Steady-State Error, Static Error Constants & System Type

For unity negative feedback, error is

$$E(s)=\frac{R(s)}{1+L(s)}=S(s)R(s).$$

The final-value theorem can calculate steady error only when every pole of $sE(s)$ lies in the open left half-plane, except any pole removed by the multiplication by $s$. Applying it to an unstable or oscillatory closed loop can produce a finite-looking number that is not a limit.

Each integrator embeds one additional polynomial order in the loop. This structural benefit is conditional on closed-loop stability and available actuator authority.

Static error constants #

For a stable unity-feedback loop, define

$$K_p=\lim_{s\to0}L(s),\qquad K_v=\lim_{s\to0}sL(s),\qquad K_a=\lim_{s\to0}s^2L(s).$$

Then, for unit-amplitude standard inputs:

$$e_{ss,step}=\frac{1}{1+K_p},\qquad e_{ss,ramp}=\frac{1}{K_v},\qquad e_{ss,parabola}=\frac{1}{K_a},$$

with $1/\infty=0$ and $1/0=\infty$ in the ideal classification.

For amplitude $A$ or ramp slope $v$, scale the result accordingly. For a parabolic input written $r(t)=\tfrac12at^2$, steady error is $a/K_a$.

System type #

System type is the number of uncancelled poles at the origin in $L(s)$.

This table assumes the corresponding closed loop is stable. Adding integrators raises low-frequency gain and improves polynomial tracking, but also adds phase lag, state, windup exposure, and possibly slow transients.

Do not count a pole at the origin that is exactly cancelled in the loop expression. More importantly, do not rely on cancellation of an unstable or marginal plant mode: model error restores it and can destroy internal stability.

Derivation for polynomial inputs #

Suppose near the origin

$$L(s)\sim\frac{K}{s^q}.$$

For a degree-$m$ polynomial reference, $R(s)\propto1/s^{m+1}$. Then

$$sE(s)=s\frac{R(s)}{1+L(s)}\sim \frac{s\,s^{-(m+1)}}{K s^{-q}}=\frac{1}{K}s^{q-m}.$$

The final value is zero if $q>m$, finite if $q=m$, and divergent if $q<m$. This is the low-order SISO form of the internal model principle: exact asymptotic tracking requires the closed loop to contain the dynamics that generate the reference.

Disturbance location changes the result #

For an input disturbance $d_i$ added before plant $P$,

$$\frac{Y}{D_i}=\frac{P}{1+PC}.$$

For an output disturbance $d_o$ added after the plant,

$$\frac{Y}{D_o}=\frac{1}{1+PC}.$$

An integrator in $C$ can reject constant disturbances in many common configurations, but the exact error and required controller structure depend on where the disturbance enters and whether it is matched to the control channel. Derive the path instead of applying the reference-error table blindly.

Nonunity feedback and calibration #

With sensor $H(s)$, the comparator error is $e=r-Hy$, not necessarily the physical tracking error $r-y$. A loop can drive $e\to0$ while $y\to r/H(0)$. Sensor scale, bias, units, and reference prefilter must be part of the equations.

Similarly, a controller bias can hold a nonzero actuator command at zero error. A proportional controller without bias may require nonzero error to generate the equilibrium input. Integral state can learn that bias, but a feedforward equilibrium model may provide it immediately and reduce windup.

Practical integrators are conditional #

Real implementations include:

• saturation and anti-windup logic;
• leakage or reset that moves the pole away from exactly $s=0$;
• finite precision and deadband;
• limited sensor resolution and bias;
• mode transitions and state initialization;
• disturbances that vary rather than remain constant.

With leaky integral action $K_i/(s+\epsilon)$, DC gain is large but finite, so exact zero error becomes small finite error. That can be a deliberate robustness tradeoff, but it should be stated rather than described as an ideal integrator.

Error versus control authority #

The final-value theorem assumes the required equilibrium input exists. If a heater is at maximum power or a valve cannot supply the required flow, the integrator can grow while the output remains below reference. Zero-error algebra does not create energy or actuator range.

Before promising zero steady error:

1. solve the plant equilibrium for the required $u_{ss}$;
2. confirm magnitude and rate limits with margin;
3. establish closed-loop stability including delay;
4. identify the reference/disturbance signal class;
5. verify sensor bias and scaling;
6. test saturation, leakage, quantization, and mode transfer.

System type is a structural prediction under assumptions. Those assumptions are the engineering work.