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A Framework for Continuous-Time Optimization with Time-Varying Cost Functions using Derivative Estimation


Core Concepts
A framework for adapting continuous-time optimization algorithms to handle time-varying cost functions by incorporating a derivative estimation scheme, providing robust performance guarantees.
Abstract
The paper presents a framework for adapting continuous-time optimization algorithms to handle time-varying cost functions. The key insights are: Lemma 1 shows how a continuous-time optimization algorithm that is input-to-state stable (ISS) for static cost functions can be adapted to handle time-varying cost functions, provided the time variations are known in the form of the derivative of the parameter vector. Theorem 1 introduces a novel derivative estimation scheme based on the "dirty-derivative" concept, and provides explicit input-to-output stability (IOS) bounds on the estimation error. Theorem 2 combines the results of Lemma 1 and Theorem 1 to show that the interconnection of the adapted optimization algorithm and the derivative estimator also results in an IOS system, providing robust performance guarantees for tracking the time-varying minimizer. The framework allows incorporating time-variation information without requiring explicit knowledge of the derivative, and the performance can be improved by tuning the estimator's gain parameter. Simulation results demonstrate the effectiveness of the approach in a time-varying optimization task.
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Deeper Inquiries

How can the proposed framework be extended to handle time-varying constraints in the optimization problem?

In order to extend the proposed framework to handle time-varying constraints in the optimization problem, we can incorporate the time-varying nature of the constraints into the continuous-time trajectory tracking problem. This can be achieved by modifying the dynamics of the system to account for the time-varying constraints in addition to the time-varying cost function. The constraints can be included in the optimization problem as additional conditions that the solution trajectory must satisfy at each time instant. By formulating the constraints as part of the optimization problem, the framework can be adapted to handle time-varying constraints seamlessly. The analysis would involve ensuring that the system remains stable and converges to the optimal solution trajectory while satisfying the time-varying constraints.

How would the analysis and results change if the time-varying parameter vector θ(t) depends on the optimization variable x(t) in a closed-loop manner?

If the time-varying parameter vector θ(t) depends on the optimization variable x(t) in a closed-loop manner, the analysis and results of the framework would need to account for this feedback relationship. This closed-loop dependency introduces a more complex dynamic interplay between the optimization variable and the parameter vector. The stability and convergence analysis would need to consider the feedback loop and ensure that the system remains stable despite the interdependence between x(t) and θ(t). The results would likely show a more intricate relationship between the tracking error and the closed-loop dynamics, potentially leading to more stringent stability conditions and convergence guarantees.

Can the derivative estimation scheme be further improved to handle higher-order derivatives or non-smooth signals?

The derivative estimation scheme can indeed be further improved to handle higher-order derivatives or non-smooth signals by extending the recursive structure of the estimator to estimate higher-order derivatives. By increasing the order of the estimation scheme, the accuracy and convergence speed of the derivative estimates can be enhanced, allowing for more precise tracking of the derivatives of the input signal. Additionally, techniques from signal processing and system identification can be incorporated to handle non-smooth signals more effectively. By utilizing advanced filtering and estimation methods, the derivative estimation scheme can be optimized to handle a wider range of signals, including those with discontinuities or irregularities, improving the overall performance and robustness of the estimation process.
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