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1. Sample Complexity Bound for Evaluating the Robust Observer’s Performance under Coprime Factors Uncertainty

   Sabau, Serban; Zhang, Yifei; Ukil, Sourav Kumar (June 2023, Proceedings of Machine Learning Research)

   Lawrence, N (Ed.)

   This paper addresses the end-to-end sample complexity bound for learning in closed loop the state estimator-based robust H2 controller for an unknown (possibly unstable) Linear Time Invariant (LTI) system, when given a fixed state-feedback gain. We build on the results from Ding et al. (1994) to bridge the gap between the parameterization of all state-estimators and the celebrated Youla parameterization. Refitting the expression of the relevant closed loop allows for the optimal linear observer problem given a fixed state feedback gain to be recast as a convex problem in the Youla parameter. The robust synthesis procedure is performed by considering bounded additive model uncertainty on the coprime factors of the plant, such that a min-max optimization problem is formulated for the robust H2 controller via an observer approach. The closed-loop identification scheme follows Zhang et al. (2021), where the nominal model of the true plant is identified by constructing a Hankel-like matrix from a single time-series of noisy, finite length input-output data by using the ordinary least squares algorithm from Sarkar et al. (2020). Finally, a H∞ bound on the estimated model error is provided, as the robust synthesis procedure requires bounded additive uncertainty on the coprime factors of the model. Reference Zhang et al. (2022b) is the extended version of this paper. 

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2. Certified Control-Oriented Learning: A Coprime Factorization Approach

   https://doi.org/10.1109/CDC51059.2022.9992821  

   Singh, Rajiv; Sznaier, Mario (December 2022, IEEE)

   This paper considers the problem of learning models to be used for controller design. Using a simple example, it argues that in this scenario the objective should reflect the closed-loop, rather than open-loop distance between the learned model and the actual plant, a task that can be accomplished by using a gap metric motivated approach. This is particularly important when identifying open-loop unstable plants, since typically in this case the open-loop distance is unbounded. In this context, the paper proposes a convex optimization approach to learn its coprime factors. This approach has a dual advantage: (1) it can easily handle open-loop unstable plants, since the coprime factors are stable, and (2) it is "self certified", since a simple norm computation on the learned factors indicates whether or not a controller designed based on these factors will stabilize the actual (unknown) plant. If this test fails, it indicates that further learning is needed. 

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3. Robust Online Convex Optimization for Disturbance Rejection

   https://doi.org/10.1109/CDC56724.2024.10886354  

   Lai, Joyce; Seiler, Peter (December 2024, IEEE)

   Online convex optimization (OCO) is a powerful tool for learning sequential data, making it ideal for high precision control applications where the disturbances are arbitrary and unknown in advance. However, the ability of OCO-based controllers to accurately learn the disturbance while maintaining closed-loop stability relies on having an accurate model of the plant. This paper studies the performance of OCO-based controllers for linear time-invariant (LTI) systems subject to disturbance and model uncertainty. The model uncertainty can cause the closed-loop to become unstable. We provide a sufficient condition for robust stability based on the small gain theorem. This condition is easily incorporated as an on-line constraint in the OCO controller. Finally, we verify via numerical simulations that imposing the robust stability condition on the OCO controller ensures closed-loop stability. 

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4. Robust Empirical Risk Minimization with Tolerance

   Bhattacharjee, Robi; Hopkins, Max; Kumar, Akash; Yu, Hantao; Chaudhuri, Kamalika (February 2023, Algorithmic Learning Theory)

   Developing simple, sample-efficient learning algorithms for robust classification is a pressing issue in today's tech-dominated world, and current theoretical techniques requiring exponential sample complexity and complicated improper learning rules fall far from answering the need. In this work we study the fundamental paradigm of (robust) empirical risk minimization (RERM), a simple process in which the learner outputs any hypothesis minimizing its training error. RERM famously fails to robustly learn VC classes (Montasser et al., 2019a), a bound we show extends even to `nice' settings such as (bounded) halfspaces. As such, we study a recent relaxation of the robust model called tolerant robust learning (Ashtiani et al., 2022) where the output classifier is compared to the best achievable error over slightly larger perturbation sets. We show that under geometric niceness conditions, a natural tolerant variant of RERM is indeed sufficient for γ-tolerant robust learning VC classes over ℝd, and requires only Õ (VC(H)dlogDγδϵ2) samples for robustness regions of (maximum) diameter D. 

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5. Optimistic robust linear quadratic dual control

   Umenberger, J and (June 2020, Proceedings of the 2nd Conference on Learning for Dynamics and Control)

   Recent work by Mania et al. (2019) has proved that certainty equivalent control achieves nearly op- timal regret for linear systems with quadratic costs. However, when parameter uncertainty is large, certainty equivalence cannot be relied upon to stabilize the true, unknown system. In this paper, we present a dual control strategy that attempts to combine the performance of certainty equivalence, with the practical utility of robustness. The formulation preserves structure in the representation of parametric uncertainty, which allows the controller to target reduction of uncertainty in the parame- ters that ‘matter most’ for the control task, while robustly stabilizing the uncertain system. Control synthesis proceeds via convex optimization, and the method is illustrated on a numerical example. Keywords: Dual control, linear systems, convex optimization 

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