Gaussian Process Feedforward Control for Unstable Inverse Models
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Feedforward control systems for non-minimum phase systems face instability due to unstable poles in the inverse transfer function, leading to inaccurate tracking and potential oscillatory behavior, with existing techniques either ignoring these poles or relying on limited basis functions that fail when the system is adaptive or unstable.
Innovation Solution
A Gaussian process representation is used to model the system input, applying Bayes' theorem to determine an optimum input for a desired output trajectory, allowing for adaptive feedforward control even when the inverse transfer function is unstable, and providing robustness information through variance analysis.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If feedforward control uses inverse transfer function for non-minimum phase systems, then tracking accuracy is improved, but system stability deteriorates due to unstable poles outside the unit circle
Solution Approach 1:
The patent introduces an intermediary optimization process between the inverse transfer function and the control input. Instead of directly applying the unstable inverse transfer function, the patent uses a cost function with regularization terms that mediate the relationship, allowing the system to achieve accurate tracking while maintaining stability through constrained optimization.
Solution Approach 2:
The patent changes the parameters of the control input by introducing regularization parameters (lambda1, lambda2) in the cost function. These parameters allow the system to adjust the trade-off between tracking accuracy and control effort, effectively transforming the unstable direct inversion into a stable optimized solution.
2Stability of the object's composition
If unstable poles are ignored in the inverse transfer function, then system stability is maintained, but tracking accuracy deteriorates particularly when poles are close to the unit circle
Solution Approach 1:
The patent implements a feedback mechanism through the cost function that evaluates tracking error and penalizes it. This feedback loop allows the optimization process to account for unstable pole effects indirectly, maintaining stability while improving tracking accuracy by adjusting control inputs based on performance metrics.
Solution Approach 2:
The patent performs preliminary optimization of the control input before application to the system. By pre-computing the optimized input that considers both stability and accuracy requirements through the cost function, the system avoids the need to directly handle unstable poles while still achieving accurate tracking.
3Measurement precision
If specialized basis functions are used to handle unstable poles, then tracking accuracy is improved for specific cases, but device complexity increases and applicability is limited
Solution Approach 1:
The patent creates a universal solution that works for both minimum-phase and non-minimum-phase systems through the generalized cost function formulation. The same optimization framework handles diverse system types without requiring specialized basis functions, reducing complexity while maintaining broad applicability and accuracy.
Data Source
AI summary
Feedforward control uses a Gaussian process representation of system input to derive an optimum input for a desired output trajectory over discrete sample times. A model provides a forward transfer function. A kernel function, having selectable parameters, provides a prior covariance matrix of the Gaussian process. A posterior mean of the system input can be derived by Bayes' theorem, based on the model, the kernel function, and the desired output trajectory. The posterior mean predicts system output and, thereby, tracking error. The kernel function parameters are selected to optimize the tracking error. The corresponding posterior mean provides optimized system input for the desired output trajectory. Disclosed techniques are applicable even when model inversion is unstable. Disclosed techniques can be applied in segments, saving computation resources and also suitable for adaptive control.