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 <math>A</math> a stable matrix (eigenvalues have negative real part), the steady state error for a step input is given by
+
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 <math>L(s)</math> and is given by
+
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

math

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

math

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

math

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

math