Chapter 0: Mathematical Foundations for Control Co-Design
The notation and calculus used throughout the course¶
Control co-design combines dynamic models, time histories, objectives, constraints, and sensitivities in a single design process. This short chapter reviews the mathematical language needed to read those formulations confidently. It is a refresher rather than a substitute for courses in calculus, differential equations, linear algebra, control, or optimization.
Learning objectives¶
After completing this chapter, you should be able to:
interpret indexed sums used for sampled trajectories, operating conditions, and aggregate objectives;
compute and interpret means, variances, and root-mean-square engineering metrics;
use first and second derivatives to describe rates, sensitivities, curvature, and local optima;
apply the product and chain rules to nested engineering models; and
recognize why differentiability and smoothness affect numerical optimization.
Why these ideas matter in CCD¶
A continuous objective such as
is often approximated on a time mesh by a weighted sum. Derivatives then describe how that objective and its constraints respond when a plant variable, controller parameter, state, or control value changes. These two ideas—aggregation and sensitivity—reappear in direct transcription, gradient-based optimization, uncertainty propagation, mesh convergence, and design interpretation.
Chapter map¶
Section 0.1 reviews summation notation and statistical summaries used for trajectories and uncertainty. Section 0.2 reviews derivatives, extrema, curvature, and smoothness for dynamic modeling and optimization. Linear algebra and differential-equation concepts are introduced where they are needed in Chapter 2 and revisited more systematically in later computational chapters.