Steady-state tracking & sys. types G(s) C(s) + - r(s ) e y(s ) plant controll er 1 0 1 1 1 1 0 0 0 0 1 0 1 0 1 2 TF from e to forris System type #factorsofsin d(s) Type 0: 0, 0, / Type 1: 0, 0, , 0, / Type 1: , 0, 0, e r ol m m n n N N n N N v a p p a v p v G G bs bs b s a s a s as N a K K K b a a a K K K b a a a a K K 0 2 , / a K b a 0 0 2 0 1 lim ( ); to step 1 1 lim ( ); to ram p 1 lim ( ); to acc p e r ss s p v e r ss s v a e r ss s a K G s e K K sG s e K K sG s e K
The root locus technique 1.Obtain closed-loop TF and char eq d(s) = 0 2.Re-arrange to get 3.Mark zeros with “o” and poles with “x” 4.High light segments of x-axis and put arrows 5.Decide #asymptotes, their angles, and x-axis meeting place: 6.Determine jw-axis crossing using Routh table 7.Compute breakaway: 8.Departure/arrival angle:
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Steady-state tracking & sys. typesG(s) C(s)+
-r(s) e y(s)
plant
1 01 1
1 1
0 0 0
0 1 0 1
0 1 2
TF from e to for r is
System type #factors of s in d(s) Type 0: 0, 0, /
Type 1: 0, 0, , 0, /
Type 1: , 0, 0,
e r ol
mm
n n N Nn N N
v a p
p a v
p v
G G
b s b s bs a s a s a s
Na K K K b a
a a K K K b a
a a a K K
0 2, /aK b a
controller0
0
2
0
1lim ( ); to step1
1lim ( ); to ramp
1lim ( ); to acc
p e r sssp
v e r sssv
a e r sssa
K G s eK
K sG s eK
K s G s eK
1 2
20
20 0
TF from to : ( ) ( ); TF from to : ( ) ( );
( ) 1; if is step: 1 / 2 ( ) 1 / (0) (0)
1 1if is ramp: ; if is acc: (0) | (0) |
e d d e
sse B e Bs
e B s e B s
e d G s G s d e G s G s
sd se dG G s G G
d dsG s G
G1(s)+
-r(s) e
G2(s)
d(s)
AB
y(s)Type w.r.t. d
1# factor in ( ).e dG ss
Type 0 rejects nothing, has finite to step d, to ramp or acc dType 1 rejects const d, has finite to ramp d, has to acc dType 2 rejects step or ramp d, has finite
ss ss
ss ss
ss
e ee e
e
to acc d
The root locus technique1. Obtain closed-loop TF and char eq d(s) = 02. Re-arrange to get 3. Mark zeros with “o” and poles with “x”4. High light segments of x-axis and put arrows5. Decide #asymptotes, their angles, and x-axis meeting