3.1 Design Variables, Objectives, and Constraints

From an engineering question to a mathematical problem¶

A useful optimization model separates what the designer may change, what must be made small or large, and what must remain acceptable. In compact form,

\begin{aligned}\min_z\quad & J(z)\\ \text{subject to}\quad & g(z)\le 0,\\ & h(z)=0,\\ & z^L\le z\le z^U.\end{aligned}

The vector $z$ contains design variables. The scalar $J$ is the objective. The functions $g$ and $h$ describe inequality and equality constraints. Bounds express simple limits on individual variables.

For the active suspension, choose

z=[k,\ F_{\max},\ K]^T,

where spring stiffness and actuator rating are plant decisions and $K$ is a feedback gain. One system objective is

J=w_rJ_{\mathrm{ride}}+w_eJ_{\mathrm{energy}}+w_m m_a(F_{\max}).

A displacement requirement may be written

g_1(z)=\max_{t\in[0,T]}|x(t;z)|-x_{\max}\le0.

This single constraint already exposes CCD coupling: $k$ changes the dynamics, $K$ changes feedback, and $F_{\max}$ limits the control that can actually be delivered.

Models are not design variables¶

A simulation output is not automatically a decision. States, forces, temperatures, and stresses are analysis quantities unless the formulation intentionally promotes their discretized values to optimization variables, as direct transcription does. Likewise, an environmental input is usually problem data, not a design variable.

Requirements are not all objectives¶

Turning every requirement into a weighted objective can obscure feasibility and make weights difficult to defend. A safety limit, packaging envelope, or actuator saturation level is usually clearer as a constraint. Use an objective for preference; use a constraint for acceptability.

Activity 3.1: classify the formulation elements¶

Activity 3.1: classify the formulation elements

For an active suspension, classify road elevation, spring stiffness, body displacement, controller gain, actuator force limit, ride RMS, and maximum suspension travel as design variables, states or outputs, problem data, objectives, or constraints. Identify any item whose role could change under a different formulation.

Quantitative audit for Design Variables, Objectives, and Constraints: compare the following two feasible-design candidates.

Candidate	Plant loss $J_p$	Control loss $J_c$	$g_{\max}$	Evaluations
A	1.09	0.72	0.96	130
B	0.92	0.84	0.91	275

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.