# FAQ: What is steady state error?

(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 $r$ (a constant) and the actual output is $y(t)$, 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 $M_p$. 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 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 and is given by