Preprints of the 8th IFAC Symposium on Advanced Control of Chemical Processes
The International Federation of Automatic Control
Furama Riverfront, Singapore, July 10-13, 2012
Effectiveness of signal excitation design
methods for ill-conditioned processes
identification ?
André Seichi Ribeiro Kuramoto ∗ Osmel Reyes Vaillant ∗
Claudio Garcia ∗
∗
Escola Politécnica da Universidade de São Paulo - Department of
Telecommunications and Control Engineering
(kuramoto@petrobras.com; osmel.reyes@usp.br; clgarcia@lac.usp.br)
Abstract: A comparison of the effectiveness between five different methods to generate
excitation input signals for the identification of ill-conditioned processes is presented. Evaluation
is made using a 2x2 MIMO model of a high purity distillation column. Results are analysed
using a proposed factor, which evaluates dispersion of the system output plane.
Keywords: input signal design; ill-conditioned process; MIMO; MPC;
1. INTRODUCTION
The success of MPC applications basically depends on the
quality of the employed model. The quality of the identification process which yields these models is influenced by
every aspect of the system identification procedure proposed in (Ljung, 1999). One crucial element contained in
that procedure is the selection or design of the signal used
to excite the system to be identified. Usually, the concept
of persistence of excitation of an input signal is considered
necessary and sufficient to guarantee the uniqueness of
identification algorithm solutions (Zhu, 2001). An input
signal persistently exciting (PE) of order n is distinguished
by having a n × n non-singular covariance matrix. This
condition is associated to the signal waveform and its
spectral characteristics.
Ill-conditioned plants can be found in the process industry.
Examples of this kind of plant are high-purity distillation
columns, heat exchanger networks, gasifiers (Rivera et al.,
2007) and also industrial paper machines, which are largescale systems (Featherstone and Braatz, 1998). A highpurity distillation column and the physical reasons for the
ill-conditioned behavior are discussed in (Skogestad and
Morari, 1987) and (Skogestad, 1997).
The particular dynamic behavior of these processes could
hinder, not only the control system design, but also their
identification (Waller, 2003), (Rivera et al., 2007). In (Zhu,
2001) it is shown that these difficulties are not caused by
the identification method or model structure, but they
are related to poor data when PRBS (pseudo-random
binary sequences) (Ljung, 1999) or GBN (generalized
binary noise) signals (Tulleken, 1990) are used as input
sequences. Identification of ill-conditioned systems could
demand excitation signals with additional features besides
those established for a PE signal excitation.
? Research project sponsored by Petrobras S.A.
© IFAC, 2012. All rights reserved.
337
Several approaches have appeared during the last years,
intended to overcome limitations of PRBS or GBN signals
when an ill-conditioned process has to be identified (Stec
and Zhu, 2001), (Lee et al., 2003), (Lee, 2006) and (Tan
et al., 2009).
The objective of this work is to compare the effectiveness
of five excitation signal design methods for ill-conditioned
process identification, which generate models intended to
be used in a MPC context. The comparison is intended
for small-scale systems (high-purity distillation columns).
Additionally, a factor to evaluate the effect of the signal
created by these methods over the output directionality
is proposed. The factor is a measure of the 2 × 2 MIMO
system output scattering on its output plane. The following approaches will be compared in this work: Two step
methods (Zhu, 2001) (Zhu and Stec, 2006); Rotated inputs
(Conner and Seborg, 2004) (Koung and MacGregor, 1993);
Ternary signals with correlated harmonics (Tan et al.,
2009) and SOH (sum-of-harmonics) signals with modified
zippered power spectrum (Lee et al., 2003).
The paper is organized as follows: in Section 2, a brief
background about ill-conditioned processes and a description of the aforementioned input signal generation methods are provided; in Section 3, a new factor to measure
the output directionality of ill-conditioned processes is
proposed; in Section 4, the results of the high purity
distillation column (Skogestad et al., 1988) identification
employing each one of the aforementioned methods are
discussed. Finally, conclusions are drawn in Section 5.
2. ILL-CONDITIONED PROCESS IDENTIFICATION
2.1 Ill-conditioned process
Ill-conditioning behavior is mainly associated to highly
interactive process dynamics. The processes that present
this behavior are characterized by a strong gain directionality, that is, different gain values depending on the input
directions.
8th IFAC Symposium on Advanced Control of Chemical Processes
Furama Riverfront, Singapore, July 10-13, 2012
Condenser
A linear MIMO system, with open-loop steady-state gain
matrix G, can be factorized through a singular value
decomposition (SVD) as:
G = U ΣV H
VT
PC
p
L
LC
MD
(1)
D, yD
The ratio between the maximum (σ) and the minimum (σ)
singular values of diag(Σ) is known as condition number
(γ). For an ill-conditioned plant γ is higher than one,
which could suggest that values of the system output
vectors will strongly depend on the direction of its input
vectors and consequently that minimum and maximum
gain directions will be found in the system response. This
criterion is scaling-dependent: thus, in order to obtain
confident results, normalized MV and CV values must be
analyzed to guarantee that they are in the same numerical
range, considering that a strongly interactive process is
always ill-conditioned, while the opposite is not always
true (Jacobsen and Skogestad, 1994).
Relative Gain Array (RGA) (Bristol, 1966) is a scalingindependent approach for interactiveness detection and
consequently another method to determine ill-conditioning
plant behavior. RGA can be expressed as:
RGA = ∧ = G × (G
−1 T
)
F,z F
LC
V
MB
B, x B
Bottom product
Reboiler
Fig. 1. Typical distillation column controlled with LV
configuration.
vector direction [1, −1]T (high-gain direction) causes larger
changes in these outputs. Fig. 2 shows the system outputs
y1 and y2 , that represent the output compositions yD and
xB , respectively, for a test with PRBS. The high and lowgain directions are also represented in Fig. 2.
1
(2)
Low gain direction
0.5
More specifically ∧ can be understood as:
RGA = ∧ = [λij ]m×m
Distillate
(3)
where λij are the ratios of gij , gain of the i -j th model of
∗
G with all the loops open, and gij
, gain of the i -j th with
h
i
gij
only i th loop open: λij = g∗ij .
For a 2 × 2 linear MIMO system and considering λ = λ11
= λ22 ; 1 - λ = λ12 = λ21 (Seborg et al., 2004) ∧ can be
expressed as:
λ
1−λ
∧=
(4)
1−λ
λ
For an interactive process λ is larger than one, which
implies an ill-conditioned plant behavior.
Effective identification of dynamic models implies that
both maximum and minimum gain directions must be
properly excited.
In this paper the problem of generating input signals for
an ill-conditioned process identification will be exemplified
using a simplified and linearized high-purity distillation
column model described in (Skogestad et al., 1988). A
simplified diagram of this distillation column, operating
in L-V configuration, is shown in Fig. 1.
The transfer function shown in (5) (Skogestad et al., 1988)
contains the open-loop simplified models of the diagram
shown in Fig. 1.
"
#
87.8
−86.4
1
G(s) =
(5)
75s + 1 108.2 −109.6
Ill-conditioned behavior of this plant makes input vector
direction [1, 1]T ( FL , VF in Fig. 1) to produce a small
effect (low-gain direction) in output compositions (yD and
xB , respectively, in Fig. 1). On the other hand, input
338
y2 0
High gain direction
-0.5
-1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y1
Fig. 2. Open-loop outputs for the ill-conditioned highpurity binary distillation column.
Input signals designed for ill-conditioned plants identification should guarantee an opposed effect to those low and
high-gain direction, and at the same time, not to exceed
operational system constraints.
2.2 Methods for input signal design for ill-conditioned
process identification
In (Koung and MacGregor, 1993) a test design method
using highly correlated input signals was presented. This
method is able to excite the process both in high-gain
and low-gain directions. That method uses rotated inputs,
in which the input vectors are strongly aligned in prespecified directions, in order to avoid the generation of
output data mostly aligned in the strong direction and to
emphasize low gain directions. The rotation of the inputs
are performed from the rotation matrix of steady-state
gain matrix SVD of the plant. A previous evaluation of rotated PRBS input design for process system identification
was presented in (Misra and Nikolaou, 2003) and (Seborg
et al., 2004).
In (Zhu, 2001) a two step method to obtain good data
for estimation of a highly interactive system of two inputs
was proposed. In the first step, a test using uncorrelated
8th IFAC Symposium on Advanced Control of Chemical Processes
Furama Riverfront, Singapore, July 10-13, 2012
binary signals is performed. These signals will provide a
good estimation of high gain direction. In the second step,
a test is performed using two binary signals, consisting of
two well defined periods: one identical for both signals with
high amplitude and the other with low amplitudes and
uncorrelated samples. The high amplitude and identical
periods will create the strong correlation needed to identify
the low gain direction. The ratio of the two amplitudes of
the periods can be determined from the model identified in
the first step. For the distillation column shown in Fig. 1,
the increasing reflux has opposite effect to increasing the
steam flow rate. Then, at the high amplitude period,
the signals have the same sign. The method can be
implemented without quantitative knowledge of the plant,
because it is considered near [1, 1]T .
In (Zhu and Stec, 2006), a second method was proposed,
consisting of the sum of the two well defined periods of
the signal generated in the previous method (Zhu, 2001).
For √
the same period of test, this approach allows reducing
by 2 the amplitude of the previous signals, remaining
the same model quality identified by the previous signals.
The effectiveness of this method was shown in (Zhu and
Stec, 2006) by comparing these results with those obtained
using GBN signals.
The idea of creating a strong time domain correlation
between the signals needed to identify the low gain direction, was modified in (Lee et al., 2003). It that work
the proposal was to create frequency domain correlation,
defining a modification in the power spectrum of a set of
SOH signals (Schroeder, 1970). The resulting signals have
orthogonal frequencies and include correlated harmonics
with high levels of power, which allow emphasizing low
gain information in the data. These correlated harmonics are “rotated” by the rotation matrix of steady-state
gain matrix SVD of the plant. A proposal of relative
power between correlated and uncorrelated harmonics was
presented in (Rivera et al., 2009). Different from (Zhu,
2001) and (Zhu and Stec, 2006), this method requires
quantitative knowledge about the low gain direction. A
priori knowledge of the steady-state gain matrix of the
plant is needed. If this information is not available, it can
be obtained from data of simple preliminary identification
test (Zhu, 2001).
An additional optimization step for SOH signals design
was introduced in (Lee et al., 2003). It was proposed to
minimize the crest factor (CF) over all output channels
by the optimization of phases and amplitudes of the
harmonics that compose the signals. For this optimization,
a dynamic model needs to be available, which rarely
occurs.
A similar idea to that presented in (Lee et al., 2003) was
proposed in (Tan et al., 2009). This approach also creates
correlation in frequency domain, but the number of signal
levels are limited to three. These pseudo-random signals,
generated arithmetically over a Galois field, are called simply PRTS (pseudo-random ternary signals) herein. There
is no flexibility to specify the spectrum due to the constraint imposed by the number of signal levels allowed. As
well as the method proposed in (Zhu, 2001), PRTS generation does not require a quantitative knowledge about
the low gain direction, but a priori knowledge of the
339
directional properties of the system is needed, in order
to emphasize the low gain directions and to avoid the
generation of output data mostly aligned in the strong
direction.
There are other proposals for input signal design to be
used to excite ill-conditioned highly iterative systems.
Somehow, they are mainly variants or evolutions of the
five basic ideas summarized in this section, such as numerical optimization. However, optimal design methods
(Rivera et al., 2009) need knowledge about system dynamics, which rarely is available in industrial plants before
identification. In (Darby and Nikolaou, 2009) an in-depth
study is presented where, starting from the (Koung and
MacGregor, 1993) proposals, the integral controllability
condition is combined with the optimal design of inputs.
The results originally proposed for 2 × 2 systems, is extended for large-scale systems. This approach also involves
the real process transfer matrix (G) and the identified
model (Ĝ). Table 1 synthesizes the main characteristics of
the five methods described in this section.
2.3 Parameterization of input signals for test
The bandwidth (BW ) of the plant can be determined by
(Rivera et al., 1994) and (Gaikwad and Rivera, 1996):
ωL =
1
αs
≤ω≤
= ωH
βs τmax
τmin
(6)
where τmax and τmin are the major and minor dominant
time constants of the MIMO system, respectively; αs is
specified to insure that sufficiently high frequency content
is available in the input signal, commensurate with how
much faster the closed-loop response is expected to be,
related to the open-loop response; βs is specified to tailor
how much low frequency information is present in the
input signal. Then, the interval BW = {ωL ; ωH } is the
frequency range where the energy of the excitation signal
must be concentrated.
The first order system under analysis represented by (5)
has only one time constant. Then, τ = τmax = τmin =
75 minutes. An estimative of the time constants can be
obtained from data of simple preliminary stair tests, if this
information is not available. Defining βs = 5 and αs = 1
results in ωL = 0.0026 rad/minutes and ωH = 0.0133
rad/minutes.
Rotated signals (Koung and MacGregor, 1993) were generated from PRBS, which were rotated and emphasized to
excite low gain directions. PRBS generation was characterized by the length Ns and the switching time Tsw . These
parameters, specified in (Gaikwad and Rivera, 1996), were
set to 426 and 8 minutes, respectively. This resulted in
426 × 8 = 3408 minutes of experiment period (Te ).
A GBN signal was employed to generate the signals
proposed in (Zhu, 2001) and (Zhu and Stec, 2006) two
step methods. The non switching probability (p) of this
GBN was adopted as 0.88 based on BW of the plant
(Chen and Yu, 1997). The scaling factor (Γ) applied to
the signal amplitude to emphasize the low gain direction
was 78. The whole experiment period was 3408 minutes
for both methods.
8th IFAC Symposium on Advanced Control of Chemical Processes
Furama Riverfront, Singapore, July 10-13, 2012
Table 1. Main characteristics of the five signal design methods.
Method
Rotated signals
(Koung and MacGregor, 1993)
Two step
method 1 (Zhu, 2001)
Two step
method 2 (Zhu and Stec, 2006)
Modified zippered
SOH signals (Rivera et al., 2009)
PRTS (Tan et al., 2009)
A priori system
knowledge required
Calculation
principle
Amplitude
characteristic
steady-state gain
matrix of the plant
signs of low
gain direction
signs of low
gain direction
steady-state gain
matrix of the plant
signs of low
gain direction
rotation by SVD
rotation matrix
four levels
concatenated signals
four levels
combined signals
four levels
Modified zippered power spectrum SOH signals, as defined
in (Lee, 2006) and (Rivera et al., 2009), demand the
following parameters for their generation: sampling time
(Ts ), number of specified harmonics (ns ) and signal length
(Ns ). The values adopted were Ts = 2 minutes, ns = 13
and Ns = 1704. This resulted in Te = 1704 × 2 = 3408
minutes. The range of scaling factor Γ applied to the signal
amplitude to emphasize the low gain direction was defined
by the steady state gain matrix (Lee, 2006) (Rivera et al.,
2009). The estimated value in this case was 78. Neither
optimization step nor constraints introduced in (Lee et al.,
2003) were applied in this work.
Generation of PRTS is characterized by the extended
Galois field GF (q m ), with q > 2 and m an integer,
one primitive sequence for each PRTS and the switching
time (Tsw ) (Tan et al., 2009). In this work, GF (132 ) was
adopted, generated by the primitive polynomial x2 +x+2,
the primitive sequences C and D of Table I of (Tan et al.,
2009) and Tsw = 8 minutes. This resulted in signals with
1344 minutes, which were cyclically repeated to complete
Te = 3408 minutes.
3. SCATTERING FACTOR
Cross correlation coefficient is proposed in this work as a
measure of the output scattering factor:
!
C(y1 , y2 )
SF = 1 − p
× 100 [%]
(7)
C(y1 , y1 )C(y2 , y2 )
where C(y1 , y2 ) is the covariance of outputs y1 and y2 .
SF is an evaluation of the outputs (y1 , y2 ) spreading in the
system output plane. If SF is high, the input signals were
able to excite the plant in low and high gain directions in
the same proportion. On the other hand, low values of SF
clearly reflect a gain directionality in the output system.
In this case, input signals excite the plant mostly in one
direction. Often this single direction is the one with high
gain.
This scattering factor allows numerically evaluating the
effectiveness of every input signal design method, independently of its nature. For the sake of the validity of this
metric, high values of SNR (signal-to-noise ratio) should
be maintained.
4. RESULTS
The identification of the plant was simulated using each
one of the five methods for signal generation. The adopted
340
phase optimization
and rotation
finite field
arithmetic
continuous
three levels
sample time was Ts = 2 minutes. In order to obtain a
fair comparison, equal length experiments (Te = 3408
minutes) were performed for the five methods. For the
same reason, the peak-to-peak amplitude of each signal
was also adjusted to the same value [−0.3; +0.3]. It is
noteworthy that each method of generation of excitation
signal can yield a different crest factor.
Filtered white Gaussian noise of variance σ12 = σ22 = 0.1
was added to the output y1 and y2 for all the simulations.
The disturbance model employed was a first order filter
with time constant τd = 100 minutes. The model structure
employed to identify the ill-conditioned distillation plant
was an ARX with the same order of the plant (5). Each
validation signal was conformed using two section GBN
signals, with two different periods. In the first section, both
signals were correlated and, in the second section, signals
were not correlated. A switching probability of 0.88 was
set for both signals.
Monte-Carlo simulations were implemented in order to
verify the repeatability of the obtained results. 100
realizations were employed for each signal generation
method. Table 2 summarizes information obtained from
the
identification runs. Scattering Factors, FIT (100 ×
2
1 − ||ŷ(i)−y(i)||
) and SNR values shown correspond to
||y(i)−y||2
the mean values of realizations for each method. As a
measure of these values dispersion, the standard deviations
are also shown.
As can be seen, a significant improvement of the identification result was found when excitation signals were
different from PRBS. The best FIT and the highest SNR
were obtained using modified zippered power spectrum
SOH signals. The FIT obtained using Zhu methods 1 and
2 are similar and worse than those obtained using rotated
signals. It is noteworthy that to design rotated signals and
modified zippered SOH signals, a priori knowledge of the
steady-state gain matrix of the plant is required (Table
1). This is a disadvantage of these methods as compared
to Zhu methods 1 and 2. The performance obtained using
PRTS was not good. Although these signals have correlated harmonics and results in high SNR values at outputs,
the output data were mostly aligned in the strong direction
and emphasized the low gain direction (similar to PRBS).
These output data are not sufficient to obtain good model
estimations in low gain directions.
Figures 3 to 7 show the excitation output plane (y1 versus
y2 that represent the output compositions yD versus xB ,
8th IFAC Symposium on Advanced Control of Chemical Processes
Furama Riverfront, Singapore, July 10-13, 2012
Table 2. Simulation result summary.
Method
SF [%]
PRBS
(Ljung, 1999)
Rotated signals
(Koung and MacGregor, 1993)
Two step method 1
(Zhu, 2001)
Two step method 2
(Zhu and Stec, 2006)
Modified zippered SOH
signals (Rivera et al., 2009)
PRTS
(Tan et al., 2009)
σSF
0.012
0.004
88.61
8.02
78.24
13.38
62.67
12.17
65.43
2.46
0.048
0.006
Output
F IT [%]
σF IT
SN R
σSN R
y1
y2
y1
y2
y1
y2
y1
y2
y1
y2
y1
y2
40.88
40.45
70.07
72.43
68.12
70.20
67.60
70.19
75.21
75.42
56.14
57.97
32.72
37.38
5.91
5.04
4.34
3.87
4.87
4.59
4.10
6.16
6.91
5.36
1.3 × 105
1.2 × 105
35.39
35.52
18.99
24.81
64.51
17.16
83.72
108.16
9.1 × 103
1.5 × 104
3.9 × 104
6.1 × 104
9.26
10.39
5.04
6.32
16.73
3.91
13.20
20.45
1.5 × 103
2.5 × 103
0.4
0.25
0.3
0.2
0.15
0.2
0.1
0.1
0.05
y2
y2
0
0
−0.1
−0.05
−0.2
−0.1
−0.3
−0.15
−0.4
−0.2
−0.5
−0.25
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
−0.4
0.3
−0.2
0
0.2
0.4
0.6
y1
y1
Fig. 3. Outputs of the open-loop test with rotated signals.
Fig. 5. Outputs of the open-loop test with two step method
2 excitation signals.
0.3
0.6
0.2
0.4
0.1
y2
y2
0.2
0
0
−0.1
−0.2
−0.2
−0.3
−0.4
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
−0.6
0.3
−0.5
y1
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
y1
Fig. 4. Outputs of the open-loop test with two step method
1 excitation signals.
respectively) of the simulated system for each one of the
five input signal generation methods. In Fig. 4, the characteristic of concatenation of two periods of signals can
clearly be seen: one excites the low gain direction at the
beginning of the experiment and a second one excites the
right gain direction at the other moment of the experiment.
A significant different scattering pattern of output plane
is observed in Fig. 5, because this method simultaneously
excites high and low gain directions, similar to rotated
signals (Fig. 3). The modified zippered SOH signals are
time continuous (Tab. 1), then the scattering pattern of
output plane shows more continuous lines than the other
methods. Fig. 7 shows that the PRTS output data are
mostly aligned in the strong direction and poorly excites
the low gain direction.
341
Fig. 6. Outputs of the open-loop test with Modified
zippered SOH excitation signals.
For the outputs data obtained with PRBS and PRTS
the SF calculated was very low (< 1%), showing that
these signals were not able to excite the plant in low
and high gain directions in the same proportion. On the
contrary, for the other four methods, the SFs were not low
and excitations in both directions were balanced. Visually,
outputs obtained with rotated signals and Zhu method
1 were almost symmetrical, then the SF results in high
values (≈ 80%). As expected, the best FIT was obtained
with the method (modified zippered SOH signals) that
establishes a trade-off between SF and SNR values.
An evaluation of signal excitation design methods for identification of the distillation column model with different
dynamics for input-output pair (Jacobsen and Skogestad,
1994) (case N2) was conducted in (Vaillant et al., 2012).
8th IFAC Symposium on Advanced Control of Chemical Processes
Furama Riverfront, Singapore, July 10-13, 2012
10
8
6
4
y2
2
0
−2
−4
−6
−8
−10
−8
−6
−4
−2
0
2
4
6
8
y1
Fig. 7. Outputs of the open-loop test with PRTS.
5. CONCLUSIONS
A comparison of the effectiveness of five excitation signal
design methods for ill-conditioned process identification is
presented. By simulation, it was shown that the methods
result in different performances depending on how output
directions are excited. To evaluate the proportion of the
excitation in high and low gain directions, a measure
(SF) was proposed to evaluate the outputs of the system.
The results show that good performance is obtained with
signals that presented good SNR and good SF outputs,
which implies a trade-off between SF and SNR.
ACKNOWLEDGEMENTS
The authors thank the support provided by the Center of Excellence for Industrial Automation Technology
(CETAI) and Petrobras S.A.
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