Effectiveness of signal excitation designmethods for ill-conditioned processes

identification ?

Andre Seichi Ribeiro Kuramoto ∗ Osmel Reyes Vaillant ∗

Claudio Garcia ∗

∗ Escola Politecnica da Universidade de Sao Paulo - Department ofTelecommunications 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 generateexcitation input signals for the identification of ill-conditioned processes is presented. Evaluationis made using a 2x2 MIMO model of a high purity distillation column. Results are analysedusing 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 thequality of the employed model. The quality of the identi-fication process which yields these models is influenced byevery aspect of the system identification procedure pro-posed in (Ljung, 1999). One crucial element contained inthat procedure is the selection or design of the signal usedto excite the system to be identified. Usually, the conceptof persistence of excitation of an input signal is considerednecessary and sufficient to guarantee the uniqueness ofidentification algorithm solutions (Zhu, 2001). An inputsignal persistently exciting (PE) of order n is distinguishedby having a n × n non-singular covariance matrix. Thiscondition is associated to the signal waveform and itsspectral characteristics.

Ill-conditioned plants can be found in the process industry.Examples of this kind of plant are high-purity distillationcolumns, heat exchanger networks, gasifiers (Rivera et al.,2007) and also industrial paper machines, which are large-scale systems (Featherstone and Braatz, 1998). A high-purity distillation column and the physical reasons for theill-conditioned behavior are discussed in (Skogestad andMorari, 1987) and (Skogestad, 1997).

The particular dynamic behavior of these processes couldhinder, not only the control system design, but also theiridentification (Waller, 2003), (Rivera et al., 2007). In (Zhu,2001) it is shown that these difficulties are not caused bythe identification method or model structure, but theyare related to poor data when PRBS (pseudo-randombinary sequences) (Ljung, 1999) or GBN (generalizedbinary noise) signals (Tulleken, 1990) are used as inputsequences. Identification of ill-conditioned systems coulddemand excitation signals with additional features besidesthose established for a PE signal excitation.

? Research project sponsored by Petrobras S.A.

Several approaches have appeared during the last years,intended to overcome limitations of PRBS or GBN signalswhen an ill-conditioned process has to be identified (Stecand Zhu, 2001), (Lee et al., 2003), (Lee, 2006) and (Tanet al., 2009).

The objective of this work is to compare the effectivenessof five excitation signal design methods for ill-conditionedprocess identification, which generate models intended tobe used in a MPC context. The comparison is intendedfor small-scale systems (high-purity distillation columns).Additionally, a factor to evaluate the effect of the signalcreated by these methods over the output directionalityis proposed. The factor is a measure of the 2 × 2 MIMOsystem output scattering on its output plane. The follow-ing approaches will be compared in this work: Two stepmethods (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 modifiedzippered power spectrum (Lee et al., 2003).

The paper is organized as follows: in Section 2, a briefbackground about ill-conditioned processes and a descrip-tion of the aforementioned input signal generation meth-ods are provided; in Section 3, a new factor to measurethe output directionality of ill-conditioned processes isproposed; in Section 4, the results of the high puritydistillation column (Skogestad et al., 1988) identificationemploying each one of the aforementioned methods arediscussed. Finally, conclusions are drawn in Section 5.

2. ILL-CONDITIONED PROCESS IDENTIFICATION

2.1 Ill-conditioned process

Ill-conditioning behavior is mainly associated to highlyinteractive process dynamics. The processes that presentthis behavior are characterized by a strong gain direction-ality, that is, different gain values depending on the inputdirections.

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© IFAC, 2012. All rights reserved. 337

A linear MIMO system, with open-loop steady-state gainmatrix G, can be factorized through a singular valuedecomposition (SVD) as:

G = UΣVH (1)

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 outputvectors will strongly depend on the direction of its inputvectors and consequently that minimum and maximumgain directions will be found in the system response. Thiscriterion is scaling-dependent: thus, in order to obtainconfident results, normalized MV and CV values must beanalyzed to guarantee that they are in the same numericalrange, considering that a strongly interactive process isalways ill-conditioned, while the opposite is not alwaystrue (Jacobsen and Skogestad, 1994).

Relative Gain Array (RGA) (Bristol, 1966) is a scaling-independent approach for interactiveness detection andconsequently another method to determine ill-conditioningplant behavior. RGA can be expressed as:

RGA = ∧ = G × (G−1)T (2)

More specifically ∧ can be understood as:

RGA = ∧ = [λij ]m×m (3)

where λij are the ratios of gij , gain of the i -j th model ofG with all the loops open, and g∗ij , gain of the i -j th with

only ith loop open: λij =[

gijg∗ij

].

For a 2× 2 linear MIMO system and considering λ = λ11= λ22; 1 - λ = λ12 = λ21 (Seborg et al., 2004) ∧ can beexpressed as:

∧ =

[λ 1− λ

1− λ λ

](4)

For an interactive process λ is larger than one, whichimplies an ill-conditioned plant behavior.

Effective identification of dynamic models implies thatboth maximum and minimum gain directions must beproperly excited.

In this paper the problem of generating input signals foran ill-conditioned process identification will be exemplifiedusing a simplified and linearized high-purity distillationcolumn model described in (Skogestad et al., 1988). Asimplified diagram of this distillation column, operatingin 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 diagramshown in Fig. 1.

G(s) =1

75s+ 1

[87.8 −86.4

108.2 −109.6

](5)

Ill-conditioned behavior of this plant makes input vectordirection [1, 1]T (L

F , VF in Fig. 1) to produce a small

effect (low-gain direction) in output compositions (yD andxB , respectively, in Fig. 1). On the other hand, input

PC

LC

F,z

VT

F

V

L LC

p

B, xB

MB

MD

Condenser

Reboiler

Distillate

Bottom product

D, yD

Fig. 1. Typical distillation column controlled with LVconfiguration.

vector direction [1,−1]T (high-gain direction) causes largerchanges in these outputs. Fig. 2 shows the system outputsy1 and y2, that represent the output compositions yD andxB , respectively, for a test with PRBS. The high and low-gain directions are also represented in Fig. 2.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.5

0

0.5

1

y1

Low gain direction

High gain direction

y2

Fig. 2. Open-loop outputs for the ill-conditioned high-purity binary distillation column.

Input signals designed for ill-conditioned plants identifica-tion should guarantee an opposed effect to those low andhigh-gain direction, and at the same time, not to exceedoperational system constraints.

2.2 Methods for input signal design for ill-conditionedprocess identification

In (Koung and MacGregor, 1993) a test design methodusing highly correlated input signals was presented. Thismethod is able to excite the process both in high-gainand low-gain directions. That method uses rotated inputs,in which the input vectors are strongly aligned in pre-specified directions, in order to avoid the generation ofoutput data mostly aligned in the strong direction and toemphasize low gain directions. The rotation of the inputsare performed from the rotation matrix of steady-stategain matrix SVD of the plant. A previous evaluation of ro-tated PRBS input design for process system identificationwas presented in (Misra and Nikolaou, 2003) and (Seborget al., 2004).

In (Zhu, 2001) a two step method to obtain good datafor estimation of a highly interactive system of two inputswas proposed. In the first step, a test using uncorrelated

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338

binary signals is performed. These signals will provide agood estimation of high gain direction. In the second step,a test is performed using two binary signals, consisting oftwo well defined periods: one identical for both signals withhigh amplitude and the other with low amplitudes anduncorrelated samples. The high amplitude and identicalperiods will create the strong correlation needed to identifythe low gain direction. The ratio of the two amplitudes ofthe periods can be determined from the model identified inthe first step. For the distillation column shown in Fig. 1,the increasing reflux has opposite effect to increasing thesteam flow rate. Then, at the high amplitude period,the signals have the same sign. The method can beimplemented 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 ofthe signal generated in the previous method (Zhu, 2001).For the same period of test, this approach allows reducingby√

2 the amplitude of the previous signals, remainingthe same model quality identified by the previous signals.The effectiveness of this method was shown in (Zhu andStec, 2006) by comparing these results with those obtainedusing GBN signals.

The idea of creating a strong time domain correlationbetween the signals needed to identify the low gain di-rection, was modified in (Lee et al., 2003). It that workthe proposal was to create frequency domain correlation,defining a modification in the power spectrum of a set ofSOH signals (Schroeder, 1970). The resulting signals haveorthogonal frequencies and include correlated harmonicswith high levels of power, which allow emphasizing lowgain information in the data. These correlated harmon-ics are “rotated” by the rotation matrix of steady-stategain matrix SVD of the plant. A proposal of relativepower between correlated and uncorrelated harmonics waspresented in (Rivera et al., 2009). Different from (Zhu,2001) and (Zhu and Stec, 2006), this method requiresquantitative knowledge about the low gain direction. Apriori knowledge of the steady-state gain matrix of theplant is needed. If this information is not available, it canbe obtained from data of simple preliminary identificationtest (Zhu, 2001).

An additional optimization step for SOH signals designwas introduced in (Lee et al., 2003). It was proposed tominimize the crest factor (CF) over all output channelsby the optimization of phases and amplitudes of theharmonics that compose the signals. For this optimization,a dynamic model needs to be available, which rarelyoccurs.

A similar idea to that presented in (Lee et al., 2003) wasproposed in (Tan et al., 2009). This approach also createscorrelation in frequency domain, but the number of signallevels are limited to three. These pseudo-random signals,generated arithmetically over a Galois field, are called sim-ply PRTS (pseudo-random ternary signals) herein. Thereis no flexibility to specify the spectrum due to the con-straint imposed by the number of signal levels allowed. Aswell as the method proposed in (Zhu, 2001), PRTS gen-eration does not require a quantitative knowledge aboutthe low gain direction, but a priori knowledge of the

directional properties of the system is needed, in orderto emphasize the low gain directions and to avoid thegeneration of output data mostly aligned in the strongdirection.

There are other proposals for input signal design to beused to excite ill-conditioned highly iterative systems.Somehow, they are mainly variants or evolutions of thefive basic ideas summarized in this section, such as nu-merical optimization. However, optimal design methods(Rivera et al., 2009) need knowledge about system dy-namics, which rarely is available in industrial plants beforeidentification. In (Darby and Nikolaou, 2009) an in-depthstudy is presented where, starting from the (Koung andMacGregor, 1993) proposals, the integral controllabilitycondition is combined with the optimal design of inputs.The results originally proposed for 2 × 2 systems, is ex-tended for large-scale systems. This approach also involvesthe real process transfer matrix (G) and the identified

model (G). Table 1 synthesizes the main characteristics ofthe 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

βsτmax≤ ω ≤ αs

τmin= ωH (6)

where τmax and τmin are the major and minor dominanttime constants of the MIMO system, respectively; αs isspecified to insure that sufficiently high frequency contentis available in the input signal, commensurate with howmuch faster the closed-loop response is expected to be,related to the open-loop response; βs is specified to tailorhow much low frequency information is present in theinput signal. Then, the interval BW = {ωL;ωH} is thefrequency range where the energy of the excitation signalmust 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 beobtained from data of simple preliminary stair tests, if thisinformation is not available. Defining βs = 5 and αs = 1results in ωL = 0.0026 rad/minutes and ωH = 0.0133rad/minutes.

Rotated signals (Koung and MacGregor, 1993) were gen-erated from PRBS, which were rotated and emphasized toexcite low gain directions. PRBS generation was character-ized by the length Ns and the switching time Tsw. Theseparameters, specified in (Gaikwad and Rivera, 1996), wereset to 426 and 8 minutes, respectively. This resulted in426× 8 = 3408 minutes of experiment period (Te).

A GBN signal was employed to generate the signalsproposed in (Zhu, 2001) and (Zhu and Stec, 2006) twostep methods. The non switching probability (p) of thisGBN was adopted as 0.88 based on BW of the plant(Chen and Yu, 1997). The scaling factor (Γ) applied tothe signal amplitude to emphasize the low gain directionwas 78. The whole experiment period was 3408 minutesfor both methods.

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Table 1. Main characteristics of the five signal design methods.

MethodA priori system

knowledge requiredCalculation

principleAmplitude

characteristic

Rotated signals(Koung and MacGregor, 1993)

steady-state gainmatrix of the plant

rotation by SVDrotation matrix

four levels

Two stepmethod 1 (Zhu, 2001)

signs of lowgain direction

concatenated signals four levels

Two stepmethod 2 (Zhu and Stec, 2006)

signs of lowgain direction

combined signals four levels

Modified zipperedSOH signals (Rivera et al., 2009)

steady-state gainmatrix of the plant

phase optimizationand rotation

continuous

PRTS (Tan et al., 2009)signs of low

gain directionfinite fieldarithmetic

three levels

Modified zippered power spectrum SOH signals, as definedin (Lee, 2006) and (Rivera et al., 2009), demand thefollowing parameters for their generation: sampling time(Ts), number of specified harmonics (ns) and signal length(Ns). The values adopted were Ts = 2 minutes, ns = 13and Ns = 1704. This resulted in Te = 1704 × 2 = 3408minutes. The range of scaling factor Γ applied to the signalamplitude to emphasize the low gain direction was definedby the steady state gain matrix (Lee, 2006) (Rivera et al.,2009). The estimated value in this case was 78. Neitheroptimization step nor constraints introduced in (Lee et al.,2003) were applied in this work.

Generation of PRTS is characterized by the extendedGalois field GF (qm), with q > 2 and m an integer,one primitive sequence for each PRTS and the switchingtime (Tsw) (Tan et al., 2009). In this work, GF (132) wasadopted, 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 with1344 minutes, which were cyclically repeated to completeTe = 3408 minutes.

3. SCATTERING FACTOR

Cross correlation coefficient is proposed in this work as ameasure of the output scattering factor:

SF =

(1− C(y1, y2)√

C(y1, y1)C(y2, y2)

)× 100 [%] (7)

where C(y1, y2) is the covariance of outputs y1 and y2.

SF is an evaluation of the outputs (y1, y2) spreading in thesystem output plane. If SF is high, the input signals wereable to excite the plant in low and high gain directions inthe same proportion. On the other hand, low values of SFclearly reflect a gain directionality in the output system.In this case, input signals excite the plant mostly in onedirection. Often this single direction is the one with highgain.

This scattering factor allows numerically evaluating theeffectiveness of every input signal design method, inde-pendently of its nature. For the sake of the validity of thismetric, high values of SNR (signal-to-noise ratio) shouldbe maintained.

4. RESULTS

The identification of the plant was simulated using eachone of the five methods for signal generation. The adopted

sample time was Ts = 2 minutes. In order to obtain afair comparison, equal length experiments (Te = 3408minutes) were performed for the five methods. For thesame reason, the peak-to-peak amplitude of each signalwas also adjusted to the same value [−0.3; +0.3]. It isnoteworthy that each method of generation of excitationsignal can yield a different crest factor.

Filtered white Gaussian noise of variance σ21 = σ2

2 = 0.1was added to the output y1 and y2 for all the simulations.The disturbance model employed was a first order filterwith time constant τd = 100 minutes. The model structureemployed to identify the ill-conditioned distillation plantwas an ARX with the same order of the plant (5). Eachvalidation signal was conformed using two section GBNsignals, with two different periods. In the first section, bothsignals were correlated and, in the second section, signalswere not correlated. A switching probability of 0.88 wasset for both signals.

Monte-Carlo simulations were implemented in order toverify the repeatability of the obtained results. 100realizations were employed for each signal generationmethod. Table 2 summarizes information obtained fromthe identification runs. Scattering Factors, FIT (100 ×(

1− ||y(i)−y(i)||2||y(i)−y||2

)) and SNR values shown correspond to

the mean values of realizations for each method. As ameasure of these values dispersion, the standard deviationsare also shown.

As can be seen, a significant improvement of the iden-tification result was found when excitation signals weredifferent from PRBS. The best FIT and the highest SNRwere obtained using modified zippered power spectrumSOH signals. The FIT obtained using Zhu methods 1 and2 are similar and worse than those obtained using rotatedsignals. It is noteworthy that to design rotated signals andmodified zippered SOH signals, a priori knowledge of thesteady-state gain matrix of the plant is required (Table1). This is a disadvantage of these methods as comparedto Zhu methods 1 and 2. The performance obtained usingPRTS was not good. Although these signals have corre-lated harmonics and results in high SNR values at outputs,the output data were mostly aligned in the strong directionand emphasized the low gain direction (similar to PRBS).These output data are not sufficient to obtain good modelestimations in low gain directions.

Figures 3 to 7 show the excitation output plane (y1 versusy2 that represent the output compositions yD versus xB ,

8th IFAC Symposium on Advanced Control of Chemical ProcessesFurama Riverfront, Singapore, July 10-13, 2012

340

Table 2. Simulation result summary.

Method SF [%] σSF Output FIT [%] σFIT SNR σSNR

PRBS(Ljung, 1999)

0.012 0.004y1 40.88 32.72 1.3 × 105 3.9 × 104

y2 40.45 37.38 1.2 × 105 6.1 × 104

Rotated signals(Koung and MacGregor, 1993)

88.61 8.02y1 70.07 5.91 35.39 9.26y2 72.43 5.04 35.52 10.39

Two step method 1(Zhu, 2001)

78.24 13.38y1 68.12 4.34 18.99 5.04y2 70.20 3.87 24.81 6.32

Two step method 2(Zhu and Stec, 2006)

62.67 12.17y1 67.60 4.87 64.51 16.73y2 70.19 4.59 17.16 3.91

Modified zippered SOHsignals (Rivera et al., 2009)

65.43 2.46y1 75.21 4.10 83.72 13.20y2 75.42 6.16 108.16 20.45

PRTS(Tan et al., 2009)

0.048 0.006y1 56.14 6.91 9.1 × 103 1.5 × 103

y2 57.97 5.36 1.5 × 104 2.5 × 103

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

y1

y2

Fig. 3. Outputs of the open-loop test with rotated signals.

−0.3 −0.2 −0.1 0 0.1 0.2 0.3

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

y1

y2

Fig. 4. Outputs of the open-loop test with two step method1 excitation signals.

respectively) of the simulated system for each one of thefive input signal generation methods. In Fig. 4, the char-acteristic of concatenation of two periods of signals canclearly be seen: one excites the low gain direction at thebeginning of the experiment and a second one excites theright gain direction at the other moment of the experiment.A significant different scattering pattern of output planeis observed in Fig. 5, because this method simultaneouslyexcites high and low gain directions, similar to rotatedsignals (Fig. 3). The modified zippered SOH signals aretime continuous (Tab. 1), then the scattering pattern ofoutput plane shows more continuous lines than the othermethods. Fig. 7 shows that the PRTS output data aremostly aligned in the strong direction and poorly excitesthe low gain direction.

−0.4 −0.2 0 0.2 0.4 0.6

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25

y1

y2

Fig. 5. Outputs of the open-loop test with two step method2 excitation signals.

−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5−0.6

−0.4

−0.2

0

0.2

0.4

0.6

y1

y2

Fig. 6. Outputs of the open-loop test with Modifiedzippered SOH excitation signals.

For the outputs data obtained with PRBS and PRTSthe SF calculated was very low (< 1%), showing thatthese signals were not able to excite the plant in lowand high gain directions in the same proportion. On thecontrary, for the other four methods, the SFs were not lowand excitations in both directions were balanced. Visually,outputs obtained with rotated signals and Zhu method1 were almost symmetrical, then the SF results in highvalues (≈ 80%). As expected, the best FIT was obtainedwith the method (modified zippered SOH signals) thatestablishes a trade-off between SF and SNR values.

An evaluation of signal excitation design methods for iden-tification of the distillation column model with differentdynamics for input-output pair (Jacobsen and Skogestad,1994) (case N2) was conducted in (Vaillant et al., 2012).

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−8 −6 −4 −2 0 2 4 6 8−10

−8

−6

−4

−2

0

2

4

6

8

10

y1

y2

Fig. 7. Outputs of the open-loop test with PRTS.

5. CONCLUSIONS

A comparison of the effectiveness of five excitation signaldesign methods for ill-conditioned process identification ispresented. By simulation, it was shown that the methodsresult in different performances depending on how outputdirections are excited. To evaluate the proportion of theexcitation 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 withsignals 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 Cen-ter of Excellence for Industrial Automation Technology(CETAI) and Petrobras S.A.

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