The Design of Feedback Control System
Introduction
The performance of a feedback control system is primary importance. What is a suitable control system? -- It is stable. -- It results in an acceptable response to input commands. -- It is less sensitive to system parameter changes. changes. -- It results in a minimum steady-state error for input. -- It is able to reduce the effect of undesirable disturbances. disturbances.
Performance specifications performance specifications in the time domain
Overshoot Setting time
%
t s
Steady-state error
e ss
y (t ) overshoot
0
t
t p
t s
Performance specifications performance specifications specifications in the frequency domain
Closed-loop
Open-loop
Resonant peak M r
Gain-crossover frequency
Resonant frequency Bandwidth
r
b
Gain margin
c
h / Lh
Phase margin
20 lg G M r
0.707M (0)
c
0
h G
r
b
Performance specifications Typical complex domain indices indices are represented by the location of the dominant poles j
% e
1 2
100% 100
n
t s
3.5
n
or
t s 3T
1
T
n
o
Example What is compensation or correction of a control system ? G( s)
For example :
K s(Ts 1)( s 1)
make this closed - loop systemcan be be stable
Solution: According to Routh - Hurwitz criterion, we can get : K T 1 1 1 T
But if : G( s)
K s (Ts 1) 2
T
(K 0 T 0)
According to Routh - Hurwitz Criterion, this closed - loop systemcan not be stable only only varyi varying K or T .
If we make : G( s)
K ( s 1) 2 s (Ts 1)
τ
T
This closed-loop system can be stable.
We make the system stable by increasing a component.
Compensation & Compensator
Increasing a component ,which makes the system s performance to be improved, other than only varying the system s parameters, this procedure is called ’
’
the compensation compensation or correction of the system.
The compensating device may be electric, mechanical, hydraulic, pneumatic, or some other type of device or network and is often called a compensator .
A compensator compensator is an additional additional component that is inserted into a control system to compensate for a deficient performance.
Compensation & Compensator Example : G( s)
K s (Ts 1) 2
, to increase component ( s 1) ,
t he systemcan be stable, ( s 1) is a compensator.
The compensator can be placed in a suitable location within the structure of the system. The compensator placed in forward path is called a cascade (or series) compensator.
R( s)
GC ( s )
Controll er
G( s)
controlled process
Y ( s)
Compensation & Compensator Similarly, the other compensation schemes are called feedback, output, input and disturbance compensation.
R( s)
Y ( s)
R( s)
G( s)
GC ( s)
R( s) GC ( s)
Y ( s)
G( s)
H ( s)
G( s)
H ( s)
Gn ( s)
Y ( s)
N ( s)
R( s) GC ( s)
H ( s)
GC ( s)
G( s)
Y ( s)
Approaches to Compensation
In the following sections, we will assume that the process has been improved as much as possible and that the G(s) representing the process is unalterable. For frequency response methods, we are concerned with altering the system so that the frequency response of the compensated system will satisfy the system specifications. Alternatively the the design of a control control system can be accomplished in the s-plane by root locus methods. For the case of the s-plane, the designer wishes to alter and reshape the root locus so that the roots of the system will lie in the desired position in the s-plane. We shall consider the addition of so-called so -called phase-lead , phase-lag and phase lag-lead compensation network ,and describe the design of the network by frequency response techniques.
Phase-lead Compensation Network Consider the first-order compensator with the transfer function Gc ( s )
Ts 1
Ts 1
( 1)
The design problem then becomes the selection of parameters and T , in order to provide a suitable performance.
Gc ( j ) tan1 T tan1 T The maximum value of the phase lead occurs at frequency m The maximum phase lead is m arcsin
1
20 lg G( j m ) 10 lg
T
L
1
The frequency m is the geometric mean of z 1/ T and p 1 / T .
1
[+20] 10 lg
0
Gc
0
m 1
1
1
Phase-lead Compensation Network Gc ( s )
Ts 1
( 1)
Ts 1
1 900 0 m arcsin 1 0
( 1)
The above equation is very useful for calculating a necessary ratio between the pole and zero of a compensator in order to provide a required maximum phase lead. C
R1
V 1
R2
Example : Example : Phase-lead electric network compensation Gc ( s)
V 2 ( s) V 1 ( s)
R2 R1 R2
R1Cs 1 1 Ts 1 R2 Ts 1 R1Cs 1 R1 R2
R1 R2 R2
T
R1 R2 R1 R2
C
V 2
Summary of Effects of Phase-lead Compensation Advantages and disadvantages of Phase-lead controller on performance are : Advantages and 1. Improving damping and reducing maximum overshoot. 2. Improving h(Lh) and γ. 3. Increasing Wc. 4. Reducing setting time because of increasing Wc 5. Possibly accentuating noise at higher frequencies.
Black curve 1 ― controlled process G(s) Bode plot Red curve ― controller GC(s) Bode plot Green curve 2 ― Compensated system G C(s)G(s) Bode plot
Example: A phase-lead compensator design for a second-order system using the Bode diagram Let us consider a single-loop feedback control system, where
G( s)
K
Y ( s )
R( s) GC ( s)
s(0.1 s 1)
G( s)
We want to have steady-state error ess=0.01 for an unit ramp input. Furthermore, we desire that the phase margin of the system be at least 450 and the gain crossover frequency be at least 40 rad/s. e ss
1
K
0.01
100 K 100
Example: A phase-lead compensator design for a second-order system using the Bode diagram The first step is to plot the Bode diagram of the uncompensated uncompensated transfer function. 20 lg G
dec 20dB / dec
dec 40dB / dec 44
0
c 31 rad / s
10
17.90
c and don' t satisfy the specificat ions
22 31
88
6dB
Example: A phase-lead compensator design for a second-order system using the Bode diagram 20 lg G
G( s)
dec 20dB / dec
100 100
s(0.1 s 1) dec 40dB / dec
Gc ( s)
0.04544 s 1 0.01136 s 1
Gc ( s )
here
44
T1
Ts 1
Ts 1
G( s)Gc ( s)
1 T
0
(
22,
10
22 31
88
6dB
1)
T2
1 T
88
44rad / s c 0 s(0.1 s 1)0.01136 s 1 49.8 100 100(0.04544 s 1)
Black line : 20 log G( j ) Green line : 20 log G( j )Gc ( j )
Example: A phase-lead compensator design for a second-order system using the Bode diagram
Blue line : controlled process Bode plot 20 log G( j )
and G( j )
Green line : Compensated system Bode plot 20 log G( j )G ( j ) and G( j )G ( j )
Phase-lag Compensation Network Consider the first-order compensator with the transfer function Gc s
1 Ts 1 Ts
1
Y ( s )
R ( s ) GC ( s )
G ( s )
The design problem then becomes the selection of parameters and T , in order to provide a suitable performance. L +
1/T
1/
0 -
T
-20
0
0 T 1 1 1 L( ) 20 lg T T ( T ) 20 lg 1 ( ) T
Gc ( j ) tan1 T tan1 T
The phase of this compensator is always negative. Thus it is called a phase-lag compensator.
Phase-lag Compensation Network L +
1/T 0
1/
T
20 lg
-20
-
0
The phase-lag compensation transfer function can be obtained with the network shown in the following Figure: Gc ( s)
R1
vi
R2 C
v0
V o ( s)
V i ( s)
R2Cs 1 ( R1 R2 )Cs 1
R2 R1 R2
Gc s
1 Ts 1 Ts
T ( R1 R2 )C
1
Summary of Effects of Phase-lag Compensation Advantages and disadvantages of Phase-lag controller on performance are : Advantages and 1. Improving damping and reducing maximum overshoot. 2. Improving h(L h) and γ. 3. Filtering out high-frequency noise (lessening noise at higher frequencies). 4. Decreasing Wc. 5. Increasing settling time time because of decreasing Wc .
1 T
Black curve 1 ― controlled process G(s) Bode plot Red curve ― controller GC(s) Bode plot Green curve 2 ― Compensated system G (s)G(s) Bode plot
1 T
The phase lag-lead compensator
The phase-lead compensator improves settling time, phase margin and increase the bandwidth. However, phase-lag compensator when applied properly improves phase margin but usually results in a longer settling time.
Therefore, each of these control schemes has its advantages, disadvantages,and limitations, and there are many systems that cannot be satisfactorily compensated by either scheme acting alone. It is natural, therefore, therefore, whenever whenever necessary, necessary, to consider consider using a combination of the lead-lag compensator, so that the advantages of both schemes are utilized.
The phase lag-lead compensator The transfer function of a lag-lead compensator can be written as
1 T 1 s 1 aT 2 s 1 T 1 s 1 T 2 s
Gc ( s) Gc1 ( s)Gc 2 ( s)
lag
( 1, 0 1)
lead
It is usually assumed that the two break frequencies of the lag portion are lower than the two break frequencies of the lead portion. L( )
20 lg | G | 0
1
1
1
1
T 1
T 1
T 2
T 2
( ) G
The phase lag-lead compensator Phase lag-lead electric network compensation
i ei
R1 C 1
R2 e0 C 2
PID controllers in the frequency domain The PID controller provides a proportional term, an integral term, and a derivative term. We have the PID controller transfer function as Gc ( s) K P
K I s
K D s Effects are similar to phase lag-lead compensation.
If we set K D 0 , we have the PI controller Gc ( s) K P
If we set
K I
Effects are similar to phase-lag compensation.
s
K I 0
, we have the PD controller
Gc ( s) K P K D s
Effects are similar to phase-lead compensation.
PD controller Effects are similar to phase-lead compensation transfer function : Gc ( s) K p K D s ( )
Assuming :
G( s )
n
2
s( s 2 n )
K p K D s
G (
)
( )
G C ( s )
n ( K P K D s ) 2
The open - loop transfer function of the compensated systemis : Gc ( s)G( s )
s( s 2 n )
It shows that the PD controller is equivalent to adding a open loop zero at : s
Advantages and disadvantages of PD controller on the performance are : Advantages and 1. Improving damping and reducing maximum overshoot. 2. Improving h(Lh) and γ. 3. Increasing Wc. 4. Reducing setting time because of increasing Wc . 5 Possibly accentuating noise at higher frequencies.
K P K D
PI controller Effects are similar to phase-lag compensation Transfer function : Gc ( s ) K p K I
Assuming : G ( s )
n
1
R (s )
s
K p
2
G(s )
K I
C (s )
1 s
G C ( s )
s ( s 2 n )
T he open loop transfer function of t he compensated systemis : 1 2 n ( K P K I ) s Gc ( s )G ( s ) s ( s 2 n )
2
n ( K P s K I )
s 2 ( s 2 n )
PI controller is equivalent to adding a open loop zero at : s
K I K P
and a pol p ole e at : s 0
Advantages and disadvantages of PI controller on the performance are : Advantages and 1. Improving damping and reducing maximum overshoot. 2. Improving h(L h) and γ. 3. Filtering out high-frequency noise (lessening noise at higher frequencies). 4. Decreasing Wc.
PID controller Effects are similar to phase lag-lead compensation
(s ) K p K I Transfer function: Gc(s)
-
K p K I
1
s
K D s
+
1 s
K D s
GC ( s )
PID controller have advantages both of PI and PD.
Circuits of PI , PD and PID R2 u r
R1
C U 0 ( s ) R2 (1 R1C 1 s) U R ( s ) R 1
R2
_ u 0
u
+
R1
r
_ u 0
PI controller C
+ PD controller
1 U 0 ( s ) R2 ) (1 U R ( s ) R R Cs 1 2 R2 u r
R1
C2
U 0 ( s ) ? U R ( s )
_ u 0
C
+
1
PID controller
