control//controller design//time-optimal control

2026-03-09

Time-optimal control is the problem of driving a system from one state to another in the shortest possible time with a bounded input, and its answer for most mechanical systems is to use the input at its limits only, switching from full effort one way to full effort the other at computed instants: the bang-bang solution. It sets the performance ceiling for any positioning task (a pick-and-place axis, a hard disk's head, a crane moving a load, a drone dashing between two hover points) and is the shape an MPC approaches when asked to arrive as soon as possible.

The simplest case shows everything. A cart on a rail is a double integrator, position driven by a force with acceleration bounded by aaa; to move a distance DDD and stop, it accelerates at full force for half the distance and brakes at full force for the other half. The time is

Tmin⁡=2D/a,T_{\min}=2\sqrt{D/a},Tmin=2D/a,

so 1 m at 2 m/s22\ \text{m/s}^22 m/s2 takes about 1.41 s, and no controller with the same force can do it faster. As feedback, the rule is a switching curve in the plane of position error and velocity: brake at full force when the remaining distance equals the stopping distance at the current speed, e=v∣v∣/(2a)e=v\lvert v\rvert/(2a)e=v∣v∣/(2a), push at full force otherwise. Pontryagin's maximum principle is the theory that proves the extremes are optimal, and for a linear system of order nnn with real poles it allows at most n−1n-1n−1 switches.

The optimum leaves no margin.

It uses the whole force for the whole move, so a load heavier than modelled, a motor slightly weaker or a delay in the switch makes it overshoot, with no authority left to correct; near the switching curve, noise makes it chatter between the extremes. Practice uses near-time-optimal versions: the profile softened into trapezoids or S-curves with a limit on jerk (minimum-snap trajectory in drones), or a linear controller taking over close to the target (disk drives switch to a linear loop for the last few tracks).

It shares only the word bang-bang with on-off control. A thermostat switches between its two states around a setpoint because its actuator has nothing in between, and it switches whenever the error leaves a band; time-optimal control switches a fully proportional actuator between its extremes at instants computed from a model, a few times per move.

It needs the limits to be real and known. The time-optimal plan is built on the actuator's saturation, which drifts with battery voltage, temperature and wear, so it is computed with a derated limit and the difference is the margin (actuator).

It sits in the controller design map as the limit case of constrained optimal control: an LQR penalizes effort and never saturates on purpose, an MPC with a time-weighted cost moves toward the bang-bang shape as the effort penalty goes to zero.