Cheap Control

Cheap control is the control-theoretic problem of designing a controller that achieves satisfactory performance with minimal control effort — or, more precisely, with control effort that is small relative to the system's natural dynamics. In singular perturbation terms, cheap control is the limit where the control cost coefficient approaches zero, making the controller infinitely fast relative to the plant. The controller becomes the fast subsystem, and the plant becomes the slow subsystem.

The problem was first formulated in the context of linear-quadratic regulators, where the cheap control limit reveals the structural properties of the system that are independent of control cost. In the limit, the controller drives the controlled outputs to zero instantaneously, and the remaining dynamics evolve on the nullspace of the output map. These remaining dynamics are the cheap control subspace, and they are the slow manifold of the controlled system.

The cheap control limit is not merely a mathematical curiosity. It is the formalization of the engineering intuition that a good controller should be faster than the plant it controls. The separation of timescales between controller and plant is a design goal, and singular perturbation theory provides the tools for analyzing the resulting two-timescale system. The fast controller dynamics stabilize the output; the slow plant dynamics govern the internal behavior.

In Robotics and Model Predictive Control, cheap control ideas appear in the design of high-gain feedback loops and in the analysis of control architectures with multiple time constants. The Receding Horizon Control framework is a form of cheap control in discrete time: the controller optimizes over a short horizon (fast) while the plant evolves over a long horizon (slow). The separation is not exact but approximate, and the approximation is justified by the same singular perturbation arguments that justify adiabatic elimination in physics.

The cheap control limit reveals that control is not about forcing a system to do what you want. It is about designing a fast subsystem that collapses the slow dynamics onto a manifold where the desired behavior is natural. The controller does not fight the plant; it redesigns the plant's slow manifold.