Extremum Seeking Control with State-Dependent Perturbation Scaling
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Solution Overview
Problem
Conventional extremum seeking control systems face challenges in achieving fast convergence to the correct extremum while maintaining stability, often resulting in oscillations and losses due to fixed perturbation signal amplitudes, which can fail to excite the plant effectively, especially in applications with rapid variations like photovoltaic cells.
Innovation Solution
A state-dependent parameter extremum seeking control system that adjusts the perturbation signal amplitude based on the variance of the state signal, using a scaling unit to multiply the initial amplitude by a factor proportional to e^(-var(x)) to reduce oscillations and losses once the desired value is approached, ensuring sufficient excitation without introducing instability.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Speed
If a fixed amplitude perturbation signal is used in extremum seeking control, then the system can maintain stability and simplicity, but the convergence speed is reduced and oscillations persist even near the extremum
Solution Approach 1:
The perturbation signal amplitude is made dynamic rather than fixed. The scaling unit continuously adjusts the amplitude based on real-time feedback from the plant output, increasing amplitude when far from extremum for fast convergence and decreasing amplitude when near extremum to reduce oscillations. This dynamic adaptation resolves the contradiction between convergence speed and stability.
Solution Approach 2:
A feedback mechanism is introduced where the plant output is continuously monitored and fed back to the scaling unit. The scaling unit uses this feedback to compute the appropriate perturbation amplitude, creating a closed-loop system that automatically adjusts excitation levels based on proximity to the extremum, thereby achieving both fast convergence and stability.
2Speed
If a high amplitude perturbation signal is used, then faster convergence to the extremum is achieved, but oscillations and energy losses increase
Solution Approach 1:
The perturbation amplitude transitions from a static high value to a dynamic value that automatically decreases as the system approaches the extremum. This dynamic reduction minimizes unnecessary oscillations and energy losses during the steady-state operation while maintaining high convergence speed during the transient phase.
Solution Approach 2:
The amplitude parameter of the perturbation signal is changed from a constant design-time parameter to a runtime-adjustable parameter. The scaling unit modifies this parameter based on the instantaneous system state, allowing the system to operate with high amplitude during convergence and low amplitude during steady-state, thereby reducing energy losses.
3Loss of energy
If a low amplitude perturbation signal is used, then oscillations and energy losses are reduced, but the plant may fail to be excited effectively
Solution Approach 1:
The perturbation amplitude is dynamically adjusted to match the system's needs: high when plant excitation is required for convergence, and low when the plant is already near the extremum. This dynamic adaptation ensures effective plant excitation only when necessary, reducing energy losses during steady-state operation.
4Device complexity
If conventional extremum seeking control is used, then the system structure remains simple, but the ability to handle rapid variations in plant output is limited
Solution Approach 1:
A feedback loop is added to continuously monitor plant output and adjust perturbation amplitude in real-time. This feedback mechanism enables the system to respond rapidly to variations in plant behavior while maintaining a relatively simple overall structure, as the additional components are focused specifically on amplitude adaptation.
Data Source
AI summary
A control system (1) for controlling a plant (2) comprises a feedback loop including an integrator (7); a signal generator (32); and a scaling unit (10). The feedback loop comprises an input suitable for connection to an output (18) of the plant. The integrator integrates a signal received from the input to generate a state signal x. The signal generator generates a periodic base perturbation signal (34) with an initial amplitude. The scaling unit generates a scaling factor (30) having a first value if the variance of the state signal var(x) is zero, or a second value if the variance of the state signal is non-zero, wherein the second value is proportional to (formulae 1) The scaling unit is arranged to multiply (16) the initial amplitude of the periodic base perturbation signal by the scaling factor to produce a state dependent perturbation signal (35, 36), which is applied to an input of the plant.