Types of control objectives
A feedback controller is designed to meet a goal, and the goals have different names. Stabilization makes an equilibrium asymptotically stable. Regulation holds an output at a constant set point despite disturbances. Tracking makes an output follow a reference that changes over time.
Definitions
Consider a system that does not stay at a desired resting state by itself, like a broom balanced on your hand. Stabilization is designing a controller that makes the system settle at that resting state and stay near it.
Consider a control system with an equilibrium (or a set) that is to be made stable. Stabilization is the control problem of finding a feedback law that makes the equilibrium of the closed-loop system asymptotically stable.
Examples
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Balancing an inverted pendulum upright
Balancing the pendulum is stabilization because the upright position is an unstable equilibrium that feedback must make asymptotically stable.
Missing definition: regulation
Tracking is making a quantity follow a changing target, as a driver steers a car to follow a road.
In control theory, tracking is a control problem with the goal of making the output $y(t)$ follow a time-varying reference $r(t)$, so that the tracking error
\[e(t) = r(t) - y(t)\]converges to zero (or stays small).
Examples
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A robot arm following a moving conveyor belt
Following the conveyor belt is tracking because the reference position changes over time.
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A spacecraft following a planned trajectory
Following a planned trajectory is tracking because the reference is a time-varying path.
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An aircraft approaching a tanker aircraft and then holding position behind the tanker to refuel in flight
An aircraft refueling in flight solves a tracking problem while refueling, because the aircraft must hold its position relative to the moving tanker at every time.
Non-examples
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Cruise control holding 100 km/h on hilly roads
Holding $100$ km/h is not tracking, because the reference does not change.
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Balancing an inverted pendulum upright
Balancing an inverted pendulum upright is not tracking, because the reference does not change over time.
Properties
- Tracking a reference can be posed as stabilizing the origin of the error dynamics.
Examples
| Example | stabilization | regulation | tracking | Why |
|---|---|---|---|---|
| Balancing an inverted pendulum upright | ✓ | ✓ | ✗ | |
| Cruise control holding 100 km/h on hilly roads | – | ✓ | ✗ | |
| A robot arm following a moving conveyor belt | – | ✗ | ✓ | |
| A spacecraft following a planned trajectory | – | ✗ | ✓ |
Notes
- Tracking can be posed as stabilization: the goal is to make the origin of the error dynamics, in $e = r - y$, asymptotically stable. For a time-varying reference, the error dynamics are time-varying.
- In output regulation theory (Francis and Wonham), “regulation” includes tracking references and rejecting disturbances generated by a known exosystem. The internal model principle says the controller must contain a model of that exosystem, such as an integrator for constant references.