Worked CCD Formulations

Example 4.1: mass–spring–damper CCD¶

Choose spring stiffness $k$ and a proportional–derivative controller

to reduce motion and effort for a system with fixed $m$ and $c$.

Decisions:

States and dynamics: With $x_1=x$ and $x_2=\dot{x}$,

Substituting the controller produces a closed-loop model in $k$, $K_p$, and $K_d$.

Objective:

Bounds and path constraint:

This is a complete, if small, CCD formulation.

Example 4.2: active suspension¶

For a quarter-car suspension, let

A possible objective is

where $a_b$ is body acceleration and $z_s$ is suspension deflection. The quarter-car equations enforce the dynamics; path constraints can limit suspension travel, tire load, and actuator force. Plant and controller variables are optimized together while the state trajectory obeys the model.

Example 4.2 revisited: physically concrete plant constraints¶

The abstract stiffness-and-damping version above is a reasonable first pass, but real suspension co-design studies replace $k_s$ and $c_s$ with the physical geometry that produces them, and replace the generic path constraint $\mathbf{g}(\cdot)\leq\mathbf{0}$ with a list of named engineering limits. In one such study, the quarter-car plant vector is the helical-spring and telescopic-damper geometry

where $d$ is spring wire diameter, $D$ is spring coil diameter, $p$ is spring pitch, $N_a$ is the number of active coils, $D_0$ is the damper valve diameter, $D_p$ is the damper piston diameter, and $D_s$ is the damper stroke; spring stiffness $k_s$ and damping rate $c_s$ become dependent quantities computed from this geometry rather than independent design variables. The Lagrange objective weighs road holding, ride comfort, and control cost,

with weights on the order of $r_1=10^5$, $r_2=0.5$, and $r_3=10^{-5}$ chosen so that all three terms have comparable numerical magnitude—an ordinary but essential scaling step for any multi-term Lagrange objective.

The physical-constraint set $\mathbf{g}_p(\mathbf{x}_p)\leq\mathbf{0}$ for this system is not one generic bound but a list of named engineering limits, including:

• a buckling constraint relating free spring length to coil diameter;

• packaging constraints requiring the spring and damper to fit within a fixed pocket length (about 0.40 m) and outer diameter, with clearance around the coaxial damper;

• a rattlespace constraint bounding peak suspension travel, evaluated from a full vehicle simulation over a ramp road input rather than from a closed-form bound;

• a spring stress and fatigue constraint using a Soderberg criterion that combines mean and alternating shear stress computed from the mean and amplitude of the axial spring force under a rough-road simulation;

• a damper thermal constraint limiting the maximum damper fluid temperature (about 390 K), predicted by a lumped heat-transfer model of the damper fluid, steel housing, and surrounding air, to prevent fluid fade and seal damage;

• a damper pressure constraint limiting maximum damper pressure (about $4.75\times10^{6}$ Pa) to protect the seals; and

• a damper velocity constraint limiting piston velocity to protect internal valve components.

Several of these—rattlespace, fatigue, and the thermal limit—depend on the full state trajectory under a specified road disturbance, not merely on $\mathbf{x}_p$ in isolation, so they are best classified as path constraints evaluated along a simulated response rather than as simple bounds. The general lesson is that a physically faithful CCD formulation of even a “simple” quarter-car suspension has more than a dozen named engineering constraints, most of them unit-bearing and traceable to a specific failure mode, rather than one abstract $\mathbf{g}(\cdot)\leq\mathbf{0}$.

Example 4.3: classify an objective¶

Suppose

where $P(t_f)$ is final delivered energy, $\theta_p$ is blade-pitch angle, and $M_t$ is tower-base moment. The terminal term $-P(t_f)$ is Mayer; the integral is Lagrange; together they form a Bolza objective.