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A Matrix Finsler's Lemma with Applications to Data-Driven Control
TL;DR: In this article, a matrix version of the classical Finsler's lemma has been shown to provide a tractable condition under which all matrix solutions to a quadratic equality also satisfy a quadrinomial inequality.
Abstract: In a recent paper it was shown how a matrix S-lemma can be applied to construct controllers from noisy data. The current paper complements these results by proving a matrix version of the classical Finsler's lemma. This matrix Finsler's lemma provides a tractable condition under which all matrix solutions to a quadratic equality also satisfy a quadratic inequality. We will apply this result to bridge known data-driven control design techniques for both exact and noisy data, thereby revealing a more general theory. The result is also applied to data-driven control of Lur'e systems.
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11 Sep 2020
TL;DR: The approach leads to linear matrix inequality (LMI) based feasibility criteria which guarantee stability, $\mathcal{H}_2$-performance, or quadratic performance robustly for all closed-loop systems consistent with the prior knowledge and the available data.
Abstract: 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 simple data-dependent uncertainty bound which can be employed for controller design using dualization arguments and S-procedure-based LMI relaxations. While most parts of the paper focus on input-state measurements, we also provide an extension to robust output-feedback design based on noisy input-output data. Finally, we apply sum-of-squares methods to construct relaxation hierarchies for the considered robust controller design problem which are asymptotically exact. We illustrate through various 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 purely data-driven controller design.
60 citations
TL;DR: Data-driven analysis, signal processing, and control methods as mentioned in this paper can be broadly classified as implicit and explicit approaches, with the implicit approach being more robust to uncertainty and robustness to noise.
38 citations
TL;DR: In this paper , the authors 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, which leads to linear matrix inequality based feasibility criteria which guarantee stability and performance robustly for all closed-loop systems consistent with the prior knowledge and the available data.
Abstract: 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.
6 citations
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24 Sep 2021TL;DR: In this paper, the authors apply Petersen's lemma to the problem of data-driven control with noisy data and obtain necessary and sufficient conditions for linear systems through a linear matrix inequality.
Abstract: We address the problem of designing a stabilizing closed-loop control law directly from input and state measurements collected in an open-loop experiment. In the presence of noise in data, we have that a set of dynamics could have generated the collected data and we need the designed controller to stabilize such set of data-consistent dynamics robustly. For this problem of data-driven control with noisy data, we advocate the use of a popular tool from robust control, Petersen's lemma. In the cases of data generated by linear and polynomial systems, we conveniently express the uncertainty captured in the set of data-consistent dynamics through a matrix ellipsoid, and we show that a specific form of this matrix ellipsoid makes it possible to apply Petersen's lemma to all of the mentioned cases. In this way, we obtain necessary and sufficient conditions for data-driven stabilization of linear systems through a linear matrix inequality. The matrix ellipsoid representation enables insights and interpretations of the designed control laws. In the same way, we also obtain sufficient conditions for data-driven stabilization of polynomial systems through (convex) sum-of-squares programs. The findings are illustrated numerically.
6 citations
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10 Jun 2021
TL;DR: In this article, a robust bilevel formulation for data-driven adaptive building control with measured process noise and unknown measurement noise via a robust BLEW formulation is proposed. But the authors do not consider the effect of noise on the prediction quality.
Abstract: In the era of digitalization, utilization of data-driven control approaches to minimize energy consumption of residential/commercial building is of far-reaching significance. Meanwhile, A number of recent approaches based on the application of Willems' fundamental lemma for data-driven controller design from input/output measurements are very promising for deterministic LTI systems. This paper addresses the key noise-free assumption, and extends these data-driven control schemes to adaptive building control with measured process noise and unknown measurement noise via a robust bilevel formulation, whose upper level ensures robustness and whose lower level guarantees prediction quality. Corresponding numerical improvements and an active excitation mechanism are proposed to enable a computationally efficient reliable operation. The efficacy of the proposed scheme is validated by a numerical example and a real-world experiment on a lecture hall on EPFL campus.
5 citations
References
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TL;DR: It is proved that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior.
615 citations
TL;DR: This paper proposes a multi-stage procedure that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate, and provides end-to-end bounds on the relative error in control cost.
Abstract: This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a quasi-convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.
489 citations
TL;DR: This research presents a meta-modelling framework that automates the very labor-intensive and therefore time-heavy and expensive process of manually cataloging and cataloging individual neurons to provide real-time information about their levels of activity.
Abstract: Recent successes in the field of machine learning, as well as the availability of increased sensing and computational capabilities in modern control systems, have led to a growing interest in learn...
405 citations
TL;DR: The presented results provide the first (theoretical) analysis of closed-loop properties, resulting from a simple, purely data-driven MPC scheme, including a slack variable with regularization in the cost.
Abstract: We propose a robust data-driven model predictive control (MPC) scheme to control linear time-invariant systems. The scheme uses an implicit model description based on behavioral systems theory and past measured trajectories. In particular, it does not require any prior identification step, but only an initially measured input–output trajectory as well as an upper bound on the order of the unknown system. First, we prove exponential stability of a nominal data-driven MPC scheme with terminal equality constraints in the case of no measurement noise. For bounded additive output measurement noise, we propose a robust modification of the scheme, including a slack variable with regularization in the cost. We prove that the application of this robust MPC scheme in a multistep fashion leads to practical exponential stability of the closed loop w.r.t. the noise level. The presented results provide the first (theoretical) analysis of closed-loop properties, resulting from a simple, purely data-driven MPC scheme.
381 citations