Chapter 7: Numerical Methods for Control Co-Design

Discretization, transcription, sparse optimization, derivatives, and verification

The numerical method, mesh, scaling, derivatives, and verification procedure can change the design reported as optimal.

CCD problems are often infinite-dimensional because states and controls are functions of time. Computers cannot optimize arbitrary continuous functions directly. Numerical methods replace them with finite variables and algebraic constraints, producing a nonlinear program (NLP).

This chapter develops time discretization, direct and multiple shooting, direct transcription, collocation and defect constraints, sparse NLP structure, derivative computation, scaling, initialization, mesh refinement, and numerical verification.

Learning objectives¶

After completing this chapter, you should be able to:

1. convert a continuous CCD problem into a finite NLP;

2. construct and assess a computational mesh;

3. compare direct shooting, multiple shooting, and transcription;

4. write Euler, trapezoidal, and Hermite–Simpson defects;

5. explain direct-transcription sparsity;

6. compare numerical and exact derivative methods;

7. perform mesh-refinement and independent-simulation checks; and

8. organize a reliable CCD implementation workflow.

Chapter roadmap¶

Continuous problem $\rightarrow$ Mesh $\rightarrow$ Shooting or transcription $\rightarrow$ Sparse NLP $\rightarrow$ Derivatives $\rightarrow$ Refinement and verification