FAQ: What is steady state error?

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(Contributed by Richard Murray (with corrections by B. Grunloh), 15 November 2008)

Steady state error is a property of the input/output response for a linear system. If the desired value of the output for a system is <math>r</math> (a constant) and the actual output is <math>y(t)</math>, the steady state error is defined as


The steady state error for a step response is often reported as a percentage of the input magnitude, similar to the overshoot <math>M_p</math>. Steady state error can also be defined for other types of signals, such as ramps, as long as the error converges to a constant. The steady state error is only defined for a stable system.

For a SISO linear system with state space dynamics


with math a stable matrix (eigenvalues have negative real part), the steady state error for a step input is given by


In the frequency domain, the steady state error for a step input to a unity gain, negative feedback system can be computed from the loop transfer function math and is given by

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