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Page 1Dr. WONG, Lik-Kin
PID Controllers
Control Engineeringby Dr. L. K. Wong
Page 2Dr. WONG, Lik-Kin
Output Feedback Control Systems
• Feed back only the output signal – Easy access
– Obtainable in practice
Controller PlantReference
Output+−
Page 3Dr. WONG, Lik-Kin
PID Controllers
• Proportional controllers– pure gain or attenuation
• Integral controllers– integrate error
• Derivative controllers– differentiate error
Page 4Dr. WONG, Lik-Kin
Proportional Controller
eKu p=
• Controller input is error (reference − output)• Controller output is control signal
• P controller involves only a proportional gain (or attenuation)
Page 5Dr. WONG, Lik-Kin
Integral Controller
• Integral of error with a constant gain• Increase system type by 1
– Infinity steady-state gain
– Eliminate steady-state error for a unit step input
dteKu i ∫=
Page 6Dr. WONG, Lik-Kin
Integral Controller
)(1)(
)(
)()()(
)(1
)(
)()(
sGsR
sE
sGsEsY
sG
sG
sR
sY
p
p
p
p
+=
=
+=
01
1)(1
1lim
)(1)(
lim)(lim)(lim000
=∞+
=+
=+
===→→→∞→ sGsG
ssRssEtee
ps
psstss
Page 7Dr. WONG, Lik-Kin
Derivative Control
• Differentiation of error with a constant gain• Reduce overshoot and oscillation
• Do not affect steady-state response• Sensitive to noise
dt
deKu d=
Page 8Dr. WONG, Lik-Kin
Controller Structure
• Single controller– P controller, I controller, D controller
• Combination of controllers– PI controller, PD controller
– PID controller
Page 9Dr. WONG, Lik-Kin
PID Controller
• PI controller
• PD controller
• PID controller
dteKeKu ip ∫+=
dt
deKeKu dp +=
dt
deKdteKeKu dip ++= ∫
Page 10Dr. WONG, Lik-Kin
PID Controller
• PI controller
• PD controller
• PID controller
)()()( sEs
KKsU i
p +=
)()()( sEsKKsU dp +=
)()()( sEsKs
KKsU d
ip ++=
Page 11Dr. WONG, Lik-Kin
Controller Performance
• P controller• PI controller
• PD controller• PID controller
Page 12Dr. WONG, Lik-Kin
Block Diagram
PIDController
PlantReference
Output+−
Page 13Dr. WONG, Lik-Kin
P Controller
11
)( 2 ++=
sssGp
)()( sEKsU p=
12 ++=
ss
KOLTF p
p
p
Kss
KCLTF
+++=
12
Page 14Dr. WONG, Lik-Kin
P Controller
• Increase in gain – upgrade both steady-state and transient
responses
– reduce steady-state error
– reduce stability
Page 15Dr. WONG, Lik-Kin
Time (sec.)
Am
plit
ude
S t ep Response
0 2 4 6 8 10 120
0.5
1
1.5From: U(1)
To:
Y(1
)
P Controller
Kp = 10
Kp = 1
Kp = 5
Kp = 2
Page 16Dr. WONG, Lik-Kin
P Controller
Time (sec.)
Am
plit
ude
S t ep Response
0 2 4 6 8 10 120
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2From: U(1)
To:
Y(1
) Kp = 1000
Page 17Dr. WONG, Lik-Kin
Proportional Controller
Frequency (rad/sec)
Pha
se (
deg)
; M
agni
tude
(dB
)
Bode Diagrams
-150
-100
-50
0
50From: U(1)
10 -1 100 101 102 103-200
-150
-100
-50
0
To:
Y(1
)
)100)(1(1
)(2 ++=
sssL
)100)(1(1000
)(3 ++=
sssL
Page 18Dr. WONG, Lik-Kin
PI Controller
)()(
)()()()(
)()()(
sGs
KsK
sGs
KK
sE
sY
s
KK
sE
sU
pip
pi
p
ip
+=
+=
+=
ip
ip
KsKss
KsKCLTF
+++++
=)1(23
Page 19Dr. WONG, Lik-Kin
Time (sec.)
Am
plit
ude
S t ep Response
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6From: U(1)
To:
Y(1
)
PI Controller
Ki = 0
Ki = 1
Ki = 0.5
Ki = 2
Kp = 2
Page 20Dr. WONG, Lik-Kin
PD Controller
)()()( sEsKKsU dp +=
12 +++
=ss
KsKOLTF pd
)1()1(2pd
pd
KsKs
KsKCLTF
+++++
=
Page 21Dr. WONG, Lik-Kin
PD Controller
Time (sec.)
Am
plit
ude
S t ep Response
0 5 10 150
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1From: U(1)
To:
Y(1
)
Kd = 0
Kd = 1
Kd = 10
Kd = 2
Kp = 2
Page 22Dr. WONG, Lik-Kin
PID Controller
)(
)()()(
2
sEs
KsKsK
sEsKs
KKsU
ipd
di
p
++=
++=
sss
KsKsKOLTF ipd
++++
=23
2
ipd
ipd
KsKsKs
KsKsKCLTF
+++++++
=)1()1( 23
2
Page 23Dr. WONG, Lik-Kin
Time (sec.)
Am
plit
ude
S t ep Response
0 2 4 6 8 10 12 140
0.2
0.4
0.6
0.8
1
1.2
1.4From: U(1)
To:
Y(1
)
PID Controller
Kp = 2, Ki = 1, Kd = 2,
Kp = 2, Ki = 2, Kd = 2,
Kp = 5, Ki = 5, Kd = 5Kp = 5, Ki = 5, Kd = 2
Page 24Dr. WONG, Lik-Kin
Design of PID Controllers
• Based on the knowledge of P, I and D– trial and error
– manual tuning
– simulation
Page 25Dr. WONG, Lik-Kin
Design of PID Controllers
• Ziegler-Nichols method– based on a open-loop process
– based on a critical gain
Page 26Dr. WONG, Lik-Kin
Ziegler-Nichols Method 1
Time delay L = τd
Slope R = K/τ
τ
K1
)(+
=−
s
KesG
sd
τ
τ
Page 27Dr. WONG, Lik-Kin
Ziegler-Nichols Method 1
• P controller– Kp = 1/RL
• PI controller– Kp = 0.9/RL, Ki = 0.27/RL2
• PID controller– Kp = 1.2/RL, Ki = 0.6/RL2, Kd = 0.6/R
Page 28Dr. WONG, Lik-Kin
Ziegler-Nichols Method 2
• Increase a pure gain Ku of a closed-loop system until the system is marginally stable
• Measure the period of oscillation Pu (unit is second)
Page 29Dr. WONG, Lik-Kin
Ziegler-Nichols Method 2
• P controller– Kp = 0.5Ku
• PI controller– Kp = 0.45Ku, Ki = 0.54Ku/Pu
• PID controller– Kp = 0.6Ku, Ki = 1.2Ku/Pu, Kd = 0.075KuPu
Page 30Dr. WONG, Lik-Kin
Digital P and D Controller
)()( zEKzU p=
Tkeke
ku
dt
tdetu
)1()()(
)()(
−−≈
=
Page 31Dr. WONG, Lik-Kin
Digital I Controller
[ ]
[ ]
)(11
2)(
)()(2
)()(
)()1(2
)(
)()()1(
)()()(
)1(
00
zEz
zTzU
zEzzET
zUzzU
kekeT
ku
dekuku
detutu
Tk
kT
t
t
−+
=
++=
+++=
+=+
+=
∫
∫+
ττ
ττ
Page 32Dr. WONG, Lik-Kin
Digital PID Controller
pK
11
2 −+
z
zTKi
Tz
zKd
)1( −
+ U(z)E(z)
Page 33Dr. WONG, Lik-Kin
Conclusion
• Properties of P, I, D, PI, PD, and PID controllers
• Design of PID controllers• Digital PID controllers
Page 34Dr. WONG, Lik-Kin
Reference
• M. Gopal, Digital Control Engineering. John Wiley & Sons.
• B. C. Kuo, Automatic Control System. Englewood Cliffs, N.J.: Prentice Hall, 1995.
• G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of Dynamic Systems. Singapore: Addison-Wesley, 1988.