Continuous-Time Optimization Control With Finite-Time Convergence
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Solution Overview
Problem
Current optimization algorithms lack efficient convergence to optimal solutions in finite time, especially in real-life applications like robotics and artificial intelligence, and are not robust against uncertainties and time-varying cost functions.
Innovation Solution
Designing a family of discontinuous flows based on Lyapunov theory for finite-time convergence, incorporating differential inclusions and barrier Lyapunov functions, which allows for robustification against uncertainties and handling of time-varying cost functions, and extending these methods to constrained optimization problems.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Loss of time
If conventional optimization algorithms are used, then the system can find optimal solutions, but the convergence time is not finite and computation load is high
Solution Approach 1:
The patent transforms the optimization problem by changing the parameterization of the solution trajectory. Instead of directly optimizing parameters, it uses time-scaling transformations and reparameterization techniques to convert the original optimization problem into an equivalent form that can be solved by integrating a differential inclusion over a fixed time interval, achieving finite-time convergence with reduced computation.
Solution Approach 2:
The patent replaces traditional iterative numerical optimization methods with a dynamical systems approach. By formulating the optimization problem as a differential inclusion and using Lyapunov-based control theory, it substitutes mechanical iterative computation with a continuous-time dynamical system that naturally converges in finite time, reducing overall computation load.
2Speed
If discontinuous flows are used for finite-time convergence, then convergence speed is improved, but robustness against uncertainties deteriorates
Solution Approach 1:
The patent introduces a differential inclusion as an intermediary mathematical framework that bridges the gap between discontinuous flows and robustness requirements. The differential inclusion formulation with set-valued right-hand side allows the system to handle uncertainties and disturbances while maintaining finite-time convergence properties, acting as a mediator between speed and reliability requirements.
Solution Approach 2:
The patent incorporates robustification by designing the discontinuous flow to account for bounded additive uncertainties in advance. By using Lyapunov-based analysis and differential inclusions that explicitly consider uncertainty bounds, the system prepares for potential disturbances beforehand, ensuring that finite-time convergence is maintained even when uncertainties are present.
3Loss of time
If Lyapunov-based finite-time control is applied, then finite-time convergence is achieved, but handling of time-varying cost functions and constraints becomes complex
Solution Approach 1:
The patent creates a universal framework using differential inclusions that can handle multiple types of optimization problems simultaneously - including time-varying cost functions, equality constraints, and inequality constraints. The barrier Lyapunov function approach provides a unified method that works across different problem types, reducing the complexity that would otherwise arise from treating each case separately.
Solution Approach 2:
The patent employs asymmetric barrier Lyapunov functions that are designed to handle constraints in a non-uniform manner. By using asymmetric barrier functions that grow differently in different directions, the method efficiently handles inequality constraints and time-varying cost functions without requiring symmetric treatment of all constraints, simplifying the overall complexity.
Data Source
AI summary
A controller for controlling a system is provided. The controller performs measuring variables via an interface to generate a vector of variables, providing a cost function, with respect to the system, based on the vector variables using weighting factors, wherein the vector variables are represented by a time-step, computing first-derivative of the cost function at an initial time-step, obtaining a convergence time from the first-derivative of the cost function, computing second derivative of the cost function and generating an optimization differential equation based on the first and second derivatives of the cost function, proceeding, starting with the initial time-step, to obtain a value of the optimization differential equation by solving the optimization differential equation, in an iteration manner, with a predetermined time step being multiplied with the value of the solved differential equation to obtain next vector variables corresponding to a next iteration time-step, until the time-step reaches the convergence time, and outputting optimal values of the vector of variables and the cost function.