Difference between revisions of "FAQ: What is steady state error?"
| Line 1: | Line 1: | ||
{{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 = | ||
| Line 14: | Line 14: | ||
\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
