Steady-State Error and System Type
Why a proportional controller can never quite arrive, and why counting integrators answers the whole question.
Skip to the animationA proportional controller must retain an error to produce any output, so it never quite arrives — and the fix is structural rather than a matter of gain: each integrator in the open loop removes the error for one more order of input, at a cost of 90° of phase margin.
Why proportional control cannot arrive
With u = K_p·e, zero error means zero output — so the loop settles at whatever error produces exactly the drive the plant needs. For a step, e_ss = 1/(1 + K_p).
Raising the gain shrinks the error and approaches zero only as gain approaches infinity — and long before that the loop rings and then goes unstable. Gain is the wrong lever; something structural has to change.
The integrator
An integrator's output can be non-zero while its input is zero, because it holds what it has accumulated. So the drive persists with no error sustaining it, and the loop settles at exactly the target. This is precisely the argument the proportional term could not make.
System type
System type is the number of poles at the origin in the open-loop transfer function — the number of integrators. Nothing else about G(s) affects steady-state error.
| Type | Step | Ramp | Parabola |
|---|---|---|---|
| 0 | finite | ∞ | ∞ |
| 1 | zero | finite | ∞ |
| 2 | zero | zero | finite |
The diagonal structure is the whole content: type n tracks inputs up to order n exactly, lags order n+1 by a constant, and cannot follow order n+2. Each integrator buys exactly one order of input.
Why not simply use type 2 everywhere
Each integrator contributes −90° of phase lag, eating the phase margin that keeps the loop stable. Type 2 systems are hard to stabilise and type 3 essentially impractical.
This is why PID pairs its integral term with derivative action: the D term returns phase, paying for the accuracy the I term bought. Accuracy and stability are traded against each other, and this is the exchange rate.
The numbers you will be asked for
- Position error constant
K_p = lim(s→0) G(s)H(s)
Step error = 1/(1 + K_p).
- Velocity error constant
K_v = lim(s→0) s·G(s)H(s)
Ramp error = 1/K_v.
- Acceleration error constant
K_a = lim(s→0) s²·G(s)H(s)
Parabola error = 1/K_a.
- System type
the power of s in the denominator at the origin
Count the integrators.
- Final value theorem
e_ss = lim(s→0) s·E(s)
Valid only if the system is stable.
Advantages and disadvantages
Advantages
- Steady-state error is answered by counting integrators, with no response computed.
- An integrator removes the error exactly, not approximately.
- The error constants give the number directly for standard inputs.
- It explains the I term in PID structurally rather than empirically.
Disadvantages
- Each integrator costs 90° of phase margin.
- The final value theorem is invalid for an unstable system, and applying it anyway gives a plausible wrong answer.
- It says nothing about the transient, only the destination.
- Integrators introduce windup when the actuator saturates.
Watch it work
Check yourself
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