Final-Value Theorem — Steady-State Error
Steady-state error of a unity-feedback loop from system type and static error constant — step, ramp, or parabolic input.
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The engineering
The final-value theorem reads the t → ∞ behavior of a signal straight off its Laplace transform: e(∞) = lim s→0 s·E(s), no inverse transform needed. For a unity-feedback loop the whole answer collapses into the static error constants Kp, Kv, Ka and the system type — the number of pure integrators in the open-loop G(s). Type 0 tracks a step with finite offset; type 1 zeroes the step error and follows a ramp with finite lag; type 2 zeroes both and only shows finite error against a parabola.
Two gotchas. First, the theorem is only valid when s·E(s) has all poles strictly in the left half-plane — if the closed loop is unstable or marginally stable, the limit is a lie, so verify stability (Routh, root locus) before trusting the number. Second, the K entered here is lim s→0 sᴺ·G(s), the finite nonzero constant, not the raw forward-path gain — cancel the integrators first.
Quick sanity check: a type-1 loop with Kv = 100 tracking a unit ramp lags by exactly 0.01 — ten times the Kv, one tenth the error. If you need less steady-state error, raise gain or add an integrator; both moves cost phase margin, which is why PI controllers exist.
Where this math comes from
The machinery is Oliver Heaviside's. Working alone in the 1880s–90s on telegraph-cable problems, he treated d/dt as an algebraic operator p and manipulated it with a physicist's nerve and no proofs — the initial- and final-value shortcuts fall out of his operational calculus. Mathematicians were appalled; his famous retort was that he did not refuse his dinner simply because he did not fully understand digestion. Gustav Doetsch's 1937 treatise finally put the Laplace transform, and the precise pole conditions under which the final-value theorem holds, on rigorous footing.
The static-error-constant packaging on this card is a product of the wartime servo labs. Gordon Brown's Servomechanisms Laboratory at MIT turned fire-control tracking error into the Kp/Kv/Ka and system-type framework, codified in Brown and Campbell's 1948 Principles of Servomechanisms — and it has anchored the steady-state-error chapter of every controls text since, including Nise, whose treatment this card follows.
- 1782Pierre-Simon LaplaceIntroduces the transform in work on probability — the s-domain is born.
- 1892Oliver HeavisideOperational calculus for circuit transients; the value theorems in embryonic form.
- 1937Gustav DoetschTheorie und Anwendung der Laplace-Transformation — rigorous conditions for the final-value theorem.
- 1948G. S. Brown & D. P. CampbellPrinciples of Servomechanisms formalizes system type and the Kp/Kv/Ka static error constants.
- 2010Norman NiseControl Systems Engineering (6th ed.) — the standard classroom statement this card implements.
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