Black and Gray-box Identification of a Hydraulic ... - Semantic Scholar

IEEE TRANS. CONTR. SYST. TECHNOL.

IEEE Transactions on Control Systems Technology, 19(2):398–406, 2011. DOI: 10.1109/TCST.2010.2042600. 1

Black and Gray-box Identification of a Hydraulic Pumping System Bruno H. G. Barbosa, Luis A. Aguirre, Carlos B. Martinez, and Antônio P. Braga

Abstract—The use of auxiliary information during the identification of nonlinear systems can be handled in different ways and at different levels. In this paper, static information of a 15kW hydraulic pumping system is used as a priori knowledge in the parameters estimation of polynomial models which are compared to polynomial and neural models obtained by blackbox techniques. The aim is to find models with good performance in both transient and steady-state regimes. The paper presents a novel bi-objective problem that uses free-run simulation and a new decision-maker. The optimization problem is solved using a genetic algorithm. Compared with other techniques, the proposed approach can lead to models with better dynamic and static performance. Index Terms—identification, NARMAX, gray-box, multiobjective, static function, hydraulic system, genetic algorithms.

I. I NTRODUCTION

M

ODELLING techniques can be classified into three categories [1]: i. white-box modelling: the model is obtained taking into account physical equations that govern the process. In this class, a deep knowledge of the system is necessary; ii. gray-box modelling: prior or auxiliary knowledge of the system is used. Such auxiliary knowledge may be available in the form of steady-state data. iii. black-box modelling: the model is identified only using the data set acquired from the process during a dynamical test. In this case, no other source of knowledge is used. This paper is concerned with black and gray-box procedures using different model classes. It is often desirable to find parsimonious models with good static and dynamical responses [2]. The estimation of nonlinear models with such features is quite hard mainly because static and dynamic information are not equally weighed in a single set of data. In this respect static and dynamic information can be thought of as being conflicting. Flexible black-box structures are able to accurately fit a single piece of data. However there are two main drawbacks with most of such structures. First, once such models are estimated, the static information (e.g. static nonlinearity) is not readily available analytically. Second, not all such model structures and algorithms have been adapted to permit the effective use of static B. H. G. Barbosa is with the Departamento de Engenharia, Universidade Federal de Lavras, CP 3037, 37200-000, Lavras, MG, Brazil, email:[email protected]. L. A. Aguirre and A. P. Braga are with the Departamento de Engenharia Eletrônica, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, 31270-901, Belo Horizonte, MG, Brazil e-mail: {aguirre,apbraga}@cpdee.ufmg.br. C. B. Martinez is with the Departamento de Engenharia Hidráulica e Recursos H´ıdricos, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, 31270-901, Belo Horizonte, MG, Brazil e-mail:[email protected].

information during training (parameter estimation). It should be noticed that black-box identification does not necessarily guarantee correct steady-state performance when the model is nonlinear [3]. When the data sets are conflicting in some way, it is advisable to use multi-objective approaches which yield a set of optimal solutions called the Pareto set. Bi-objective algorithms have proved to be quite useful in combining both static and dynamic data during model identification [4]. This work aims to identify models of a 15 kW hydraulic pumping system. There has been a clear increase of variable frequency drives as the final control element for such systems. This has enabled the implementation of fast and automatic control systems. Models of such systems are highly desirable for characterization and control. Such models should, ideally, represent the system accurately both in transient and steadystate regimes over a wide range of operating conditions. This requires, more often than not, the use of nonlinear models. Because the aim of this work is to obtain models that perform well both in transient and steady-state regimes, different identification approaches were implemented to “guarantee” a good balance between such features. In order to improve the model steady-state performance, the measured static curve of the pumping system was used as auxiliary information. Such information was used in different intensities, depending on the model representation used. A novel bi-objective identification approach is presented and a new decision-maker is defined. In this paper we used and compared polynomial and neural NARMAX (Non linear Auto Regressive with Moving Average and eXogenous variables) models. This paper is organized as follows. In Section II, the hydraulic pumping system is briefly presented together with the measured static nonlinearity and the dynamical data. Section III quickly reviews the main points of the system modelling and a new proposed approach to estimate parameters of polynomial models regarding a priori information of the system were explained. In Section IV, NARMAX models identified without the use of prior knowledge are presented. A priori information about the system static curve was used in the parameters estimation task as also shown in Section IV. Finally, the conclusions of this work and future work are summarized in Section V. II. T HE H YDRAULIC P ROCESS AND THE DATA In a full-scale hydroelectric power plant (over 80% of Brazilian electrical energy is produced in such plants) the water head can be considered constant over reasonably long


2

24 22 20 18

Pressure (mlc)

periods of time. At testing plants, however, the turbines are fed by powerful hydraulic systems and not by a water head. Because of the characteristics of the centrifugal pumps used in such plants, the pressure on the turbine decreases as the water flow increases. Therefore, in realistic testing plants, pressure must be controlled over a wide range of operating conditions. Mathematical models are desired to simulate and to design the closed-loop control of the real pumping system, where the models output is the system pressure and the models input is the pumps reference speed. The hydraulic plant described in this section is composed by two centrifugal pumps that feed a hydraulic turbine. The hydraulic plant should be seen by the turbine as a water head. The static and dynamic data used in this paper were measured from this plant, composed by two centrifugal pumps coupled to induction motors of 7.5 kW and variable speed drive systems (Fig. 1). The pumps can be operated alone, in parallel or in a series configuration, always at the same speed. In this work, the pumps were set in a parallel configuration working at the same instantaneous speed with a Francis turbine as load [5]. The modelling data presented in this work were collected from a data acquisition system. The piezoresistive pressure transmitter error is ±0.175 mlc (meter of liquid column).

16 14 12 10 8 6 4 2 700

900

1100

1300

1500

Pump speed (rpm)

1700

Fig. 2. Static curve of the hydraulic pumping system, where (∗) is the measured data and the drawn curve is the second order polynomial approximation (1).

B. Dynamical data One important task that has to be developed during the identification process is the input signal selection as it can influence not only parameter estimation, but also structure selection in the case of nonlinear systems [6]. Since the presence of a “variable time-constant” in the pumping system dynamics was verified in an earlier work [5], the input signal was chosen to excite the system at different operating points using different step sizes. The sampling time Ts = 50 ms was selected according to the criterion defined in [7]. Examples of input-output data are shown in Fig. 3. In this work N = 3200 data points from the dynamical data set were used for model identification and N = 800 were used for validation. III. BACKGROUND A. NARMAX models The NARMAX model [8] can be represented by:

Fig. 1.

y(k) = F ! [y(k − 1), . . . , y(k − ny ), u(k − τd ), . . .

u(k − τd − nu ), e(k), e(k − 1), . . . , e(k − ne )],

Water pumping system.

A. Static behavior of the system The static curve of the system was measured by: i. setting the turbine distributor blade to 50%; ii. maintaining the pumps speed fixed at the chosen values – the speed references of both pumps were maintained the same during this procedure. After transients died out, the output pressure was recorded for each reference speed. During this test, the pumps speed was varied from 750 to 1650 rpm. The static curve is shown in Fig. 2 as well as the second order polynomial approximation: H(u) = βu2 + αu + κ,

where ny , nu and ne are the maximum lags considered for the output y(k), input u(k) and noise e(k), respectively, τ d is the delay and F is a non-linear function with nonlinearity degree % ∈ Z+ . The deterministic part of a polynomial NARMAX model (i.e. a NARX model) can be expanded as the summation of terms with degrees of nonlinearity in the range 0 ≤ m ≤ %. Each m-th order term can contain a p-th order factor in y(k − n i ) and a (m − p)-th order factor in u(k − n i ) and is multiplied by a coefficient c p,m−p (n1 , . . . , nm ) as follows y(k) =

(1)

with β = 7.2652 × 10−6 , α = 1.4933 × 10−3 , κ = −1.3312, and where H is the pressure in the output pipe and u is the steady-state pump speed. This static curve will be useful during the gray-box modelling and will also be used to evaluate the identified models.

(2)

× where

y ,nu ! ! m n! !

cp,m−p (n1 , . . . , nm )

m=0 p=0 n1 ,nm m "

i=p+1

p "

i=1

u(k − ni ) ,

ny ,nu

!

n1 ,nm

≡

ny ny ! !

n1 =1 n2 =1

y(k − ni ) (3)

···

nu !

nm =1

,

(4)


3

(a)

extended least-squares (ELS) estimator [13]. Other black-box models considered are the neural NARMAX model, a feedforward multilayer perceptron with weights estimated using the Leverberg-Marquardt algorithm available in Norgaard’s toolbox [14].

Pump speed (rpm)

1700

1500

1300

B. Gray-box Modelling

1100

900

700 0

10

20

time (s)

30

40

50

30

40

50

(b) 20

Pressure (mlc)

18 16 14 12 10 8 6 4 0

10

20

time (s)

Fig. 3. Dynamical data (a) pumps speed reference and (b) system output pressure.

and the upper limit is n y if the summation refers to factors in y(k − ni ) or nu for factors in u(k − n i ). Assuming stability, then in steady-state for constant inputs we may write y¯ = y(k − 1) = y(k − 2) = . . . = y(k − ny ), u¯ = u(k − 1) = u(k − 2) = . . . = u(k − nu ) and equation (3) can be rewritten as y ,nu ! !−m ! ! n! y¯ = cp,m (n1 , . . . , nm )¯ yp u ¯m , (5) m=0 p=0 n1 ,nm

#n ,nu where the constants ny1 ,nm cp,m−p (n1 , . . . , nm ) are the coefficients of the term clusters Ωyp um−p , which contain terms of the form y p (k − i)um (k − j) for m+p ≤ %. Such coefficients are called cluster coefficients and are represented as Σ yp um . If max[p] = 1 in the dynamical model (3), such a model is closely related to a Hammerstein type [9] and the steady-state output can be expressed as [10] #! Σ0 + Σu u¯ + m=2 Σum u¯m . (6) y¯ = #!−1 1 − Σy − m=1 Σyum u ¯m

Hence if the model has a term cluster of the form Ωyum , m = 1, 2, . . . , %, then the static function is rational, if not it is polynomial. The clusters coefficients are useful to write the models static functions and to implement the graybox modelling techniques as shown in the Section IV. In this paper, the model structure of the NARMAX polynomials are automatically chosen using the Error Reduction Ratio (ERR) criterion [11], [12]. In the context of black-box modeling the parameters of such models are obtained by the

In gray-box modelling, a priori information is used for model building. There are many ways to use prior knowledge about a system during its identification [15], [16]. In this paper, the steady-state data presented in Fig. 2 will be considered the prior knowledge and will be used in all gray-box approaches to estimate the model parameters. One simple and efficient way to use static data as auxiliary information in identification problems is the implementation of the constrained least squares (CLS) algorithm [17] following the procedure presented in [18]. The use of steady-state data to constrain parameter estimation by a total least squares algorithm, in the context of Takagi-Sugeno fuzzy models, has been recently discussed in [2]. Nonetheless, due to practical constraints, very often there is no unique solution that has the best dynamic and static response simultaneously [4]. In such cases, it is advantageous to define the identification procedure as a bi-objective optimization problem such as [19]: $ θˆ = arg min J(θ) θ (7) subject to: θ ∈ Rn , with J(θ) = [J1 (θ) JSF (θ)], where J1 and JSF are defined as the one step ahead (prediction) mean squared error and the static function mean squared error of the model, respectively. Should objective-functions be conflicting, instead of arriving at one solution, a set of solutions, namely Pareto-optimal solutions, is reached: ˆ J(θ) &= J(θ)}. ˆ (8) Θ = {θˆ ∈ Rn : &∃ θ ∈ Rn | J(θ) ≤ J(θ), Considering that both J 1 and JSF are convex functions, the weighted sum approach may be applied [20]. Thus, the bi-objective problem (7), namely a prediction error (PE) biobjective problem, can be rewritten as θˆ = arg min λ1 J1 (θ) + λ2 JSF (θ), θ

(9)

where λ1,2 ≥ 0, λ1 + λ2 = 1. Since the deterministic part of the NARMAX model (2) can be written as y = Ψθˆ + ξ,

(10)

where y ∈ RN and Ψ ∈ RN ×n is the regressors matrix, the PE bi-objective problem (9) can be solved using the following formulation [19]: θˆ =[λ1 ΨT Ψ + λ2 (QR)T (QR)]−1 × [λ1 ΨT y + λ2 (QR)T y],

(11)

where Q = [q1 . . . qNSF ], qi = [1 y i y 2i . . . y !i ui u2i . . . u!i Fyu ]T ,

(12)


4

where NSF is the number of available steady-state points (ui , yi ), Fyu represents the model non-linear terms that involve y and u, and R is a constant matrix of 0’s and 1’s that maps the parameter vector to the cluster coefficients such that the ˆ ˆ = QRθ. estimated static points are given by y Instead of using J 1 as objective-function of the bi-objective problem proposed in [19], another possible choice is the use of free-run simulation error function since this criterion may be seen as dynamically more representative and more robust than the one step ahead prediction error as discussed in some recent works [6], [21]–[23]. In this way, the new bi-objective problem can be defined as θˆ = arg min λ1 JS (θ) + λ2 JSF (θ), θ

Following the general guidelines pointed out in the previous paragraph, the NARMAX model with the best dynamical performance was: y(k) = θˆ1 y(k − 1) + θˆ2 y(k − 4) + θˆ3 u(k − 4)u(k − 6) + θˆ4 y(k − 2) + θˆ5 u(k − 2)y(k − 6) + θˆ6 u(k − 2)y(k − 5) + θˆ7 u(k − 2)y(k − 1) + θˆ8 u(k − 2)y(k − 3) + θˆ9 u(k − 4)u(k − 5) + θˆ10 u(k − 6) + θˆ11 + θˆ12 u(k − 6)2 + θˆ13 u(k − 2)y(k − 4) + θˆ14 y(k − 6)

+ θˆ15 u(k − 2)u(k − 5) + θˆ16 u(k − 2)2

(13)

where JS (θ) is defined as the free-run (simulation) mean squared error of a model with parameters set θ. We shall refer to (13) as the simulation error (SE) bi-objective problem. To solve (13), genetic algorithms (GAs) [24] were implemented using the stochastic universal sampling selection procedure [25], the heuristic crossover and gaussian mutation. Using (9) and (13), one arrives at Pareto sets. The next step is the decision stage, during which a model from the Pareto is selected. Procedures such as the minimal correlation criterion have been put forward in [4], [19]. In this paper a different procedure is suggested, based on the fact that the pressure transmitter does not have infinite precision. In other words, the measurement uncertainty will be taken into account in the decision stage. We therefore propose to choose the model from the Pareto set that satisfies: $ θˆ = arg min JS (θ) θ (14) DSF (Θ) ! ˆ ∈ B, subject to: JSF (θ) where B is uncertainty interval associated to the pressure transmitter. In this work B = [0 0.031] mlc 2 since the transmitter error limits are ±0.175 mlc. All criteria are based on the mean squared errors (MSE).

+ θˆ17 u(k − 4) +

2 ! j=1

θˆj∗ ξ(k − j) + ξ(k),

(15)

where θˆj∗ (j = 1, 2) are the parameters of the MA part. The parameters of the NARX part are θî (i = 1, . . . , 17) and the estimated values are shown in Table II. The steady-state relation of the estimated model can be given in terms of cluster coefficients as (see Eq. 6):

y

=

y

=

Σ0 + Σu u + Σu2 u2 (16) 1 − Σy − Σuy u −0.4770 + 8.925 × 10−4 u − 1.2727 × 10−7 u2 , 1 − 0.9264 − 2.7797 × 10−5 u

where the terms were grouped as: constant term (Σ 0 = −0.4770); linear terms in u (Σ u = 8.925 × 10−4 ); quadratic terms in u (Σu2 = −1.2727 × 10−7 ); linear terms in y (Σy = −0.9264); and cross-terms (Σ uy = −2.7797 × 10−5 ). The static function of this model is presented in Fig. 4. Table I presents a comparison among all models identified in this work.

IV. R ESULTS

24

A. Black-box modelling

22

All identified black-box models of this work are polynomial or neural NARMAX models. The performance of the estimated models was quantified using three different data sets: steady-state data, dynamical identification data and dynamical validation data. Regarding the pressure transmitter accuracy, if the MSE for a given model was less than (0.175) 2 mlc2 , then it was considered zero. Based on the static process characteristics, the degree of nonlinearity was chosen to be % = 2 (see Eq. 3). The maximum lags used were ny = 6, nu = 6 and ne = 2. These values define a large set of candidate model parameters. The terms to include in the models were automatically selected using the ERR criterion (Sec. III-A). The moving average (MA) part of the models was used to reduce bias during parameter estimation but was not further used in the simulations. Parameters were estimated using the extended least-squares estimator.

18

Pressure (mlc)

20

16 14 12 10 8 6 4 2 700

900

1100

1300

Pump speed (rpm)

1500

1700

Fig. 4. (∗) Steady-state measured data; (· · · ) static curve of model (15), which is Eq. 16 and (- -) static curve of model (17), which is Eq. 19.

Attempting to improve the performance of model (15), a more flexible model structure was allowed by increasing the degree of nonlinearity to % = 3. Proceeding as before, the


5

model with the best dynamical performance was: y(k) = θˆ1 y(k − 1) + θˆ2 u(k − 6)2 u(k − 4) + θˆ3 y(k − 4) + θˆ4 u(k − 4)u(k − 2) + θˆ5 y(k − 2) + θˆ6 y(k − 3) + θˆ7 y(k − 6) + θˆ8 y(k − 5) + θˆ9 u(k − 5)y(k − 1) + θˆ10 u(k − 5)y(k − 3)y(k − 1)

+ θˆ11 u(k − 4)u(k − 2)y(k − 1) + θˆ12 y(k − 2)2 y(k − 3) + θˆ13 u(k − 5)3

+ θˆ14 u(k − 5)2 y(k − 2) + θˆ15 u(k − 5)y(k − 2)2 + θˆ16 u(k − 5)2 + θˆ17 y(k − 2)2 + θˆ18 u(k − 5)y(k − 2)

+ θˆ19 u(k − 5) + θˆ20 u(k − 6)y(k − 6)2 + θˆ21 u(k − 2)y(k − 1)y(k − 4) + θˆ22 u(k − 2)2 y(k − 6) + θˆ23 u(k − 2)u(k − 4)y(k − 4)

+

2 ! j=1

θˆj∗ ξ(k − j) + ξ(k),

(17)

where θˆj∗ (j = 1, 2) are the parameters of the MA part – not used in the simulations –, and the other parameters θî (i = 1, . . . , 23) are shown in Table III. Using the ERR it is possible to produce several models similar to (17). This model was chosen because it had the smallest mean squared error over dynamical data. A price to be paid for the additional cubic terms in model (17) is that such a model has now three stable equilibria, which are solutions of the following expression: y¯ =Σy3 y¯3 + Σy2 y¯2 + Σy y¯ + Σy2 u y¯2 u ¯ + Σyu2 y¯u ¯2 + Σyu y¯u ¯ + Σu3 u ¯3 + Σu2 u ¯2 + Σu u¯.

(18)

Fortunately, as it usually happens, only one of the three possible solutions falls within the range of the steady-state data (Fig. 2) and the choice of the equilibrium is not ambiguous. The equilibrium can be found using the trigonometric solution to cubic polynomials [26] as: ) *   −ς % √ 3 arccos + 4π   δ 2 (−$ /27 −)  − , (19) cos  y=2   3 3 3

where

)

=

ς

=

δ

=

γ

=

λ =

δ2 , 3 2δ 3 − 9δγ , λ+ 27 −Σy2 − Σy2 u u , −Σy3 1 − Σy − Σyu u − Σyu2 u2 , −Σy3 −Σu u − Σu2 u2 − Σu3 u3 , −Σy3 γ−

(20)

with constants Σy = 1.0299; Σ u = -1.4475 × 10 −3 ; Σy2 = 2.6151 × 10 −2 ; Σu2 = 5.6331 × 10 −6 ; Σy3 = 4.3873 × 10 −4 ;

Σu3 = -3.4869 × 10 −9 ; Σyu = -7.4151 × 10 −4 ; Σy2 u = 3.4620 × 10 −5 ; Σyu2 = 6.5291 × 10 −7 . The static curve of model (17) is given by Eq. 19 and is shown in Fig. 4. Model (17) outperforms model (15) in all criteria shown in Table I. As a final black-box modelling approach, a neural NARMAX model was identified using the function nnnarmax2.m available in Norgaard’s toolbox [14]. The implemented neural network has the following features: n u = ny = 6, ne = 2, seven non-linear nodes in the hidden layer with hyperbolic tangent activation function and one linear node in the output layer. The neural network weights were adjusted over 60 epochs. As shown in Table I, the black-box neural model outperformed the polynomial black-box models in all criteria, at the cost of having more than four times the number of parameters. Besides, a closed expression for the network static function is not easily obtained. B. Gray-box modelling Although the neural model and the polynomial model (17) arrived at better static performance than model (15), the latter has a simple static function (16), that could prove useful in controller design– ∂ y¯/∂ u ¯ gives the process gain as a function of the input. Therefore, in what follows the steady-state information about the process will be imposed on the deterministic part of model (15) and the parameters will be reestimated. From a parameter bias point of view, the constraints will play the role of the MA part – which will not be used in gray-box modeling – with the added advantage of imposing the desired steady-state behavior. The procedure presented in [18] will be used to improve the steady state performance of model (15) by imposing that the model static function Eq. 16 should match the polynomial approximation of the process static function (1) thus Σ0 + Σu u + Σu2 u2 = β u2 + α u + κ. 1 − Σy − Σuy u

(21)

One possible solution to this equality yields the following set of parameter constraints: Σyu

=

0,

Σy Σ0

= =

Σu Σu2

= =

Σ∗y , (1 − Σ∗y )κ,

(1 − Σ∗y )α, (1 − Σ∗y )β,

(22)

where Σ∗y is the sum of the linear terms in y of (15). The set of constraints (22) was used in parameter estimation as suggested in [18]. The resulting model thus estimated will have the specified static function by design. Its estimated parameters are shown in Table II (see the CLS column). As it can be seen from Table I and Fig. 5, with the use of constraints taken from the static curve, the steady state performance of the resulting model improved at a rather acceptable loss of dynamical performance. In this case, an interesting approach is the implementation of a bi-objective optimization problem. As suggested by [19], the conflicting objective-functions might be the one step ahead


parameters of this model are shown in Table II. 12 10 8

JSF(e)

PE

6 4 ELS

2 SE

CLS

0 1.8

2

2.2

2.4

2.6

2.8

3

J (e)

3.2

3.4

3.6

S

Fig. 5. Pareto sets of the bi-objective parameters estimation approaches (9) and (13) using the structure given by (15) evaluated (MSE) on the identification data. The models with parameters estimated by ELS and CLS are indicated by square and cross respectively, whereas the dense set of asterisks and circles indicate those obtained by PE (9) and SE (13), respectively.

22 20 18

Pressure (mlc)

prediction error, J 1 , and the static function error, J SF (the PE bi-objective problem). So, a PE Pareto set composed by 100 models with the same structure given by (15) without the MA part was generated using (11), where λ 1 ∈ [10−6 , 1] in a logarithmic scale. This scale was used to obtain a representative Pareto set. Figure 5 presents the achieved PE Pareto optimal solutions using the prediction error bi-objective approach (9). The models were evaluated by the J S and JSF objective-functions. The parameters were estimated as in [19] using J 1 and JSF as objective-functions. Comparing the PE Pareto optimal solutions obtained by (9) with the gray-box model reached by CLS using the constraints in (22), some Pareto solutions arrived at better performance in both objective-functions, although the improvement in terms of JSF is marginal compared to CLS. Therefore, the PE biobjective approach may be seen as a better and more general problem solver than the CLS solution, although the latter demands less computing time. Comparing the PE Pareto optimal solutions and the ELS solution, it can be seen (Fig. 5) that the latter achieved the best dynamical performance, however, at the cost of having a higher static function error. As expected, a steadystate improvement can be gained using the PE bi-objective approach. After finding the Pareto set the decision-making approach D SF was implemented as described in Section III-B. The performance of the PE selected model and its estimated parameters are shown in Table I and Table II, respectively. Although this model has an improved static function, its dynamic response is worse than the ELS solution to (15) in the identification and validation data sets. Instead of using J 1 in the bi-objective problem proposed in [19], other possible choice is the use of J S as discussed in Section III-B (the SE bi-objective problem). In this way, the optimisation problem (13) was solved by GAs, where λ 1 ∈ [10−2 , 1] in a logarithmic scale. The SE Pareto solutions (100) were obtained using the following GAs parameters: population size of 200 individuals, crossover probability of 0.9 (ratio 1.2) and mutation probability of 0.1 (standard deviation of 5% of the variable value). The GAs were run 100 times (one for each weight pair (λ1 , λ2 ), starting from λ 1 = 1 to λ1 = 10−2 ) and the solution of each previous run was included into the initial population of the algorithm together with a random population. The random population models (individuals) parameters were defined using the least squared estimator applied to 30 randomly selected samples of the dynamic data. The achieved Pareto optimal solutions are also shown in Fig. 5. Comparing SE Pareto set with the other approaches (see Fig. 5), it is clear that it outperformed all other solutions in both objectives. The SE bi-objective approach proposed in this work arrived at a solution set that dominates, in a Pareto optimality sense, all solutions obtained by the blackbox and other gray-box modelling approaches, however at a high computing cost. Table I shows the performance of the SE Pareto set model selected by the decision-maker D SF and Fig. 6 presents its free-run simulation on the validation data set. The estimated

6

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time (s)

25

30

35

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Fig. 6. Free-run simulation of (—) the system output pressure and (- -) the polynomial model with ! = 2 and with parameters estimated by the SE biobjective approach (13) and selected by the decision-maker DSF .

The same bi-objective approaches (and algorithm parameters) applied to the deterministic part of the NARMAX model (15), % = 2, were also applied to the deterministic part of the NARMAX model (17), % = 3, in order to further improve the static performance of the black-box model. Table III presents the estimated parameters of the bi-objective approaches selected models. Figure 7 shows the identification performance criteria of the ELS solution to (17) and the two Pareto sets (SE and PE bi-objective approaches). As in the aforementioned results (% = 2), the Pareto set obtained by means of the SE biobjective approach dominates all models estimated by the other methodologies, including the PE bi-objective approach. It is also interesting to note that comparing the solutions obtained by a pure SE method and a pure PE method, the former provided a model with much better static performance (see the Pareto sets in the Fig. 5 and 7). The free-run simulation of the SE Pareto set model selected by the decision-maker D SF from the simulation error Pareto


7

set is shown in Fig. 8 (see Table I). The selected model achieved better static function and dynamic performance in the identification data than the ELS solution to (17) but was worse over the dynamic validation data set. 0.5

0.4

JSF(e)

PE

0.3

0.2

SE

0.1

ELS

0 1

1.2

1.4

1.6

1.8

2

JS(e)

2.2

2.4

2.6

Fig. 7. Pareto sets of the bi-objective parameters estimation approaches (9) and (13) using the structure given by (17) evaluated (MSE) on the identification data. The model with parameters estimated by ELS is indicated by square, whereas the dense set of asterisks and circles indicate those estimated by PE (9) and SE (13), respectively.

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18 16 14 12 10 8 6 4 0

5

10

15

20

time (s)

25

30

35

40

Fig. 8. Free-run simulation of (—) the system output pressure and (- -) the polynomial model with ! = 3 and with parameters estimated by the SE biobjective approach (13) and selected by the decision-maker DSF . TABLE I C OMPARISON OF THE IDENTIFIED MODELS . W HERE Np IS THE NUMBER OF PARAMETERS OF EACH MODEL . MSE LESS THAN 0.031 mlc2 SHOULD BE CONSIDERED ZERO DUE TO THE UNCERTAINTY INTERVAL ASSOCIATED TO THE PRESSURE TRANSMITTER .

Model ELS (15) CLS (22) PE (DSF ) SE (DSF ) ELS (17) PE (DSF ) SE (DSF ) Neural

MSE (mlc2 ) JS (Ident.) JS (Val.) 2.611 2.116 3.221 2.909 2.632 2.389 2.173 1.751 1.510 1.104 2.296 1.857 1.208 1.306 1.09 0.536

Np JSF 1.6040 0.0019 0.0009 0.0259 0.0603 0.0012 0.0291 0.0496

17 17 17 17 23 23 23 99

V. C ONCLUSIONS The use of a priori information to identify non-linear systems is usually justified when the system is not well represented in all operating points by the available dynamical data sets, which often occurs in practical situations. For instance, [3] show that information about the static curve of a system can be useful during the dynamic model identification process when this information is not completely available in the dynamic data. Nevertheless, in this paper measured static curves and dynamic data were used even though the dynamic data set might supply by itself enough information to arrive at models with good approximation of the static curve of the system. Thus, these data sets could be seen as carrying redundant information. This paper addressed the problem of identification of nonlinear systems using different methods that use auxiliary information in various degrees. Using data from a 15 kW pumping plant, it was shown that steady-state information and free-run simulation error criteria can be useful during the identification process. In this paper a novel multi-objective approach to system identification was proposed: it uses the static curve as the additional source of information and the simulation error criterion instead of the prediction error criterion. Besides, a new decision-maker that takes into account the measurement uncertainty was also introduced. This approach arrived at models with better static curve and dynamic response, being possible to find a model that outperformed the black-box counterpart in the dynamic and static performance criteria. As far as the simulation error bi-objective approach is concerned, it would be interesting to develop an algorithm to find the Pareto set without having to use the free-run simulation which is very computationally demanding. In spite of its high computational cost, it is also desired in future work to apply the simulation error criterion to detect the model structure of the process studied in this work as in [6]. ACKNOWLEDGEMENTS The authors gratefully acknowledge financial support by CNPq. R EFERENCES [1] J. Sjöberg, Q. Zhang, L. Ljung, A. Beneviste, B. Delyon, P. Glorennec, H. Hjalmarsson, and A. Juditsky, “Non-linear black-box modeling in system identification: A unified overview,” Automatica, pp. 31–1961, 1995. [2] S. Jakubek, C. Hametner, and N. Keuth, “Total least squares in fuzzy system identification: An application to an industrial engine,” Engineering Applications of Artificial Intelligence, vol. 21, pp. 1277–1288, 2008. [3] L. A. Aguirre, P. F. Donoso-Garcia, and R. Santos-Filho, “Use of a priori information in the identification of global nonlinear models — A case study using a Buck converter,” IEEE Trans. Circuits Syst. I, vol. 47, no. 7, pp. 1081–1085, 2000. [4] M. S. F. Barroso, R. H. C. Takahashi, and L. A. Aguirre, “Multi-objective parameter estimation via minimal correlation criterion,” Journal of Process Control, vol. 17, no. 4, pp. 321–332, 2007. [5] B. H. Barbosa, “Instrumentation, Modelling, Control and Supervision of a Hydraulic Pumping System and Turbine–Generator Module,” Master’s thesis, Federal University of Minas Gerais, School of Electrical Engineering, 2006, (in Portuguese).


[6] L. Piroddi and W. Spinelli, “An identification algorithm for polynomial NARX models based on simulation error minimization,” International Journal of Control, vol. 76, no. 17, pp. 1767–1781, 2003. [7] L. A. Aguirre, “A nonlinear correlation function for selecting the delay time in dynamical reconstructions,” Physics Letters, vol. 203A, no. 2,3, pp. 88–94, 1995. [8] I. J. Leontaritis and S. A. Billings, “Input-output parametric models for non-linear systems Part II: Deterministic non-linear system,” International Journal of Control, vol. 41, no. 2, pp. 329–344, 1985. [9] L. A. Aguirre, M. C. S. Coelho, and M. V. Corrêa, “On the interpretation and practice of dynamical differences between Hammerstein and Wiener models,” Proc. IEE Part D: Control Theory and Applications, vol. 152, no. 4, pp. 349–356, 2005. [10] M. V. Corrêa, L. A. Aguirre, and R. R. Saldanha, “Using steady-state prior knowledge to constrain parameter estimates in nonlinear system identification,” IEEE Trans. Circuits Syst. I, vol. 49, no. 9, pp. 1376– 1381, september 2002. [11] S. A. Billings, S. Chen, and M. J. Korenberg, “Identification of MIMO nonlinear systems using a forward-regression orthogonal estimator,” International Journal of Control, vol. 49, no. 6, pp. 2157–2189, 1989. [12] S. Chen, S. A. Billings, and W. Luo, “Orthogonal least squares methods and their application to nonlinear system identification,” International Journal of Control, vol. 50, no. 5, pp. 1973–1896, 1989. [13] S. A. Billings and W. S. F. Voon, “Least squares parameter estimation algorithms for nonlinear systems,” Int. J. Syst. Sci., vol. 15, no. 6, pp. 601–615, 1984. [14] M. Norgaard, “Neural network based system identification - TOOLBOX,” Thecnical University of Denmark, Thecnical report 97-E-851, 1997. [15] T. Abdelazim and O. Malik, “Identification of nonlinear systems by takagi-sugeno fuzzy logic grey box modeling for real-time control,” Control Engineering Practice, vol. 13, no. 12, pp. 1489 – 1498, 2005. [16] C. Ghiaus, A. Chicinas, and C. Inard, “Grey-box identification of airhandling unit elements,” Control Engineering Practice, vol. 15, no. 4, pp. 421 – 433, 2007. [17] N. R. Draper and H. Smith, Applied Regression Analysis, 3rd ed. New York: Wiley, 1998. [18] L. A. Aguirre, M. F. S. Barroso, R. R. Saldanha, and E. M. A. M. Mendes, “Imposing steady-state performance on identified nonlinear polynomial models by means of constrained parameter estimation,” Proc. IEE Part D: Control Theory and Applications, vol. 151, no. 2, pp. 174–179, 2004. [19] E. G. Nepomuceno, R. H. C. Takahashi, and L. A. Aguirre, “Multiobjective parameter estimation: Affine information and least-squares formulation,” International Journal of Control, vol. 80, no. 6, pp. 863– 871, 2007. [20] V. Chankong and Y. Y. Haimes, Multiobjective decision making: theory and methodology. New York: North-Holland (Elsevier), 1983. [21] P. Connally, K. Li, and G. W. Irwing, “Prediction- and simulation-error based perceptron training: Solution space analysis and a novel combined training scheme,” Neurocomputing, vol. 70, pp. 819–827, 2007. [22] L. Piroddi, “Simulation error minimization methods for NARX model identification,” Int. J. Modelling, Identification and Control, vol. 3, no. 4, pp. 392 – 403, 2008. [23] Y. Pan and J. H. Lee, “Modified subspace identification for long-range prediction model for inferential control,” Control Engineering Practice, vol. 16, no. 12, pp. 1487 – 1500, 2008. [24] D. E. Goldberg, Genetic algorithms in search, optimization and machine learning. New York: Addison-Wesley, 1989. [25] J. E. Baker, “Reducing bias and inefficiency in the selection algorithm,” in Proceedings of the Second International Conference on Genetic Algorithms on Genetic algorithms and their application. Mahwah, NJ, USA: Lawrence Erlbaum Associates, Inc., 1987, pp. 14–21. [26] D. Zwillinger, Standard Mathematical Tables And Formulae, 31st ed. Chapman & Hall/CRC, 2002.

VI. A PPENDIX The greatly different orders of magnitude of the estimated parameters deserves a remark. It must be realized that the parameters multiply regressor variables which, in this case, are usually nonlinear. A large average value of a variable that appears to the cubic power will require a much smaller parameter value to compensate. For all the models in this

8

TABLE II I DENTIFIED MODELS PARAMETERS (l = 2), MODEL STRUCTURE PRESENTED IN E Q . 15. T HE MA TERMS PARAMETERS ARE NOT SHOWN .

θˆ1 θˆ2 θˆ3 (×10−7 ) θˆ4 θˆ5 (×10−4 ) θˆ6 (×10−4 ) θˆ7 (×10−4 ) θˆ8 (×10−4 ) θˆ9 (×10−6 ) θˆ10 (×10−3 ) θˆ11 θˆ12 (×10−6 ) θˆ13 (×10−4 ) θˆ14 θˆ15 (×10−7 ) θˆ16 (×10−7 ) θˆ17 (×10−3 )

ELS 0.6303 -0.0264 0.0608 0.4388 2.3013 -1.6981 2.4569 -1.5936 0.5191 1.2475 -0.4770 -0.4134 -1.1884 -0.1163 -3.7493 1.3597 -0.3549

CLS (22) 0.9667 -0.3122 6.8554 0.2219 -0.5289 -0.6784 -0.8580 0.0761 -0.4136 0.1051 -0.0980 -0.1309 1.9893 0.0526 6.1912 -2.2525 0.0048

PE (DSF ) 1.0113 -0.1258 7.6177 0.1603 0.8248 -0.7551 0.1681 -0.4746 -0.6799 -0.5049 0.0767 -0.1136 0.1347 -0.0739 8.5091 -3.3413 0.2257

SE (DSF ) 0.6011 0.8246 0.3383 0.1005 4.6482 0.5876 5.1472 -1.8397 -1.1277 -0.3730 -0.3355 -1.5104 -8.3407 -0.5735 -9.9186 0.0097 1.1591

TABLE III I DENTIFIED MODELS PARAMETERS (l = 3), MODEL STRUCTURE PRESENTED IN E Q . 17. T HE MA PARAMETERS ARE NOT SHOWN .

θˆ1 θˆ2 (×10−11 ) θˆ3 (×10−1 ) θˆ4 (×10−7 ) θˆ5 θˆ6 (×10−1 ) θˆ7 (×10−1 ) θˆ8 θˆ9 (×10−4 ) θˆ10 (×10−5 ) θˆ11 (×10−7 ) θˆ12 (×10−4 ) θˆ13 (×10−9 ) θˆ14 (×10−7 ) θˆ15 (×10−5 ) θˆ16 (×10−6 ) θˆ17 (×10−2 ) θˆ18 (×10−4 ) θˆ19 (×10−3 ) θˆ20 (×10−6 ) θˆ21 (×10−6 ) θˆ22 (×10−8 ) θˆ23 (×10−8 )

ELS 0.7418 5.8941 -0.8754 -2.0096 0.5012 -0.2666 0.3065 -0.1296 0.9340 -0.5657 0.6199 4.3873 -3.5459 0.0601 -2.6161 5.8340 2.6150 -8.3492 -1.4475 1.0767 -3.8788 1.8040 -2.8455

PE (DSF ) 1.0122 5.3573 0.4728 -2.6422 -0.0486 -0.6566 -0.0946 -0.0599 -0.9738 -0.5847 1.3015 3.8520 -1.0059 1.0050 -0.4154 1.5441 0.3420 0.8918 -0.0916 0.5628 -4.7493 2.4458 -9.6289

SE (DSF ) 1.3356 0.9251 -3.4779 1.3142 0.0256 1.6692 2.7295 -0.3300 -4.1309 -1.1723 0.5955 1.4885 -3.0426 5.7248 -1.5776 5.2165 2.2269 -3.4293 -1.7539 2.5579 0.3530 -0.2781 -6.5877

paper, the contribution of each term multiplied by the respective parameter is of the same order of magnitude. One way of avoiding this situation is to normalize the data. This was not done in this paper in order to maintain the engineering units and therefore to facilitate a physical interpretation of the simulated data.