Extremum-Seeking Control with Adaptive Step-Size Convergence
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Solution Overview
Problem
Extremum seeking controllers (ESCs) face slow convergence to optimal equilibrium due to the separation of time-scales between system dynamics and gradient estimation, which hinders real-time optimization of dynamic systems.
Innovation Solution
Implementing a step-size adaptation mechanism based on the quality of the estimated gradient, using a noise-to-signal ratio to adjust the step-size, allowing the ESC to switch between exploration and exploitation modes, and incorporating tracking error and system output changes to reduce the need for time-scale separation, thereby accelerating convergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If separation of time-scales is used between system dynamics and gradient estimation, then the system can settle and cost can be measured near desired equilibrium, but the system exhibits slow convergence to the desired equilibrium
Solution Approach 1:
The patent applies dynamics by making the step-size adaptive rather than fixed. The step-size adaptation mechanism dynamically adjusts the exploration-exploitation balance based on the estimated gradient quality and system state, allowing the system to converge faster while maintaining measurement accuracy. This resolves the contradiction by enabling the system to operate in different regimes (fast convergence vs. accurate measurement) as needed during the optimization process.
Solution Approach 2:
The patent changes the parameter of step-size from a fixed value to an adaptive parameter that evolves over time. By modifying the step-size based on gradient estimates and system performance, the system can achieve faster convergence初期 while ensuring accurate cost measurements later, thus resolving the time-accuracy tradeoff inherent in fixed step-size approaches.
2Measurement precision
If gradient estimator has slower bandwidth than the system, then the system can settle and cost can be measured, but convergence to desired equilibrium is slow
Solution Approach 1:
The patent makes the gradient estimation process dynamic through step-size adaptation. Instead of using a fixed slow bandwidth gradient estimator, the system adapts its estimation speed and accuracy requirements based on the current optimization state, allowing faster convergence while maintaining measurement precision when needed.
Solution Approach 2:
The patent implements feedback by using the estimated gradient and system state information to continuously adjust the step-size. This feedback mechanism allows the system to accelerate convergence when the gradient estimate is reliable while ensuring accurate cost measurements when the system is near equilibrium, thus resolving the speed-precision contradiction.
3Reliability
If controller has slower bandwidth than the gradient estimator, then the gradient estimate can converge, but the system exhibits slow convergence to the desired equilibrium
Solution Approach 1:
The patent resolves this contradiction by making the controller bandwidth dynamic through adaptive step-size. The effective controller bandwidth adjusts based on the gradient estimate quality and system state, allowing the system to achieve both reliable gradient convergence and fast equilibrium convergence by operating in different regimes as needed.
Data Source
AI summary
An extremum seeking control system and a method for controlling a system is provided. The method comprises receiving a tracking error, consecutive measurements of a system output, and consecutive measurements of a cost function of the system, and estimating a gradient based on the tracking error, the consecutive measurements of the system output, and the consecutive measurements of the cost function of the system. The method further comprises determining a step-size based on a ratio of a covariance of the gradient to the gradient, and determining a set-point optimizing performance of the system, based on the gradient and the step-size. The method further comprises controlling the system based on the set-point optimizing performance of the system.