On the Sample Complexity of the Linear Quadratic Regulator

A convex parameterization of internally stabilizing controllers is fundamental for many controller synthesis procedures. The celebrated Youla parameterization relies on a doublycoprime factorization of the system, while the recent system-level and input-output characterizations require no doubly-coprime factorization but a set of equality constraints for achievable closed-loop responses. In this paper, we present explicit affine mappings among Youla, system-level and input-output parameterizations. Two direct implications of the affine mappings are 1) any convex problem in Youla, system level, or input-output parameters can be equivalently and convexly formulated in any other one of these frameworks, including the convex system-level synthesis (SLS); 2) the condition of quadratic invariance (QI) is sufficient and necessary for the classical distributed control problem to admit an equivalent convex reformulation in terms of Youla, system-level, or input-output parameters.

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“…Proof. We show that any controller in (29) is an internally stabilizing controller in (13). The other direction is similar.…”

Section: A Simplified Parameterizations For State Feedbackmentioning

confidence: 64%

“…Theorem 1 presents explicit affine mappings between Youla parameterization and IOP: any element in the Youla parameterization (5) corresponds to an element in the IOP (13), and they represent the same controller. The following result presents explicit affine mappings between SLP and IOP.…”

Section: A Explicit Equivalencementioning

confidence: 99%

On the Equivalence of Youla, System-Level, and Input–Output Parameterizations

Zheng

Furieri

Papachristodoulou

et al. 2021

IEEE Trans. Automat. Contr.

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“…Additionally, the data-driven LQR problem is popular in the machine learning community, where it is typically assumed that the system is influenced by (Gaussian) process noise, see e.g. [51].…”

Section: Future Workmentioning

confidence: 99%

Data Informativity: A New Perspective on Data-Driven Analysis and Control

Waarde

Eising

Trentelman

et al. 2020

IEEE Trans. Automat. Contr.

334

354

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The use of persistently exciting data has recently been popularized in the context of data-driven analysis and control. Such data have been used to assess system theoretic properties and to construct control laws, without using a system model. Persistency of excitation is a strong condition that also allows unique identification of the underlying dynamical system from the data within a given model class. In this paper, we develop a new framework in order to work with data that are not necessarily persistently exciting. Within this framework, we investigate necessary and sufficient conditions on the informativity of data for several data-driven analysis and control problems. For certain analysis and design problems, our results reveal that persistency of excitation is not necessary. In fact, in these cases data-driven analysis/control is possible while the combination of (unique) system identification and model-based control is not. For certain other control problems, our results justify the use of persistently exciting data as data-driven control is possible only with data that are informative for system identification.

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“…This is a classical problem, so there is a very large literature on robust stochastic control and its application to learning-theoretic methods; see e.g. [11,4,17,18,6,5,22,14,2,23]. A rather comprehensive literature review is presented in [13], and we refer the reader to the literature review therein.…”

Section: Literature Review and Contributionsmentioning

confidence: 99%

Robustness to Incorrect Models in Average-Cost Optimal Stochastic Control

Kara

Raginsky

Yüksel

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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We study continuity properties of infinite-horizon average expected cost problems with respect to (controlled) transition kernels, and applications of these results to the problem of robustness of control policies designed for incorrect models applied to systems with incomplete models. We show that sufficient conditions presented in the literature for discounted-cost problems are in general not sufficient to ensure robustness for average-cost problems. However, we show that the average optimal cost is continuous in the convergences of controlled transition kernel models where convergence of models entails (i) continuous weak convergence in state and actions, and (ii) continuous setwise convergence in the actions for every fixed state variable, in addition to uniform ergodicity and regularity conditions. Using such continuity results, we establish that the mismatch error due to the application of a control policy designed for an incorrectly estimated model to the true model decreases to zero as the incorrect model approaches the true model under the stated convergence criteria. Our findings significantly relax related studies in the literature which have primarily considered the more restrictive total variation convergence criteria.

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On the Sample Complexity of the Linear Quadratic Regulator

Cited by 341 publications

References 51 publications

On the Equivalence of Youla, System-Level, and Input–Output Parameterizations

On the Equivalence of Youla, System-Level, and Input–Output Parameterizations

Data Informativity: A New Perspective on Data-Driven Analysis and Control

Robustness to Incorrect Models in Average-Cost Optimal Stochastic Control

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