Iterative Chebyshev approximation method for optimal control problems

Area of Science:

• Numerical Analysis
• Control Theory

Background:

• Nonlinear constrained optimal control problems (NCOCPs) are complex to solve directly.
• Conventional collocation methods can introduce errors at non-collocation points.

Purpose of the Study:

• To develop a novel numerical approach for solving NCOCPs with high precision.
• To reduce approximation errors inherent in existing methods.

Main Methods:

• Linearize constraints and dynamic systems to create sub-problems.
• Employ Chebyshev polynomials to estimate state and control vectors.
• Estimate coefficient functions using Chebyshev polynomials to eliminate non-collocation errors.

Main Results:

• The proposed method transforms sub-problems into nonlinear optimization problems with linear equality constraints.
• Achieved lower approximation error compared to the Chebyshev pseudo-spectral method.
• Demonstrated efficacy through three example problems.

Conclusions:

• The novel method offers a high-precision approximation for NCOCPs.
• Suitable for applications demanding accuracy, like aerospace and precision manufacturing.