INTRODUCTION
The stringent environmental and safety requirements and the growing competition in the global market have put a tremendous challenge on process industries. On the other hand, the rapid development in computer and software technology, the advancement of instrumentation and data acquisition facilities and the sustained achievements in the formulation of efficient computational algorithms have brought incredible opportunities. The meeting of these strong challenges and resourceful opportunities has led to the birth of several model based technologies, like model based control systems, online process optimizations and fault detection and diagnosis, to mention a few. At the centre of all these technologies is the mathematical model of the system.
Models are extensively used in advanced process control design and implementations.
Nearly all optimal control design techniques rely on the use of model of the
system to be controlled. In model predictive control (MPC), models are used
to predict the future values of the output which is used in calculating the
optimal control move. The process of developing models from experimental data
is known as system identification. Currently, there is a significant interest
towards orthonormal basis filter (OBF) models because of their advantage both
during the development and implementation stages of the models. They are parsimonious
in their parameters, the parameters can be easily calculated using linear least
square method, they are consistent in parameters and time delays can be easily
estimated and incorporated into the model (Van den Hof et
al., 2005; Heuberger et al., 1995; Ninness
and Gustafsson, 1997; Patwardhan and Shah, 2005;
Nelles, 2001).
Control relevant system identification can be carried out using inputoutput
data either from openloop or closedloop tests. When a system identification
test is carried out in open loop, in most cases, the noise sequence is not correlated
to the input sequence and OBF model identification is carried in a straight
forward manner. However, when the system identification test is carried out
from closed loop data the input sequence is correlated to the noise sequence
(Nelles, 2001; Ljung, 1999; Gilson
and van den Hof, 2005). In such cases, if conventional OBF model development
technique (Nelles, 2001) is used, the resulting model
will not be consistent in parameters (Nelles, 2001; Ljung,
1999; Gilson and van den Hof, 2005; Gaspar
et al., 1999; Ananth and Chidambaram, 1999).
In addition, conventional OBF structures do not include explicit noise model
and any unmeasured disturbance cannot be modeled. There are several reasons
to prefer to conduct the identification tests in closedloop. Some of the most
compelling reasons are (Nelles, 2001; Ljung,
1999):
• 
Feedback controller is required to stabilize the process 
• 
Safety and cost consideration may not allow the process to run openloop 
• 
The model is to be used for the design of improved controller 
The Orthonormal Basis Filter plus Autoregressive Moving Average with Exogenous
Input (OBFARMAX) structure proposed in this study leads to consistent deterministic
model with explicit noise model.
ORTHONORMAL BASIS FILTER MODELS
The basis of OBF models is the orthonormal filters. There are several OBF filters allowing one or more of the system poles to be introduced into the model. The most commonly used filters are Laguerre, Kautz and Generalized Orthonormal Basis Filters (GOBF). They are defined as follows:
Laguerre:
Kautz:
Where:
1 < a < 1 and 1 < b < 1 n = 1, 2, …
GOBF:
where, p ≡ {p_{j }: j = 1, 2, 3, …} is an arbitrary sequence
of poles inside the unit circle appearing in complex conjugate pairs.
Lemma et al. (2010) showed that a Box Jenkins
(BJ) type structure with the OBF deterministic component and an Autoregressive
(AR) or Autoregressive Moving Average (ARMA) component can be easily developed.
They demonstrated that both weaklydamped and welldamped systems with unmeasured
disturbances can be easily modeled using the proposed model structure. The OBFAR
and the OBFARMA structures proposed by Lemma et al.
(2010) are given by Eq. 4 and 5, respectively.
In AIChE conference, Lemma et al. (2009) showed
that OBF based process models with explicit noise model can be successfully
identified from closedloop identification data. Two structures were proposed;
one for indirect identification and another for direct identification. The indirect
identification was based on decorrelation of the input and noise sequences.
The structure used in the indirect identification was OBFARMA. The direct identification
method was based on the fact that the OBFARX structure assumes the input and
noise sequences are correlated and leads to model that is consistent in parameters.
The OBFARX structure is given in Eq. 6.
In this study, OBFARMAX structure that can be used for direct closedloop identification of both openloop stable and openloop unstable processes that are stabilized by feedback controllers is proposed. It provides more flexible noise model than the OBFARX model.
Present work: The proposed OBFARMAX structure has a common denominator
dynamics for the input and noise model and is represented by Eq.
7. Figure 1 shows the block diagram representation of
the OBFARMAX structure. As it is pointed out by Ljung (1999),
such a model results in a stable predictor even though the system is unstable.
This renders the proposed structure the ability to capture the dynamics of both
openstable and openunstable systems that are stabilized by a feedback controller.
The OBFARMAX structure confers more flexible noise model than the OBFARX structure.

Fig. 1: 
Blockdiagram of a OBFARMAX model 
The algorithm for estimating the model parameters and the multistep ahead
prediction are formulated in the following parts.
The OBFARMAX structure is given by:
Model parameters estimation:
Rearranging Eq. 7,
With A(q) and C(q) monic, expanding Eq. 8,
From Eq. 9 the regressor matrix is formulated for orders m, n, p:
Where:
m 
: 
Order of the OBF model 
n 
: 
Order of the A(q) 
p 
: 
Order of C(q) 
mx 
: 
Max ( n, m, p) + 1

u_{fi} 
: 
Input u filtered by the corresponding OBF filter f_{i} 
e(i) 
: 
The prediction error 
To develop an OBFARMAX model, first an OBFARX model with high A(q) order
is developed. The prediction error e (k) is estimated from this OBFARX model
and used to form the regressor matrix (Eq. 10). OBFARX model
development is discussed in detail in (Lemma et al.,
2009). The parameters of the OBFARMAX model are, then, estimated using
the regressor matrix (Eq. 10) in the least square Eq.
11. The prediction error and consequently the OBFARMAX parameters can be
improved by estimating the parameters of the OBFARMAX model iteratively.
Multi –step ahead prediction formula: Consider the OBFARMAX model Eq. 12:
The istep ahead prediction is obtained by replacing k with k + i:
To calculate the istep ahead prediction, the error term should be divided into current and future parts:
Since e(k) is assumed to be a white noise with mean zero, the mean of E_{i}(q) e(k+i) is equal to zero and therefore Eq. 14 can be simplified to:
Rearranging Eq. 15:
Comparing Eq. 13 to 16, F_{i}
and E_{i} can be calculated by solving the Diophantine equation:
Rearranging Eq. 12:
Using Eq. 18 in 15 to eliminate e(k):
Rearranging the Diophantine Eq. 17:
Finally using Eq. 20 in 19, the usable
form of the istep ahead prediction formula, Eq. 21 is obtained:
When OBFARMAX model is used for modeling openloop unstable processes that are stabilized by a feedback controller, the common denominator A(q) that contains the unstable pole does not appear in the predictor Eq. 21. Therefore, the predictor is stable regardless of the presence of unstable poles in the OBFARMAX model, as long as the noise model is invertible. Inevitability is required because C(q) appears in the denominator.
CASE STUDIES
Closed loop identification of openloop stable systems using OBFARMAX model:
In this closedloop identification simulation case study, an OBFARMAX model
of an openloop stable system is identified from closedloop test data using
the direct identification approach. The deterministic and stochastic components
of the system are given by Eq. 22a and b,
respectively:
A feedback proportional controller, with K_{c}=1.0 is used to control
the system. The controller gain is chosen so that the closed loop response is
stable and gives not more than 25% overshoot. The block diagram of the feedback
controlled system is shown in Fig. 2. A white noise sequence
with mean 0.0070 and standard deviation 0.0993 is introduced into the system.
The Signal to Noise Ratio (SNR) is 6.7350. An external excitation signal, r_{1},
is used for the purpose of identification.
The excitation signal, r_{1}, is a ‘PRBS’ signal generated using the MATLAB function ‘idinput’ with band (0 0.02) and level ( 2 2). Four thousand data points are generated and 3000 of these data points are used for identification while the remaining 1000 data points are used for validation. The changes in the external excitation signal, r_{1}, system input, u(k) and system output, y(k), are shown in Fig. 3.
OBFARMAX model
Pole and number of parameters selection: The dominant pole method
to develop parsimonious OBF models is not applicable for OBFARMAX structure
since the two structures are different. However, the best pole and number of
OBF parameters can be estimated by comparing the Percentage Prediction Error
(PPE) of various poles and number of OBF parameters. The results of the comparison
are presented in Table 1. From Table 1 it
is observed that the minimum PPE for OBF4 (the most parsimonious among the
tested) is 6.7929 while the smallest PPE in all the tabulated values is 6.7440
for OBF7.

Fig. 2: 
Block diagram of the closed loop system 

Fig. 3: 
Data used for system identification of system Eq.
22 

Fig. 4: 
Output of the simulation model compared to the output of system
(Eq. 22) for the validation data points 
Table 1: 
PPE for various poles and number of OBF parameters for n_{D}
= n_{C} = 2 of system (Eq. 22) 

The difference between the two percentage prediction errors is less
than 0.05% which is insignificant and OBF4 is selected.
The OBF parameters for 4 Laguerre filters and pole of 0.90 is:
l = (0.0068 0.0079 0.0010 2.5302e004) 
The denominator polynomial A(q):
The noise model is:
Model validation: The output of the deterministic component the OBFARMAX
model compared to the system’s output for the validation data points is
shown in Fig. 4. The PPE of the simulation model compared
to the systems output is 16.7642. The spectrum of the noise model (Eq.
22b) compared to the spectrum of the system’s noise transfer function
(Eq. 23) is shown in Fig. 5. The standard
deviation of the residuals of the OBFARMAX model is 0.0992. The PPE of the
spectrum of the noise model compared to the system’s noise transfer function
is 2.3659.
The one stepaheadprediction using the OBFARMAX model compared to the system’s
output for the validation data points (30013500) is shown in Fig.
6 and the PPE is 6.7929.

Fig. 5: 
Spectrum of the noise model compared to the noise transfer
function of system (Eq. 22) 

Fig. 6: 
Onestepahead prediction of the OBFARMAX model compared
to the system output for the validation data points for the system (Eq.
22) 
In this case, only 500 points are presented to show the closeness of the system
output and the predicted data more clearly.
Residual analysis: The qqplot and of the residuals with respect to the white noise added to the system is shown in Fig. 7. It is observed from the figure that almost all the points on the qqplot lie on a straight line with slope equal to one. This shows that the residuals have nearly the same distribution as the white noise. Figure 8 shows the distribution of the residuals compared to the white noise. It is noted from the figure that the two distributions are nearly the same.
The correlation among the residuals for τ =10:
The correlation among the residuals which is close to zero also shows that
there is no significant correlation among the residuals and the residuals can
be considered white noise.

Fig. 7: 
qqplot of the residual compared to the white noise for system
( Eq. 22) 

Fig. 8: 
Distribution of the residuals compared to the white noise
for the system ( Eq. 22) 
This confirms that the OBFARMAX model has captured the model with good accuracy.
Closeloop identification of openloop unstable process: In this case
study, an OBF model with ARMAX structure is used to identify an openloop unstable
process which is stabilized by a feedback control system. The plant and noise
transfer functions of the system are given by Eq. 24a and
b:
The plant transfer function has one RHS pole, 1/15, therefore is openloop
unstable.

Fig. 9: 
System stabilized by feedback control system 

Fig. 10: 
Data sequences used for identification of system (Eq.
24) 
The system is stabilized using a proportional feedback controller
with K_{c} = 11. A white noise sequence with mean 0.0049 and standard
deviation 0.1989 is added to the system and the SNR is 9.9702. An external excitation signal, r_{1}, is introduced into the system
to conduct the identification. The excitation signal is, a PRBS signal generated
using the MATLAB function ‘idinput’ with band (0 0.02) and level of
(1 1). The block diagram of the feedback control system to be identified is
shown in Fig. 9.
The excitation signal, the plant input and the plant output used for identification are shown in Fig. 10. Four thousand data points, with a sampling interval of one time unit, are generated using SIMULINK and 3000 of them were used for identification and the remaining 1000 are used for validation.
Pole and number of parameters selection: The same procedure as the previous case is used to determine the number of OBFparameters and the OBFpole. OBF model with four numbers of parameters and pole equal to 0.7 is chosen to develop the OBFARMAX model (Table 2).
OBFARMAX model: The OBF model has four Laguerre filters with pole 0.4
and parameters:
l = (0.0028 0.0014 2.9540e004 9.5944e004) 
Table 2: 
PPE for various poles and number of OBF parameters for n_{D}
= n_{C }= 2 for system (Eq. 24) 

The denominator polynomial A(q):
The noise model is:
The discrete poles of the noise model that are also shared by the plant model are 1.0590 and 0.3106. It is observed that one of the poles, 1.0590, is outside the unit circle hence it is the unstable pole shared by the plant model and the noise model as the theory requires. Note that the poles of the noise model are shared by the plant model as defined by (Eq. 7).
Model validation: The onestepahead prediction by the OBFARMAX model compared to the output of the stabilized system for the validation data points is shown in Fig. 9 and the corresponding PPE is 6.1931.
The simulation model and the noise spectrum are irrelevant for such cases, because both are unstable. The accuracy of the model can be checked by residual analysis, as in the case of the OBFARX.
Residual analysis: The mean and standard deviation of the residuals
of the OBFARMAX model are 0.0026 and 0.2160. The qqplot of the residual of
the OBFARMAX model with respect to the white noise added to the system for
the validation data points are shown in Fig. 10. The distribution
of the residuals of the OBFARMAX model compared to the white noise added to
the system is shown in Fig. 11. It is noted from the figures
that, the residual is a normally distributed signal with mean around zero, similar
to the white noise. However, it can also be observed that there is small deviation
at the intercept qqplot.

Fig. 11: 
Onestepahead prediction of the OBFARMAX model compared
to the output of the system for the validation data points 

Fig. 12: 
The qqplot of the residual of the OBFARMAX model with respect
to the white noise added into system (Eq. 24) 
This is due to a small increase in the standard deviation
of the residual as compared to the white noise as can be seen in Fig.
11.
The qqplant and distribution of the residual of the OBFARMAX model are shown
in Fig. 12 and 13.
The correlation among the residuals is given by:
In this case study also both the prediction of the validation data and the residual analysis confirm that the OBFARMAX model is successfully used to identify openloop unstable systems that are stabilized by a feedback controller.

Fig. 13: 
Distribution of the residual of the OBFARMAX model compared
to the white noise added into system (Eq. 24) 
CONCLUSION
A closedloop identification scheme based on orthonormal basis filters with OBFARMAX structure is presented. The procedure for estimating the model parameters is and the multistep ahead prediction are derived. It is demonstrated by simulation case studies that the proposed scheme is effective for closedloop identification of both openloop stable and openloop unstable systems that are stabilized by a feedback control system.