Difference between revisions of "Errata: Impulse response does not include the direct term in the proper manner"

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{{errata page
 
{{errata page
| chapter =  
+
| chapter = Linear Systems
| page =  
+
| page = 147
| line =  
+
| line = 10--15
| contributor = R. M. Murray
+
| contributor = C. Rowley
| date =  
+
| date = 18 Feb 2013
| rev = 2.10e
+
| rev = 2.11b
 
| version = Third printing
 
| version = Third printing
| text =  
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| text = In equation (5.18) and the subsequent text, the impulse response does not properly account for the direct (<math>D</math>) term.  The corrected text should read:
 +
<center><math>
 +
  h(t) = \int_0^t C e^{A(t-\tau)} B \delta(\tau)\, d\tau + {\color{blue} D \delta(t)}
 +
    = C e^{At} B + D \delta(t),
 +
</math></center>
 +
where the second equality follows from the fact that $\delta(t)$ is
 +
zero everywhere except the origin and its integral is identically 1.
 +
We can now write the convolution equation in terms of the
 +
initial condition response <font color=blue>and</font> the convolution of the impulse response
 +
and the input signal<s>, and the direct term</s>:
 +
<center>
 +
<math>  y(t) = C e^{At} x(0) + \int_0^t h(t-\tau) u(\tau)\, d\tau</math> + <s>D u(t)</s>
 +
</center>
 
}}
 
}}

Latest revision as of 12:23, 28 August 2016

Return to Errata page

Location: page 147, line 10--15

In equation (5.18) and the subsequent text, the impulse response does not properly account for the direct () term. The corrected text should read:

where the second equality follows from the fact that $\delta(t)$ is zero everywhere except the origin and its integral is identically 1. We can now write the convolution equation in terms of the initial condition response and the convolution of the impulse response and the input signal, and the direct term:

+ D u(t)

(Contributed by C. Rowley, 18 Feb 2013)