Tracking, Disturbance Rejection & the Internal Model Principle

Tracking makes an output follow a reference. Disturbance rejection prevents an unwanted input from changing that output. The two share a structural fact: persistent signals cannot be rejected robustly by accidental cancellation. The controller must reproduce the dynamics that generate the signal class.

Francis and Wonham's internal model principle states this precisely for robust regulation: a stabilizing controller that maintains regulation under plant perturbations must include a suitably replicated model of the reference/disturbance generator in the feedback path.

Integral action models constants #

A constant signal is generated by

$$\dot w=0.$$

An integrator has a pole at the origin and is therefore an internal model of constant references and disturbances. With a stable closed loop and sufficient actuator authority,

$$\dot z=r-y,\qquad u=K_p(r-y)+K_i z$$

can drive steady-state error to zero for a constant target or matched constant load disturbance.

That conclusion fails if the actuator saturates, the plant has an incompatible zero, the loop is unstable, the measured output is biased, or the disturbance enters a path the controller cannot influence.

State-space servo augmentation #

Integral action becomes an ordinary state when the controlled output is part of a state-space model. Assume $D=0$ and define

$$\dot x=Ax+Bu,\qquad y=Cx,$$

$$\dot\eta=y-r=Cx-r.$$

The sign convention makes positive $\eta$ mean accumulated output-above-reference error. With augmented state $x_a=[x^T\;\eta^T]^T$,

$$\dot x_a= \underbrace{\begin{bmatrix}A&0\\C&0\end{bmatrix}}_{A_a}x_a+ \underbrace{\begin{bmatrix}B\\0\end{bmatrix}}_{B_a}u+ \underbrace{\begin{bmatrix}0\\-I\end{bmatrix}}_{E_r}r.$$

Then choose

$$u=-K_xx-K_\eta\eta=-K_ax_a.$$

The regulator poles belong to $A_a-B_aK_a$. They may be assigned only if the augmented pair $(A_a,B_a)$ is controllable, or at least stabilizable for a stabilization objective. This test catches an important failure that “add an integrator” hides: a plant zero at the reference-generator mode can make the servo augmentation structurally impossible to stabilize.

For a discrete plant, an accumulated error state can be defined as

$$\eta_{k+1}=\eta_k+T_s(y_k-r_k),$$

or without $T_s$ if the gain convention absorbs it. The chosen convention changes the numerical integral gain and must be recorded. Delay, saturation, anti-windup, and reset remain part of the implemented augmented system.

Other signal classes need other models #

• A ramp is generated by two integrations; type-2 tracking needs two poles at the origin in the relevant loop.
• A sinusoid at known $\omega_0$ is generated by $\ddot w+\omega_0^2w=0$; resonant controller dynamics at $\pm j\omega_0$ can reject or track it.
• A periodic signal with several harmonics requires internal models of those harmonics or a repetitive-control structure.

flowchart TD
  G["reference / disturbance generator"] --> S["signal class"]
  S --> P["plant and measured error"]
  P --> C["controller containing model of G"]
  C --> P

An internal model gives infinite or very high loop gain at selected frequencies. That improves regulation there while increasing sensitivity to model mismatch, noise, saturation and nearby unmodeled dynamics.

Servo architecture #

A tracking system needs more than a position error gain. Typical elements are:

• reference shaping to respect velocity, acceleration and jerk limits;
• feedforward from a plant/reference model;
• feedback for disturbance rejection and uncertainty;
• state estimation when velocity or disturbance is not measured;
• integral or resonant states for persistent error;
• anti-windup and constraint handling;
• bumpless mode transfer and reference initialization.

For small-motion tracking, friction, backlash, quantization and sensor resolution can dominate linear dynamics. Increasing gain may create limit cycles rather than better accuracy.

Invariance and disturbance decoupling #

“Invariant control” often means the regulated output is insensitive to a disturbance within a model. Perfect feedforward cancellation would require knowing the disturbance and exact inverse path. Feedback can reduce sensitivity without measuring the disturbance but is limited by stability, delay and non-minimum-phase behavior.

Use the sensitivity function to state the achieved attenuation, not a categorical claim of invariance. If disturbance-to-output transfer is $G_d$,

$$y_d=S G_d d,\qquad S=\frac{1}{1+L}.$$

Small $S$ over the disturbance spectrum yields attenuation. It cannot be small at all frequencies in a causal uncertain system.

Design questions #

1. What reference and disturbance signal classes must be handled?
2. Through which channels do they enter, and what can the actuator influence?
3. Which generator modes must the controller contain?
4. Can the augmented closed loop be stabilized with adequate margins?
5. What control effort and transient arise when the internal-model state accumulates?
6. How are saturation, reset, mode changes and sensor bias handled?

The internal model principle explains why integral action works and why it sometimes fails. Zero steady-state error is a structural closed-loop result with assumptions, not a property of adding an I term.