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Transcript of Eee3420 lecture01 rev2011
1Week © Vocational Training Council, Hong Kong.
│ Lecture 1 │
Introduction to Control Systems
EEC3420 Industrial ControlDepartment of Electrical Engineering
2Week© Vocational Training Council, Hong Kong.
EEE3420 Industrial Control
Learning objectives Understand the basic concept in control systems. Know what is a Transfer Function. Appreciate the PID control process. Know what criteria leading to a stable control system.
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EEE3420 Industrial Control
Open-loop control system
An open-loop control system is one in which the control signal of the process is independent of the process output
control accuracy is determined by the calibration of the plant
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EEE3420 Industrial Control
Advantages & disadvantage of open-loop control system Advantages
simple and inexpensive no stability problem
Disadvantages cannot compensate for any disturbances that
add to the controller’s driving signal
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EEE3420 Industrial Control
Closed-loop control system
A closed-loop control system depends on the output of the process to adjust the signal controlling the closed loop
process output is compared to the user command and an output from the plant
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EEE3420 Industrial Control
Advantages & disadvantages of closed-loop control system
Advantages less sensitive to noise, disturbances and
changes in the environment transient response and steady-state error can
be controlled more conveniently and with greater flexibility
Disadvantages relatively expansive may be unstable if not properly designed
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EEE3420 Industrial Control
Architecture of a closed-loop control system
Controlled Variable (CV).
Set point. Error = set point –
current value of CV. Manipulated Variable. Feedback Loop.
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EEE3420 Industrial Control
Feedback control real-time scheduling Choices for control variables, manipulated variables, set
points Choice of appropriate control functions Stability problem of feedback control in the context of
real-time scheduling? How to tune control parameters? How significant is the overhead and how to minimize it? How to integrate a runtime analysis of time constraints
with scheduling algorithms?
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EEE3420 Industrial Control
Using negative feedback control system
typically more stable less sensitive to variation in component values more immune to noise
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EEE3420 Industrial Control
Transfer function of a control system
The transfer function of a control system is defined as the ratio of the output to the input
predict how the system will perform if the transfer function is known
output depends on both the present input and the past history of the input, so the output c(t) is a convolution product of the input r(t) and the system g(t)
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EEE3420 Industrial Control
Transfer function of a control system
c(t) = r(t) * g(t)
0
)()( dgtr
with the use of Laplace transform, we get
0 00)()()()]([)( dtedgtrdtetctcLsC stst
12Week© Vocational Training Council, Hong Kong.
EEE3420 Industrial Control
Transfer function of a control system
Let t’ = t – τ then t = t’ + τ
the convolution product of r(t)*g(t) is now transformed into an algebraic product of R(s).G(s)
)()()(')'(
')()'()()()(
0 0
'
0
)'(
00 0
sGsRdegdtetr
dtedgtrdtedgtrsC
sst
tsst
13Week© Vocational Training Council, Hong Kong.
EEE3420 Industrial Control
Transfer function of a control system
time-domain output c(t) may be obtained by the inverse Laplace operation
open-loop control system can be represented in frequency domain as shown above
the output C(s) is given by G(s) x R(s).
jk
jk
st dsesCj
sCLtc )(21)]([)( 1
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EEE3420 Industrial Control
PID Control System PID – Controller is the most widely used control strategy
in industry used for various control problems such as automated
systems or plants consists of three different elements
P Proportional control I Integral control D Derivative control
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EEE3420 Industrial Control
PID Control System
for control loop to work properly, the PID loop must be properly tuned
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EEE3420 Industrial Control
The PID transfer function
)(teKP P
dtteKI I )(
dttedKD D))((
P Proportional control,
D Derivative control,
I Integral control,
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EEE3420 Industrial Control
The PID transfer function
The total controller output,
We get,
Use the Laplace transform,
dttedKdtteKteKtu DIP))(()()()(
sxdtx
sdtsx
dtdxs
dtd ;1;;
)1()()()(
sKsKKsG
sEsU
DI
PC
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EEE3420 Industrial Control
The PID transfer function
Re-arrange to get,sT
sTsTTKsGI
IDIPC
)1()(2
Where KP is the proportional gain
TI is the integral time constant
TD is the derivative time constant
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EEE3420 Industrial Control
The PID transfer function
the three different adjustments (KP, TI, TD) interact with each other
it can be very difficult and time consuming to tune these three values in order to get the best performance according to the design specifications of the system.
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EEE3420 Industrial Control
A Thermal Control System
• electrical heater of heat capacity Ch & thermal resistance Rho
• oven of heat capacity Co & thermal resistance Ro • environment temperature Te, set-point temperature Ts• temperature controller adjusts the power W by
comparing To with Ts
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EEE3420 Industrial Control
On-Off Control of the Thermal System• the simplest form of control• when the oven is cooler
than the set-point temperature the heater is turned on at maximum power M
• once the oven is hotter than the set-point temperature the heater is switched off completely
Red line: set-point temperatureGreen line: actual temperatureBlue line: Delivered Power
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EEE3420 Industrial Control
Proportional Control of the Thermal System
• proportional controller attempts to perform better than the On-Off type by applying power W to the heater in proportion to the difference in temperature between the oven and the set-point
• KP is known as the proportional gain of the controller
OSP TTKW
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EEE3420 Industrial Control
PD Control of the Thermal System
• add D-Control (proportional to the time-derivative of the error signal) to mitigate the stability and overshoot problems that arise from a high gain proportional controller
• adjust KD (the damping constant) to achieve a critically damped response
dtTTdKTTKW OS
DOSP
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EEE3420 Industrial Control
PID Control of the Thermal System• add I-Cotnrol (proportional to
the time-integral of the error signal) to change the heater power until the time-averaged value of the temperature error is zero
• KI is the integral gain parameter
dtTTdKdtTTKTTKW OS
DDSIOSP
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EEE3420 Industrial Control
Analysis of a Control System using Transfer Function
given that,6116
10)(: 23
ssssGprocessThe P
and the feedback path: H(s) = 1
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EEE3420 Industrial Control
Analysis of a Control System using Transfer Function
for P-control only, if KP = 3, then GC(s) = KP =3, and
3611630
)()(1)()(
23
ssssGsG
sGsG
PC
PC
i
o
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EEE3420 Industrial Control
Analysis of a Control System using Transfer Function
for PI-control, if KP = 2.7 and TI = 1.5, then
and the transfer function is:
275.495.1695.1275.40
)()(1)()(
234
sssss
sGsGsGsG
PC
PC
i
o
ss
sTsTK
sTKsG
I
IP
IPC 5.1
)15.1(7.2)1(11)(
28Week© Vocational Training Council, Hong Kong.
EEE3420 Industrial Control
Analysis of a Control System using Transfer Function
for PID-control, if KP = 2,
TI = 0.9 and TD = 0.6, then
and the transfer function is:
204.237.204.59.020188.10
)()(1)()(
234
2
ssssss
sGsGsGsG
PC
PC
i
o
sss
sTsTsTTKsG
I
IDIPC 9.0
)19.054.0(2)1()(22
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EEE3420 Industrial Control
Analysis of a Control System using Transfer Function
The transfer functions give the step responses as shown on the right
• for P-control (the red curve) – a steady state error occurs
• for PI-control (the blue curve) – the response becomes more oscillatory and needs longer to settle, the error disappears
• for PID-control (the green curve) – the overshoot and the number of oscillatory cycles are much reduced
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EEE3420 Industrial Control
Concluding remarks of the PID Effect
• In general, we may observe that• P term is used to adjust the speed of response. • I term provides zero error. • D term introduces damping.
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EEE3420 Industrial Control
3. Stability of Control System• a control system responds to an input by undergoing a
transient response before reaching a steady-state• the total response of a system consists of two parts,
namely, the natural response and the forced response • natural response describes the way the system
dissipates or acquires energy, the nature of this response is dependent only on the system
• the nature of the forced response is dependent on the input
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EEE3420 Industrial Control
3. Stability of Control System• for a linear system, we can write
Total response = Natural response + Forced response• for a control system to be useful, the natural response
must eventually approach zero, thus leaving only the forced response
• a stable control system will always return to a stable operating state
• in an unstable system, any disturbance will result in oscillations building up until some parts fails
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EEE3420 Industrial Control
3. Stability of Control System
• Oscillatory System• between the stable state and the unstable state
lies the conditionally stable system in which oscillations neither increase nor decrease
• each cycle being identical to the previous one and results in sustainable oscillation.
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EEE3420 Industrial Control
Stability of First Order SystemConsider the response of a control
system with transfer function (s+2)/(s+5) under a step input with R(s) = 1/s
• a pole on the real axis generates an exponential response of the form e-αt, where –α is the pole location on the real axis.
• if α is positive, the transient response will decay to zero
• if α is negative, then the transient response will grow and the system will be unstable.
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EEE3420 Industrial Control
Stability of Second Order System
As long as the poles of the output function lies on the left hand side of the complex plane
• the system output will not grow without bound
• it will be stable
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EEE3420 Industrial Control
Stability of Second Order System
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EEE3420 Industrial Control
Stability of Higher Order System
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EEE3420 Industrial Control
Stability of Higher Order System - Routh table The Routh table method will
yield the stability information of a system without the need to solve for the system poles.
The method requires two steps:(1) generate a Routh table For a fourth order system given
on the right, Construct the table for the
denominator by using the formulae shown on the right
Similar formulae are used for system with order higher than 4.
s4 a4 a2 a0
s3 a3 a1 0
s2 b1=(a2*a3-a1*a4)/a3 b2=(a0*a3- 0*a4)/a3 0
s1 c1=(a1*b1-a3*b2)/b1 0 0
s0 d1=(b2*c1-0*b1)/c1 0 0
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EEE3420 Industrial Control
Stability of Higher Order System - Routh table (2) interpret the Routh table The number of poles that
are in the right half plane is equal to the number of sign change in the first column of the Routh table.
Note: For the special cases such as the element in the first column is equal to zero or the elements in the entire row are equal to zero will not be treated here, please refer to descriptions in books dealing with control theory.
s4 a4 a2 a0
s3 a3 a1 0
s2 b1=(a2*a3-a1*a4)/a3 b2=(a0*a3- 0*a4)/a3 0
s1 c1=(a1*b1-a3*b2)/b1 0 0
s0 d1=(b2*c1-0*b1)/c1 0 0
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EEE3420 Industrial Control
Stability of Higher Order System - Routh table Example Determine the stability of
the system on the right by Routh table method.
SolutionAs there are two sign changes
in the first column of the Routh table, so there are two poles lying in the right half plane and hence the system is not stable.
s3 a3 = 1 a1 = 31 0
s2 a2 = 10 a0 = 1030 0
s1 b1=(31*10-1*1030)/10
= -72
0 0
s0 c1 = (1030*(-72)-0*10)/(-72)
= 1030
0 0
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EEE3420 Industrial Control
Summary of Introduction to Control System• Closed-loop control system is less sensitive to noise,
disturbances and changes in the environment.• The transfer function of a control system is the ratio of the output
to the input.• Proportional control is used to adjust the speed of response. • Integral control provides zero error.• Differential control introduces damping.• If the poles of the output function lies on the left hand side of the
complex plane, the system will be stable.
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EEE3420 Industrial Control
Introduction to Control System
End of Lecture 1
RevisionNorman S. Nise, Control Systems Engineering, Fourth Edition, Johne Wiley & Sons, Inc., page 177 to page 183.