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Proportional−Integral−Derivative Controller Performance
Assessment and Retuning Based on General Process Response Data
Sheng Yu and Xiangshun Li*
Cite This: https://doi.org/10.1021/acsomega.1c00523
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ABSTRACT: In this paper, the current research status of controller
performance assessment is reviewed in brief. Solving the problem of
proportional−integral−derivative performance assessment usually requires
step response data, and several methods are combined and extended. Using
the integral of signals, implicit model information contained in process
response data becomes explicit, and then the least squares approach is
adopted to construct a detailed low-order process model based on process
response data in more general types. A one-dimensional search algorithm is
used to attain better estimation of process time delay, and integral equation
approach is extended to be useful for more general process response. Based
on the obtained model, a performance benchmark is established by
simulating model output. Appropriate retuning methods are selected when
the index of absolute integral error (IAE) indicates bad performance.
Simulations and experiments verify the effectiveness of the proposed
method. Issues about estimation of process time delay, data preprocessing, and parameter selection are studied and discussed.
1. INTRODUCTION
With quality standards and functional demands of products
getting higher and higher, industrial processes are becoming
increasingly complex, and demands for control performance
are also stricter. According to statistics, after controllers are
put into operation for a period of time, around 60% of
controllers have performance degradation issues due to
inappropriate controller parameters, wear of actuator, and
change of the external environment.
1
However, an engineer
usually maintains 201−500 control loops,
2
and operators
need time and experience skills to maintain controllers. In
addition, as system complexity continues to increase, the
maintenance costs of the system cannot be neglected.
The goal of controller performance assessment (CPA) is to
assess how far the current controller performance from the
desired benchmark and also retune controller parameters with
routine operating data, which could provide operators with
controller health status and related suggestions. CPA has
attracted great attention and research in the past 30 years
since Harris proposed the minimum variance control (MVC)
index.
3
Shockingly, the 2016 survey showed that the control
loop problem was the same as in 1989, and the problem of
inappropriate parameters is still prominent.
2
CPA can be divided into model-based methods and data-
driven methods. The performance assessment method based
on historical benchmark is a kind of data-based approach.
First, select or train a benchmark model with satisfying
process data from daily operation. When a new data set
comes, compare it with the trained model or trained
threshold to determine whether its performance is good or
not.
4
The method of evaluating controller performance based
on historical benchmarks has been successfully applied in
single-loop and multiloop control systems of industrial
processes.
5
Multivariate statistical process monitoring has been a
research hotspot in the past 25 years, and partial least
squares and principal component analysis methods are the
most commonly used.
2
Yu and Qin (2008) propose statistical
methods based on generalized eigenvalue analysis for
performance monitoring of multiple input multiple output
processes, which could locate the bottom control loops that
cause performance degradation.
6,7
Considering the controller
constraints and robustness, the trade-offcurve of integrated
squared error and total squared variation of controller actions
is used to assess the current controller state.
8
The tuned
proportional−integral−derivative parameters are directly
obtained by solving the convex optimization problem of
Received: January 28, 2021
Accepted: March 26, 2021
Article
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approximating a reference model through set-point changes in
data.
Most of the data-driven methods are used to complete the
task of control performance monitoring at the system level
and require a lot of process variables to determine whether
the system status is normal or abnormal. However, data-
driven methods mostly build a black box system, and it is
difficult to analyze and master the mechanisms of the actual
system. In addition, this kind of data-driven method is usually
not helpful for proportional−integral−derivative (PID)
tuning. While, model-based methods are clear and direct
for PID tuning, and there are scores of mature research
studies on model-based controller tuning.
The model-based method mostly considers the perform-
ance evaluation of a single control loop. Integrals or sums of
control loop variables such as integrated absolute error (IAE)
are usually selected as performance benchmarks, and these
indicators can be fused, such as by the way of producing or
weighting, to form an overall benchmark. The linear
quadratic Gaussian (LQG) method determines the perform-
ance benchmark in the form of a trade-offcurve by balancing
control performance and controller effort. Now the LQG
method, generally implemented based on predictive control,
has been extended to discrete processes.
9
However, this
method requires an explicit model and a relatively high
computation. Moreover, the LQG performance benchmark is
not applicable to PID directly due to the limitation of the
controller structure.
8
For the typical FBC/FFC control
structure, Huang et al. (1999, 2000) solve the problem
because of the feedforward controller or the feedback
controller when the current control effect is not good.
10,11
The performance assessment of the PID controller follows
the concepts of the abovementioned method, but it should be
simple and practical. Swanda and Seborg simply adopt a well-
designed IMC controller as the performance benchmark and
compare the actual PID controller with it to evaluate the
control performance.
12
The method of reachable PID MVC
performance assessment draws on the idea of LQG
9
and
takes into account the constraints of the controller. Its lower
bound is much larger than that of MVC, but it can be
realized.
13
This method also requires a process model or an
impulse response sequence. In actual applications, the process
model may be not available.
14
In order to obtain process model, the integration of
predefined variables with step response data can be used to
estimate parameters of approximate low-order models, and
the performance benchmark is established according to
obtained model parameters.
15,16
A semi-nonparametric
approach is proposed to estimate the parameters of unknown
process models and the indicators of the IAE, and the total
variation of control signals (TV) are calculated to assess the
performance of liner controllers.
17
The integral equation
approach (IEA) is proposed for identification of continuous-
time models from step responses.
18
The effectiveness of these
modeling methods
15,16,18
shows that implicit model informa-
tion contained in daily dynamic data can be explicit by using
the integrals of control loop signals. A review on process
identification from step or relay feedback test is presented.
19
For retuning PID when the control performance is poor,
simple internal model control (SIMC) tuning
20
and Direct
Synthesis Design for Disturbance Rejection (DS-d) tuning
21
have been used.
15,16
There are abundant research studies on
PID controller parameter tuning, such as the IMC tuning
method, AMIGO tuning method, and so on.
22
These tuning
methods focus on different performances, and each has its
own characteristics.
This kind of model-based PID CPA usually uses low-order
models to approximate high-order processes
20
and applies the
obtained models to attain performance evaluation and
parameter tuning. To get low-order models, the data used
for parameters estimation are usually obtained through
identification tests or extracted from daily operation data.
23
However, in practice, it is generally not allowed to add
identification test signals to prevent the impact on industrial
production and safety, and the identification test consumes
considerable time and resources.
16
In addition, some model-
based methods for CPA are derived from the ideal step
response, but the actual signal is generally not the step type.
Furthermore, the step response is not always available in the
daily operation. Some systems may be stable in a working
condition for a long time, which is unfavorable for online
evaluation of controller performance.
The proposed approach combines and extends several
methods to accomplish the task of deterministic performance
assessment and retuning of PID controllers based on process
response data in more general types. The general process
dynamic data means response data of a closed-loop control
system stimulated by step input or nonideal step input,
meaning system response with constrained controller outputs
or under measurement noise. With integral signals, the least
squares approach is adopted to construct a detailed low-order
process model with process dynamic data in more general
types. Because of the using of integral signals, the form of
exciting signals is not so important. A one-dimensional search
algorithm is used to attain better estimation of process time
delay, and IEA is extended to be useful for more general
process response. Based on the obtained model, PID
parameters are determined and performance benchmark
used for performance assessment is established by simulating
model output. Appropriate retuning methods are selected
when the index of absolute integral error (IAE) indicates bad
performance. Because the presented method is based integral
of signals, it is inherently robust to uncertain noise. It is
worth noting that controller performance can be divided into
deterministic performance and stochastic performance from
the perspective of control tasks.
24
Moreover, the proposed
method in this article focuses on the deterministic perform-
ance.
The organizational structure of this paper is as follows.
Section 2 introduces the principle of process modeling. PID
tuning and performance assessment are expounded in Section
3.Section 4 proves the effectiveness of the algorithm through
several simulations. The applications of temperature control
on the Tennessee Eastman process (TEP) and water level
control on intelligent process control test facility (IPC-TF)
verify the validity of the method in Section 5. The last
section summarizes the work of this paper.
2. PROCESS MODELING
The proposed method is mainly divided into two parts, one
is process modeling, and the other is PID tuning and
performance evaluation. First, the principles and specific
implementation of process modeling will be introduced.
2.1. Problem Formulation. A simple unity-feedback
control system is shown in Figure 1.C(s) is the controller in
the form of PID, and D(s) is the disturbance process. P(s)is
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the controlled process, which is a single-in single-out self-
regulating process. The signal ris the set point, dis the
measured process variable, eis the control error, uis the
controller output, and yis the disturbance.
There are several forms of PID controllers, and different
forms can be transformed. The form adopted in this article is
the ideal form as eq 1. Of course, a filter should be added to
the differential part in practical applications as Ts
s1
T
N
D
D+, and N
can be selected as 10.
i
k
j
j
j
j
j
y
{
z
z
z
z
z
Cs K Ts Ts() 1 1
P
I
D
=++ (1)
In order to conveniently study PID tuning and perform-
ance evaluation, for the real process as eq 2, it can be
approximated to the first-order plus dead time (FOPDT)
model (3) or second order plus dead time (SOPDT) model
(4). For actual low-order systems such as first-order plants,
FOPDT model should be used. For a more complex high-
order system, “Half Rule”can be used to approximate it to a
SOPDT model.
20
P
s
Ts
Ts
() (1)
(1)
e
jj
ii
s
0
0
0
=∏+
∏+
θ−
(2)
P
ss
() 1e
s
1
μ
τ
=+
θ−
(3)
P
sss
() (1)(1)
e
s
2
12
μ
ττ
=++
θ−
(4)
In practical applications, if the process model is known in a
high-order form, it can be approximated as a low-order
system through approximation rules. Otherwise, it can be
considered that the model structure is known, and the model
parameters should be solved.
2.2. Estimation of Process Time Delay. The correct
estimation of process delay has a great influence on the
accuracy of process modeling. Process time delay is common,
but 90% of industrial control loops in practical applications
are PID control types without delay compensation, which will
cause the actual PID control performance far from theoretical
benchmarks such as MVC-based benchmarks, no matter how
the parameters are tuned. The problem of process delay
estimation is solved by the fixed model variable regressors
proposed by Elnaggar.
25
For process dynamic response with
sampling period T, the original process output sequence yo(k)
and process input sequence uo(k) are obtained, k=1,2,3,
...M, and Mis recommended above 500. First, subtract the
initial values from the original data to get y(k) and u(k).
With the preprocessed data, d, the number of sampling
intervals corresponding to the process delay can be solved
with the eq 5,whereErepresents the mathematical
expectation, and krepresents the k-th sampling point.
Multiply dcorresponding to the maximum E1by the
sampling period Tto obtain process time delay θ.
E E yk yk uk d
dE
dT
(() ( 1)) ( )
max
d
1
1
θ
=[ − − · − ]
̂=
=̂·(5)
2.3. Estimation of Process Gain and Time Constants.
For the SOPDT model, the estimation of process gain and
time constant will be described below. In addition, the
procedures for the FOPDT model can be easily derived
similarly. After preprocessing of raw data and estimating
process time delay as above, take the sequence of u(1), ...,
u(M−d
̂), and y(1 + d
̂), ..., y(M) to construct the process
response without time delay to estimate parameters of eq 6.
Then, eq 6 can be transformed into the form of the integral
equation as eq 7, and then, eq 8 is obtained through the
inverse Laplace transform, which can be written as the matrix
form as eq 9.
18
P
sYs
Us
b
sasa
() ()
()
20
210
==
++ (6)
Y
sa
Ys
saYs
sbUs
s
() () () ()
10
202
++=
(7)
y
taytaytbut() () () ()
110202
++=
[] [] []
(8)
z
H=
∂
(9)
The meaning of symbols are as in eqs 10 and 11. Using the
least squares method to solve the available parameters a0,a1,
and b0, the unbiased estimation vector ∂is shown in eq 12.
yt yrr
yt y r
ut u r
() ()d
() ( )d d
() ( )d d
t
tr
tr
10
200
200
∫
∫∫
∫∫
ρρ
ρρ
=
=
=
[]
[]
[] (10)
Hhnhn hL
ht yt ytut
zynyn yL
aab
( ) ( 1)... ( )
() () () ()
( ) ( 1)... ( )
T
12
2T
T
010
=[ + ]
=[− − ]
=[ + ]
∂=[ ]
[] [] []
(11)
HH Hz()
T1T
∂=
−(12)
The process time constant τ1and τ2and the process static
gain μcan be obtained by factorization and simple
conversion as eq 13.τ1≥τ2will be ensured by exchanging
them if the result is not obtained.
Figure 1. Simple control system.
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C
b
a
aaa
a
a
4
2
1
0
0
2
11
20
0
1
02
μ
τ
ττ
=
=−−
=(13)
After the model is established, the model output sequence
ŷcan be simulated. Model accuracy can be evaluated by the
model fitness index Qas eq 14, where yrepresents the actual
measured output, ŷrepresents the model output, y̅represents
the average value of the actual output, and Mrepresents the
length of data sequence. When Qis close to 1, it indicates
that the established model is accurate. The length of the data
sequence is recommended to be greater than 500.
Q
yi yi
yi y
1() ()
()
i
M
i
M
1
2
1
2
=−∑[−
̂]
∑[−
̅]
=
=(14)
Remark 1. Note that the data sequences uand yused in
the above calculation are the deviations of the actual input
and output data, that is, the initial value has been subtracted.
This can avoid the influence of the initial state of the system
on the parameters estimation. For a transient response
without a clear system initial steady-state value, the way of
preprocessing is to consider the initial steady-state value as an
unknown value for transient response. By integrating eq 8 on
both sides to increase the number of equations, the unknown
steady-state value is solved.
18
However, multiple integrations
will increase the error, and the solution effect may be not
good.
Remark2. The accurate estimation of time delay is the key
to ensuring the accuracy of the process model because the
estimation of time delay will affect the correct estimation of
process gain and time constants. Another empirical approach
is to estimate process delay from step response. The process
delay is the time when the measurements of process variable
first satisfy the equation ym> 0.02As+ NB, where Asis the
step size and NB is the noise band.
26
Step response may be
not available, and step response can be obtained from
nonstep response with a filter as in eq 15.
18
However, this
kind filter is hard to be implemented when the controller
output Uis not an explicit expression.
Ys Ps
sYssU s
Fs sU s
() ()
1() 1
()
() 1
()
step ==
=(15)
In order to ensure a more accurate estimation of the
process time delay θ, a one-dimensional search algorithm can
be used to search downward. The reason for the downward
search is that many trials have shown that the direct
estimated time delay θ0is generally big. After the direct
estimated time delay θ0is obtained, ddecrements from θ0to
θ0/2 and the corresponding process gain and time constants
can be calculated by the IEA, and the corresponding model
fitness index Q(d) is calculated. Finally, dcorresponding to
the maximum Qis taken as the best process time delay
sample.
Remark 3. Implicit model information contained in daily
dynamic data can be explicit by using the integral equation
approach (IEA), without designing test signals to conduct
system identification tests. The proposed modeling method is
the upgraded version of approaches
26
and more general. It
can be seen from the derivation process that this method is
easy to be extended to higher-order systems such as the third
order and fourth order. In addition, this method is not only
effective for step response data but also for more general
process dynamic data, and the identification accuracy is
enough. In practice, the integral is usually approximated by
the sum of rectangular divisions, and trapezoidal divisions can
be used to obtain higher calculation accuracy.
Since integration can eliminate the influence of white
noise, this method is inherently robust. Regarding colored
noise, variants of the least squares method such as the
instrumental variable method can be used to solve model
parameters.
18
Remark 4. In practical applications, if there is a process
model, it can be approximated as a low-order system through
approximation rules. If there is no process model, the
FOPDT model should be used for modeling first; if the
model fitness Qexceeds 85%, FOPDT is appropriate to be
used. Otherwise, then try to use the SOPDT model for
modeling. Systems of the third order and above can be
extended by the above method, but the complexity and
computation will also increase. In fact, in order to analyze
and study process dynamics accurately, more precise process
models may be required. Then, the model order should be
determined first. For example, after estimating the process
time delay, the number of system poles from 1 to 10 and the
number of system zeros from 1 to 10 are arranged and
combined, and the process model can be solved cyclically.
The number of poles and zeros of the process model can be
determined when the model fits Qis the maximum, and the
parameters of the model can be determined simultaneously.
Remark 5. Here is a brief introduction on how to use the
obtained model to simulate the output. For MATLAB, you
can use the built-in function “lsim”to implement easily. For
the python implementation, use the “inverse_laplace_trans-
form”module in the SYMPY library to perform the inverse
Laplace transform, and then use the “signal.convolve”
function in the SCIPY library to facilitate the implementation.
Note that the input parameters supposed to be of the same
data type of one function should be guaranteed in the same
data type; otherwise, the running speed of the program will
be very slow. For other language implementations, you need
to construct the function yourself. Use the known transfer
function to perform the inverse Laplace transform to obtain
the impulse response sequence, and convolve the input
sequence with the impulse response function to obtain the
model output sequence.
3. PID TUNING AND PERFORMANCE ASSESSMENT
In practical applications, the control tasks of set point
tracking and load disturbance rejection are very common. In
order to facilitate understanding and direct application of this
method in practice, the following will specifically introduce
the performance assessment of set point tracking response
and load disturbance rejecting response with the SOPDT
model. In addition, the procedures for the FOPDT model
can be easily derived similarly.
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3.1. Set Point Tracking Task. After obtaining the
process model such as the SOPDT model, the SIMC tuning
method
20
can be used to make the desired closed-loop
transfer function as eq 16. In practical applications, τccan be
selected according to specific performance requirements. The
literature
17,26
describes some research studies on the selection
of this parameter. A second-order reference model was
proposed to achieve better control performance.
8
Without considering the disturbance process, the closed-
loop transfer function in Figure 1 is given as eq 17.In
addition, the PID benchmark parameters can be determined
with e−θs=1−θsfor the SOPDT model as given in eq 18.
i
k
j
j
j
j
y
{
z
z
z
z
Ys
Rs s
()
()
e
1
s
dc
τ
=+
θ−
(16)
Ys
Rs
PsCs
PsCs
()
()
() ()
1()()
2
2
=+(17)
l
m
o
o
o
o
o
o
o
o
o
o
n
o
o
o
o
o
o
o
o
o
o
l
m
o
o
o
o
o
o
o
o
o
o
o
o
n
o
o
o
o
o
o
o
o
o
o
o
o
i
k
j
j
j
j
j
y
{
z
z
z
z
z
K
T
T
K
T
T
4( )
()
4( )
()
14( )
4( )
1
1c
P12
c
I12
D12
12
1c
P1
c
2
c
Ic 2
D2
4( )
2
c
ττθ
ττ
τθμ
ττ
ττ
ττ
ττθ
τ
τθμ
τ
τθ
τθ τ
τ
≤+
=+
+
=+
=+
≥+
=+++
=++
=+τ
τθ+(18)
For the FOPDT model, the PI controller can be
determined (19).
K
T
()
min , 4( )
P
c
Ic
τ
μτ θ
ττθ
=+
={ +} (19)
In order to make a balance between set point tracking task
and load disturbance rejecting requirement, performance, and
robustness, a compromise parameter can be selected as τc=
θ.
For step response with amplitude As, the theoretical
approximate absolute integral error benchmark IAEdcan be
derived (eq 20). This approximate expression is easy to
implement and has a low computation. A more accurate IAE
can be obtained by simulating model outputs.
rt yt t AIAE ( ) ( ) d 2
d0s
∫
θ
=|−|≈
∞
(20)
A
e
IAE
IAE
2
()d
t
SF
d
a
s
0
∫
η
θ
νν
==
||
(21)
The set point tracking performance index ηris calculated
(21), where tis the time for the process to reach a steady
state. When ηris lower than the preset threshold, it indicates
poor set point following performance. If the performance is
poor, the PID retuning is applied (18).
Note that for nonideal step signals such as set point
changes of ramp-up first and then saturation, it is difficult to
use the formula to calculate the approximate performance
benchmark IAEd. At this time, the desired closed-loop
transfer function model can be used to simulate the expected
output sequence. This method is more reasonable in practice
because the input is generally not an ideal step signal.
3.2. Load Disturbance Rejecting Task. For step
disturbance with an amplitude of Ad, the literature
21
gives
the DS-d tuning method for the PID controller. For the
SOPDT model, in order to achieve the desired disturbance
closed-loop transfer function as in eq 22, the PID parameter
can be obtained (eq 23) with τc=θ. Use the desired closed-
loop transfer function to predict the expected output,
calculate the ideal IAE as the evaluation criterion of load
disturbance rejecting performance, and compare it with the
actual response to obtain the performance index ηd.In
practice, the size of step disturbance may be unknown and
A
e()d
K
T
t
d0
P
I∫νν=− can be used to estimate the size of the
step disturbance.
i
k
j
j
j
j
y
{
z
z
z
z
Ys
Ds
T
K
s
s
()
()
e
(1)
s
d
I
Pc3
τ
=+
θ−
(22)
K
T
T
4( ) 4 4
8
4( ) 4 4
()
7()
4( ) 4 4
P
212 12 3
3
I
212 12 3
12 1 2 2
D
212 312 4
212 12 3
θτ τ θττ θ
θμ
θτ τ θττ θ
ττ θ τ τ θ
θττ θ τ τ θ
θτ τ θττ θ
=++ −
=++ −
+++
=−+−
++ −
(23)
Figure 2. Flow chart of the proposed method.
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The literature
26
gives a simple and useable performance
evaluation benchmark for step disturbance, which is easy to
apply in engineering practice. That is, the integral error value
IE is used as the performance index benchmark IAEdbecause
the integral error value is less than or equal to the absolute
integral error value (24).
et t et tIAE ( ) d ( )d IE
00
∫∫
=||≥ =
∞∞
(24)
Remark 6. For set point tracking task, the PID tuning rules
of SIMC
20
and DS-d
21
are the same when the condition τ1≤
4(τc+θ) is true for SOPDT model. However, SIMC tuning
considers more when the condition is false. For load
disturbance rejecting task, the simulations in Section 4
show that DS-d method is much better than SIMC.
Therefore, SIMC tuning is suggested for the control task of
set point tracking, and DS-d tuning is advised for load
disturbance rejecting task. Besides, for the sake of safety and
practical operation, the way of selecting τcis presented as
follows. First, τccanbesolvedtomakethetuned
proportional coefficient KPapproach to the original propor-
tional coefficient. Then, change τca little at a time. Apply the
tuned PID to check whether control performance becomes
better or not. A smaller τccan attain fast speed of response
and good disturbance rejection, while a larger one is better
for small input variation and can enhance stability and
robustness of the system.
3.3. General Procedure. The procedures of process
modeling and PID performance assessment and retuning are
depicted as Figure 2.
After obtaining the process model such as the SOPDT
model, PID tuning methods can be selected according to
actual demands. For the dynamic response data of the
process with the sampling period T, set point value ro,
process variable yo, and controller output uousually can be
accessed. After the PID benchmark parameters and process
parameters are determined, the control loop can be used to
simulate the output. The difference between the set point
value and the process variable, namely, the control error
signal, is taken as the input sequence, and the theoretical
output of PID controller can be simulated. For those
exceeding the controller’s constraints, the output values are
taken as the limit values. Then, the process output is
simulated with the process model. The control performance
assessment can be attained by comparing the IAE of actual
variables with that of simulated variables.
When the control performance assessment index ηis close
to 1 (or higher than the preset threshold), it indicates that
the control performance is good. The threshold is generally
set to 0.6. When it is close to 0 (or below the preset
threshold), it indicates poor control performance. Then, the
controller parameters need to be adjusted.
4. SIMULATIONS
In order to verify the effectiveness of the method proposed in
this paper, the feasibility and effectiveness are verified by the
simulation below. The third-order process is adopted as
follows.
26
P
ssss
() 1
(10 1)(5 1)( 1) e
s
4
=+++
−
(25)
4.1. Set Point Tracking Case. The initial parameters of
the ideal form PID used in the simulation are Kp= 1.1, Ti=
11.0, and Td= 0.9091, which are consistent with the serial
form PID as Kp=1,Ti= 10, and Td= 1, respectively.
26
In
order to be consistent with practical situation, the controller
output is limited between −1 and 3.
4.1.1. Step without Measurement Noise. Without
considering the measurement noise, the sampling period is
set to 0.1 s, and the total simulation time is 200 s. The set
point input rchanges from 0 to 1 after a sampling period,
and the disturbance input signal dremains at zero. Using the
Figure 3. Measured output and the FOPDT model output.
Table 1. Models for Set Ppoint Tracking Response without
Measurement Noise
modeling methods model
fitness
(%)
proposed method with the
FOPOT model s
1.0067
16.2655 1 es4.8
+
−77.85
proposed method with the
SOPOT model ss
0.9999
(9.7047 1)(5.5998 1) es4.8
++
−99.05
referenced method
26
s
0.9852
11.0134 1 es6
+
−72.43
Figure 4. Measured output and model outputs.
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obtained step response data, the FOPDT model is first used
for process modeling. The model obtained is
P
s() e
s
s
1.0067
16.2655 1
4.8
=+
−. Based on the obtained model, the
model output can be simulated with the model input and the
model fit index Q= 77.85%. Figure 3 shows the measured
process variable and the model output. It can be seen that
the trend is basically consistent, but the accuracy of the
model needs to be further improved.
Using the SOPDT model for process modeling, the model
obtained is
P
s() e
ss
s
0.9999
(9.7047 1)(5.5998 1)
4.8
=++
−. Based on the
obtained model, the model output can be simulated with the
model input and the model fit index Q= 99.05%, indicating
that the established model is more accurate and much better
than the FOPDT model. Note that the one-dimensional
search algorithm for process time delay is adopted as default.
The method
26
is also used, and
P
s() e
s
s
0.9852
11.0134 1
6
=+
−is
obtained with model fitness 72.43%. The information of these
models are concluded in Table 1, and the corresponding
curves of model outputs are shown in Figure 4.
Choosing τc=θ, the set point tracking performance index
is calculated as ηr= 0.5005, indicating that the control
performance is poor. The actual output and expected output
are shown in Figure 5, consistent with the performance
indicator.
After adjusting the PID parameters to Kp= 1.5944, Ti=
15.3045, and Td= 3.5509 with τc=θin eq 18,ηr= 0.7118 is
obtained. The results show that the set point tracking
performance has been significantly improved after retuning
the controller parameters, as shown in Figure 6.
Based on different models, corresponding tuning methods
can be chosen to determine retuned PID parameters as Table
2. The curves of process variable and control variable of
different PID are shown in Figure 6.
The result shows that the established SOPDT model is
closest to the actual process, and PID parameters determined
by the proposed method with the SOPDT model also
Figure 5. Actual process output and expected output.
Figure 6. System response corresponds to different PID without measurement noise.
Table 2. Retuned PID and IAE for Set Point Tracking
Response without Measurement Noise
type KpTiTdIAE
initial PID 1.1 11 0.9091 19.2808
proposed method with the
FOPOT model 1.6830 16.2655 21.5905
proposed method with the
SOPOT model 1.5944 15.3045 3.5509 13.5579
referenced method with
retuning algorithm
26
0.9315 11.0134 2.7528 17.5367
referenced method with
formula directly
26
0.9315 11.0134 19.8570
Table 3. Models for Set Point Tracking Response with
Measurement Noise
modeling methods model
fitness
(%)
proposed method with the
FOPOT model s
0.9728
15.8555 1 es4.9
+
−77.65
proposed method with the
SOPOT model ss
0.9666
(8.7288 1)(6.1916 1) es4.9
++
−96.63
referenced method
26
s
0.9784
9.5524 1e
s
6.2
+
−41.19
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perform best. The model established by the referenced
method is not accurate as models obtained by the proposed
method. The PID parameters determined by the proposed
method with the FOPDT model seem to be the worst, even
poorer than the initial PID. The reason is that the real
process is a third-order system, and the FOPDT model
cannot approximate the actual process dynamics very well.
Also, it seems that PI controller parameters determined by
the proposed method are not better than the referenced
method, although its model is with higher fitness. However,
the result may be different if another process is adopted
because we use the same tuning ruler for the PI controller,
and the model built by the proposed method is with better
accuracy.
4.1.2. Step with Measurement Noise. In practical
applications, measurement noise is very common. This part
will study the effectiveness of the proposed method under
measurement noise.
After adding the measurement noise, repeat the above
procedures; the model information is concluded in Table 3,
and the corresponding curves of model outputs are shown in
Figure 7.
Based on different models, corresponding tuning methods
can be chosen to determine retuned PID parameters, as
shown in Table 4. The curves of the process variable and
control variable of different PID are shown in Figure 8.
The result shows that proposed method is still effective
under measurement noise. The model built by the proposed
method keeps good model fits.However,themodel
established by the referenced method is obviously poorer
under the same measurement noise and the time constant is
too small.
4.1.3. Ramp without Measurement Noise. Because the
ideal step signal may be not practical in industry processes,
set point changes of ramp-up first and then saturation are
selected as system input in this test. The information of
models is concluded in Table 5, and the corresponding
curves of model outputs are shown in Figure 9.
Based on different models, corresponding tuning methods
can be chosen to determine retuned PID parameters, as
shown in Table 6. The curves of the process variable and
control variable of different PIDs are shown in Figure 10.
The result shows that the proposed method is still
applicable under set point changes of ramp-up first and
then saturation, while the referenced method is totally
ineffective.
4.1.4. Sine without Measurement Noise. Any kind of
signal can be decomposed into superposition of sine waves by
Fourier transform. To validate the effectiveness of the
proposed method, sine signal is selected as the system
input in below test. The information of models is concluded
Figure 7. Measured output and model outputs.
Table 4. Retuned PID for Set Point Tracking Response
with Measurement Noise
type KpTiTdIAE
initial PID 1.1 11 0.9091 19.9525
proposed method with the
FOPOT model 1.6631 15.8555 22.3445
proposed method with the
SOPOT model 1.5751 14.9205 3.6223 14.6018
referenced method with
retuning algorithm
26
0.7873 9.5524 2.2484 20.2663
referenced method with
formula directly
26
0.7873 9.5524 21.7565
Figure 8. Process variable and the control variable correspond to different PIDs.
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in Table 7, and the corresponding curves of model outputs
are shown in Figure 11.
Based on different models, corresponding tuning methods
can be chosen to determine retuned PID parameters as Table
Table 5. Models for Set Point Tracking Response without Measurement Noise
modeling methods model fitness (%)
proposed method with the FOPOT model s
1.0079
16.9034 1 es4.2
+
−77.11
proposed method with the SOPOT model
is is
0.9998
(7.886 1.2455 ) 1 (7.886 1.2455 ) 1 e
s4.2
[+ +][− +]
−
97.99
referenced method
26
Figure 9. Measured output and model outputs.
Table 6. Retuned PID for Set Point Tracking Response
without Measurement Noise
type KpTiTdIAE
initial PID 1.1 11 0.9091 18.4063
proposed method with the
FOPOT model 1.9966 16.9034 26.0805
proposed method with the
SOPOT model 1.878 15.7719 4.0413 9.7971
Figure 10. Process variable and the control variable correspond to different PIDs.
Table 7. Models for Set Point Tracking Response without
Measurement Noise
modeling methods model
fitness
(%)
proposed method with the
FOPOT model s
1.0731
18.3381 1 es6.6
+
−75.33
proposed method with the
SOPOT model ss
1.0009
(9.4619 1)(5.7194 1) es4.8
++
−99.01
referenced method
26
Figure 11. Measured output and model outputs.
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8. The curves of process variable and control variable of
different PID are shown in Figure 12.
The result shows that the proposed method is still
applicable under set point changes of the sine type, while
the referenced method is totally ineffective.
4.2. Load Disturbance Rejecting Case. 4.2.1. Step
without Measurement Noise. Without considering the
measurement noise, the set point input rremains at zero,
and the disturbance input signal dchanges from 0 to 1 after a
sampling period.
It is worth noting that the data of process dynamic
response used for process modeling is the input and output
of the controlled process. The information of models are
concluded in Table 9, and the corresponding curves of model
outputs are shown in Figure 13.
Based on different models, corresponding tuning methods
can be chosen to determine retuned PID parameters, as
shown in Table 10. The curves of the process variable and
control variable of different PIDs are shown in Figure 14.
The result shows that proposed method is also effective for
step load disturbance rejecting response even when the
controller outputs are limited. The process models built by
the proposed method is good and even better than ones
established with set point tracking response. The model
cannot be established by the referenced method
26
when the
controller output is limited. With no controller constraints,
the referenced method is applicable and the obtained model
is as good as the model built by the proposed method with
the FOPDT model. However, Figure 14 shows the proposed
method outweighs the referenced method.
4.2.2. Step with Measurement Noise. After adding the
measurement noise, repeat the above procedure; the
information of models is concluded in Table 11, and the
corresponding curves of model outputs are shown in Figure
15.
Based on different models, corresponding tuning methods
can be chosen to determine retuned PID parameters, as
shown in Table 12. The curves of the process variable and
control variable of different PIDs are shown in Figure 16.
The proposed method is still effective under both
controller constraints and measurement noise, and the
referenced method cannot accomplish the task of perform-
Table 8. Retuned PID for Set Point Tracking Response
without Measurement Noise
type KpTiTdIAE
initial PID 1.1 11 0.9091 178.7505
proposed method with the
FOPOT model 1.2946 18.3381 182.0859
proposed method with the
SOPOT model 1.5800 15.1813 3.5647 173.6559
Figure 12. Process variable and the control variable correspond to different PID.
Table 9. Models for Load Disturbance Rejecting Response
without Measurement Noise
modeling methods model
fitness
(%)
proposed method with the
FOPOT model s
1.0050
15.2817 1 es5.9
+
−82.30
proposed method with the
SOPOT model ss
0.9962
(9.7337 1)(5.3877 1) es4.9
++
−99.26
referenced method
26
(with
controller limits)
referenced method
26
(no
controller limits) s
0.9847
13.5700 1 es6.4
+
−83.68
Figure 13. Measured output and model outputs.
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ance assessment and retuning of the PID controller under the
same situation.
4.2.3. Sine without Measurement Noise. Any kind of
signal can be decomposed into a superposition of sine waves
by the Fourier transform. To validate the effectiveness of the
proposed method, the sine signal is selected as system
disturbance input in the below test.
The information of models are concluded in Table 13.
Figure 17 shows the measured process variable and the
model output. It can be seen that the trend is consistent.
Based on the obtained SOPDT model, tuning methods as
shown in eq 23 can be chosen to determine retuned PID
parameters as shown in Table 14. The curves of the process
variable and control variable of different PIDs are shown in
Figure 18.
The result shows that the proposed method is still
applicable under the disturbance signal in the sine type.
5. APPLICATIONS
5.1. Tennessee Eastman Process. Based on the actual
chemical reaction process, Eastman Chemical Company of
the United States has developed an open and challenging
chemical simulation platform,TEP.
27
The process data
generated by it is time-varying, strong-coupling, and non-
linear. In addition, it is widely used in control tests and fault
diagnosis.
The controlled process studied in this paper is the reactor
temperature control loop shown in Figure 19. The PI
controller is used to control the reactor temperature
Table 10. Retuned PID for Set Point Tracking Response
without Measurement Noise
type KpTiTdIAE
initial PID 1.1 11 0.9091 13.2101
proposed method with the
FOPOT model 1.6841 19.9725 11.9736
proposed method with the
SOPOT model 2.1433 13.3482 3.2139 8.6177
referenced method for the
PID algorithm
26
2.0046 16.2717 1.8401 9.7486
referenced method for PI
26
1.3610 10.9957 12.4324
Figure 14. Process variable and the control variable correspond to different PIDs.
Table 11. Models for Load Disturbance Rejecting
Response with Measurement Noise
modeling methods model
fitness
(%)
proposed method with the
FOPOT model s
1.0049
15.9437 1 es5.4
+
−77.76
proposed method with the
SOPOT model ss
0.9945
(7.99 1)(7.03 1) es4.9
++
−92.22
referenced method
26
s
57.6212
0.1717 1e
s
0.8
−
+
−
Figure 15. Measured output and model outputs.
Table 12. Retuned PID for Load Disturbance Rejecting
Response with Measurement Noise
type KpTiTdIAE
initial PID 1.1 11 0.9091 14.2778
proposed method with the
FOPOT model 1.9549 21.2155 12.3672
proposed method with the
SOPOT model 2.2142 13.4790 3.4236 9.6588
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XMEAS9 by adjusting the opening XMV10 of the reactor
cooling water valve.
The initial temperature of the TE reactor is 122.9°, the
sampling period is 0.0005 s, the total running time is 50 s,
and the set point value is changed from 122.9 to 130°at 0.02
s. The temperature change curve of TE reactor is shown in
Figure 20. The random fluctuation after 25 s is caused by
fault 11. Fault 11 corresponds to the cooling water inlet
temperature of the reactor changing randomly.
The first 1201 sampling points of the original data are
shown in Figure 24, demonstrating the set point tracking
process. At the beginning, the reactor temperature follows set
point changes and soon stabilizes at the set value. The TE
reactor temperature process uses a PI controller, and the
FOPDT model is suitable for PI controller design. Therefore,
using the FOPDT model for process modeling, the model
obtained by parameter estimation is
P
s() e
s
s
3.1223
0.1078 1
0.009
=−
+
−.
Based on the obtained model, the model output can be
simulated with the model input and the model fit index Q=
32.35%.
Figure 21 shows the measured process variable and the
model output. It can be seen that the trend is nearly
consistent, but the difference between model output and
measured output is large because the TE reactor temperature
process is a high-order system and the FOPDT model cannot
reach a good model fit.
To validate the effectiveness of modeling algorithm, the
number of system poles from 1 to 10 and the number of
zeros from 1 to 10 are arranged and combined, and the
process model is solved cyclically. The number of poles and
zeros of the process model can be determined when the
model fitQis the maximum. The result shows that model
with four poles could attain good fitness, which indicates the
real process probably is a fourth-order system, and the
parameters of the process can be determined as
P
s() ss s
sss s
3.989 2113 39,130 85,020
136.1 1070 21,950 10,990
32
432
=−− −
++++
. The model fit index
Q= 98.08%. The model has great accuracy. The process also
is modeled by the later irregularly dynamic data showed in
Figure 20. The result is
P
s()
sss
ss s s
2.045 0.1011 3.403 0.91
17.6 167.3 0.7541 29.29
32
43 2
=
+−+
++ − −
,
and the model fit is 49.85%. It can be seen from Figure 22
that the trend is basically consistent.
The performance index is calculated η= 0.3430, indicating
that the control performance is quite poor. The actual output
and expected output are shown in Figure 23, consistent with
the performance indicator.
Figure 16. Process variable and the control variable correspond to different PIDs.
Table 13. Models for Set Point Tracking Response without
Measurement Noise
modeling methods model
fitness
(%)
proposed method with the
FOPOT model
proposed method with the
SOPOT model
ss
0.9993
(8.5284 1)(6.484 1) e
s
4.7
++
−
99.01
referenced method
26
Figure 17. Measured output and model output.
Table 14. Retuned PID for Load Disturbance Rejecting
Response without Measurement Noise
type KpTiTdIAE
initial PID 1.1 11 0.9091 96.9553
proposed method with the
SOPOT model 2.3503 13.1859 3.3341 69.1404
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It can be seen from the response curve that the tracking
response is slow, and it will take a long time to stabilize at
the set value. In order to speed up the system response
speed, a small τc, twice the sampling period, is selected for PI
tuning. With eq 19, we can get Kp=−3.4539 and Ti=
0.1078. After adjusting the PID parameters, we get η=
0.5211. The results show that the control performance has
been improved after retuning the controller parameters, as
shown in Figure 24.
The successful application of the TEP shows the
effectiveness of the proposed method. In addition, the
referenced method is also tried to be used on the above
reactor temperature control system, but the model obtained
is obviously wrong. The reason may be that the TEP has
measurement noise, and the referenced method is invalid.
5.2. Intelligent Process Control Test Facility. The
IPC-TF is established at the Wuhan University of
Technology, which is developed based on the NPCTF
(Nuclear Process Control Test Facility)
28
of the CIES
laboratory at the University of Western Ontario. The IPC-TF
is a physical simulator that simulates typical dual-loop nuclear
power plants and simplified physical processes in the general
Figure 18. Process variable and the control variable correspond to different PIDs.
Figure 19. Reactor temperature control loop in the TEP.
Figure 20. Curve of the reactor temperature.
Figure 21. Actual process output and model output.
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process industry, so it can be applied in research in the fields
of modeling, control, and fault diagnosis. The IPC-TF
platform is shown in Figure 25.
In this study, the water level control loop used is
highlighted with a red square, as shown in Figure 26. The
water in the bottom water tank is pumped out by pump 2,
and it flows into the spherical water tank through the water
inlet valve CV1-17. By adjusting the opening of the outlet
valve CV-15, the liquid level of the spherical water tank is
controlled.
The initial parameters of the PID controller are Kp=−8
and Ti= 50. It is worth noting that the derivative part cannot
be used in this water-level control system because the
transmitter of the water level is a wireless device, and the
measurement value of water level changes a lot. If the
derivative part is applied, valve actuation is too frequent and
abnormal voice occurs. After the water level is stable at 24
cm, the set point value changes from 24 to 20.
Using the step response data, the SOPDT model is
obtained with
P
s() ss
2.0516
(1701.7 1)(13.6768 1)
=−
++
and a good fitness
of 90.91%. Figure 27 shows the measured process variable
and the model output. It can be seen that the trend is well
consistent.
Figure 22. Measured output and fourth-order model output with
later irregular data.
Figure 23. Actual and expected response.
Figure 24. Temperature and valve opening correspond to initial and proposed PID.
Figure 25. Intelligent process control-test facility.
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Because the process time delay is zero, τc=θcannot be
directly used for retuning PID, and the set point tracking
process is very slow. Choosing τc= 50, the set point tracking
performance index is calculated ηr= 0.3777, indicating that
the control performance is poor. The actual output and
expected output are shown in Figure 28, consistent with the
performance indicator.
After adjusting the PI parameters to Kp=−17.7235 and Ti
= 213.6768 (eq 18), ηr= 0.6567 is obtained. The results
show that the set point tracking performance has been
significantly improved after retuning the controller parame-
ters, as shown in Figure 29.
The successful application on water level control of IPC-
TF shows the effectiveness of the proposed method. In
addition, the referenced method is also tried to be used with
the same step response data, but the model obtained is
obviously wrong. The reason is that the level transmitter uses
wireless device and a water level without small measurement
noise. Figure 29 shows the measurement value of the water
level.
6. CONCLUSIONS
In this article, to solve the problem of PID performance
assessment that usually requires step response data, several
methods are combined and extended. Using the integral
signals, implicit model information contained in process
response data becomes explicit, and then, least squares
approach is adopted to construct the detailed low-order
Figure 26. Water level control loop in red square.
Figure 27. Measured output and model output. Figure 28. Actual process output and expected output.
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process model based on process response data in more
general types. Because of the use of integral signals, the form
of exciting signals is not so important. A one-dimensional
search algorithm is used to attain better estimation of process
time delay, and IEA is extended to be useful for more general
process response. The general process response data means
dynamic data of closed loop control system stimulated by
step input or nonideal step input, which means system
response with constrained controller outputs or under
measurement noise. Based on the obtained model, PID
parameters are determined and the performance benchmark
used for performance assessment is established by simulating
the model output. PID performance assessment can be
attained by comparing the actual performance index and the
expected one, and PID controller will be retuned when the
performance is poor. By comparing with the referenced
method, the simulation and experiment verify the effective-
ness of the proposed method. The proposed method may be
more practical than existing approaches in actual applications
of PID CPA because step response may not happen when
CPA is needed. Therefore, this proposed method may be
helpful to accomplish the tasks of online PID CPA.
■AUTHOR INFORMATION
Corresponding Author
Xiangshun Li −Wuhan University of Technology, Wuhan
430070, P. R. China; Email: lixiangshun@whut.edu.cn
Author
Sheng Yu −Wuhan University of Technology, Wuhan
430070, P. R. China; orcid.org/0000-0002-6722-3706
Complete contact information is available at:
https://pubs.acs.org/10.1021/acsomega.1c00523
Notes
The authors declare no competing financial interest.
■NOMENCLATURE
CPA controller performance assessment
IAE absolute integral error
IMC internal model control
TEP Tennessee Eastman process
IPC-TF intelligent process control test facility
IEA the integral equation approach
SOPDT second-order plus dead time
umanipulated variable
ycontrolled variable
θprocess time delay
μprocess static gain
τ1τ2process constants
Qmodel fitness index
Mlength of data sequence
τccoefficient used for PID tuning
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