Q7Control System

Regulator Problem

In the regulator problem, the desired state (or output) of the system is fixed, typically taken as the origin (zero) after a coordinate shift, and the objective of the controller is to bring the system state back to this equilibrium and hold it there despite initial disturbances or perturbations, while minimizing a cost functional that penalizes deviation of the state and the control effort used, e.g.:

This leads to the classical Linear Quadratic Regulator (LQR) formulation, where the optimal control is a linear state-feedback law u = -Kx obtained from the algebraic Riccati equation. Typical applications include stabilizing an aircraft at a fixed attitude, maintaining a fixed process temperature/pressure set-point in an industrial plant, or keeping an inverted pendulum balanced upright — situations where the reference is constant (often zero after shifting coordinates) and the concern is rejecting disturbances and returning to equilibrium.

Tracking Problem

In the tracking problem, the desired output is not fixed but is a prescribed, generally time-varying reference signal r(t) (e.g. a ramp, sinusoid, or an arbitrary trajectory), and the controller's objective is to force the actual output y(t) to follow r(t) as closely as possible over time, minimizing the tracking error e(t) = r(t) - y(t), typically via a cost functional such as:

This is solved by augmenting the state with the reference dynamics (or using a servo/tracking-augmented Riccati formulation, or a two-degree-of-freedom controller with feedforward plus feedback), so that the resulting control law includes both a state-feedback term (for stability/disturbance rejection) and a feedforward term driven by the reference trajectory. Typical applications include a robot arm following a prescribed path, an aircraft autopilot following a commanded flight path, or a CNC machine tool following a programmed cutting trajectory — situations where the target itself is moving and the system must dynamically track it rather than merely hold a fixed point.

In essence, the regulator problem is a special (degenerate) case of the tracking problem where the reference r(t) = 0 (or constant) for all time; the general tracking problem is more complex because it must additionally handle the dynamics and frequency content of a time-varying reference, often requiring internal-model-principle-based designs (e.g. including an integrator to track step references with zero steady-state error, or a resonant/internal model to track sinusoidal references exactly).