Second Order Time Response

UWE Bristol

Dynamics and Control UFMFM8-30-3

 Aerospace Systems, Avionics and Control Control UFMFB7-30-2

Control UFMF7-!"-2

#ect$re %& Second 'rder  (ime )esponse and t*e Bloc+ diaram approac*

(odays #ect$re . #ast /ee+&   1np$ts& Step, )amp and $lse   )esponse in t*e (ime (ime Domain First  First order4   Final al$e (*eorem

. (*is /ee+&   Second order time response   1ntrod$ction to 5loc+ diaram ale5ra

Second 'rder Systems . C*aracterised 5y ζ  dampin ratio4 and ωn $ndamped nat$ral 6re$ency4 . (*e standard 6orm may 5e epressed in t9o di66erent 9ays 2 n

1

G ( s ) =

1+

2ζ  s

ω n

+

 s

2

ω n2

ω  = 2 2  s + 2ζω n s + ω n

Second 'rder Step response . /or+ o$t t*e step response as per last 9ee+& 1

G ( s) =

1+ C ( s ) =

1  s

2ζ  s

ω n

+

 s

2

=

ω n2  s 2

ω n2 1

* 1+

+ 2ζω n s + ω n2

2ζ  s

ω n

+

 s

2

response to unit step input

ω n2

ω n2 C ( s ) = * 2  s  s + 2ζω n s + ω n2 1

   s + 2ζω n     →  Part . Fract . −  2 2    s   s + 2ζω n s + ω n   1

Second 'rder Step response Further manipulati on to fit transf  orm tables... .

 ( s + ζω n ) + ζω n     C(s) = −  2 2    s  ( s + ζω n ) + ω  D   1

where ω  D

Transform

  −ζω  t   ζω n C (t ) = 1 −  e Cosω  D t  + ω     D   n

   Sinω  D t        

= ω n

1 − ζ  2

Second 'rder Step response       ζω  − ζω  t  n So, 9*at does t*is C (t ) = 1 −  e Cosω  D t  + Sinω  D t        ω  response loo+ li+e:   D      n

y t 4

Under-damped ζ