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

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{{FAQ page | {{FAQ page | ||

| chapter = Linear Systems | | chapter = Linear Systems | ||

− | | contributor = Richard Murray | + | | contributor = Richard Murray (with corrections by B. Grunloh) |

| date = 15 November 2008 | | date = 15 November 2008 | ||

| answer = | | answer = | ||

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

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

− | with < | + | with <amsmath>A</amsmath> a stable matrix (eigenvalues have negative real part), the steady state error for a step input is given by |

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

− | e_\text{ss} = 1 - C A^{-1} B + D. | + | e_\text{ss} = 1 - \left. (C (sI - A)^{-1} B + D) \right|_{s = 0} = 1 + C A^{-1} B + D. |

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

− | In the frequency domain, the steady state error for a step input to a [[unity gain feedback system|unity gain, negative feedback system]] can be computed from the loop transfer function < | + | In the frequency domain, the steady state error for a step input to a [[unity gain feedback system|unity gain, negative feedback system]] can be computed from the loop transfer function <amsmath>L(s)</amsmath> and is given by |

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

e_\text{ss} = \frac{1}{1+ L(0)}. | e_\text{ss} = \frac{1}{1+ L(0)}. |

## Latest revision as of 19:37, 13 October 2012

(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 (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