Industrial Control Systems - PID Controllers

Industrial Control
Behzad Samadi
Department of Electrical Engineering
Amirkabir University of Technology
Winter 2010
Tehran, Iran
Behzad Samadi (Amirkabir University) Industrial Control 1 / 95

Feedback Control Loop
r: reference signal
y: process (controlled) variable
u: manipulated (control) variable
e: control error
d: load disturbance signal
n: measurement noise signal
F: feedforward ﬁlter
C: controller
P: plant
[Visioli, 2006]

On-Oﬀ Control
One of the simplest control laws:
u =
{
umax if e > 0
umin if e < 0

On-Oﬀ Control
One of the simplest control laws:
u =
{
umax if e > 0
umin if e < 0
Disadvantage: persistent oscillation of the process variable
P =
1
10s + 1
e−2s
, umax = 2, umin = 0

On-Off Control
a) Ideal on-off controller
b) Modified with a dead zone
c) Modified with hysteresys
[Visioli, 2006]

PID Control
1 Proportional action
2 Integral action
3 Derivative action
[Visioli, 2006]

Proportional Action
Proportional control action:
u(t) = Kpe(t) = Kp(r(t) − y(t)),
[Visioli, 2006]

Proportional Action
u(t) = Kpe(t) = Kp(r(t) − y(t)),
Kp: proportional gain
[Visioli, 2006]

Proportional Action
u(t) = Kpe(t) = Kp(r(t) − y(t)),
Controller transfer function:
C(s) = Kp
[Visioli, 2006]

Proportional Action
u(t) = Kpe(t) = Kp(r(t) − y(t)),
C(s) = Kp
Advantage: small control signal for a small error signal
[Visioli, 2006]

Proportional Action
u(t) = Kpe(t) = Kp(r(t) − y(t)),
C(s) = Kp
Advantage: small control signal for a small error signal
Disadvantage: steady state error
[Visioli, 2006]

Proportional Action
Steady state error occurs even if the process presents an integrating
dynamics, in case a constant load disturbance occurs.
[Visioli, 2006]

Proportional Action
Steady state error occurs even if the process presents an integrating
dynamics, in case a constant load disturbance occurs.
Adding a bias (or reset) term:
u(t) = Kpe + ub
The value of ub can be ﬁxed or can be adjusted manually until the
steady state error is zero.
[Visioli, 2006]

Proportional Action
Proportional Band (PB):
PB =
100%
Kp
[Astrom and Hagglund, 1995]

Integral Action
Integral control action:
u(t) = Ki
∫ t
0
e(𝜏)d𝜏,
[Visioli, 2006]

Integral Action
u(t) = Ki
∫ t
0
e(𝜏)d𝜏,
Ki : integral gain
[Visioli, 2006]

Integral Action
u(t) = Ki
∫ t
0
e(𝜏)d𝜏,
Ki : integral gain
C(s) =
Ki
s
[Visioli, 2006]

Integral Action
u(t) = Ki
∫ t
0
e(𝜏)d𝜏,
Ki : integral gain
C(s) =
Ki
s
Advantage: zero steady state error
[Visioli, 2006]

Integral Action
u(t) = Ki
∫ t
0
e(𝜏)d𝜏,
Ki : integral gain
C(s) =
Ki
s
Advantage: zero steady state error
Disadvantage: integrator windup in the presence of saturation
[Visioli, 2006]

PI Controller
Proportional Integrator Controller:
Transfer function:
C(s) = Kp(1 +
1
Ti s
)
[Visioli, 2006]

PI Controller
Transfer function:
C(s) = Kp(1 +
1
Ti s
)
Integral action is able to set automatically the value of ub.
[Visioli, 2006]

PI Controller
Transfer function:
C(s) = Kp(1 +
1
Ti s
)
Integral action is able to set automatically the value of ub.
The integral action is also called automatic reset.
[Visioli, 2006]

Derivative Action
Derivative control action:
u(t) = Kd
de(t)
dt
,
[Visioli, 2006]

Derivative Action
u(t) = Kd
de(t)
dt
,
Kd : derivative gain
[Visioli, 2006]

Derivative Action
u(t) = Kd
de(t)
dt
,
C(s) = Kd s
[Visioli, 2006]

Derivative Action
u(t) = Kd
de(t)
dt
,
C(s) = Kd s
Advantage: Derivative action is an instance of predictive control.
[Visioli, 2006]

Derivative Action
u(t) = Kd
de(t)
dt
,
C(s) = Kd s
Advantage: Derivative action is an instance of predictive control.
Disadvantage: Sensitive to the measurement noise in the manipulated
variable
[Visioli, 2006]

Derivative Action
Derivative action is an instance of predictive control.
[Visioli, 2006]

Derivative Action
Taylor series expansion of the control error at time Td ahead:
e(t + Td ) ≈ e(t) + Td
de(t)
dt
[Visioli, 2006]

Derivative Action
e(t + Td ) ≈ e(t) + Td
de(t)
dt
A control law proportional to e(t + Td )
u(t) = Kp
(
e(t) + Td
de(t)
dt
)
[Visioli, 2006]

Derivative Action
e(t + Td ) ≈ e(t) + Td
de(t)
dt
A control law proportional to e(t + Td )
u(t) = Kp
(
e(t) + Td
de(t)
dt
)
Derivative action is also called anticipatory control, or rate action, or
pre-act.
[Visioli, 2006]

PID Controller
Transfer function:
C(s) = Kp
(
1 +
1
Ti s
+ Td s
)
Time windows:
Proportional action responds to current error.
Integrator action responds to accumulated past error.
Derivative action anticipated future error.
Peter Woolf umich.edu

PID Controller
Transfer function:
C(s) = Kp +
Ki
s
+ Kd s
Frequency band:
Proportional action: all-band
Integrator action: low pass
Derivative action: high pass
[Li et al., 2006]

PID Controller
Transfer function:
C(s) = Kp +
Ki
s
+ Kd s
[Li et al., 2006]

PID Controller
Implementation methods:
Pneumatic
Hydraulic
Electronic
Digital

PID Controller
Structures of PID controllers:
Ideal or noninteracting form:
Ci (s) = Kp
(
1 +
1
Ti s
+ Td s
)
[Visioli, 2006] and [Astrom and Hagglund, 1995]

PID Controller
Ci (s) = Kp
(
1 +
1
Ti s
+ Td s
)
Series or interacting from:
Cs(s) = K
′
p
(
1 +
1
T
′
i s
)
(1 + T
′
d s)

PID Controller
Ci (s) = Kp
(
1 +
1
Ti s
+ Td s
)
Series or interacting from:
Cs(s) = K
′
p
(
1 +
1
T
′
i s
)
(1 + T
′
d s)
Parallel form:
Ci (s) = Kp +
Ki
s
+ Kd s

PID Controller

PID Controller
Series to ideal form conversion:
Kp =K
′
p
T
′
i + T
′
d
T
′
i
Ti =T
′
i + Td
′
Td =
T
′
i T
′
d
T
′
i + T
′
d

PID Controller
Ideal to series form conversion: Only if Ti ≥ 4Td
K
′
p =
Kp
2
(
1 +
√
1 − 4
Td
Ti
)
T
′
i =
Ti
2
(
1 +
√
1 − 4
Td
Ti
)
T
′
d =
Ti
2
(
1 −
√
1 − 4
Td
Ti
)
[Visioli, 2006]

PID Controller
Alternative series form:
Cs(s) = Kp(1 +
1
𝛼Ti s
)(𝛼 + Td s)
Ideal to alternative series form conversion: Only if Ti ≥ 4Td
𝛼 =
1 ±
√
1 − 4Td
Ti
2
> 0
[Li et al., 2006]

PID Controller
A PID controller has two zeros and one pole at the origin.
Ti > 4Td : two real zeros
Ti = 4Td : two coincident zeros
Ti < 4Td : two complex conjugate zeros
[Visioli, 2006]

Problems with the Derivative Action
Noise:
n(t) = A sin(𝜔t)
[Visioli, 2006]

Noise:
n(t) = A sin(𝜔t)
Derivative action:
u(t) = A𝜔 cos(𝜔t)
[Visioli, 2006]

Noise:
n(t) = A sin(𝜔t)
Derivative action:
u(t) is large for high frequencies.
[Visioli, 2006]

Noise:
n(t) = A sin(𝜔t)
Derivative action:
u(t) is large for high frequencies.
In practice, a (very) noisy control signal might lead to a damage of
the actuator.
[Visioli, 2006]

Modiﬁed Derivative Action
Modiﬁed ideal form:
Ci1a(s) = Kp
(
1 +
1
Ti s
+
Td s
Td
N s + 1
)
[Visioli, 2006]Behzad Samadi (Amirkabir University) Industrial Control 24 / 95

Ci1a(s) = Kp
(
1 +
1
Ti s
+
Td s
Td
N s + 1
)
Gerry and Shinskey, 2005:
Ci1b(s) = Kp
⎛
⎜
⎝1 +
1
Ti s
+
Td s
1 + Td
N s + 0.5
(
Td
N s
)2
⎞
⎟
⎠

Ci1a(s) = Kp
(
1 +
1
Ti s
+
Td s
Td
N s + 1
)
Ci1b(s) = Kp
⎛
⎜
⎝1 +
1
Ti s
+
Td s
1 + Td
N s + 0.5
(
Td
N s
)2
⎞
⎟
⎠
Modiﬁed series form:
Cs(s) = K
′
p
(
1 +
1
T
′
i s
)
⎛
⎝T
′
d s + 1
T
′
d
N s + 1
⎞
⎠

Ci1a(s) = Kp
(
1 +
1
Ti s
+
Td s
Td
N s + 1
)
Ci1b(s) = Kp
⎛
⎜
⎝1 +
1
Ti s
+
Td s
1 + Td
N s + 0.5
(
Td
N s
)2
⎞
⎟
⎠
Modiﬁed series form:
Cs(s) = K
′
p
(
1 +
1
T
′
i s
)
⎛
⎝T
′
d s + 1
T
′
d
N s + 1
⎞
⎠
N generally assumes a value between 1 and 33, although in the
majority of the practical cases its setting falls between 8 and 16 (Ang
et al., 2005).

Alternative modiﬁed ideal form:
Ci2a(s) = Kp
(
1 +
1
Ti s
+ Td s
)
1
Tf s + 1
[Visioli, 2006]

Ci2a(s) = Kp
(
1 +
1
Ti s
+ Td s
)
1
Tf s + 1
˚Astr¨om and H¨agglund, 2004:
Ci2b(s) = Kp
(
1 +
1
Ti s
+ Td s
)
1
(Tf s + 1)2
[Visioli, 2006]

Ci2a(s) = Kp
(
1 +
1
Ti s
+ Td s
)
1
Tf s + 1
Ci2b(s) = Kp
(
1 +
1
Ti s
+ Td s
)
1
(Tf s + 1)2
Derivative kick: A spike in the control signal due to an abrupt
(stepwise) change of the set-point signal.
[Visioli, 2006]

Ci2a(s) = Kp
(
1 +
1
Ti s
+ Td s
)
1
Tf s + 1
Ci2b(s) = Kp
(
1 +
1
Ti s
+ Td s
)
1
(Tf s + 1)2
Derivative kick: A spike in the control signal due to an abrupt
(stepwise) change of the set-point signal.
If the set-point is constant, the derivative action can be applied only
to the process variable:
u(t) = −Kd
dy(t)
dt
[Visioli, 2006]

Derivative Action
80% of the employed PID controllers have the derivative part
switched-oﬀ (Ang et al., 2005).
[Visioli, 2006]

Derivative Action
Derivative action is the most diﬃcult to tune, why?
[Visioli, 2006]

Derivative Action
Consider a ﬁrst-order-plus-dead-time (FOPDT) plant:
P(s) =
K
Ts + 1
e−Ls
and a PD controller:
C(s) = Kp(1 + Td s)
[Visioli, 2006]

Derivative Action
Consider a ﬁrst-order-plus-dead-time (FOPDT) plant:
P(s) =
K
Ts + 1
e−Ls
and a PD controller:
C(s) = Kp(1 + Td s)
Open loop frequency response:
∣C(j𝜔)P(j𝜔)∣ = KKp
√
1 + T2
d 𝜔2
1 + T2𝜔2
[Visioli, 2006]

Derivative Action
KKp
√
1 + T2
d 𝜔2
1 + T2𝜔2
≥ KKp min
(
1,
Td
T
)
[Visioli, 2006]

Derivative Action
KKp
√
1 + T2
d 𝜔2
1 + T2𝜔2
≥ KKp min
(
1,
Td
T
)
Td ≥ T ⇒ min
(
1, Td
T
)
= 1
[Visioli, 2006]

Derivative Action
KKp
√
1 + T2
d 𝜔2
1 + T2𝜔2
≥ KKp min
(
1,
Td
T
)
Td ≥ T ⇒ min
(
1, Td
T
)
= 1
If Td ≥ T and KKp > 1, then
∣C(j𝜔)P(j𝜔)∣ ≥ 1
[Visioli, 2006]

Derivative Action
KKp
√
1 + T2
d 𝜔2
1 + T2𝜔2
≥ KKp min
(
1,
Td
T
)
Td ≥ T ⇒ min
(
1, Td
T
)
= 1
∣C(j𝜔)P(j𝜔)∣ ≥ 1
Td ≤ T ⇒ min
(
1, Td
T
)
= Td
T
[Visioli, 2006]

Derivative Action
KKp
√
1 + T2
d 𝜔2
1 + T2𝜔2
≥ KKp min
(
1,
Td
T
)
Td ≥ T ⇒ min
(
1, Td
T
)
= 1
∣C(j𝜔)P(j𝜔)∣ ≥ 1
Td ≤ T ⇒ min
(
1, Td
T
)
= Td
T
If Td ≤ T and KKp
Td
T > 1, then
∣C(j𝜔)P(j𝜔)∣ ≥ 1
[Visioli, 2006]

Frequency Response
The magnitude of the open-loop
transfer function is not less than
0 dB. As a consequence, since
the phase decreases when the
frequency increases because of
the time delay, the closed-loop
system will be unstable.
[Visioli, 2006]

Derivative Action
Consider the following process:
P(s) =
2
s + 1
e−0.2s
controlled by a PID controller in series form with Kp = 1 and Ti = 1.

Derivative Action
P(s) =
2
s + 1
e−0.2s
If Td = 0.01, GM=12.3dB, PM=68.2 deg.

Derivative Action
P(s) =
2
s + 1
e−0.2s
If Td = 0.01, GM=12.3dB, PM=68.2 deg.
If Td = 0.05, GM=13.2dB, PM=72.7 deg.

Derivative Action
P(s) =
2
s + 1
e−0.2s
If Td = 0.01, GM=12.3dB, PM=68.2 deg.
If Td = 0.05, GM=13.2dB, PM=72.7 deg.
If Td = 0.5, the system stability is lost!

Derivative Action
P(s) =
2
s + 1
e−0.2s
If Td = 0.01, GM=12.3dB, PM=68.2 deg.
If Td = 0.05, GM=13.2dB, PM=72.7 deg.
If Td = 0.5, the system stability is lost!
In summary:
Sensitive to noise
Hard to tune (4 parameters)
[Visioli, 2006]

Integral Windup

Integral Windup
Solid:
process output,
Dashed:
process input,
Dotted:
integral term
[Visioli, 2006]

Conditional Integration
The integral term is limited to a predeﬁned value.
[Visioli, 2006]

The integration is stopped when the error is greater than a predeﬁned
threshold, namely, when the process variable value is far from the
setpoint value.
[Visioli, 2006]

setpoint value.
The integration is stopped when the control variable saturates, i.e.,
when u
′
∕= u.
[Visioli, 2006]

setpoint value.
The integration is stopped when the control variable saturates, i.e.,
when u
′
∕= u.
The integration is stopped when the control variable saturates and
the control error and the control variable have the same sign (i.e.,
when ue > 0).
[Visioli, 2006]

Anti-windup for Automatic Reset Conﬁguration
Automatic reset:
[Visioli, 2006]

Automatic reset:
Limiting the controller output:
[Visioli, 2006]

Automatic reset:
Limiting the controller output:
Limiting the integrator output:
[Visioli, 2006]

Back-calculation
Integrator input:
ei =
Kp
Ti
e +
1
Tt
(u
′
− u)

Back-calculation
Integrator input:
ei =
Kp
Ti
e +
1
Tt
(u
′
− u)
Tuning rule for Tt (˚Astr¨om and H¨agglund 1995):
Tt =
√
Td Ti

Back-calculation
Integrator input:
ei =
Kp
Ti
e +
1
Tt
(u
′
− u)
Tuning rule for Tt (˚Astr¨om and H¨agglund 1995):
Tt =
√
Td Ti
Not useful for a PI controller

Back-calculation
Integrator input:
ei =
Kp
Ti
e +
1
Tt
(u
′
− u)
[Visioli, 2006]

Back-calculation
Integrator input:
ei =
Kp
Ti
e +
1
Tt
(u
′
− u)
Bohn and Atherton, 1995 suggest Tt = Ti .
[Visioli, 2006]

Back-calculation
Integrator input:
ei =
Kp
Ti
e +
1
Tt
(u
′
− u)
Bohn and Atherton, 1995 suggest Tt = Ti .
Conditioning technique (Hanus et al., 1987;Walgama et al., 1991):
This is a tracking rule (u tracks u
′
). In this framework:
Tt = Kp
[Visioli, 2006]

PID with Tracking Input
SP: Setpoint, MV: Manipulated Variable, TR: Tracking Input

Bumpless Transfer
Bumpless Transfer between Manual (M) and Automatic (A)
A is to track M in this case.
[Visioli, 2006]

Bumpless Transfer

Bumpless Transfer
Incremental manual input

Bumpless Transfer
Manual Control Module:

Bumpless Transfer
PID with Manual Switch:

PID Controller Design
In process industries, more than 97% of the regulatory controllers are
of the PID type.
[Yu, 2007]

of the PID type.
Most loops are actually under PI control (as a result of the large
number of ﬂow loops).
[Yu, 2007]

of the PID type.
Pulp and paper industry over 2000 loops:
[Yu, 2007]

of the PID type.
Only 20% of loops worked well (i.e. less variability in the automatic
mode over the manual mode).
[Yu, 2007]

of the PID type.
30% gave poor performance due to poor controller tuning.
[Yu, 2007]

of the PID type.
30% gave poor performance due to control valve problems (e.g. control
valve stick-slip, dead band, backlash).
[Yu, 2007]

of the PID type.
30% gave poor performance due to control valve problems (e.g. control
valve stick-slip, dead band, backlash).
20% gave poor performance due to process and/or control system
design problems.
[Yu, 2007]

Process industries:
[Yu, 2007]

Process industries:
30% of loops operated on manual mode.
[Yu, 2007]

Process industries:
20% of controllers used factory tuning.
[Yu, 2007]

Process industries:
30% gave poor performance due to sensor and control valve problems.
[Yu, 2007]

Process industries:
Chemical process industry:
[Yu, 2007]

Process industries:
Half of the control valves needed to be ﬁxed (results of the Fisher diagnostic
valve package).
[Yu, 2007]

Process industries:
valve package).
Most poor tuning was due to control valve problems.
[Yu, 2007]

Process industries:
valve package).
Reﬁning, chemicals, and pulp and paper industries over 26,000 controllers:
[Yu, 2007]

Process industries:
valve package).
Only 32% of loops were classiﬁed as excellent or acceptable.
[Yu, 2007]

Process industries:
valve package).
32% of controllers were classiﬁed as fair or poor, which indicates
unacceptably sluggish or oscillatory responses.
[Yu, 2007]

Process industries:
valve package).
36% of controllers were on open- loop, which implies that the controllers
were either in manual or virtually saturated.
[Yu, 2007]

Process industries:
valve package).
36% of controllers were on open- loop, which implies that the controllers
were either in manual or virtually saturated.
PID algorithms are used in vast majority of applications (97%). For the rare
cases of complex dynamics or signiﬁcant dead time, other algorithms are
used.
[Yu, 2007]

Ziegler-Nichols PID Tuning Method:
Ziegler-Nichols closed-loop tuning method
Ziegler-Nichols open-loop tuning method
Ziegler, Nichols, “Optimum settings for automatic controllers”, Trans.
ASME, 64, pp. 759-768, 1942
[Woolf, 2007]

Ziegler-Nichols closed-loop tuning method:
1 Remove integral and derivative action.

2 Create a small disturbance in the loop by changing the set point.
Adjust the proportional, increasing and/or decreasing, the gain until
the oscillations have constant amplitude.

3 Record the gain value (Ku) and period of oscillation (Pu).

3 Record the gain value (Ku) and period of oscillation (Pu).
[Woolf, 2007]

[Love, 2007]

Ziegler-Nichols closed-loop tuning method (ultimate cycle method):
C(s) = Kp(1 +
1
Ti s
+ Td s)
Kp Ti Td
P Controller Ku/2 - -
PI Controller Ku/2.2 Pu/1.2 -
PID Controller Ku/1.7 Pu/2 Pu/8
[Woolf, 2007]

Advantages:
[Woolf, 2007]

Advantages:
Easy experiment; only need to change the P controller
[Woolf, 2007]

Advantages:
Includes dynamics of whole process, which gives a more accurate
picture of how the system is behaving
[Woolf, 2007]

Advantages:
Disadvantages:
[Woolf, 2007]

Advantages:
Disadvantages:
Experiment can be time consuming
[Woolf, 2007]

Advantages:
Disadvantages:
Can venture into unstable regions while testing the P controller, which
could cause the system to become out of control
[Woolf, 2007]

Advantages:
Disadvantages:
Can venture into unstable regions while testing the P controller, which
could cause the system to become out of control
It does not hold for I, D and PD controllers.
[Woolf, 2007]

Process Reaction Curve:
P: the size of the step disturbance in the setpoint
L: the time taken from the moment the disturbance was introduced
to the ﬁrst sign of change in the output signal
ΔCp: the change in output signal in response to the initial step
disturbance
T: the time taken for this change to occur
[Woolf, 2007]

Process Reaction Curve:
N =
ΔCp
T : reaction rate
[Woolf, 2007]

Ziegler-Nichols open-loop tuning method (process reaction method):
C(s) = Kp(1 +
1
Ti s
+ Td s)
Kp Ti Td
P Controller K - -
PI Controller 0.9K L/0.3 -
PID Controller 1.2K 2L 0.5L
K =
P
NL
[Woolf, 2007]

Ziegler-Nichols open-loop tuning method:
Advantages:
[Woolf, 2007]

Advantages:
Quick and easier to use than other methods
[Woolf, 2007]

Advantages:
It is a robust and popular method
[Woolf, 2007]

Advantages:
Of these two techniques, the Process Reaction Method is the easiest
and least disruptive to implement
[Woolf, 2007]

Advantages:
Disadvantages:
[Woolf, 2007]

Advantages:
Disadvantages:
It depends upon purely proportional measurement to estimate I and D
controllers.
[Woolf, 2007]

Advantages:
Disadvantages:
controllers.
Approximations for the Kc , Ti , and Td values might not be entirely
accurate for diﬀerent systems.
[Woolf, 2007]

Advantages:
Disadvantages:
controllers.
Approximations for the Kc , Ti , and Td values might not be entirely
accurate for diﬀerent systems.
It does not hold for I, D and PD controllers.
[Woolf, 2007]

Summary:
Ziegler-Nichols closed-loop tuning method: Gain margin of 2
Ziegler-Nichols open-loop tuning method: Decay ratio of 0.25

Ziegler-Nichols closed-loop tuning methods:
G(s) =
Ke−Ds
𝜏s + 1
[Yu, 2007]

Ziegler-Nichols closed-loop tuning methods:
G(s) =
Ke−td s
𝜏s + 1
Quarter decay ration for 0 < td
𝜏 < 1 [O’Dwyer, 2002]
[Chau, 2002]

Quantities to characterize the error:
Maximum error: max e(t)

Integrated Absolute Error: IAE =
∫ ∞
0 ∣e(t)∣dt

∫ ∞
0 ∣e(t)∣dt
Integrated Error for non-oscillatory processes: IE =
∫ ∞
0 e(t)dt

∫ ∞
0 ∣e(t)∣dt
∫ ∞
0 e(t)dt
Integrated Squared Error: ISE =
∫ ∞
0 e2(t)dt

∫ ∞
0 ∣e(t)∣dt
∫ ∞
0 e(t)dt
∫ ∞
0 e2(t)dt
Integrated Time Absolute Error: ITAE =
∫ ∞
0 t∣e(t)∣dt

∫ ∞
0 ∣e(t)∣dt
∫ ∞
0 e(t)dt
∫ ∞
0 e2(t)dt
∫ ∞
0 t∣e(t)∣dt
Integrated Time Error: ITE =
∫ ∞
0 te(t)dt

∫ ∞
0 ∣e(t)∣dt
∫ ∞
0 e(t)dt
∫ ∞
0 e2(t)dt
∫ ∞
0 t∣e(t)∣dt
∫ ∞
0 te(t)dt
Integrated Time Squared Error: ITSE =
∫ ∞
0 te2(t)dt

∫ ∞
0 ∣e(t)∣dt
∫ ∞
0 e(t)dt
∫ ∞
0 e2(t)dt
∫ ∞
0 t∣e(t)∣dt
∫ ∞
0 te(t)dt
Integrated Time Squared Error: ITSE =
∫ ∞
0 te2(t)dt
Integrated Squared Time Error: ISTE =
∫ ∞
0 t2e2(t)dt

ITAE optimization:
G(s) =
Ke−td s
𝜏s + 1
[Chau, 2002]

Ciancone and Marlin Tuning:
G(s) =
Ke−td s
𝜏s + 1
Fractional dead time: Tf = td
td +𝜏
Using Tf , compute 𝜇CM and 𝜏CM:
Tf 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
𝜇CM 1.1 1.1 1.8 1.1 1.0 0.8 0.59 0.42 0.32
𝜏CM 0.23 0.23 0.23 0.72 0.72 0.70 0.67 0.60 0.53
Compute the controller gains:
Kp =
𝜇CM
K
, Ti = 𝜏CM(td + 𝜏)
Minimizing IAE or ISE [Chau, 2002]

Ciancone and Marlin PI Tuning:
[Chau, 2002]

Ciancone and Marlin PID Tuning:
[Chau, 2002]

Direct Synthesis:
We have:
C
R
=
GcGp
1 + GcGp
Therefore:
Gc =
1
Gp
C/R
1 − C/R
[Chau, 2002]

Direct Synthesis:
C
R
=
1
𝜏cs + 1
⇒ Gc =
1
Gp
(
1
𝜏cs
)

Direct Synthesis:
C
R
=
1
𝜏cs + 1
⇒ Gc =
1
Gp
(
1
𝜏cs
)
Example:
Gp =
Kp
𝜏ps + 1
⇒ Gc =
𝜏p
Kp𝜏c
(
1 +
1
𝜏ps
)
[Chau, 2002]

Direct Synthesis:
For delayed systems:
C
R
=
e−𝜃s
𝜏cs + 1
⇒ Gc =
1
Gp
(
e−𝜃s
(𝜏cs + 1) − e−𝜃s
)

Direct Synthesis:
C
R
=
e−𝜃s
𝜏cs + 1
⇒ Gc =
1
Gp
(
e−𝜃s
(𝜏cs + 1) − e−𝜃s
)
Considering e−𝜃s ≈ 1 − 𝜃s:
Gc ≈
1
Gp
(
e−𝜃s
(𝜏c + 𝜃)s
)

Direct Synthesis:
C
R
=
e−𝜃s
𝜏cs + 1
⇒ Gc =
1
Gp
(
e−𝜃s
(𝜏cs + 1) − e−𝜃s
)
Considering e−𝜃s ≈ 1 − 𝜃s:
Gc ≈
1
Gp
(
e−𝜃s
(𝜏c + 𝜃)s
)
Example:
Gp =
Kpe−td s
𝜏ps + 1
⇒ Gc =
𝜏p
Kp(𝜏c + 𝜃)
(
1 +
1
𝜏ps
)
for 𝜃 = td
[Chau, 2002]

Direct Synthesis:
Second order underdamped desired response:
C
R
=
1
𝜏2s2 + 2𝜁𝜏s + 1
⇒ Gc =
1
Gp
(
1
𝜏2s2 + 2𝜁𝜏s
)

Direct Synthesis:
C
R
=
1
𝜏2s2 + 2𝜁𝜏s + 1
⇒ Gc =
1
Gp
(
1
)
Example:
Gp =
Kp
(𝜏1s + 1)(𝜏2s + 1)
⇒ Gc =
(𝜏1s + 1)(𝜏2s + 1)
Kp𝜏s(𝜏s + 2𝜁)

Direct Synthesis:
C
R
=
1
𝜏2s2 + 2𝜁𝜏s + 1
⇒ Gc =
1
Gp
(
1
)
Example:
Gp =
Kp
(𝜏1s + 1)(𝜏2s + 1)
⇒ Gc =
(𝜏1s + 1)(𝜏2s + 1)
Kp𝜏s(𝜏s + 2𝜁)
Assuming 𝜏2 > 𝜏1 and 𝜏 = 2𝜁𝜏2
Gc =
𝜏1
4Kp𝜁2𝜏2
(
1 +
1
𝜏1s
)
[Chau, 2002]

Internal Model Control: Assume that we have an approximate model ˜Gp of
the process Gp
Open loop:
Gc = ˜G−1
p

Internal Model Control: Assume that we have an approximate model ˜Gp of
the process Gp
Open loop:
Gc = ˜G−1
p
Closed loop:

Internal Model Control:
P = G★
c (R − C + ˜C)
= G★
c (R − C + ˜GpP)

P = G★
c (R − C + ˜C)
= G★
c (R − C + ˜GpP)
Therefore:
P =
G★
c
1 − G★
c
˜Gp
(R−C)

P = G★
c (R − C + ˜C)
= G★
c (R − C + ˜GpP)
Therefore:
P =
G★
c
1 − G★
c
˜Gp
(R−C)
P = Gc(R − C)
Therefore:
Gc = G★
c
1−G★
c
˜Gp
[Chau, 2002]Behzad Samadi (Amirkabir University) Industrial Control 64 / 95

Closed loop transfer function:
C =
[
(1 − G★
c
˜Gp)GL
1 + G★
c (Gp − ˜Gp)
]
L +
[
GpG★
c
1 + G★
c (Gp − ˜Gp)
]
R
[Chau, 2002]

C =
[
(1 − G★
c
˜Gp)GL
1 + G★
c (Gp − ˜Gp)
]
L +
[
GpG★
c
1 + G★
c (Gp − ˜Gp)
]
R
How to choose G★
c :
˜Gp = ˜Gp+
˜Gp−
˜Gp+ contains all positive zeros if any.
[Chau, 2002]

C =
[
(1 − G★
c
˜Gp)GL
1 + G★
c (Gp − ˜Gp)
]
L +
[
GpG★
c
1 + G★
c (Gp − ˜Gp)
]
R
How to choose G★
c :
˜Gp = ˜Gp+
˜Gp−
˜Gp+ contains all positive zeros if any.
The design is based on ˜Gp− only:
G★
c =
1
˜Gp−
[
1
𝜏cs + 1
]r
r = 1, 2
[Chau, 2002]

Pad´e Approximation of Delays:
e−td s ≈ Nd (s)
Dd (s)
http://mathworld.wolfram.com/PadeApproximant.html

e−td s ≈ Nd (s)
Dd (s)
Gd1/0(s) = 1 − td s

e−td s ≈ Nd (s)
Dd (s)
Gd1/0(s) = 1 − td s
Gd0/1(s) = 1
1+td s

e−td s ≈ Nd (s)
Dd (s)
Gd1/0(s) = 1 − td s
Gd0/1(s) = 1
1+td s
Gd1/1(s) =
−
td
2
s+1
td
2
s+1

e−td s ≈ Nd (s)
Dd (s)
Gd1/0(s) = 1 − td s
Gd0/1(s) = 1
1+td s
Gd1/1(s) =
−
td
2
s+1
td
2
s+1
Gd2/2(s) =
t2
d
12
s2−
td
2
s+1
t2
d
12
s2+
td
2
s+1

Example: ˜Gp =
Kpe−td s
𝜏ps+1
˜Gp ≈
Kp
(𝜏ps + 1)(td
2 s + 1)
(
−
td
2
s + 1
)

Example: ˜Gp =
Kpe−td s
𝜏ps+1
˜Gp ≈
Kp
(𝜏ps + 1)(td
2 s + 1)
(
−
td
2
s + 1
)
˜Gp− =
Kp
(𝜏ps + 1)(td
2 s + 1)
˜Gp+ = −
td
2
s + 1
[Chau, 2002]

Example:
G★
c = ˜G−1
p−
1
𝜏cs + 1
=
(𝜏ps + 1)(td
2 s + 1)
Kp
1
𝜏cs + 1
Gc =
G★
c
1 − G★
c
˜Gp
= Kp(1 +
1
Ti s
+ Td s)

Example:
G★
c = ˜G−1
p−
1
𝜏cs + 1
=
(𝜏ps + 1)(td
2 s + 1)
Kp
1
𝜏cs + 1
Gc =
G★
c
1 − G★
c
˜Gp
= Kp(1 +
1
Ti s
+ Td s)
where:
Kc =
1
Kp
2
𝜏p
td
+ 1
2𝜏c
td
+ 1
; Ti = 𝜏p +
td
2
; Td =
𝜏p
2
𝜏p
td
+ 1
[Chau, 2002]

[Chau, 2002]

Skogestad PID Tuning Method:
S. Skogestad, “Probably the best simple PID tuning rules in the
world” Presented at AIChE Annual meeting, Reno, NV, USA, 04-09
Nov. 2001, Paper no. 276h
[Skogestad, 2001]

Skogestad PID Tuning Method:
g(s) = k
e−𝜃s
(𝜏1s + 1)(𝜏2s + 1)
[Skogestad, 2001]

Skogestad Internal Model Control (SIMC):
(
y
ys
)
desired
=
1
𝜏cs + 1
e−𝜃t

(
y
ys
)
desired
=
1
𝜏cs + 1
e−𝜃t
c(s) =
(𝜏1s + 1)(𝜏2s + 1)
k
1
(𝜏c + 𝜃)s

(
y
ys
)
desired
=
1
𝜏cs + 1
e−𝜃t
c(s) =
(𝜏1s + 1)(𝜏2s + 1)
k
1
(𝜏c + 𝜃)s
c(s) = Kc(1 +
1
𝜏I s
)(𝜏Ds + 1)

(
y
ys
)
desired
=
1
𝜏cs + 1
e−𝜃t
c(s) =
(𝜏1s + 1)(𝜏2s + 1)
k
1
(𝜏c + 𝜃)s
c(s) = Kc(1 +
1
𝜏I s
)(𝜏Ds + 1)
Tuning Rule
Kc =
1
k
𝜏1
𝜏c + 𝜃
, 𝜏I = 𝜏1. 𝜏D = 𝜏2
[Skogestad, 2001]

Tuning Rule
Kc =
1
k
𝜏1
𝜏c + 𝜃
, 𝜏I = 𝜏1, 𝜏D = 𝜏2
Eﬀective cancellation of the ﬁrst order dynamics by 𝜏I = 𝜏1
[Skogestad, 2001]

Tuning Rule
Kc =
1
k
𝜏1
𝜏c + 𝜃
, 𝜏I = 𝜏1, 𝜏D = 𝜏2
For processes with large 𝜏1, this choice results in a long settling time
for disturbance response.
[Skogestad, 2001]

Tuning Rule
Kc =
1
k
𝜏1
𝜏c + 𝜃
, 𝜏I = 𝜏1, 𝜏D = 𝜏2
For processes with large 𝜏1, this choice results in a long settling time
for disturbance response.
Consider: g(s) = k e−𝜃s
𝜏1s+1 ≈ k
𝜏1s for large 𝜏1. With a PI controller
c(s) = Kc(1 + 1
𝜏I s ), the poles of the closed loop system can be
obtained from the following equation:
𝜏I
k′
Kc
s2
+ 𝜏I s + 1 = 0
with k
′
= k
𝜏1
[Skogestad, 2001]

Comparing to s2 + 2𝜁𝜔0s + 𝜔2:
𝜔0 =
√
k′
Kc
𝜏I
, 𝜁 =
1
2
√
k′
Kc𝜏I
[Skogestad, 2001]

𝜔0 =
√
k′
Kc
𝜏I
, 𝜁 =
1
2
√
k′
Kc𝜏I
To avoid slow oscillations, consider: 𝜁 ≥ 1
[Skogestad, 2001]

𝜔0 =
√
k′
Kc
𝜏I
, 𝜁 =
1
2
√
k′
Kc𝜏I
To avoid slow oscillations, consider: 𝜁 ≥ 1
𝜏I ≥
4
k′
Kc
[Skogestad, 2001]

Disturbance rejection: Assuming input disturbance (gd (s) = g(s))
∣y(j𝜔)∣ =
∣g(j𝜔)∣
∣1 + g(j𝜔)c(j𝜔)∣
d ≤ ymax
[Skogestad, 2001]

∣y(j𝜔)∣ =
∣g(j𝜔)∣
∣1 + g(j𝜔)c(j𝜔)∣
d ≤ ymax
Assuming ∣gc∣ >> 1 at low frequencies:
∣c(j𝜔)∣ >
d
ymax
[Skogestad, 2001]

∣y(j𝜔)∣ =
∣g(j𝜔)∣
∣1 + g(j𝜔)c(j𝜔)∣
d ≤ ymax
∣c(j𝜔)∣ >
d
ymax
For P, PI and PID: c(j𝜔) ≥ Kc
[Skogestad, 2001]

∣y(j𝜔)∣ =
∣g(j𝜔)∣
∣1 + g(j𝜔)c(j𝜔)∣
d ≤ ymax
∣c(j𝜔)∣ >
d
ymax
For P, PI and PID: c(j𝜔) ≥ Kc
Kc ≥
d
ymax
[Skogestad, 2001]

Skogestad Internal Model Control (SIMC)
Kc =
1
k
𝜏1
𝜏c + 𝜃
=
1
k′
1
𝜏c + 𝜃
𝜏I = min{𝜏1,
4
k′
Kc
} = min{𝜏1, 4(𝜏c + 𝜃)}
𝜏D = 𝜏2
Tuning for fast response with good robustness
𝜏c = 𝜃
Tuning for slow response with acceptable disturbance rejection
Kc ≥
d
ymax
[Skogestad, 2001]

Rules of thumb:
[McMillan, 2001]

Digital Control
Ts: sampling time
[Astrom and Wittenmark, 1996]

Digital Control
Ts: sampling time
Analog to Digital (A-D) conversion:
y(tk) = y(kTs)

Digital Control
Ts: sampling time
Analog to Digital (A-D) conversion:
y(tk) = y(kTs)
Digital to Analog (D-A) conversion:
u(t) = u(tk) for kTs ≤ t < (k + 1)Ts

Digital Control
Diﬀerence equation:
x((k + 1)Ts) = ax(kTs) + bu(kTs)

Digital Control
Discrete time:
x[k + 1] = ax[k] + bu[k]

Digital Control
Discrete time:
x[k + 1] = ax[k] + bu[k]
Z-transform:
zX(z) = aX(z) + bU(z) ⇒
X(z)
U(z)
=
b
z − a

Digital Control
Approximating Continuous-Time Controllers:
Euler’s method (forward diﬀerence):
z = eTs s
≈ 1 + Tss ⇒ s ≈
z − 1
Ts
˙x(t) ≈
x(t + Ts) − x(t)
Ts

Digital Control
Euler’s method (forward diﬀerence):
z = eTs s
≈ 1 + Tss ⇒ s ≈
z − 1
Ts
˙x(t) ≈
x(t + Ts) − x(t)
Ts
Backward diﬀerence:
z = eTs s
≈
1
1 − Tss
⇒ s ≈
1 − z−1
Ts
˙x(t) ≈
x(t) − x(t − Ts)
Ts

Digital Control
Tustin’s approximation:
z = eTs s
≈
1 + sTs/2
1 − sTs/2
⇒ s ≈
2
Ts
z − 1
z + 1
It is also called trapezoidal method or bilinear transformation.

Digital Control
Stability: Images of the left side of the s-plane

Digital Control
Re(s) < 0
Forward
−−−−−−→
Diﬀerence
Re(z) < 1

Digital Control
Re(s) < 0
Forward
−−−−−−→
Diﬀerence
Re(z) < 1
Stable system in s domain
Forward
−−−−−−→
Diﬀerence
Stable or unstable system in z
domain

Digital Control
Re(s) < 0
Forward
−−−−−−→
Difference
Re(z) < 1
Forward
−−−−−−→
Difference
domain
Re(s) < 0
Backward
−−−−−−→
Difference
(Re(z) − 1
2)2 + Im(z)2 < 1
4

Digital Control
Re(s) < 0
Forward
−−−−−−→
Difference
Re(z) < 1
Forward
−−−−−−→
Difference
domain
Re(s) < 0
Backward
−−−−−−→
Difference
(Re(z) − 1
2)2 + Im(z)2 < 1
4
Stable or unstable system in s domain
Backward
−−−−−−→
Difference
Stable system in z
domain

Digital Control
Re(s) < 0
Forward
−−−−−−→
Difference
Re(z) < 1
Forward
−−−−−−→
Difference
domain
Re(s) < 0
Backward
−−−−−−→
Difference
(Re(z) − 1
2)2 + Im(z)2 < 1
4
Backward
−−−−−−→
Difference
Stable system in z
domain
Re(s) < 0
Tustin
−−−−−−−−→
Approximation
∣z∣ < 1

Digital Control
Re(s) < 0
Forward
−−−−−−→
Difference
Re(z) < 1
Forward
−−−−−−→
Difference
domain
Re(s) < 0
Backward
−−−−−−→
Difference
(Re(z) − 1
2)2 + Im(z)2 < 1
4
Backward
−−−−−−→
Difference
Stable system in z
domain
Re(s) < 0
Tustin
−−−−−−−−→
Approximation
∣z∣ < 1
Tustin
−−−−−−−−→
Approximation
Stable system in z domain

Digital Control
Digital PID:
Proportional: P(t) = Kpe(t)
P[k] = Kpe[k]

Digital Control
Digital PID:
P[k] = Kpe[k]
Integral: I(t) =
Kp
Ti
∫ t
0 e(𝜏)d𝜏
I[k + 1] = I[k] +
KTs
Ti
e[k]

Digital Control
Digital PID:
P[k] = Kpe[k]
Integral: I(t) =
Kp
Ti
∫ t
0 e(𝜏)d𝜏
I[k + 1] = I[k] +
KTs
Ti
e[k]
Derivative: Td
N
dD
dt + D = −KpTd
dy
dt
D[k] =
Td
Td + NTs
D[k − 1] −
KpTd N
Td + NTs
(y[k] − y[k − 1])

Digital Control
Typical sampling time:
Type of variable Sampling time (sec)
Flow 1-3
Level 5-10
Pressure 1-5
Temperature 10-20

Digital Control
Type of variable Sampling time (sec)
Flow 1-3
Level 5-10
Pressure 1-5
Temperature 10-20
Rule of thumb for PI controllers:
Ts
Ti
≈ 0.1 to 0.3

Digital Control
Rule of thumb for PID controllers:
NTs
Td
≈ 0.2 to 0.6

Digital Control
Inverse Response: When the initial response of a dynamic system is in a
direction opposite to the ﬁnal outcome, it is called an inverse response.
[Skogestad, 2001]

Digital Control
Obtaining the eﬀective delay (Half rule):
Eﬀective delay =True delay + inverse response time constant(s)
+ half of the largest neglected time constant
+ all smaller higher order time constants
[Skogestad, 2001]

Digital Control
Obtaining the eﬀective delay (Half rule):
Eﬀective delay =True delay + inverse response time constant(s)
+ half of the largest neglected time constant
+ all smaller higher order time constants
Time constant:
𝜏 = the largest time constant +
1
2
the second largest time constant
{
𝜏1 = the largest time constant,
𝜏2 = the second largest time constant + 1
2the third largest time constant
[Skogestad, 2001]

Digital Control
Plant transfer function:
G(s) =
∏
j (−Tj0s + 1)
∏
j (𝜏i0s + 1)
e−td s
First order approximation:
ˆG1(s) =
e−𝜃s
𝜏s + 1
𝜃 =td +
∑
j
Tj0 +
𝜏20
2
+
∑
j≥3
𝜏i0 +
Ts
2
𝜏 =𝜏10 +
𝜏20
2
[Skogestad, 2001]

Digital Control
Second order approximation:
ˆG2(s) =
e−𝜃s
(𝜏1s + 1)(𝜏2s + 1)
𝜃 =td +
∑
j
Tj0 +
𝜏30
2
+
∑
j≥4
𝜏i0 +
Ts
2
𝜏1 =𝜏10
𝜏2 =𝜏20 +
𝜏30
2
[Skogestad, 2001]

Digital Control
Series form of the PID controller:
U(s) = Kp(1 + 1
𝜏i s )(1 + 𝜏d s)E(s)

Digital Control
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
𝜏i s )(E(s) + 𝜏d sE(s))

Digital Control
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
𝜏i s )(E(s) − 𝜏d sY (s))

Digital Control
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
𝜏i s )(R(s) − (1 + 𝜏d s)Y (s))

Digital Control
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
U(s) = Kp(1 + 1
𝜏i s )(R(s) − (1 + 𝜏d s)Y (s))
U(s) = Kp(1 + 1
𝜏i s )Ed (s)

Digital Control

Digital Control
Antiwindup digital PID:
public class SimplePID {
private double u,e,v,y;
private double K,Ti,Td,Beta,Tr,N,h;
private double ad,bd;
private double D,I,yOld;
public SimplePID(double nK, double nTi, double NTd,
double nBeta, double nTr, double nN, double nh) {
updateParameters(nK,nTi,nTd,nBeta,nTr,nN,nh);
}
ARTIST Graduate Course on Embedded Control Systems

Digital Control
public void updateParameters(double nK, double nTi,
double NTd, double nBeta, double nTr, double nN,
double nh) {
K = nK; Ti = nTi; Td = nTd; Beta = nBeta;
Tr = nTr
N = nN;
h = nh;
ad = Td / (Td + N*h);
bd = K*ad*N;
}

Digital Control
public double calculateOutput(double yref, double newY) {
y = newY;
e = yref - y;
D = ad*D - bd*(y - yOld);
v = K*(Beta*yref - y) + I + D;
return v;
}
public void updateState(double u) {
I = I + (K*h/Ti)*e + (h/Tr)*(u - v);
yOld = y;
}
}

Digital Control
public class Regul extends Thread {
private SimplePID pid;
public Regul() { pid = new SimplePID(1,10,0,1,10,5,0.1); }
public void run() {
// Other stuff
while (true) {
y = getY();
yref = getYref():
u = pid.calculateOutput(yref,y);
u = limit(u);
setU(u);
pid.updateState(u);
// Timing Code
}}}

Astrom, K. J. and Hagglund, T. (1995).
PID Controllers: Theory, Design, and Tuning.
Instrument Society of America.
Astrom, K. J. and Wittenmark, B. (1996).
Computer-Controlled Systems: Theory and Design.
Prentice Hall, 3 edition.
Chau, P. C. (2002).
Process Control: A First Course with MATLAB (Cambridge Series in
Chemical Engineering).
Cambridge University Press, 1 edition.
Li, Y., Ang, K. H., and Chong, G. C. Y. (2006).
Pid control system analysis and design.
IEEE Control Syst. Mag., 26(1):32–41.
Love, J. (2007).
Process Automation Handbook: A Guide to Theory and Practice.
Springer, 1 edition.

McMillan, G. K. (2001).
Good Tuning: A Pocket Guide.
The Instrumentation, Systems, and Automation Society (ISA).
O’Dwyer (2002).
Handbook of PI & Pid Controller Tuning Rules.
World Scientiﬁc Publishing.
Skogestad, S. (2001).
Probably the best simple pid tuning rules in the world.
In AIChE Annual meeting, Reno, NV, USA.
Visioli, A. (2006).
Practical PID Control (Advances in Industrial Control).
Springer, 1 edition.
Woolf, P., editor (2007).
The Michigan Chemical Process Dynamics and Controls Open Text
Book.
Yu, C.-C. (2007).

Autotuning of PID Controllers: A Relay Feedback Approach.
Springer.

Industrial Control Systems - PID Controllers

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