3.5 Equality and Inequality Constraints

Equalities define consistency¶

An equality constraint $h(z)=0$ usually expresses a law or required consistency: static equilibrium, a geometric relation, an algebraic coupling between disciplines, or a discretized dynamic equation. In CCD, the state equation

\dot x-f(t,x,u,p)=0

is a time-dependent equality constraint. In a simultaneous transcription, its discrete defects appear explicitly in the nonlinear program.

Inequalities define acceptable sides¶

An inequality $g(z)\le0$ defines an allowed side of a boundary. Examples include

|x(t)|-x_{\max}\le0,\quad |u(t)|-F_{\max}\le0,\quad T(t)-T_{\max}\le0.

Absolute-value constraints are commonly implemented as two smooth inequalities. A constraint is active at a design when $g_i(z^*)=0$ and inactive when $g_i(z^*)<0$ .

Objective contours meeting active inequality constraints.

Active constraints explain why an optimum sits where it does. If actuator saturation and suspension travel are both active, more feedback gain cannot improve ride without increasing actuator size or changing the plant. The physical bottleneck is visible in the optimization structure.

Path, boundary, and design constraints¶

A path constraint must hold throughout time. A boundary constraint applies at the initial or final time. A design constraint depends only on time-independent variables. Checking a path limit only at a few output samples can miss an inter-sample violation.

Activity 3.5: write the constraints¶

Activity 3.5: write the constraints

Write smooth inequalities for $|x(t)|\le0.08$ m and $|u(t)|\le F_{\max}$ . Classify the dynamic equations, zero initial conditions, final settling requirement, actuator packaging limit, and stability margin as equality, inequality, path, boundary, or design constraints.

Quantitative audit for Equality and Inequality Constraints: compare the following two feasible-design candidates.

Candidate	Plant loss $J_p$	Control loss $J_c$	$g_{\max}$	Evaluations
A	1.13	0.72	0.96	170
B	0.94	0.84	0.91	335

Use $J=0.6J_p+0.4J_c$ and require $g_{\max}\leq1$ .

Calculate objective values, feasibility margins, and percentage improvement.
Increase B plant loss by 8% for independent validation and repeat the ranking.
Compute objective improvement per 100 additional evaluations and state which result you would retain.