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This paper derives fundamental limits on disturbance rejection for linear time-invariant (LTI) feedback systems with bandwidth and robustness constraints. First, we provide a closed-form expression for the minimal, achievable output energy for a given disturbance energy spectral density along with the corresponding optimal sensitivity gain. This fundamental performance limit is based on a finite-band Bode sensitivity integral. Second, we present a numerical method to compute the fundamental performance limit when the disturbance is realized from a finite set of possible values. This numerical method also incorporates robustness constraints on the peak sensitivity. Finally, we provide a simple example to illustrate these results. These fundamental performance limits provide a benchmark for the performance that can be achieved by LTI control. They can be used to determine when LTI controllers are insufficient and hence motivate the use of more advanced nonlinear and/or time-varying controllers. 

Disturbance rejection in high-precision control applications can be significantly improved upon via online convex optimization (OCO). This includes classical techniques such as recursive least squares (RLS) and more recent, regret-based formulations. However, these methods can cause instabilities in the presence of model uncertainty. This paper introduces a safety filter for systems with OCO in the form of adaptive finite impulse response (FIR) filtering to ensure robust disturbance rejection. The safety filter enforces a robust stability constraint on the FIR coefficients while minimally altering the OCO command in the ∞-norm cost. Additionally, we show that the induced 𝓁∞-norm allows for easy online implementation of the safety filter by directly limiting the OCO command. The constraint can be tuned to trade off robustness and performance. We provide a simple example to demonstrate the safety filter. 

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.