Original Pseudospectral Method Code Implementation

The pseudospectral method is a widely used numerical approach for solving optimal control problems, particularly effective for handling complex multi-phase optimal control scenarios. The core concept involves transforming continuous-time optimal control problems into nonlinear programming (NLP) problems through discretization and polynomial approximation for efficient solution.

### Fundamental Principles of Pseudospectral Method The pseudospectral method employs orthogonal polynomials (such as Chebyshev or Legendre polynomials) as basis functions for numerical approximation at collocation points. The implementation typically follows these steps: Time Discretization: Dividing the continuous time domain into multiple segments, especially suitable for multi-phase problems. State and Control Variable Parameterization: Using polynomial approximations to represent states and control variables, thereby reducing optimization variable dimensions. Dynamic Constraint Transformation: Converting differential equation constraints into algebraic constraints through differentiation matrices for numerical processing. Objective Function and Constraint Optimization: Combining Gaussian quadrature for numerical integration to formulate the final nonlinear programming problem.

### hp-Adaptive Strategy and Gaussian Quadrature The hp-adaptive strategy combines mesh refinement (h-refinement) and polynomial order enhancement (p-refinement), dynamically adjusting computational precision based on local problem characteristics to improve solution efficiency. Gaussian quadrature ensures accurate computation of integral-type objective functions, guaranteeing numerical solution precision.

### MATLAB Implementation Advantages MATLAB-based pseudospectral method frameworks typically integrate optimization toolboxes (such as fmincon) to efficiently handle nonlinear programming problems. Implementation often involves key functions like: - `chebfun` or `legpoly` for polynomial basis generation - Collocation point calculation using `chebpts` or similar functions - Differentiation matrix computation via `diffmat` - Gaussian quadrature weights calculation for accurate integration This software approach is particularly suitable for aerospace trajectory optimization, robotic path planning, and represents a classical optimal control solution tool.

Due to its high precision and computational efficiency, the pseudospectral method has become essential in optimal control领域, while the combination of hp-adaptive techniques and Gaussian quadrature further enhances its applicability and reliability.