Feedback Control: Open Loop, Closed Loop & Error Dynamics

Open-loop control chooses an input without measuring the outcome it is meant to regulate. Feedback control measures an output, compares it with a reference and uses the difference to change the input. Feedforward measures or predicts a disturbance and compensates before its effect appears. Practical systems often combine all three.

flowchart TD
  R["reference r"] --> J(("Σ"))
  Y["measured y"] -->|"− feedback"| J
  J -->|"error e"| C["controller C"]
  C -->|"control u"| P["plant P"]
  D["disturbance d"] --> P
  P --> O["output"]
  O --> M["sensor"] --> Y

The sign at the summing junction is load-bearing. A reversed sensor or actuator direction turns intended negative feedback into positive feedback.

Closed-loop equations #

For a linear loop with plant $P(s)$ and controller $C(s)$, loop transfer is $L=PC$. With unity measurement and an input disturbance $d$,

$$y=\frac{L}{1+L}r+\frac{P}{1+L}d.$$

Define sensitivity and complementary sensitivity:

$$S=\frac{1}{1+L},\qquad T=\frac{L}{1+L},\qquad S+T=1.$$

Large loop gain can make $|S|$ small, reducing tracking error and sensitivity to plant variation in that frequency region. But $T$ then approaches one, so measurement noise and unmodeled high-frequency dynamics become important through their own paths. Feedback reallocates sensitivity; it does not erase uncertainty.

Why feedback works #

If a plant gain changes, an open-loop command calibrated for the old gain produces proportional output error. In a stable feedback loop with large low-frequency gain, the measured error drives corrective action. Integral action can make steady error to a constant reference or disturbance zero when the closed loop remains stable and the actuator has enough authority.

Feedback can also:

• stabilize an unstable or poorly damped process;
• shape transient response and bandwidth;
• reject disturbances that were not measured directly;
• make behavior less sensitive to parameter variation;
• coordinate components around a system-level variable.

Every benefit is conditional on a stable closed loop and a model of delays, constraints and measurement quality.

Stability is a property of the loop #

A stable plant and stable controller can form an unstable interconnection. Closed-loop poles solve $1+L(s)=0$. Delay adds phase lag; high gain can move the loop to the point where correction arrives with the wrong phase and reinforces oscillation.

Linear analysis predicts local behavior. Saturation changes the loop, integral state can wind up, dead zones create limit cycles, and mode switches can invalidate one transfer function. Verify both the nominal linear design and nonlinear implementation.

Error is not the whole objective #

Minimizing $e=r-y$ without constraints can demand impossible actuator speed or magnitude, amplify noise, consume excessive energy or wear equipment. A control design balances:

• tracking and disturbance rejection;
• stability margins and model uncertainty;
• control effort, rate and saturation;
• noise sensitivity;
• transient overshoot and settling;
• safety and allowable states.

An observed output close to reference is also not proof that feedback caused it. The reference may be constant, disturbances small, or the plant naturally stable. Test with controlled reference and disturbance changes, record actuator effort, and compare against an open-loop or feedforward prediction.

A commissioning sequence #

1. Verify sensor and actuator signs, units, ranges and fail states with the loop open.
2. Identify the operating point, dominant dynamics, delay and constraints.
3. Choose a conservative controller and predicted margins.
4. Close the loop at low authority; apply bounded tests.
5. Inspect tracking, disturbance response, noise, control effort and saturation.
6. Test startup, shutdown, mode transfer, sensor failure and actuator limits.
7. Record the validated operating envelope and retuning triggers.

Feedback is powerful precisely because it connects consequences back to action. That same connection means every hidden delay, sign, constraint and measurement error becomes part of the controller.