Journal of Process Control 16 (2006) 1013–1020
www.elsevier.com/locate/jprocont
Performance assessment using a normalized gradient
Ari Ingimundarson
*
Edifici TR11 (Campus Terrassa), Rambla de Sant Nebridi, 10, Technical University of Catalonia (UPC), 08222 Terrassa, Spain
Received 4 March 2006; received in revised form 24 July 2006; accepted 24 July 2006
Abstract
The paper presents a performance assessment method whose purpose is to make controller maintenance based on adjusting the controller gain, more systematic. A normalized gradient of a quadratic cost function is calculated with regard to the loop gain. It is shown
that this statistic is very easy to calculate and that the information can be useful to improve performance. Calculations of the gradient for
two loops from a pulp and paper mill that had performance problems show the validity of the method in an industrial setting.
2006 Elsevier Ltd. All rights reserved.
Keywords: Control performance assessment; PID control; Loop management
1. Introduction
Controller tuning has been reported as an important
cause of bad controller performance in the process industry, see [1]. The advent of academic performance assessment methods has not improved much that situation, see
[2]. It is common that loops are either tuned too aggressively and thus tend to oscillate (30% of loops according
to [1]) or they are detuned so that the response is sluggish,
see [3]. Loops might be commissioned in an appropriate
manner but as the performance decays with time the
importance of effective maintenance is clear.
One of the more frequent acts of maintenance of PID
controllers is to adjust the loop gain. The reason for the
parameter adjustment is usually abnormal variability that
the operator thinks can be fixed by an adjustment. Abnormal variability can for example occur when the loop is
working in a new operating region.
What is shown in this article is how to estimate in a simple way a normalized gradient of a quadratic cost function
with respect to controller loop gain. The aim is to make this
already existing maintenance approach more systematic.
*
Tel.: +34 93 739 8290; fax: +34 93 739 8628.
E-mail address: ari.ingimundarson@upc.edu
0959-1524/$ - see front matter 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jprocont.2006.07.004
The gradient gives information about the spectral content
of the disturbances affecting the loop compared to the
desired disturbance rejection bandwidth. This information
is compressed into one number by calculating the gradient.
Decisions to adjust the loop gain can be made with more
information if the gradient is available. The method is
not restricted to PID controllers but as 97% of controllers
in the process industry are of that type, the method is presented with that structure in mind.
The desired disturbance rejection bandwidth is
expressed with the desired complementary sensitivity function of the closed loop, denoted Td(q1). Most tuning rules
either explicitly or implicitly depend on a model of the process which in addition to the controller can be used to calculate the complementary sensitivity function. Examples of
tuning methodologies where the complementary sensitivity
function is explicitly specified are the internal model control (IMC) methodology, see [4], and k-tuning, both of
which are common control design methodologies in the
process industry, see [5].
The normalized gradient will be interpreted in a simple
way from the sensitivity function Sd(q1) that follows
directly from the relation 1 = Td + Sd. It will be
introduced through the iterative feedback control (IFT)
framework. It should be noted that similar gradient
calculations are frequent within the adaptive control
1014
A. Ingimundarson / Journal of Process Control 16 (2006) 1013–1020
community, see [6], as well as within the IFT framework
where the gradients are used to update the controller
parameters.
Performance assessment has received a great deal of
attention in recent years both in the academic and industrial world. Most major providers of process control technology offer products for performance assessment, see [7].
The use of control loop specifications to assess performance has been proposed in [8–10]. For an overview of
commercial tools for performance assessment, see [7].
A variant of the proposed method was presented in [11].
Other authors have not proposed the use of gradients for
the purpose of performance assessment to the knowledge
of the author. A performance assessment method based
on similar ideas was proposed in [12]. There the closed loop
transfer function was identified using an excitation signal
and the result compared to control specifications. Metrics
to automate the comparison are mentioned but no details
are given. The method presented in this article does not
need excitation of the process.
The article is structured in the following manner. In
Section 2 the IFT framework is introduced and the relevant equations for the calculations of the gradient are presented as well as an interpretation using Parseval’s
relation. In Section 2.4 the normalized gradient is introduced. The usage of the normalized gradient is detailed
in Section 3. In Section 4 an industrial evaluation of the
method is presented and in Section 5, conclusions are
drawn.
2.1. Iterative feedback tuning
Iterative feedback tuning has recently emerged as a technology to tune fixed order controllers like the PID by performing experiments on the closed-loop system. The tuning
is performed by calculating the partial derivative of a quadratic cost function with respect to controller parameters
and modifying the parameters in the descent direction of
this cost function. A reference covering most aspects of
IFT is [13].
The method deals with SISO linear systems on the form
ð1Þ
where w(k) is the process disturbance. The controller is assumed to be of one degree of freedom.
uðkÞ ¼ Cðq; q1 ÞðrðkÞ yðkÞÞ
yðkÞ ¼
CðqÞP
1
rðkÞ þ
wðkÞ
1 þ CðqÞP
1 þ CðqÞP
¼ T ðqÞrðkÞ þ SðqÞwðkÞ
ð3Þ
T(q) is the complementary sensitivity function while S(q) is
the sensitivity function. It is easy to check that T + S = 1.
The time argument of the signals will now be omitted as
well but again the signals will be written as a function of
controller parameters q.
Let yd be the desired response to the reference signal r.
Given the desired closed loop transfer function Td, then
yd = Tdr. Putting ~y ¼ y y d , the cost function that is
monitored is of quadratic type
"
#
N
X
1
2
~y ðqÞ
J ðqÞ ¼ E
ð4Þ
N
t¼1
For large N, J is an estimate of the variance of ~y . The letter
E denotes the expectation with respect to the weakly stationary disturbance w.
It was shown in [14] that an estimate of this gradient
could be computed from signals obtained from closed-loop
experiments. Elementary calculus gives
"
#
N
X
oJ
2
o~y ðqÞ
~y ðqÞ
¼ E
ð5Þ
oq N
oq
t¼1
It is assumed that
o~y ðqÞ oyðqÞ
¼
oq
oq
2. Synthetic gradients
yðkÞ ¼ P ðq1 ÞuðkÞ þ wðkÞ
The closed-loop system is then given by
ð2Þ
where q is the parameter for which the gradient will be calculated and q1 is the back shift operator. r(k) and y(k) are
the reference signal and process variable, respectively. u(k)
is the manipulated variable. In what follows, the dependence of transfer functions on the back shift operator q1
will be omitted and instead the dependence on parameter
q will be distinguished.
as response yd does not depend on the parameter q. From
Eq. (3) one can see that (dropping the argument of q)
!
oy
P
oC
P 2C
oC
P
oC
¼
w
r
2
2
oq
1 þ PC oq ð1 þ PCÞ oq
ð1 þ CP Þ oq
¼
1 oC
1 oC 2
Tr T r þ TSw
C oq
C oq
ð6Þ
Notice that
T 2 r þ TSw ¼ Ty
Using this in Eq. (6) results in the following equation for
oy(q)/oq:
oy
1 oC
¼
T ðr yÞ
oq C oq
ð7Þ
The above equation implies that it is possible to calculate
oy/oq by filtering r y through the complementary sensitivity function. One way to obtain this term is to sample
and store a series of data, r y and apply it as a reference
value to the real process and sample the output yielding a
second measurement series. Using the two series, an estimate of the gradient can be calculated. This is the approach
within the IFT methodology.
Another approach is to estimate T with system identification techniques, and filtering r y with this identified
A. Ingimundarson / Journal of Process Control 16 (2006) 1013–1020
transfer function to generate the second measurement series. This is the approach taken in [15]. A third approach,
and the one used in this paper, is to use T = Td obtained
from a design method. The gradient calculated by using
the last two approaches is referred to as synthetic.
In the synthetic gradient case, an estimate of oJ/oq is
formed by using one measurement series from the real process and evaluate Eq. (5). A perfect realization of ~y ðkÞ can
be obtained from the same series by calculating ~y ðkÞ ¼
yðkÞ T d ðq1 ÞrðkÞ. Eq. (5) can now be written as
N
oJ
2 X
1 oC
~y ðkÞ
¼
T d eðkÞ
oq N k¼1
C oq
ð8Þ
where e(k) = r(k) y(k).
2.2. The gradient for loop gain
The equation for the synthetic gradient is now further
evaluated for controller loop gain. The main transfer function to be evaluated is
1 oC 1
ðq Þ
C oq
ð9Þ
A general form for a discrete one degree of freedom controller with loop gain K as a parameter is
uðkÞ ¼ KGc ðq1 ÞðrðkÞ yðkÞÞ
ð10Þ
The equation for the gradient becomes particularly simple
in this case as
1015
delay introduced by sampling and general properties of
feedback systems, the amplitude is close to or above 1 for
higher frequencies. The Bode sensitivity integral for example dictates that amplitude larger than 1 is always obtained
if the relative degree is 2 or larger. The frequency that separates these two intervals of low amplitude and high amplitude of the sensitivity function is often referred to as the
bandwidth of the system.
The controller parameters affect the sensitivity function
by lifting up or dragging down the amplitude on specific
frequency intervals. If the power spectrum of the disturbances is concentrated in a frequency region where an
increase in a parameter increases the amplitude of the sensitivity function, the gradient with regard to this parameter
will be positive. The gradient will be negative if the sensitivity function amplitude is reduced with a positive change in
the parameter. It is assumed that an increase in gain causes
a decrease of the amplitude of the sensitivity function at
low frequencies.
The two extreme cases of poor performance due to tuning, i.e. when the loop has a sluggish response and when it
oscillates due to an aggressive tuning, will be discussed with
Eq. (12) in mind. Notice that either condition of the loop
might occur without any operator actually having changed
the controller parameters. Changes in gain of a loop can
occur when the loop is operating at different operating
regions, due to aging and later renewal of process equipment and other changes in the controlled process.
Insight into what information the gradient gives can be
obtained from Parseval’s relation which relates the cost
function in Eq. (4) with a frequency domain expression.
Assuming that the reference value is constant and the error
is explained solely by the disturbance w(k) which has spectra Uww, then the cost function can be expressed as
Z p
1
2
jS d ðK; eixh Þj Uww ðxÞ dx
ð12Þ
J ðKÞ 2p p
2.3.1. Sluggish response
A sluggish response means that the process variable will
stay away from the set point for large periods of time, see
[16]. This causes the disturbance spectra to be concentrated
at low frequencies compared to the closed loop bandwidth,
specifically at those frequencies where the amplitude of the
sensitivity function will be reduced by increasing gain. The
gradient can therefore be expected to be negative for sluggishly tuned loops where low frequency disturbances are
dominant.
On the other hand, the gradient being negative does not
imply that the loop is tuned conservatively as it can be arbitrarily close to zero. A zero gradient means that variance is
minimized as a function of the loop gain. Minimum variance control is generally considered to have poor robustness, see [17] and thus, a negative gradient can mean that
the loop is quite aggressively tuned. The issue of quantifying the gradient will be addressed with the normalized gradient in a later section but before, the case when the loop
oscillates due to aggressive tuning is discussed.
where Sd(K, q1) is the sensitivity function written as a
function of loop gain and the back shift operator. h is
the sampling time.
For single-input–single-output feedback loops, the sensitivity function has similar characteristics irrespective of
what control design methodology is used. It has small
amplitude at low frequencies to reject low frequency disturbances. Due to fundamental limitations such as the time
2.3.2. Aggressive tuning
The disturbance spectra of a loop that oscillates due to
an aggressive tuning will be concentrated close to the ultimate frequency of the loop transfer function (where phase
is 180). The variance of loops that are oscillating due to
aggressive tuning can usually be reduced by reducing the
loop gain. This indicates that if a positive synthetic
1 oC 1
1
ðq Þ ¼
C oK
K
ð11Þ
It is possible to calculate the synthetic gradient for any controller parameter using the above relation but attention will
be devoted only to loop gain K as this parameter affects all
other controller parameters and is strongly correlated with
performance and robustness measures.
2.3. Interpretation of the synthetic gradient
1016
A. Ingimundarson / Journal of Process Control 16 (2006) 1013–1020
gradient is encountered, the loop is tuned aggressively. It
was mentioned before that the gradient can be negative
for an aggressively tuned loop but this section will focus
on the case when the gradient is positive.
To elaborate the analysis of the situation when a positive gradient is encountered it is assumed that the oscillation is explained by a disturbance affecting the loop that
is concentrated at one dominating frequency x0. As a sampled system is considered it is assumed that x0 is smaller
than the Nyquist frequency. Supposing that the disturbance approaches a pure sinusoidal, the spectra of the disturbance is a sum of delta functions
Uww ¼ apðdðx x0 Þ þ dðx þ x0 ÞÞ
see for example [18]. The factor a determines the amplitude
of the sinusoid. Evaluating the integral in Eq. (12) with the
delta functions as the disturbance spectra gives J ¼
ajSðK; eix0 h Þj2 .
In Appendix A.1 it is shown that a positive gradient for
a frequency x0 means that jSðK; eix0 h Þj > 1. This means
that if the gradient is positive the sensitivity function is
actually amplifying the disturbances affecting the loop.
An appropriate action in this case is to reduce the loop gain
as indicated by the gradient.
Summarizing the above analysis, if the synthetic gradient is positive the loop gain should be reduced. On the
other hand, if the gradient is negative, the loop might still
be aggressively tuned as it can be arbitrarily close to 0. To
be able to draw conclusions from a negative gradient, in the
following section, normalization is introduced which
allows the numeric value of the gradient to be interpreted.
2.4. The normalized gradient
The number returned from the gradient estimation in
Eq. (8) depends on the sensitivity function Sd and also on
the value of the cost function which is an estimate of the
variance. It is of interest to obtain a more absolute number,
which does not depend on the variance but rather, when
calculated for a new closed-loop system; similar numerical
values should mean the same thing. One way to do this is to
normalize the gradient obtained with Eq. (8). Denoting the
normalized gradient as bK, a straight forward way of doing
this is to normalize the gradient as follows:
bK ¼
oJ K
oK J
ð13Þ
Considering Eqs. (8) and (11) it is easy to see that this normalized gradient is independent of loop gain K.
As a dimensionless number it is informative to know
what numerical values this number attains and what they
mean. The interpretation of a positive gradient in Section
2.3 is valid for the normalized gradient as it is only a scaling of the original one. A negative gradient is on the other
hand more difficult to interpret.
In Appendix A.2 it is shown that under general assumptions on the loop transfer function, if the disturbance spectra is a delta function and the disturbance frequency x0
goes to zero, the normalized gradient obtains the value
2. This value of the normalized gradient thus indicates
that the loop is affected by a low frequency disturbance.
For well behaving complementary sensitivity functions,
the normalized gradient will not be smaller than 2. If
the value of the normalized gradient is close to 2, most
of the energy of the disturbances are concentrated on the
low frequency region of the sensitivity function. This
means that variance can be improved by increasing loop
gain.
3. Using the normalized gradient
With the properties of the normalized gradient presented in the last section in mind, its use in performance
assessment will now be detailed. To give a more practical
explanation of the information that the normalized gradient gives, a simulation study is presented.
In Table 1 three plants, their controllers and the model
the controller design was based on are introduced. In the
first two cases the controller is a PID while in the last case
the controller is a dead-time compensator.
The disturbance model is an integrator exited by white
noise a(k):
wðkÞ ¼
1
aðkÞ
1 q1
Assume that the gain of the real process P(s) changes with
operating region so that
P ðsÞ ¼ K p ðyÞP ðsÞ
Table 1
Plants, models and controllers for the simulation study
System number
1
Tuning method
Ziegler–Nichols step response
Plant P ðsÞ
1
ðs þ 1Þ3
2
AMIGO
3
PPI dead-time compensator
1
ðs þ 1Þð0:1s þ 1Þð0:01s þ 1Þð0:001s þ 1Þ
es
ð0:1s þ 1Þ
2
Examples 1 and 3 can be found in [5] while number 2 can be found in [19].
Model Pe ðsÞ
Controller C(s)
0:22e0:6s
s
e0:073s
1:03s þ 1
3:57s þ 8:86s þ 5:5
1:61s
0:083s2 þ 2:3s þ 6:6
0:35s
1es
0:2s þ 1
0:2s þ 1
0:2s þ 1 es
2
Sampling time
0.1
0.01
0.1
A. Ingimundarson / Journal of Process Control 16 (2006) 1013–1020
1.5
y(k)
1
0.5
0
–0.5
e(k)
0.2
0
–0.2
3
K
p
2
1
0
0
2000
4000
6000
8000
Time
10000
12000
14000
16000
Fig. 1. Simulation example 1. Top graph: y(k). Middle graph: e(k).
Bottom graph: control error Kp.
10–1
J
10–2
10–3
10–4
0
2000
4000
6000
8000
10000
12000
14000
16000
2000
4000
6000
8000
Time
10000
12000
14000
16000
2
1
0
βK
The gain Kp depends on the output so that when y = 1,
Kp = Am where Am is the amplitude margin of the loop
transfer function P C. When y = 0, the gain is considerably
reduced so that the effectiveness of the closed-loop system
to reject the disturbance is severely limited and variance increases greatly. Gain Kp increases linearly between y = 0
and y = 1. Due to low gain at y = 0, there the loop behaves
in a sluggish manner with the corresponding increase in
variance. Due to the high loop gain at y = 1, the loop is
marginally unstable, again with an increase in variance.
A simulation scenario of system 1 is shown in Fig. 1.
From time 0 to 4000 the set point stays at 0. Then a slow
ramp is added to the set point so that it reaches 1 at time
12,000. The set point stays at 1 for the rest of the time.
The system thus goes from having a sluggish response to
a condition where it is marginally unstable but with the
same slow disturbance. The marginally unstable system will
amplify frequencies close to the critical frequency, causing
the frequency content of y(k) to change. After time 12,000
the system becomes quite oscillatory.
In Fig. 2 the normalized gradient for loop gain, calculated with Eq. (8) and cost function J calculated with Eq.
(4), are shown for the example. The number of data points
N was 1500. The discrete desired complementary sensitivity
function Td(q1) was found by sampling the model Pe ðsÞ
and controller C(s) and using Eq. (3). The discussion will
focus on distinct time intervals of the scenario.
The normalized gradient bK is close to 2 for time interval 0–4000. The reason is that due to the low value of Kp(y),
the loop cannot effectively reject the low frequency disturbance that affects the loop. This is captured by the normalized gradient, which in this way indicates opportunities to
improve performance by increasing the gain.
In the current example the loop has a sluggish response
in the beginning of the scenario due to a reduced loop gain.
It could also simply be affected by disturbances of very low
frequency. The value of bK does not distinguish between
1017
–1
–2
–3
0
Fig. 2. Simulation example 1. Top graph: cost function J. Bottom graph:
normalized gradient bK. Both are calculated with N = 1500.
these cases. However, in either case, an opportunity exist
to reduce variance by increasing loop gain.
From time 12,000 to 16,000 the system is quite oscillatory and has a high value of the cost function due to the
high value of Kp(y). The gradient has a positive value indicating that the loop is actually amplifying the disturbances
affecting the loop. This indicates a clear opportunity to
reduce the cost function by reducing loop gain.
In the current example the oscillations and increase in
the cost function are caused by the increase in gain Kp(y).
A positive gradient could also be due to oscillatory disturbances originating from neighboring loops or other
sources. In either case a reduction in loop gain is reasonable as the current sensitivity function is amplifying the disturbances, causing an increase in variance.
On the interval between 4000 and 12,000 it can be
seen that bK increases from 2 to 1. The cost function
drops down after time 4000 while bK is negative and
then start to rise again when bK becomes positive. The minimum of the cost function is obtained close to bK = 0 as
expected.
Systems 2 and 3 in Table 1 were simulated in a similar
manner as system 1 with the corresponding gain change
bringing the closed loop from having a sluggish response
to being marginally unstable. In Fig. 3 the cost function
as a function of bK is shown for the three systems. It can
be seen that when the normalized gradient has a value of
2, the cost function is at least three times larger than at
the minimum. The minimum is generally close to bK = 0
even though in the case of the Z–N tuning rule, the minima
seems to be shifted slightly to the left.
It is also seen that the cost function can rise very quickly
as bK becomes larger. The value for which the cost function
rises is not the same. In the case of the AMIGO tuning rule,
the cost function increases very rapidly for bK = 0.4. This
leads to the conclusion that for positive bK the system
might be sensitive to small changes bK.
1018
Normalized cost function
10
10
10
10
A. Ingimundarson / Journal of Process Control 16 (2006) 1013–1020
3
2
1
0
–2
–1.5
–1
–0.5
0
βK
0.5
1
1.5
2
Fig. 3. Value of cost function J as a function of bK for the three systems
presented in Table 1. The variance is normalized by the lowest value over
the simulation scenario. s is Ziegler–Nichols. · is the dead-time
compensator, and + is the AMIGO tuning rule.
The figure also shows that there is little difference in variance on the interval [1, 0]. It is therefore recommended
that the normalized gradient should have a value on the
interval [1, 0.5]. A value of bK equal to 0 is not recommended because of the little difference in the cost function
on the interval [1, 0] and high sensitivity of the cost function for positive values bK.
Remark. It is common that the tuning of a loop ends up
being the most conservative over all operating regions. In
the current example, if the loop would be expected to work
at y = 1 the tuning of the loop would be too aggressive as
indicated with the positive gradient at that operating
region. On the other hand, it is obvious that decreasing
loop gain to make the loop less aggressive would result in
increased variability at the operating region y = 0 as
the disturbance affecting the loop there is very slow. With
this in mind perhaps a more suitable solution for this
loop would be gain scheduling. Generally, if the gradient
changes substantially and in a consistent manner in a loop
as the operating region changes, there might exist opportunities of reducing variance with gain scheduling.
3.1. Summary of use
It is suggested that Eq. (8) is used to calculate the gradient with respect to the loop gain. Eq. (4) is used to calculate
J and the normalized gradient is calculated with Eq. (13). If
a design methodology in continuous time is used, the plant
and controller are sampled to calculate Td(q1) according
to Eq. (3), see [17].
It is possible to use the gradient presented here to initiate adjustment of the loop gain to lower the cost function.
But no automatic adjustment is considered. An operator is
always thought to be ‘‘in the loop’’ of parameter adjustment. Another way to use the method is to apply it after
poor performance has been detected by other methods.
The gradient information can then be collected and monitored over time to provide a valuable overview over the
state of the loop and the disturbances affecting it. For
example if the gradient is watched over many operating
regions it might indicate opportunities for gain scheduling.
If the relation between the variation in the gradient and
changes between operating regions is not clear, it is possible to calculate correlations between the gradient and other
process variables to discover other scheduling variable for
the controller parameters to improve performance.
If the normalized gradient is consistently found on the
interval [1, 0.5] then the spectral composition of the disturbances is such that few opportunities exist to improve
performance by changing the loop gain. If the gradient is
outside this interval for prolonged periods, opportunities
for performance improvements might exist by increasing
the loop gain if the bK < 1 and decreasing it if
bK > 0.5. The presented statistic can be easily implemented in a recursive way, see [11].
3.2. Practical issues
As with other performance assessment methods, the
validity of the results depends on the linearity of the plant.
Methods to quantify nonlinear behavior of a loop such as
the one presented in [20] should be applied to screen out
highly nonlinear loops before using the presented method.
Appropriate noise filtering is of importance as well. The
number of data points to form the estimates should be
between 1000 and 1500.
4. Industrial evaluation
Two applications of the normalized gradient on real
industrial data from a pulp and paper mill will be
presented. One will deal with detecting aggressive tuning
of loops while the other addresses detection of sluggish
loops.
4.1. Detecting aggressively tuned loops
In Fig. 4 the gradient is shown for a pressure control
loop that was k-tuned, see [5], with k = 6 and L = 3. Also
shown in the figure is the controller error.
The normalized gradient bK is close to zero or positive
over the whole interval. A loop with positive gradient presents a very clear opportunity to reduce variability in the
plant by decreasing the loop gain. The bottom graph shows
the controller error for a smaller interval so that the characteristics of the signal can be determined. The loop seems
to suffer from robustness problems or oscillations due to
high gain, a conclusion supported by the operators in the
factory.
A. Ingimundarson / Journal of Process Control 16 (2006) 1013–1020
disturbances affecting the loop. The statistic is an estimate
of the partial derivative of the variance, with regard to the
controller loop gain.
It has been shown that the evaluation of this gradient is
very simple for controllers commonly used in the process
industry. A normalization of the gradient has been presented as well and it has been shown that its numerical values should lie between 1 and 0.5. The gradient was
calculated for industrial data from a pulp and paper factory where it was confirmed by operators that values out
of range indicated performance problems.
βK
0.5
0
–0.5
e(k)
1
0
–1
0
100
200
300
Minutes
400
1019
500
e(k)
0.2
Acknowledgements
0
–0.2
120
121
122
123
124
125
Minutes
Fig. 4. Pressure control loop. Top graph: bK. Middle and bottom graph:
e(k).
The author acknowledges the support received from the
Ramon y Cajal program of the Spanish government and
the Research Commission of the ‘‘Generalitat de Catalunya’’ (group SAC ref.2001/SGR/00236).
βK
Appendix A
0.5
0
–0.5
–1
–1.5
–2
A.1
Assume that the disturbance spectra is confined to a single sinusoidal frequency x0. Then the cost function is given
2
by J ¼ ajSðK; eix0 h Þj . The loop transfer function without
gain K is given by Lðix0 Þ ¼ Gc ðeix0 h ÞP ðeix0 h Þ. Let r(x0)
and u(x0) denote the amplitude and phase of L(ix0), i.e.
Lðix0 Þ ¼ rðx0 Þeiuðx0 Þ .
It will be shown that if oJ/oK > 0 then jSðK; eix0 h Þj > 1.
As the sensitivity function is written as
e(k)
0.1
0
–0.1
100
200
300
Minutes
400
500
e(k)
0.05
SðK; eix0 h Þ ¼
0
–0.05
100
105
110
115
Minutes
120
125
130
Fig. 5. Flow control loop. Top graph: bK. Middle and bottom graph: e(k).
4.2. Detecting sluggish loops
1
1 þ Krðx0 Þeiuðx0 Þ
the cost function can be written as
J¼
a
1 þ 2Krðx0 Þ cosðuðx0 ÞÞ þ K 2 rðx0 Þ
2
The derivative with regard to K is
2
2a rðx0 Þ cosðuðx0 ÞÞ þ Krðx0 Þ
oJ
¼
2
oK
1 þ 2Krðx0 Þ cosðuðx0 ÞÞ þ K 2 rðx0 Þ2
In Fig. 5 the normalized gradient is shown for a flow
control loop that was k-tuned with k = 5.4 and L = 1.5.
The index has a value around 1.7 for the whole scenario. This is a very small value of the gradient and indicates that most disturbances are of low frequency
character. Looking at the bottom graph it is seen that the
control error stays away from 0 for many minutes at a
time, a behavior not consistent with the tuning of the loop.
In this case, operators confirmed that loop gain could have
been increased to improve performance.
For cos(u(x0)) to be negative, u(x0) is on the interval
½p2 ; 3p
. If the amplitude of the total loop transfer function
2
fulfills 0 < Kr(x0) < cos(u(x0)) it is easy to see that
j1 + KL(ix0)j < 1 which is equivalent to jSðK; eix0 h Þj > 1.
5. Conclusions
A.2
A statistic has been presented which gives useful information regarding the specified sensitivity function and the
Assume the amplitude of the loop transfer function r(x)
goes to infinity as x goes to zero. For example, any
ðA:1Þ
The condition for positivity of the gradient is that
Krðx0 Þ < cosðuðx0 ÞÞ
1020
A. Ingimundarson / Journal of Process Control 16 (2006) 1013–1020
controller with integral action fulfills this condition. Using
Eq. (A.1), the expression for bK is
2
2 Krðx0 Þ cosðuðx0 ÞÞ þ K 2 rðx0 Þ
bK ¼
2
1 þ 2Krðx0 Þ cosðuðx0 ÞÞ þ K 2 rðx0 Þ
Using the fact that limx0 !0 rðx0 Þ ¼ 1, straight forward calculations give
lim bK ¼ 2:
x0 !0
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ID
690010
Title
Performance assessment using a normalized gradient
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