness, and the size of the region of attraction. In an application to a nonlinear continuous stirred tank reactor, we show that the scheme, which only requires solving a convex quadratic program online, has comparable performance to a nonlinear MPC scheme while being computationally significantly more efficient. Further, our results provide the basis for controlling nonlinear systems based on data-dependent linear prediction models, which we explore in our companion paper.

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arXiv2022

Bounding the difference between model predictive control and neural networks

There is a growing debate on whether the future of feedback control systems will be dominated by data-driven or model-driven approaches. Each of these two approaches has their own complimentary set of advantages and disadvantages, however, only limited attempts have, so far, been developed to bridge the gap between them. To address this issue, this paper introduces a method to bound the worst-case error between feedback control policies based upon model predictive control (MPC) and neural networks (NNs). This result is leveraged into an approach to automatically synthesize MPC policies minimising the worst-case error with respect to a NN. Numerical examples highlight the application of the bounds, with the goal of the paper being to encourage a more quantitative understanding of the relationship between data-driven and model-driven control.

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arXiv2022

Rollout event-triggered control: reconciling event- and time-triggered control

Event-triggered control (ETC) and time-triggered control (TTC), the classical concepts to determine the transmission instants for networked control systems, each come with drawbacks: It is difficult to tune ETC such that a certain bandwidth is respected, whereas TTC cannot adapt the sampling interval to the current state of the control system. In this article, we provide an overview over rollout ETC, a method aimed at reconciling the advantages of ETC and TTC. We unite two variants of rollout ETC under a common framework and present conditions for convergence and compliance with a predefined bandwidth limit. Furthermore, we demonstrate that rollout ETC satisfies a performance bound and that it allows for a very flexible transmission scheduling similar to classical ETC. The mentioned beneficial properties are illustrated through extensive numerical simulations.

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arXiv2022

A novel constraint tightening approach for robust data-driven predictive control

In this paper, we present a data-driven model predictive control (MPC) scheme that is capable of stabilizing unknown linear time-invariant systems under the influence of process disturbances. To this end, Willems' lemma is used to predict the future behavior of the system. This allows the entire scheme to be set up using only a priori measured data and knowledge of an upper bound on the system order. First, we develop a state-feedback MPC scheme, based on input-state data, which guarantees closed-loop practical exponential stability and recursive feasibility as well as closed-loop constraint satisfaction. The scheme is extended by a suitable constraint tightening, which can also be constructed using only data. In order to control a priori unstable systems, the presented scheme contains a pre-stabilizing controller and an associated input constraint tightening. We first present the proposed data-driven MPC scheme for the case of full state measurements, and also provide extensions for obtaining similar closed-loop guarantees in case of output feedback. The presented scheme is applied to a numerical example.

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arXiv2022

Data-driven Control of Dynamic Event-triggered Systems with Delays

This paper studies data-driven control of unknown sampled-data systems with communication delays under an event-triggering transmission mechanism. Data-based representations for time-invariant linear systems with known or unknown system input matrices are first developed, along with a novel class of dynamic triggering schemes for sampled-data systems with time delays. A model-based stability condition for the resulting event-triggered time-delay system is established using a looped-functional approach. Combining this model-based condition with the data-driven representations, data-based stability conditions are derived. Building on the data-based conditions, methods for co-designing the controller gain and the event-triggering matrix are subsequently provided for both cases with or without using the input matrix. Finally, numerical examples are presented to corroborate the usefulness of additional prior knowledge of the input matrix in reducing the conservatism of stability conditions, as well as the merits of the proposed data-driven event-triggered control schemes relative to existing results.

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