Stability and steady-state error
A response can be bounded yet inaccurate, or accurate in theory but unreachable with real actuator limits.
Show with one model that a stable loop can still miss its target.
Step metrics describe one run. Stability asks what happens as time continues; steady-state error asks how close the settled output gets to target.
Review the previous lesson: Read a step responseFirst, picture it
An upright pole is unstable without active balancing: a small tilt grows. A motor speed loop may be stable but sit below target under load. Stability and accuracy answer different questions.
Build the idea step by step
Stability and accuracy are different questions. A stable response can settle below the target. To check whether a finite-dimensional linear state model returns to equilibrium, examine all of its closed-loop eigenvalues: each must have a negative real part in continuous time. In a minimal input-output model these are the poles. Only after checking convergence should we calculate a final error. A final-value formula does not turn an unstable response into a meaningful final reading.
Read the equations
Steady-state error is the error remaining after transients decay, if a finite limit exists.
What the symbols mean
- final tracking error, in output units
- target signal, in output units
- measured output, in output units
Steady-state error exists only if the error approaches a limit. Integral control can remove a constant tracking error when the closed loop remains stable and the actuator has enough input available.
A pole's real part sets growth or decay; its imaginary part sets oscillation frequency. The boundary itself needs separate analysis.
Stable but one-third short
Use the earlier unity-feedback plant with proportional gain 2. Its closed-loop relation is two divided by s plus three.
The pole at minus 3 makes the linear response decay. For a unit step, the final output is two-thirds.
The loop is stable in this model but retains one-third of error. Raising gain may shrink it, subject to actuator and stability limits.
More detail and model assumptions
Three different questions
Internal asymptotic stability asks whether all internal deviations decay with no further input. BIBO stability asks about every bounded input and the chosen output. Steady-state accuracy asks whether the output eventually reaches a particular target. A loop can pass one question and fail another, especially when hidden modes or actuator limits matter.
Use the final-value theorem only after checking its pole conditions. A computed number from an unstable loop is not a physical final speed. For a saturated motor, the desired target may simply require more voltage than the actuator can provide.
Stable but inaccurate is possible
Consider a loaded motor under proportional-only speed control. It can settle calmly at a speed below the target: the remaining error creates exactly the extra P voltage needed to counter the load. Its motion is bounded and convergent, yet its tracking accuracy is imperfect. Adding I can remove that persistent error if the actuator has enough voltage authority.
The reverse shortcut also fails: a late-time number near the target does not prove a system is internally stable. Look for growing modes, hidden state behavior, rail events and the full time interval, especially with nonlinear limits.
Connect the example to the lab
Apply a constant load to the motor. P-only control may keep a persistent speed error; adding I can reduce it, unless the demanded voltage exceeds the limit.
Compare stability and errorMake a prediction before moving a control.
- PREDICTCan a motor settle without reaching its target?
- CHANGE ONE THINGOpen the integral lesson with I=0 and a constant load; pin it, then add I.
- OBSERVECompare final error, settling and voltage saturation.
- EXPLAINSeparate your claim about convergence from your claim about target accuracy.
'Bounded in this 12-second run' is not a mathematical proof of stability for all time and inputs.
Check your understanding+
Can a stable motor speed loop still miss the target?
Yes. It can settle with nonzero error, especially under load or saturation.Why the next lesson follows
To state input-output stability precisely, we ask whether every bounded input keeps the chosen output bounded.