Examples and Comparison - Control Co-Design

Example 6.1: active suspension¶

For a known short road disturbance, OLOC can optimize actuator force $u(t)$ to reduce body acceleration and suspension deflection. This yields a best-case force history for that road profile. The result may reveal when strong actuation is needed. A final controller instead maps measured suspension deflection and body velocity to force, and its gains are tuned or co-designed in closed loop.

Example 6.2: wind turbine¶

Optimizing blade-pitch trajectory against a known wind trace gives an ideal load-reduction benchmark. A real turbine does not know the future gust exactly, so an implementable controller uses rotor-speed and load measurements, perhaps with preview. MPC or gain-scheduled feedback is a more realistic final architecture.

Example 6.3: marine energy device¶

OLOC can optimize PTO force for a recorded wave sequence and estimate theoretical energy capture. A practical controller must operate from measured or forecast waves. Closed-loop CCD redesigns geometry or PTO hardware together with that realizable controller.

Example 6.4: prediction horizon and architecture complexity in a quarter-car study¶

A study of six quarter-car active-suspension architectures makes the OLOC-versus-MPC trade-off concrete. Each architecture connects the road input to a single translational-force actuator through a different number of auxiliary masses, springs, and dampers—from a control-only case with no additional plant design variables up to a six-design-variable architecture. All six architectures share the objective

\psi_d=\int_{t_0}^{t_f} \left( w_1(z_U-\delta)^2 +w_2\ddot z_S^2 +w_3F^2 \right)dt,

(1)

combining road-handling performance, passenger comfort (sprung-mass acceleration), and a control-effort penalty. For each architecture, the plant variables and the controller’s prediction horizon $p$ are optimized together in a nested CCD loop—once with OLOC in the inner loop, and once with MPC at $p=40,\,10,\,5$ for a fixed control sampling time.

Two findings stand out:

Plant complexity cannot fully substitute for information. Moving from a simple two-design-variable architecture to the richest six-design-variable architecture markedly improves the open-loop and long-horizon objective values. Yet even the richer plant, evaluated with a short prediction horizon, remains far from its own open-loop bound—while still outperforming the simpler architecture operating fully open loop. A more sophisticated plant design can partially compensate for limited prediction capability, but the controller’s information horizon remains a first-order factor, not a detail to optimize last.
Disturbance frequency amplifies the OLOC/MPC gap. At low road-disturbance frequencies (around 2 Hz), OLOC- and MPC-based co-design produce objective values within about 4-5% of each other and nearly identical optimized plant designs. As the disturbance frequency rises toward 20 Hz, the objective-value gap exceeds 18% and the relative difference between the two optimized plant designs grows to roughly 650%. Faster disturbance content relative to the controller’s update rate makes the prediction-horizon and sampling-time choices matter far more.

Comparison¶

Approach	Main decision	Strength	Limitation
Open-loop optimal control	Full trajectory $\mathbf{u}(t)$	Flexible benchmark	May need unrealistic future information
Controller-parameter optimization	Gain or coefficient vector $\mathbf{x}_c$	Directly implementable	Limited by chosen structure
Feedback-controller CCD	Plant and controller parameters	Realistic closed-loop design	More restrictive than free trajectories
MPC	Repeated finite-horizon plans	Feedback with constraints and prediction	Requires online optimization and a good model

Common mistakes¶

Confusing a trajectory with a controller. A planned signal is not a feedback law.
Treating OLOC as automatically implementable. It may rely on impossible information.
Ignoring the real information set. Controllers use only measured, estimated, or credibly forecast quantities.
Assuming feedback removes all uncertainty. Feedback mitigates but does not eliminate noise and mismatch.
Calling MPC purely open loop. Repeated measurement-based replanning creates feedback.
Assuming parameterized controllers are always inferior. A realizable feedback policy can outperform a nonimplementable ideal signal on the uncertain real system.
Treating the MPC prediction horizon as fixed. Prediction horizon and control sampling time can be optimized alongside plant variables in a nested CCD loop; treating them as fixed implementation details after the fact overlooks a first-order design choice.