# Difference between revisions of "FAQ: What is steady state error?"

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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. | 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 | + | For a {{SISO}} linear system with state space dynamics |

<center><amsmath> | <center><amsmath> | ||

\dot x = Ax + B r, \qquad y = C x + D | \dot x = Ax + B r, \qquad y = C x + D |

## Revision as of 18:28, 15 November 2008

(Contributed by Richard Murray, 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 (a constant) and the actual output is , 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 . 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