performance assessment of pid controllers
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PERFORMANCE ASSESSMENT OF PID
CONTROLLERS∗
W. Tan†, H. J. Marquez‡, and T. Chen§
Abstract: Criteria based on disturbance rejection and system robustness are proposed to assess
the performance of PID controllers. The robustness is measured by a two-block structured singu-
lar value, and the disturbance rejection is measured by the minimum singular value of the integral
gain matrix. Examples show that the criteria can be applied to a variety of processes, whether
they are stable, integrating or unstable; single-loop or multi-loop.
Key Words: PID Control; Tuning; Performance; Robustness; Structured Singular Value.
1 Introduction
PID controllers are widely used in the industry due to their simplicity and ease of re-tuning on-line
[1]. In the past four decades there are numerous papers dealing with the tuning of PID controllers.
∗Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, SEM,and by the Research Foundation for the Doctoral Program, NCEPU
†Department of Automation, North China Electric Power University, Zhuxinzhuang, Dewai, Beijing 102206,China. E-mail: wtan@ieee.org
‡Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada.E-mail: marquez@ece.ualberta.ca
§Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada.E-mail: tchen@ece.ualberta.ca
1
See, for example, references [2–9] for stable processes; references [10–17] for integrating and
unstable processes; and references [18–23] for multivariable processes.
A natural question arises: How can the PID settings obtained by different methods be com-
pared? Or more generally, how can the performance of a controller be assessed? In process
control, minimum variance has been used as a criterion for assessing closed-loop performance
for decades [24, 25]. This criterion is a valuable measure of system performance but it pays lit-
tle attention to the traditional performance such as setpoint tracking and disturbance rejection.
Besides, another important factor of system performance – robustness is not addressed directly.
Clearly a criterion that can be used for stable, integrating or unstable; single- and/or multi-loop
processes would be highly desirable. This criterion should include time domain property as well
as frequency domain robustness specification.
For single-loop processes, the integral error is a good measure of system performance and the
gain-phase margin is a good robustness measure. Thus, a combination of these two elements can
serve as a criterion for system performance assessment. A comparison of the gain-phase margins
of some well-known PID tuning methods has been reported in [26]. But unfortunately gain and
phase margins are not suitable for multiloop processes.
In this paper we will propose criteria to assess system performance. The criteria reflect distur-
bance rejection performance and system robustness. The robustness is measured by a two-block
structured singular value, and the disturbance rejection is measured by the minimum singular
value of the integral gain matrix. Examples show that the criteria can be applied to a variety of
processes, whether they are stable, integrating or unstable; single-loop or multi-loop.
2
2 Performance Assessment of Closed-loop Systems
It is well-known that a well-designed control system should meet the following requirements
besides nominal stability:
• Disturbance attenuation
• Setpoint tracking
• Robust stability and/or robust performance
The first two requirements are traditionally referred to as ‘performance’ and the third, ‘robust-
ness’ of a control system.
2.1 Performance
The integral error is a good measure for evaluating the setpoint and disturbance response. The
followings are some commonly used criteria based on the integral error for a step setpoint or
disturbance response:
IAE :∫ ∞
0 |e(t)|dt
ITAE :∫ ∞
0 t|e(t)|dt
ISE :∫ ∞
0 e(t)2dt
ITSE :∫ ∞
0 te(t)2dt
ISTE :∫ ∞
0 t2e(t)2dt
These criteria, however, are not suitable for multivariable processes, since each criterion is de-
fined for a single-loop process.
3
Consider the unity feedback system (possibly multi-loop) shown in Fig. 1. Since disturbance
rejection is more common in industrial processes than setpoint tracking, the performance of the
system may be evaluated by its ability to reject disturbance.
K Gr y
_
Gd
e
d
+
Figure 1: Typical unity feedback configuration
The transfer function from d to y is
Tyd = (I +GK)−1Gd (1)
Assume our controller K has integral action, we can decompose it as
K(s) = Ki/s+Km(s) (2)
where Ki is the integral gain and Km is the part of the controller without integral action. Then at
low frequency, we have
σ((I +GK)−1Gd)( jω)) ≤ | jω|σ((G( jω)Ki)−1Gd( jω))
≤ | jω| σ(Gd( jω))σ(G( jω))σ(Ki)
(3)
4
where σ(·) and σ(·) denote the maximum and minimum singular value of a matrix, respectively.
In industrial processes, the disturbance usually occurs at low frequency, so to reject a disturbance,
the most important element of a controller is its integral gain, or specifically, the minimum sin-
gular value of the integral gain, thus it can serve as a measure of system performance.
As pointed out in [1] for a single-loop process,
∫ ∞
0e(t)dt =
1Ki
so the integral gain is related to the integral of the error (IE). Moreover, if the response is critically
damped, IE would be equal to IAE. So the minimum singular value of the integral gain is a natural
extension as a performance measure to multi-loop processes.
2.2 Robustness
For robust stability, a common choice of representing uncertainty for a multivariable system is
the multiplicative perturbation, and the maximum singular value of the complementary sensitivity
matrix is a measure of robustness against this kind of uncertainty, which is usually frequency-
dependent, and suited for the unmodeled dynamics instead of parameter variations. The coprime
factor uncertainty can represent model uncertainty in a better way [27]. The uncertain model is
represented as:
G∆ = (M +∆M)−1(N +∆N) (4)
where G = M−1N is a left normalized coprime factorization of the nominal plant model, and the
uncertainty structure is
∆ = [ ∆M ∆N ], ‖∆‖∞ < γ (5)
5
Then the system is robustly stable if and only if
ε :=
∥∥∥∥∥∥∥
⎡⎢⎣ I
K
⎤⎥⎦(I +GK)−1M−1
∥∥∥∥∥∥∥∞
≤ 1γ
(6)
So ε can serve as a measure of system robustness.
However, we note that this uncertainty clearly ignores the structure of ∆M and ∆N . Suppose
∆M = W1∆1,∆N = W2∆2 (7)
and define
∆ =
⎡⎢⎣ ∆1 0
0 ∆2
⎤⎥⎦ (8)
then
∆ = [ W1 W2 ]∆ (9)
For this uncertainty structure, we have
(I +G∆K)−1
=
⎛⎜⎝I +
[W1 W2
]⎡⎢⎣ ∆1 0
0 ∆2
⎤⎥⎦
⎡⎢⎣ I
K
⎤⎥⎦(I +GK)−1M−1
⎞⎟⎠
−1
·
(I +GK)−1(1+ M−1W1∆1) (10)
By the definition of structured singular values [28], the closed-loop system is robustly stable for
6
all ‖∆‖∞ < γ if and only if
µ∆
⎛⎜⎝
⎡⎢⎣ I
K
⎤⎥⎦(I +GK)−1M−1[ W1 W2 ]
⎞⎟⎠ <
1γ
(11)
If we choose special weightings as follows:
W1 = M;W2 = N (12)
and define
εm := µ∆
⎛⎜⎝
⎡⎢⎣ I
K
⎤⎥⎦(I +GK)−1[ I G ]
⎞⎟⎠ (13)
Then εm is a better measure of system robustness. We note that now the class of uncertain plants
can be represented as
G∆ = (M + M∆1)−1(N + N∆2) = (I +∆1)−1G(I +∆2) (14)
so it can represent simultaneous input multiplicative and inverse output multiplicative uncertainty.
If we treat a disturbance as a model uncertainty, then it can also represent simultaneous input and
output disturbance.
For a single-loop system, it can be shown that
εm = maxω
(|S( jω)|+ |T ( jω)|) (15)
where S and T are the sensitivity and complementary sensitivity functions of the closed-loop
system, respectively. The value approaches 1 at low and high frequencies and the maximum
7
occurs at the mid-range frequencies. Compared with the usual indicator such as Ms, the peak of
the sensitivity function, or Mp, the peak of the complementary sensitivity function, the measure
is more appropriate since it bounds both Ms and Mp simultaneously.
In summary we can assess the performance of a controller by evaluating the minimum sin-
gular value of its integral gain matrix, and assess the robustness by the robustness measure εm
defined in (13). We mainly concern with the disturbance response. The setpoint response can
always be improved by using a setpoint filter or a setpoint weighting.
The discussion above suggests that we can design an ‘optimal’ PID controller by solving the
following optimization problem:
maxσ(Ki) (16)
under the constraint
µ∆
⎛⎜⎝
⎡⎢⎣ I
K
⎤⎥⎦(I +GK)−1[ I G ]
⎞⎟⎠ < γm (17)
where γm is a given robust stability requirement. The problem amounts to maximizing the integral
action under the constraint of a certain degree of robust stability, a generalization of the idea used
in [29, 30] for single-loop processes.
The problem proposed is a nonconvex optimization problem thus it is not easy to solve di-
rectly. However, the loop-shaping H∞ approach provides a solution to a suboptimal problem.
Details can be found in [31].
3 Illustrative Examples
In this section, we will apply the criteria proposed in the previous section to analyze the PID
controller settings for some typical processes.
8
3.1 A first-order plus deadtime (FOPDT) process
Consider a process with the following model:
P(s) =1
s+1e−0.5s (18)
Table 1 shows the PID settings tuned by the following well-known tuning rules:
1) Ziegler-Nichols (Z-N) [2].
2) Cohen-Coon (C-C) [3].
3) Internal model control (IMC) [5]. The IMC method has a tuning parameter. The smaller it
is, the better performance the closed-loop system will have, and the less robust the closed-
loop system is. Here the tuning parameter λ is chosen as 0.25 of the delay, the smallest
value as suggested by [5].
4) Gain-phase margin (GPM) [4, 8]. Since different pair of gain-phase margin will result in
different PID settings, here we choose the tuning formula given in [7] where the gain-phase
margin is optimized.
5) Optimum integral error for load disturbance (ISE-load, ISTE-load, ITAE-load) [7, 32].
6) Optimum integral error for setpoint change (ISE-setpoint, ISTE-setpoint, ITAE-setpoint)
[7, 32].
It can be observed that the resulting controllers can be divided into three groups:
i) Controllers tuned by IMC, GPM, ISTE-setpoint and IAE-setpoint methods have small in-
tegral gains and small robustness measures.
9
Table 1: PID settings for example 1
Kp Ti Td Ki εm
IMC 2 1.25 0.2 1.6 3.193GPM 1.938 1.164 0.208 1.665 3.092
ISE-setpoint 1.952 0.989 0.264 1.973 3.646ISTE-setpoint 1.940 1.152 0.206 1.684 3.078IAE-setpoint 1.674 1.204 0.192 1.390 2.544
ITAE-setpoint 2.393 1.201 0.175 1.992 4.054Z-N 2.285 0.855 0.214 2.617 3.912C-C 2.917 1.209 0.167 2.833 6.024
ISE-load 2.885 0.532 0.285 5.422 9.852ISTE-load 2.876 0.642 0.231 4.477 6.698IAE-load 2.673 0.852 0.20 3.139 5.188
ITAE-load 2.475 0.813 0.183 3.045 4.258
ii) Controllers tuned by Z-N, ITAE-setpoint, ITAE-load and ISE-setpoint methods have medium
integral gains and medium robustness measures.
iii) Controllers tuned by C-C, ISE-load, ISTE-load and IAE-load methods have large integral
gains and large robustness measures.
Fig. 2 shows the the closed-loop system responses of all the PID controllers for a step setpoint
change of magnitude 1 at t = 0 following a step load disturbance of magnitude 1 at t = 10 for the
nominal model and for the perturbed case that the deadtime increases by 20%. It is observed that
the integral gains and robustness measures given by the first group are too small, thus the closed-
loop systems are very robust but the load rejection performance can be further improved. The
integral gains and robustness measures given by the third group are too large, thus the closed-loop
systems show oscillatory responses and are not robust. The second group gives proper integral
gains and robustness measures.
10
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Pro
cess
Out
put
IMCG−PISTE−setpointIAE−setpoint
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Pro
cess
Out
put
IMCG−PISTE−setpointIAE−setpoint
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
Time
Pro
cess
Out
put
Z−NITAE−setpointITAE−loadISE−setpoint
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Pro
cess
Out
put
Z−NITAE−setpointITAE−loadISE−setpoint
0 2 4 6 8 10 12 14 16 18 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Time
Pro
cess
Out
put
C−CISE−loadISTE−loadIAE−load
0 2 4 6 8 10 12 14 16 18 200
0.5
1
1.5
2
2.5
Time
Pro
cess
Out
put
C−CISE−loadISTE−loadIAE−load
(a) Nominal model (b) Deadtime increases by 20%
Figure 2: Responses of different PID settings for example 1
11
3.2 A process with complex poles
Second-order plus deadtime processes (SOPDT) are harder to tune than FOPDT processes due to
the existence of (possibly) underdamped complex poles. Nonetheless, the criteria apply also to
such processes. To illustrate, consider a process with the following model:
P(s) =1
s2 +0.1s+1e−0.2s (19)
It has a very small damping ratio, thus represents a heavily oscillatory process.
Table 2: PID settings for example 2
Kp Ti Td Ki εm
γm = 3 1.996 1.419 0.985 1.406 2.942γm = 4 3.825 1.550 0.710 2.468 3.920γm = 5 5.365 1.575 0.608 3.405 4.970
Table 2 shows the PID settings tuned by solving the (sub)optimization problem proposed at
the end of the previous section with different values of γm. We do not claim that the solutions are
optimal. The suboptimal solutions are just used to illustrate the impact of the robustness measure
on system performance.
Fig. 2 shows the the closed-loop system responses of all the PID controllers for a step setpoint
change of magnitude 1 at t = 0 following a step load disturbance of magnitude 1 at t = 20 for the
nominal model and for the perturbed case that the deadtime increases by 30%. It is observed that
the integral gain and robustness measure computed with γm = 3 are too small, thus the closed-
loop systems are very robust but the load rejection performance can be further improved. The
integral gain and robustness measure computed with γm = 5 are too large, thus the response of
the closed-loop system is oscillatory and not robust. γm = 4 gives a good trade-off.
Extensive simulations show that for a stable process the robustness measure εm defined in (13)
12
0 5 10 15 20 25 30 35 400
0.2
0.4
0.6
0.8
1
1.2
1.4
Time
Pro
cess
Out
put
0 5 10 15 20 25 30 35 400
0.5
1
1.5
Time
Pro
cess
Out
put
(a) Nominal (b) Deadtime increases by 30%
Figure 3: Responses of different PID settings for example 2(dashed: γm = 3; solid: γm = 4; dashdotted: γm = 5)
should lie between 3 and 5 to have a good compromise between performance and robustness.
3.3 A first-order delayed unstable process (FODUP)
PID tuning for integrating and unstable processes is much harder than that for stable processes.
There are few simple tuning formulas as those in the case of stable processes available in the
literature. Here we consider a first-order delayed unstable process as an illustrating example:
P(s) =1
s−1e−0.4s (20)
Table 3 shows the PID settings tuned by typical methods found in the literature, and the PID
setting tuned by solving the optimization problem proposed at the end of the previous section
with γm = 5. Also shown are the corresponding integral gain and the robustness measure for
each PID setting. From the table we observe that only the controllers tuned by R-L, H-C, IMC
and the proposed methods are robust enough. Now a good compromise between performance
13
Table 3: PID settings for example 3
Kp Ti Td Ki εm
De Poar and O’Malley (P-M) [10] 1.459 2.667 0.25 0.547 10.46Rotstein and Lewin (R-L) [11] 2.25 5.76 0.2 0.391 4.01
Poulin and Pomerleau (P-P) [12] 2.025 4.738 0 0.427 9.04Huang and Chen (H-C) [13] 2.636 5.673 0.118 0.465 5.70
Tan et.al. [15] 2.428 2.381 0.098 1.02 7.06IMC [16] 2.634 2.52 0.154 1.045 5.29
ITSE-setpoint [17] 3.148 1.806 0.21 1.743 9.76Proposed 2.467 4.08 0.15 0.604 4.64
and robustness requires that the robustness measure for the unstable process lies between 4 and
6 compared with between 3 and 5 for the stable processes. The IMC method has the best load
rejection, as shown by the closed-loop system responses in Fig. 4(a) for a step setpoint change of
magnitude 1 at t = 0 following a setp load disturbance of magnitude 1 at t = 20 for the nominal
model. The PID controller by the H-C method was shown to be worse than that by the IMC
method in performance and robustness [16], so its response is omitted here. The PID controller
tuned by the R-L method has the best robustness, as shown by the closed-loop system responses
in Fig. 4(b) for a perturbed model with the deadtime increased by 20%. The new setting by the
proposed method has the best compromise between performance and robustness.
3.4 A multivariable process
The criteria can also be used to compare PID settings for multivariable processes. To illustrate,
consider the distillation column model reported by Wood and Berry [33]:
G(s) =
⎡⎢⎣
12.8e−s
16.7s+1−18.9e−3s
21s+1
6.6e−7s
10.9s+1−19.4e−3s
14.4s+1
⎤⎥⎦ (21)
14
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
Time
Pro
cess
Out
put
0 5 10 15 20 25 30 35 400
0.5
1
1.5
2
2.5
3
Time
Pro
cess
Out
put
(a) Nominal (b) Deadtime increases by 20%
Figure 4: Responses of different PID settings for example 3(solid: proposed; dashed: R-L; dashdotted: IMC)
The process is highly coupled and attracts much attention in the literature.
Table 4 shows the PID settings tuned by various methods found in the literature, and the PID
setting designed by solving the optimization problem proposed at the end of the previous section
with γm = 4. It is clear that the PID controllers given in [23, 33] have very large robustness
measures, and those given in [18, 22] have too small integral actions.
For the rest settings, the proposed PID has the best disturbance ejection, which can be shown
in Fig. 5. To test the robust performance of the controllers, suppose the process delays change,
and the perturbed model becomes
Gp(s) =
⎡⎢⎣
12.8e−2s
1+16.7s−18.9e−4s
1+21s
6.6e−10s
1+10.9s−19.4e−4s
1+14.4s
⎤⎥⎦ (22)
The disturbance responses for all the controllers are shown in Fig. 6. Again, the new setting by
the proposed method has the best compromise between performance and robustness.
15
Table 4: PID controller settings for example 4
PID controller σ(Ki) εm
[21]
[0.184+ 0.0469
s −0.0102− 0.0229s +0.0082s
−0.0674+ 0.0159s −0.537s −0.0660− 0.0155
s
]0.0111 2.647
[22]
[0.2833+ 0.0285
s −(0.04105+ 0.02185s )
0.09154+ 0.00971s −(0.121+ 0.0148
s )
]0.0056 3.927
[18]
[0.375(1+ 1
8.29s) 00 −0.075(1+ 1
23.6s)
]0.0032 4.130
[20]
[0.183(1+ 1
10.7s) 00 −0.072(1+ 1
10.7s)
]0.0111 4.152
[33]
[0.2(1+ 1
4.44s) 00 −0.004(1+ 1
2.67s)
]0.0150 7.551
[23]
[0.637(1+ 1
3.84s) 00 −0.096(1+ 1
7.40s)
]0.0130 8.756
[31]
[0.2796+ 0.0334
s −(0.0085s +0.0981s)
−(0.0381+ 0.0001s + 0.4913s
5s+1 ) −(0.1089+ 0.0136s )
]0.0131 3.988
0 10 20 30 40 50 60 70 80−1
−0.5
0
0.5
1
1.5
2
2.5
3
Time
Pro
cess
Out
put
0 10 20 30 40 50 60 70 80−6
−5
−4
−3
−2
−1
0
1
Time
Pro
cess
Out
put
(a) Input disturbance at 1st channel (b) Input disturbance at 2nd channel
Figure 5: Process disturbance responses for example 4: nominal case(solid: proposed; dashdotted: [21]; dashed: [20])
16
0 10 20 30 40 50 60 70 80−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
3.5
Time
Pro
cess
Out
put
0 10 20 30 40 50 60 70 80−7
−6
−5
−4
−3
−2
−1
0
1
2
3
TimeP
roce
ss O
utpu
t
(a) Input disturbance at 1st channel (b) Input disturbance at 2nd channel
Figure 6: Process disturbance responses for example 4: perturbed case(solid: proposed; dashdotted: [21]; dashed: [20])
4 Conclusions
Criteria based on disturbance rejection and system robustness were proposed to assess the perfor-
mance of PID controllers. The robustness is measured by a two-block structured singular value,
and the disturbance rejection is measured by the minimum singular value of the integral gain
matrix. Examples showed that the criteria can be applied to a variety of processes, whether they
are stable, integrating or unstable; single-loop or multi-loop. It was also observed that robustness
measure should lie between 3 and 5 to have a better compromise on performance and robustness
for stable processes, and between 4 and 6 for unstable and integrating processes.
17
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Biographies
Wen Tan received his B.Sc. degree in applied mathematics and M.Sc. degree in systems sci-
ence from the Xiamen University, China, and Ph.D. degree in automation from the South China
University of Technology, China, in 1990, 1993, and 1996, respectively. From October 1994
to February 1996, he was a Research Assistant with the Department of Mechanical Engineering
and Electronic Engineering, Hong Kong Polytechnic University, Hong Kong. After June 1996,
he joined the faculty of the Power Engineering Department at the North China Electric Power
University, China, where he was a Lecturer until December 1998 and an Associate Professor
from January 1999. From January 2000 to December 2001, he was a Postdoctoral Fellow in the
Department of Electrical and Computer Engineering at the University of Alberta, Canada. He is
currently a Professor with the Automation Department of the North China Electric Power Uni-
versity (Beijing), China. His research interests include robust and H∞ control with applications
in industrial processes.
Horacio J. Marquez
Tongwen Chen
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