1. 5.4 Disturbance rejection
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
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.
D(s)
M(s) +
C(s)
Gp(s)
Gd(s)
plant Gc – H
Gd(s)D(s)
R(s)
Methods to reduce Td(j)
make Gd(s) small
increase loop gain by increasing Gc
reduced D(s)
use feed forward compensation
C(s)= Gp(s)M(s)+Gd(s)D(s)
The control system should minimize Gd(s)D(s).

2. 5.4 Disturbance rejection
Feedforward compensation
Feedforward compensation can be applied if
the disturbance can be measured.
We choose Gcd(s)Gc(s) such that Td(s) will
as small as possible.
C(s)
D(s)
M(s) +
Gp(s)
Gd(s)
plant Gc – H
Gd(s)D(s)
Gcd(s)
R(s)
The addition of compensator Gcd(s) does not effect T(s) the TF from R(s) to C(s) but does for Td(s)

3. 5.5 Steady State Accuracy The steady state error ess is defined as
C(s)
M(s)
Gp(s) Gc + –
R(s)
E(s)
Step input, R(s) = 1/s hence
We will examine the steady state error ess for different system types. Assume H=1. hence
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
We can express
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
N is integer and the system is called system type N.
N is also the number of integrator in GcGp
we will demonstrate its important

4. 5.5 Steady State Accuracy ess ess The steady state error ess system
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
Parabolic input, R(s) = 1/s3 hence
ess
For N < 2 then Kv = 0 and ess = 
For N = 2 then Ka = Ka  and ess = 1/ Ka
For N > 2 then Kv =  and ess = 0
type 0 system
ess
The steady state error ess
system
type input
N /s 1/s2 1/s3
/(1+KP)  
/Kv  ess
/Ka
type 1 system
type 2 system

5. 5.5 Steady State Accuracy H Non-unity-Gain feedback R(s)+ – H
R(s)
The non-unity gain feedback above has equivalent block diagram as follow provided that H(s) is constant. + –
Ru(s)
E(s)
C(s)
Where Ru(s) = R(s)/H.
Hence the position, velocity, and acceleration
error constant can be calculated in the same manner

6. Input Gp(s) Gd(s) Gc(s) edss
Disturbance Input Error
Let us consider steady state error due to
disturbance input.
Recall that the output of disturbed plant
C(s) =T(s)R(s)+Td(s)C(s)=Cr(s)+Cd(s)
The system error is then
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
(3)
To draw general conclusion we must consider
the type number of Gp(s) Gc(s) and Gd(s).
Type number of
Input
Gp(s)
Gd(s)
Gc(s)
edss
Unit step
Finite 1
ramp  2
(1)
where (assuming H = 1)
(2)
Let us model the disturbance input as
step function, D(s)=B/s. From (1) we have

7. TRANSIENT RESPONSE The system TF can be written as
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
The response for input R(s) is
The factor
is called the modes
The nature or form of each term is determined by the pole location of pi.
Where Cr(s) is the part that originate from the poles of R(s)
The inverse LT of this equation is

8. TRANSIENT RESPONSE
For each real pole, a time constant is associated with it with the value
The amplitude of each term is
For each set complex conjugate poles of values
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.
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.

9. CLOSED LOOP FREQUENCY RESPONSE
The input output characteristics are determined by CL frequency response
(1).
|T()|  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