Preprints of the 10th IFAC International Symposium on Dynamics and Control of Process Systems

The International Federation of Automatic Control

December 18-20, 2013. Mumbai, India

Optimal PID-Control on First Order Plus

Time Delay Systems & Verification of the
SIMC Rules
Chriss Grimholt Sigurd Skogestad∗

Department of Chemical Engineering, Norwegian University of Science

and Technology (NTNU), Trondheim, Norway
∗
e-mail: skoge@ntnu.no

Abstract: Optimal PID-settings are found for first-order with delay processes for specified levels
of robustness (MS -value) and compared with an extended SIMC-rule. Optimality (performance)
is defined in terms of the integrated absolute error (IAE) for combined step changes in load
output and input disturbances. The SIMC-rules gives a PI-controller for first order systems and
no recommendation is given for tuning the derivative part. We propose an extended SIMC-rule
where the the time delay is counteracted by introducing derivative action with τD = θ/3. The
modification was found to give surprisingly good settings with near Pareto-optimal performance.
However, to obtain the improvement over PI control τc should be reduced to about half of the
recommended value τc = θ.

Keywords: Optimality, PI controllers, Pareto-optimal, robustness, performance evaluation, and

closed-loop model approximation.

1. INTRODUCTION the controller characteristics is not significantly changed,

which for the controller (3) implies τF < τD /3, approxi-
We investigate optimal tunings for a first-order plus time mately. If the filter time constant (τF ) is selected to en-
delay process, hance performance, then we are no longer talking about a
k PID-controller, but a PIDF-controller with four adjustable
G(s) = · e−θs (1) parameters.
(τ1 s + 1)
where k is the process gain, τ1 is the time constants, and Optimality is generally difficult to define as there are many
θ is the time delay. We consider only the cascade form issues to consider, including:
PID-controller   • Output performance
τI s + 1 • Robustness
K(s) = Kc (τD s + 1) (2)
τI s • Input usage
where Kc , τI and τD are the controller gain, integral time • Noise sensitivity
and derivative time. For other notation, see Figure 1. This may be considered a multiobjective optimization
In practice, the measurement is usually filtered. For exam- problem, but we consider only the main dimension of the
ple, by use of the controller trade-off space, namely high versus low controller gain.

τI s + 1

τD s + 1
 High controller gain favours good output performance,
K(s) = Kc (3) whereas low controller gain favours the three other ob-
τI s τF s + 1 jectives listed above. We can then simplify and say that
Measurement filtering is not included as a part of our tun- there are two main objectives:
ing problem. For a PID-controller, the measurement filter (1) Output performance
time constant (τF ) should anyway be selected such that (2) Robustness, input usage and noise sensitivity
di do Pareto-optimality applies to multiobjective problems, and
means that no further improvement can be made in
objective 1 without sacrificing objective 2. The idea is
ys +- + q -y
-+ c - K u- c? G - c? then to find the Pareto-optimal controller, and compare
- + + with the SIMC-tuning.
6
ym c?
+ n The SIMC method for PID-controller tuning (Skogestad,
+ 2003) has already found wide industrial usage. The SIMC-
Fig. 1. Block diagram of the feedback control system. We rules are analytically derived, and from a first or second
may treat an output disturbance (do ) as a special case order process model we can easily find PI- and PID-
of setpoint change (ys ) controller settings, respectively. The rules has one tuning

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Table 1. Optimal PID-controllers (Ms = 1.59) and corresponding IAE-values for four processes.

Output disturbance Input disturbance Optimal combined (minimize J)

Process Kc τI τD IAE◦do Kc τI τD IAE◦di Kc τI τD IAEdo IAEdi J Ms
e−s 0.20 0.32 0 1.61 0.20 0.32 0 1.61 0.20 0.32 0 1.61 1.61 1 1.59
e−s
s+1
0.43 0.62 0.62 1.56 0.42 0.59 0.59 1.46 0.42 0.61 0.61 1.57 1.47 1.01 1.59
e−s
8s+1
4.97 8.00 0.32 1.61 3.75 1.56 0.59 0.58 4.35 2.52 0.48 2.35 0.63 1.27 1.59
e−s
s
0.62 ∞ 0.32 1.61 0.51 2.33 0.53 6.37 0.54 3.24 0.48 3.02 6.81 1.47 1.59
IAEdo and IAEdi are for a unit step load change on output (y) and input (u), respectively.

parameter, the closed-loop time constant τc , which can be To ensure robust reference controllers, they are required
used to trade off between performance (“tight” control) to have MS = 1.59 1 , and the resulting weighting factors
and robustness (“smooth” control). are given for four processes in Table 1.
In a previous paper, we studied the optimal PI-controller It may be argued that a two-degree of freedom controller
on the same first order process (1), where we compared with a setpoint filter can be used to enhance setpoint
the SIMC-tuned PI-controller with the “optimal” PI- performance, and thus we only need to consider input
controller (Grimholt and Skogestad, 2012). disturbances. But note that although a step load change
on the output do , as mentioned, is equivalent to a setpoint
The SIMC rules do not cover the tuning of PID-controllers step-change ys for the setup in Figure 1, it is not affected
(τD ) for first-order processes. In this work, we propose an by the setpoint filter. In summary, we consider disturbance
extension of SIMC, and we find that adding τD = θ/3 rejection which, can only be handled by the feedback
gives a close-to optimal controller. controller K (Figure 1). The optimal controller will depend
This paper is structured as follows. In Section 2 the perfor- on the specific disturbance model, and we chose to consider
mance/robustness trade-off is quantified. The optimization disturbances at the plant output (do ) and plant input (di ).
problem is defined in Section 3. Optimal PI- and PID- To get a good balance, we weigh the both equally as given
controllers are presented in Section 4, and the extended in (5).
SIMC-rule is presented and analysed in Section 5.
2.2 Robustness

2. EVALUATION OF PERFORMANCE AND Robustness (objective 2) is in this paper quantified with

ROBUSTNESS the peak of the sensitivity function MS . The sensitivity
function is defined as,
2.1 Performance 1
S= (6)
1 + GK
Performance (objective 1) is related to the difference and the MS -value is defined as its peak, or mathematically
between the measurement y(t) and the setpoint ys (Figure MS = max |S(jω)| = kSk∞ (7)
ω
1). To quantify performance in terms of a single scalar, we
choose the integrated absolute error: where k · k∞ is the H∞ -norm. In the frequency domain
Z ∞ (Nyquist plot), MS is the inverse of the closest distance
between the critical point -1 and the loop transfer function
IAE = |y(t) − ys (t)| dt (4)
0 GK. For robustness, a small MS value is desired, and
generally MS should not exceed 2. A typical “good” value
Actually, in this paper we do not consider setpoint changes is about 1.6, and notice that MS < 1.6 guarantees GM
(ys ). That is, ys may be considered constant. Hoverer, we > 2.67 and PM > 36.4◦ (Rivera et al., 1986).
do consider output disturbances do , which for the so-called
one-degree of freedom controller in Figure 1 (where there
is no filter on ys ) is equivalent to a setpoint change. 3. PROBLEM FORMULATION
To balance the servo/regulatory trade-off we choose a
For a given first order plus time delay process, the Pareto-
weighted average of IAE for a step input load disturbance
optimal curve for PI or PID control is generated by solving
di and IAE for a step output load disturbance do :
  the the following optimization problem:
IAEdo (K) IAEdi (K)
J(K) = 0.5 + (5)
IAE◦do IAE◦di
Given the set of MS -values MS = {1.1, . . . , 3},
The weighting factors IAE◦di and IAE◦do are for a reference solve for all m ∈ MS :
controller, which for the given process is the IAE-optimal
PID-controller for a step load change on input and out-
put, respectively. Note that two different controllers are
used to obtain the reference IAE-values, whereas a single 1 For those that are curious about the origin of this specific value
controller K is used to find IAEdi (K) and IAEdo (K) when MS = 1.59, it is the resulting MS -value for a SIMC tuned PI-
−s
evaluating the IAE-cost J(K). controller with τc = θ on the process G = es+1 .

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substantial 42% at MS = 1.59, compared to optimal PI-

control.
 
IAEdo (K) IAEdi (K)
min J(K) = 0.5 + 4.2 Optimal PID tuning parameters
K IAE◦do IAE◦di
s.t. MS = m The optimal tuning can be divided into to main regions:
delay dominant (τ1 /θ < 5) and lag dominant (τ1 /θ >
where K(s) is a PI- or a PID-controller. 5). For the lag dominant processes, the scaled controller
gain approaches the constant vale Kc kθ/τ1 = 0.54 for
MS = 1.59 (Figure 2, top). The same can be observed
The problem was in practice solved by gradient-free op- for integral time and derivative time which approaches to
timization on the τI and τD parameter, where in each τI θ = 3.24 and τD θ = 0.48 at the same MS -value (Figure
iteration the gain Kc was adjusted such that the MS 2, bottom). Though, the integral time converges slower.
constraint was fulfilled. For increasing MS -values (lower robustness), the optimal
controller gain increases and the optimal integral time
decreases. Interestingly, the optimal derivative time seems
4. OPTIMAL PID-CONTROL ON FIRST ORDER to be independent to the selected robustness, resulting in
PLUS TIME DELAY PROCESS τD θ = 0.48.

Four different first-order processes have been investigated, The delay dominated region can be subdivided into two
additional regions based on the controller: Equal controller
• Pure time delay (τ1 /θ = 0) zeros (τI = τD ), from approximately τ1 /θ < 2, and two
• Small time constant (τ1 /θ = 1) distinctive controllers zeros, from approximately τ1 /θ > 2.
• Intermediate time constant (τ1 /θ = 8) Setting the derivative time equal to the integral time
• Integrating (τ1 /θ = ∞) concurs with the recommendation of Ziegler and Nichols
(1942). However, this is only for a small range of first-order
In the pure time delay case, small additional poles
processes. In the upper part of the delay dominated region
1/(0.0001s + 1) were added to make the loop transfer
function proper.
Normalized controller gain, Kc kθ/τ1

1
In Table 1, the resulting optimal PID-controllers and J-
values are given for MS = 1.59 for four processes; We
MS = 2.00
note that J = 1 for a time delay process, because there 0.8
is no trade-off between disturbances for this process, and MS = 1.70
because the reference controllers have MS = 1.59. For 0.6
the other cases we have J > 1 because there is a trade-
off between input and output disturbances rejection. For
example, for the integrating process, the optimal value 0.4
MS = 1.59
of J is 1.47, mainly because we have to sacrifice output MS = 1.50
disturbance rejection. 0.2 MS = 1.20

4.1 Pareto-optimal controllers 0

0 10 20 30
A Pareto-optimal curve depicts the trade-off between Process time constant τ1 /θ
two conflicting objectives. In our case, this is the trade-
4
off between performance (J) and robustness (Ms ). The MS = 1.40
Integral and derivative time

Pareto-optimal curves for PI- and PID-control for the four MS = 1.50
MS = 1.59
processes are shown in Figure 3. Notice that we have only MS = 1.70
3
a real trade-off when there is a negative slope between the τI /θ
variables (left side of the plots). Here we have to decide
on a compromise between the two objectives. That is, if MS = 2.00
we improve one objective, the other deteriorates. We never 2
want to be in a region with zero or positive slope (right
side of the plots), because we can both improve robustness
and performance by just moving to the left. Therefore, the 1
τD /θ
minimum point in the cure represent the largest MS value
we would like to use. The deterioration in performance
at large MS -values is cased by oscillating response which 0
0 10 20 30
increases the IAE.
Process time constant τ1 /θ
For a pure time delay process there is no advantage to add
derivative action, and it is optimal to use simple PI-control Fig. 2. Pareto-optimal PID settings for five given MS -
(Figure 3, top right). As the time constant increases the −θs

benefit of using derivative action increases. For integrating values (robustness) for the process G(s) = τke1 s+1 . For
processes, using derivative action improves performance a reference τI = τ1 is also plotted (dashed line).

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4 4
e−s
Performance, J(K) G(s) = e−s G(s) = s+1

Performance, J(K)
3 3

2 2

PO-PI
PO-PI
1 1 PO-PID
PO-PID

0 0
1 1.5 2 2.5 3 1 1.5 2 2.5 3
Robustness, MS Robustness, MS
4 4
−s
e e−s
G(s) = 8s+1 G(s) = s
Performance, J(K)

Performance, J(K)
3 3

2 2
PO-PI
PO-PI

PO-PID PO-PID
1 1

0 0
1 1.5 2 2.5 3 1 1.5 2 2.5 3
Robustness, MS Robustness, MS

Fig. 3. Pareto-optimal trade-off between robustness (MS ) and performance (J) for Pareto-optimal PI- and PID-control
for four processes

the integral time is close to the process time constant form controller (8). However, as seen from Figure 5, the
(indicated by dashed line) which is in agreement with the difference between the cascade and the parallel controller
well-known IMC rule (Rivera et al., 1986). very small even for this process.

4.3 Parallel (ideal) vs. cascade PID-controller

So far we have found the optimal cascade PID-controller 4

in (2). A more general PID-controller is the parallel, or
ideal, PID-controller,
Performance, J(K)

 
0 1 0 3
Kparallel = Kc 1 + 0 + τD s (8)
τI s
The cascade controller can always be translated into the
parallel form by 2
cascade
Kc0 = Kc f, τI0 = τI f, τD 0
= τD /f (9)
where f = 1 + τD /τI . The more general parallel form (8) parallel
1
can not be translated to the cascade form (2) if it has
complex zeros.
The difference between the two forms are minor in our 0
case. For three of the processes the cascade form is optimal. 1 1.5 2 2.5 3
Only the small time constant process (τ1 /θ = 1) had Robustness, MS
optimal parallel PID-controller with complex zeros. The
optimal cascade controller (2) for this process is on the Fig. 5. Pareto-optimal cascade PID-control (blue line) and
boarder between real an complex with to coinciding real e−s
zeros, τI = τD . This compares to τI0 = 4τD0
for the parallel parallel PID-control (red line) on G(s) = s+1 .

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4 4
e−s
G(s) = e−s G(s) = s+1
SIMC-PI SIMC-PI
Performance, J(K)

Performance, J(K)
3 3

2 2 τc = 1.5θ
τc = 1.5θ
τc = θ τc = 0.5θ τc = θ
τc = 0.5θ
SIMC-PID
PO-PI
1 1 SIMC-PID
PO-PI PO-PID PO-PID

0 0
1 1.5 2 2.5 3 1 1.5 2 2.5 3
Robustness, MS Robustness, MS
4 4
SIMC-PI
e−s e−s
G(s) = 8s+1 τc = 1.5θ G(s) = s
SIMC-PI
Performance, J(K)

Performance, J(K)
3 τc = 1.5θ 3
τc = θ
τc = θ

2 τc = 0.5θ 2 τc = 0.5θ

SIMC-PID PO-PI
SIMC-PID PO-PI
1 1
PO-PID PO-PID

0 0
1 1.5 2 2.5 3 1 1.5 2 2.5 3
Robustness, MS Robustness, MS

Fig. 4. Pareto-optimal trade-off between robustness (MS ) and performance (J) for optimal and SIMC PI- and PID-
control for four processe. SIMC-PI and SIMC-PID have Kc and τI given by (10) (but the value of τc are not the
the same for a given MS ), and SIMC-PID have τD = θ/3.
Table 2. Tuning for optimal and SIMC PID-controllers with Ms = 1.59 on four processes.

Optimal PID SIMC PID

Process Kc τI τD Kc τD J MS Kc τI τD Kc τD J τc MS
e−s 0.20 0.32 0 – 1.00 1.59 0 0 0.33? – 1.02 0.62 1.59
e−s
s+1
0.43 0.61 0.61 0.26 1.01 1.59 0.62 1 0.33 0.21 1.08 0.61 1.59
e−s
8s+1
4.39 2.54 0.48 2.11 1.26 1.59 5.92 6.5 0.33 1.64 1.69 0.63 1.59
e−s
s
0.55 3.25 0.48 0.26 1.45 1.59 0.59 6.81 0.33 0.20 1.86 0.70 1.59
(?) Gives a ID-controller with I = Kc /τI = 1/(τc + θ). This is the same as a PI-controller.

4.4 Input usage controllers, which has a higher controller gain Kc , requires
more input usage than the optimal PI-controller.
Input usage is an important aspect for control. From The product Kc τD is good indication for input usage in
Figure 1 we have the high frequency range, where |KS(jω)| ≈ Kc τD ω. For
PID-controllers without a measurement filter (τF ), the
u = −T di + KS (do + n) |KS| peak goes to infinity, kKSk∞ = ∞. Therefore, it
is important to filter out the high frequency noise, and the
Thus, input usage is decided by the two transfer functions:
resulting peak will depend heavily on the selected filter.
T (from input disturbance) and KS (from output distur-
It is important that the selected filter do not influence
bance and noise).The input disturbances are not a problem
controller performance and robustness in a significant way.
because T is bound by MT which is low for our cases.
If so, we have a PIDF-controller where also the filter
KS has a peak at the intermediate frequencies which is constant should be considered a degree of freedom in the
approximately |KS(jω)| ≈ Kc MS (Åström and Hägglund, optimization problem. For this reason we recommend that
2006). Thus, with a given MS -values, the optimal PID- the filter constant should be selected no larger than τD /3.

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5. EXTENDED SIMC FOR PID-CONTROL OF frequency range and less input usage in the high frequency
FIRST-ORDER PROCESSES WITH TIME DELAY range, as can be seen from the higher Kc and lower Kc τD
values (Table 2).
The SIMC PI-settings for the first-order plus delay process
(1) are The SIMC-rules settles slower than the optimal controller
for both input and output disturbances (Figure 6). How-
1 τ1 ever, it is usually the maximum deviation that is of main
Kc = , τI = min{τ1 , 4(τc + θ)} (10)
k τc + θ concern in the industry. The SIMC-rule have roughly equal
where the desired first-order closed-loop time constant τc is peak deviation for input disturbance, and a smaller peak
the only tuning parameter. For a “fast and robust” setting, deviation for output disturbances compared with the op-
τc = θ is recommended. timal. By using SIMC-PID the peak deviation is reduced
by 26% for input disturbances, compared with SIMC-PI.
The trade-off curve for the SIMC controllers was generated
by varying the tuning parameter τc from a large to a small
value. The controllers corresponding to the three specific REFERENCES
choices Åström, K. and Hägglund, T. (2006). Advanced PID
• τc = 1.5θ (smooth tuning) Control. ISA.
• τc = θ (default value) Grimholt, C. and Skogestad, S. (2012). Optimal PI-
• τc = 0.5θ (more aggressive tuning) Control and Verification of the SIMC Tuning Rule. In
IFAC conference on Advances in PID control (PID’12),
are shown by circles. Except for the pure time delay 2.
process, the differences in performance (J) between SIMC- Rivera, D., Moriari, M., and Skogestad, S. (1986). Internal
PI and optimal-PI are within 10%, which shows that Model Control. 4. PID Controller Design. 252–265.
the SIMC PI-rules are close to optimal (Figure 4). In Skogestad, S. (2003). Simple analytic rules for model
other words, by adjusting τc we can generate the optimal reduction and PID controller tuning . 13, 291–309.
controller for a given desired robustness (Grimholt and Skogestad, S. and Postlethwaite, I. (2005). Multivariable
Skogestad, 2012). Feedback Control – Analysis and Design. John Wiley &
Sons, Ltd, 2 edition.
When considering PID-control, is commonly proposed to
Ziegler, J.G. and Nichols, N.B. (1942). Optimum settings
introduce derivative action to improve performance for
for automatic controllers. Trans. ASME, 64, 759–768.
processes with time delay, e.g. τD = 0.5θ (Rivera et al.,
1986). Based on analytical derivations and simulations,
Skogestad (2003) found that adding τD = 0.5θ only
marginally improved performance for load input distur-
bances compared with PI. However, it was also noted
that introducing derivative action improved the robustness
margins somewhat. Because of the small improvements, e−s
SIMC-PI
2 G(s) =
increased complexity and increased noise sensitivity, Sko- s
PO-PI
Outputs, y

gestad recommend to not use derivative action to coun- SIMC-PID

teract time delay for first-order plus delay processes. This do di
1
is somewhat conflicting to what we have found so far
where performance substantially improved by introducing PO-PID
derivative action (Figure 3).
0
We have found that setting the derivative time to:
τD = θ/3 (11) 0 10 20 30 40
with the SIMC-rules gives good performance, as is dis- 2
cussed in more detail bellow. This value τD = θ/3 follows PO-PID
from our previous results (Grimholt and Skogestad, 2012) 1
Inputs, u

where we derived an “improved” SIMC PI-rule. SIMC-PID

As Skogestad claimed, when comparing SIMC-PID with

0 SIMC-PI
SIMC-PI when choosing τc = θ, the robustness is some-
what improved, but performance is only marginally im- PO-PI

proved (middle circle, Figure 4). For the integrating pro- −1

cess, the MS is improved from 1.70 for PI to 1.46 for PID,
but there is only a 6% increase in performance. However, 0 10 20 30 40
due to this added robustness for PID, we can reduce τc Time, t
and significantly improve performance for a given MS -
value (36% increase for the integrating process) compared Fig. 6. Time response for optimal and SIMC PI- and PID-
to the original PI-controller. A good value for the tuning controllers (MS = 1.59) for an output disturbance (at
constant would be τc = 0.5θ, as it give approximately the time 0) and an input disturbances (at time 20), on the
same robustness as the SIMC PI-rules with τc = θ. integrating process G(s) = e−s /s. To get a proper
Compared with the optimal PID-controller, the SIMC system, a PID-controller with measurement filter (3)
PID-controller have higher input usage in the intermediate is used with τF = 0.01.

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