5.4 Disturbance rejection
C
The input to the plant we manipulated is m(t).
Plant also receives disturbance input that we
do not control.
The plant then can be modeled as follow
plant
D(s)
Gd(s)
Gd(s)D(s)
+
+
R(s)
Gc
M(s)
Gp(s)
+
C(s)
GcG p
1  GcG p H
C(s)= Gp(s)M(s)+Gd(s)D(s)
The control system should minimize Gd(s)D(s).
Gd
D
1  GcG p H
C=T(s)R(s)+Td(s)R(s)
We want Td(s) as small as possible.
Let us use frequency approach.
Td(j) can be made small for specific
frequency.
Recall that Gc(j) Gp(j) H(j) must be
large to reduce the sensitivity to plant
variation.
Td ( j) 
Gd ( j)
1  Gc ( j)G p ( j) H ( j)
–
H
R

Gd ( j)
Gc ( j)G p ( j) H ( j)
Methods to reduce Td(j)
1. make Gd(s) small
2. increase loop gain by increasing Gc
3. reduced D(s)
4. use feed forward compensation
5.4 Disturbance rejection
Td (s) 
Feedforward compensation
Feedforward compensation can be applied if
the disturbance can be measured.
D(s)
plant
Gd(s)
Gcd(s)
Gd(s)D(s)
–
+
R(s)
+
Gc M(s)
Gp(s)
+
C(s)
–
H
The addition of compensator Gcd(s) does not
effect T(s) the TF from R(s) to C(s) but does for
Td(s)
Gd (s)  Gcd (s)Gc (s)Gp (s)
1  Gc (s)Gp (s) H (s)
We choose Gcd(s)Gc(s) such that Td(s) will
as small as possible.
5.5 Steady State Accuracy
E(s)
+
R(s)
Gc
M(s)
Gp(s)
C(s)
The steady state error ess is defined as
sR
s 0 1  G G
c p
ess  lim e(t )  lim sE (s)  lim
t 
s 0
Step input, R(s) = 1/s hence
–
We will examine the steady state error ess for
different system types. Assume H=1. hence
C ( s) 
Gc (s)Gp (s)
1  Gc (s)Gp (s)
R(s)
We can express
Gc ( s)G p ( s) 
F ( s)
s N Q1 ( s)
N is integer and the system is called system
type N.
N is also the number of integrator in GcGp
we will demonstrate its importance
s 1s
1
1
ess  lim


s 0 1  G G
1  lim Gc G p 1  K p
c p
s 0
For N = 0 then Kp = Kp and ess =1/(1+ Kp)
For N > 0 then Kp =  and ess = 0.
Ramp input, R(s) = 1/s2 hence
1
1
1
ess  lim


s 0 s  sG G
lim
sG
G
Kv
c p
c p
s 0
For N = 0 then Kv = 0 and ess = 
For N = 1 then Kv = Kv  and ess = 1/ Kv
For N > 1 then Kv =  and ess = 0
5.5 Steady State Accuracy
Parabolic input, R(s) = 1/s3 hence
1
1
1
ess  lim 2 2


2
s 0 s  s G G
lim s GcG p K a
c p
s 0
Kp is called position error constant
Kv is called velocity error constant
Kv is called acceleration error constant
Plot of ramp responses of different type system
ess
For N < 2 then Kv = 0 and ess = 
type 0 system
For N = 2 then Ka = Ka  and ess = 1/ Ka
For N > 2 then Kv =  and ess = 0
ess
The steady state error ess
system
type
N
0
1/s
input
1/s2
1/(1+KP) 
type 1 system
1/s3

1
0
1/Kv

2
0
0
1/Ka
ess
type 2 system
5.5 Steady State Accuracy
K p  lim Gc G p H
Non-unity-Gain feedback
s 0
R(s) +
K v  lim sGc G p H
Gp (s)
Gc (s)
s 0
K a  lim s 2Gc G p H
–
s 0
H
The non-unity gain feedback above has
equivalent block diagram as follow provided
that H(s) is constant.
Ru(s)
E(s)
+
HGc (s)Gp (s)
C(s)
–
Where Ru(s) = R(s)/H.
Hence the position, velocity, and acceleration
error constant can be calculated in the same manner
Disturbance Input Error
Let us consider steady state error due to
disturbance input.
Recall that the output of disturbed plant
s 0
C(s) =T(s)R(s)+Td(s)C(s)=Cr(s)+Cd(s)
e(t)= r(t) – c(t) =[r(t) – cr(t)] – cd(t)
= er(t) + ed(t)
where er(t) is the error we’ve just considered
We see that ed(t)= – cd(t) and we want it to be
small. The steady state of ed(t) is
s 0
(1)
where (assuming H = 1)
Td ( s) 
Gd
Gd

1  GcG p H 1  GcG p
Let us model the disturbance input as
step function, D(s)=B/s. From (1) we have
(3)
To draw general conclusion we must consider
the type number of Gp(s) Gc(s) and Gd(s).
The system error is then
edss  lim sTd ( s ) D( s )
Gd ( s) B
s 0 1  G ( s )G ( s )
c
p
edss  lim sTd ( s) B  lim
Type number of
Input
Gp(s)
Gd(s)
Gc(s)
edss
Unit step
0
0
0
Finite
Unit step
0
0
1
0
ramp
0
0
0

ramp
0
0
2
0
(2)
TRANSIENT RESPONSE
The system TF can be written as
T ( s) 
P( s )

Q( s )
P( s )
n
an ( s  pi )
i 1
The response for input R(s) is
C ( s )  T ( s ) R( s ) 
P( s )
n
R( s )
an ( s  pi )
i 1
k1
k1
k1



 C r ( s)
s  p1 s  p1
s  p1
Where Cr(s) is the part that
originate from the poles of R(s)
The inverse LT of this equation is
c(t )  k1e p1t    k n e pnt  cr (t )
Here cr(t) is the forced response
and the rest is the natural response
or the transient response
For stable system the natural
response will decay to zero. The
way it decays to zero is important.
The transient response ctr(t) is
ctr (t )  k1e p1t    k n e pnt
pi t
e
The factor
is called the modes
The nature or form of each term
is determined by the pole location
of pi.
TRANSIENT RESPONSE
The amplitude of each term is
kj 
P( s)
n
For each set complex conjugate
poles of values
R( s)
a n  ( s  pi )
i 1
i j
For each real pole, a time constant
is associated with it with the value
τ  1  1
pi
pi
s p j
Thus the amplitude of each term
determined by other pole location,
the numerator polynomial, and the
input function.
For higher order system it is
difficult to make general assertion,
however we can consider the nature
of each term of this response.
pi  ζ i  ni  1  ζ i
2
there is associated damping ratio ,
a natural frequency n , and time
constant i =1/n. If there is one
real dominant pole the system
responds essentially as 1st order
system. If there is dominant
complex poles then the system
respond essentially as 2nd order
system.
CLOSED LOOP FREQUENCY RESPONSE
The input output characteristics are determined by CL frequency response
Gc ()G p ()
T () 
1  Gc ()G p () H ()
|T()|
(1).
B

T(0) is the system dc gain. This values determines the steady state error.
If we evaluate (1) for small  we see how the system will follow for slow varying
input. The bandwidth can be evaluated form (1). Large bandwidth indicate fast
system response.
The presence of any peaks, which denotes resonance, give indication of the
overshoot or decaying oscillation