2011 9th IEEE International Conference on Control and Automation (ICCA) Santiago, Chile, December 19-21, 2011
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Multivariable System Identification Using an Output-Injection Based Parameterization Rodrigo Alvite Romano, Felipe Pait and Claudio Garcia modes, and nonlinear estimation methods create unnecessary complications that can severely degrade the quality of the estimates. The paper is organized as follows: Section II presents the output-injection structure. The parameter estimation algorithm is developed in Section III. The proposed algorithm is evaluated based on a two-input two-output distillation column linear model in Section IV. At last, the conclusions and perspectives for future works are drawn in Section V.
Abstract— The challenge of identifying multivariable models from input/output data is a subject of great interest, either in scientific works or in industrial plants. The parameterization of multi-output models is considered to be the most crucial task in a MIMO system identification procedure. In this work, a pioneering multivariable identification method is proposed, implemented and evaluated using a linear simulated plant. It is compared to other traditional MIMO identification methods and its results outperformed the other analyzed methods. It was also tested the situation of over-dimensionality of the estimated models, through the use of Hankel singular values and again the proposed method surpassed the other ones in estimating the correct model order.
II. AN OUTPUT-INJECTION PARAMETERIZATION FOR LINEAR DYNAMIC SYSTEMS
I. INTRODUCTION It is well known that the choice of the model parameterization directly influences numerical properties of the estimation algorithms. Actually, the essential difficulty in the identification of multivariable systems is considered to be the adoption of a suitable representation of the system [2]. In this work, an algorithm is proposed for the parameter estimation of linear discrete-time multivariable systems from input-output measurements. The system is identified in a certain output-injection state space representation which was introduced in [12]. The algorithm starts by parameterizing a model structure based on a list of structural (observability) indexes. It is assumed that the observability indexes are determined in a first structure selection step of the identification procedure. The problem of observability indexes determination has been treated in several works (e.g., [3], [4], [5], [14]). Then an equivalent system description is constructed so that a linear predictor with respect to the model parameters is obtained. Finally, the model parameters are estimated solving linear least squares problems. The properties which make the parametrization particularly suitable for multivariable system identification are as follows. First, the system model presented in Section II is observable for all parameter values, and suitable for parameter estimation using linear least-squares methods, using the structure described in Section III. Second, the model is match-point controllable, that is to say, it does not exhibit pole-zero cancellations when its transfer function matches that of a process in the admissible set. Third, it is capable of matching all such transfer functions. These properties are crucial because excessive parameters, extraneous unstable
In a previous work [11] it was suggested that multivariable linear system identification could be performed using suitably defined output-injection structures. Based on the “special” canonical form described in [10], linear systems of order n, with m inputs (uk ∈ Rm ) and p outputs (yk ∈ Rp ) can be represented by the following state-space model structure: =
yk
=
(I − G(Θ))
−1
A + D(Θ) (I − G(Θ)) −1
C xk + B(Θ)uk
Cxk ,
(1)
where Θ denotes the models parameters, I is an identity matrix of suitable dimension and the pair (C, A) is observable, stable and parameter-independent. The p × p parameter matrix G(Θ) is strictly lower triangular. The other parameter matrices D(Θ) and B(Θ) take values in Rn×p and Rn×m , respectively. A possible way to construct the pair (C, A) was presented in [12] and is described as follows. Given a list of observability indexes l = {n1 , n2 , . . . , np }: 1) Adopt a stable monic polynomial α(q) of degree n = max(l), such that α(q) has a real monic factor αi (q) of degree ni . 2) For each i ∈ {1, 2, . . . , p}, it is assigned a ni dimensional observable pair (Ci , Ai ), for which αi (q) is the characteristic polynomial of Ai . Thus the matrices C and A in (1) are given by
R. A. Romano is with Escola de Engenharia Mau´a, Instituto Mau´a de Tecnologia, S˜ao Caetano do Sul, SP, Brazil; e-mail:
[email protected]. F. Pait and C. Garcia are with Escola Polit´ecnica da Universidade de S˜ao Paulo, SP, Brazil; e-mail addresses:
[email protected] and
[email protected].
978-1-4577-1476-4/11/$26.00 ©2011 IEEE
xk+1
53
C
= block diagonal{C1 , C2 , . . . , Cp }
(2)
A
= block diagonal{A1 , A2 , . . . , Ap },
(3)
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The state vector ξk is mapped to xk by a linear, matrix-valued function EI (Θ) : R(p+m)n → Rn [12], i.e.
where Ci
=
0
Ai
=
0
···
0
1
0 1 0 .. .
0 0 1
··· ··· ··· .. .
0 0 0
× × × .. .
0
0
···
1
×
(4)
1×ni
xk = EI (Θ)ξk , .
(5)
where R1 H M1 D1 (Θ) B1 (Θ) .. EI (Θ) = . . Rp H Mp Dp (Θ) Bp (Θ)
ni ×ni
The elements indicated by “×” are determined by the αi (q) coefficients. In this work the polynomial α(q) is regarded as a design variable. Note that the parameter space dimension of the model (1) is slightly greater than n (m + p), which is the number of parameters commonly encountered in other canonical forms [4], [8], due to the parameters in G(Θ). The role of this matrix is discussed in [12].
BI ,
DI ,
0(m·n)×p −−−−−−− CT 0 ··· 0 .. 0 CT . .. .. . . 0 0 · · · 0 CT
CT
0
0 .. .
CT
··· ..
0 .. .
. 0 0 · · · 0 CT −−−−−−− 0(p·n)×m
∈ R(p+m)n×m
∈ R(p+m)n×p .
−1
Ci (qI − Ai )
=
−1
(I − GI (Θ))
CI (Θ)ξk .
−1
= C (qI − A)
Mi ,
(12)
and Ri are their left inverse, i.e., Ri · Mi = I. The linear transformation H : Rn×(p+m) → Rn×n(p+m) in (11) is defined by S 7→
ΥΓT (s1 )
ΥΓT (s2 ) · · ·
ΥΓT (sp+m )
, (13)
where Υ is the inverse of observability matrix (C, A), Γ (si ) is the controllability matrix of (A, si ) and si the ith row of the matrix S. From (1), (9) and (10) it is straightforward to see that CI (Θ)
= C · EI (Θ)
(14)
GI (Θ)
= G(Θ).
(15)
The reason to employ the realization (9) becomes clear when it is rewritten as yk ξk+1 = AI ξk + [DI BI ] (16) uk yk = CI (Θ)ξk + GI (Θ)yk . (17)
(7)
The model parameters can be estimated by minimizing the mean square error between the predicted output by (16)-(17) and the measured one. Therefore, the ith row of the parameter in matrices, denoted by [CIi (Θ) GIi (Θ)], is estimated by minimizing
(8)
Vi (Θ) =
X k
where the superscript T indicates matrix transposition, the 0’s in (6)-(8) denote suitable dimension matrices filled with 0 and 0m×p is a m × p matrix of zeros. The output-injection model structure (1) with parameter independent pair (C, A) constructed as described in Section II admits a state-space realization of the form −1 ξk+1 = AI + DI (I − GI (Θ)) CI (θ) ξk + BI uk yk
(11)
The matrices [Di (Θ) Bi (Θ)] are obtained by partitioning the rows of [D(Θ) B(Θ)] according to the observability index list l, the left-invertible matrices Mi are given by
III. PARAMETER ESTIMATION ALGORITHM Instead of using an output prediction yˆk based on the model structure (1), an equivalent system description called identifier [12] is employed, in order to achieve a linear predictor with respect to the model parameters. With this in mind, let (C, A) denote a n-dimensional observable pair, such that the characteristic polynomial of A is α(q). Next define T A 0 ··· 0 .. 0 AT . ∈ R(p+m)n×(p+m)n (6) AI , . . .. .. 0 0 · · · 0 AT
(10)
CIi (Θ) GIi (Θ)
ξk yk
−
(i) yk
2 ,
(18) (i) where yk represents the ith output. As the state ξk does not depend on Θ, the parameter estimation can be seen as a set of p linear least-squares problems. Since the dynamic behavior of ξk is determined by the eigenvalues of A, the polynomial α(q) is considered to be a tuning element of the parameter estimation algorithm. The influence of the choice of α(q) is investigated using a simulated example in Subsection IV-A.
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IV. SIMULATIONS
In this Subsection, the model order is considered to be known a priori. Therefore, as the distillation column model is a fourth order system, an output-injection model structure with observability index list l = {2, 2} is adopted. This model parameterization implies in a second order polynomial α(q) which can be expressed as
The identification method described in the previous Section is evaluated based on data from a distillation column linear multivariable model. The model of the methanol/water distillation column presented in [15] is a typical two input two output plant with strong interaction between the controlled variables. The reflux and the reboiler steam flow rates are the process inputs. The compositions of the top and bottom products are the process outputs. The discretized process1 considering a sampling time of 2min, can be expressed by −1.717 1.445 (1) (1) u yk q − 0.8871 q − 0.9092 k . = 1.106 −2.516 (2) (2) uk yk q − 0.8324 q − 0.8703 (19) The proposed estimation method performance is analyzed in the presence of two different disturbance sources: measurement noise and process disturbance. In the former, the process output is corrupted by zero-mean white Gaussian noise ek . In the second situation, disturbances due to the feed composition flow rates is modeled with a colored noise vk generated through (1)
=
(2)
=
vk vk
0.4549q + 0.0224 (1) ek q − 0.8744 0.2177q + 0.4712 (2) ek . q − 0.8594
α(q) = (q − ∆1 )(q − ∆2 ). In order to evaluate the effect of the α(q) roots in the identified model fit, the estimation algorithm is executed with each of the polynomial roots (∆1 , ∆2 ) assuming 21 values equally spaced between [0, 0.95] (441 combinations). Furthermore, for each combination, 250 datasets are generated in a Monte Carlo simulation. Let F¯i denote the ith output mean fit, with the purpose of merging the results for each output, the arithmetic mean of F¯1 and F¯2 is employed as an overall fit measure. Before presenting the results achieved from noisy datasets, it is worth mentioning that if measurement noise or process disturbance are not considered, the proposed identification method provides models that match exactly with the simulated system (19). Such result do not depends on the choice of α(q) since, for any pair (∆1 , ∆2 ), F1 and F2 are 100%. The results concerning the situation that the outputs are corrupted by white noise are presented in Fig. 1. The behavior of overall model fit shows that better model estimated are achieved when the roots of α(q) approach 0.9. On the other hand, if both roots are chosen close to 0 the model fit is degraded. In other words, if the output data is corrupted by white noise, the algorithm performance is maximized when high frequency content is removed from the identifier state ξk . Although the obtained fit is closely related to the values of α(q), when both ∆1 and ∆2 are higher than 0.7, or at least one of them is higher than 0.8 such a dependence becomes negligible. As such ranges are not so narrow, the algorithm tuning can be performed without much difficulty for this situation.
(20)
Then, the estimation dataset is generated by adding vk to yk . In both disturbance cases the covariance matrix of ek is adjusted so that the SNR (Signal to Noise Ratio) in each output is 25% (in variance). The estimation data is generated by exciting both process inputs simultaneously with two distinct PRBS (Pseudo Random Binary Signal) realizations. Each 500 samples PRBS realization switches between [−0.1, 0.1] with a clock periodof 10min, i.e., 5 times the sampling period. A. Influence of α(q) choice The first issue to be investigated is the effect of the choice for α(q) in the identification algorithm performance. Such analysis is carried out based on the estimated model fit to a validation dataset which may be quantified by Fi :
ˆ
Yi − Yi
2 × 100 , Fi (%) = 1 − (21)
Yi − Y¯i 2
94
Fit to the actual outputs (%)
92
where Yi , Y¯i and Yˆi are the ith process output vector, the mean and an estimation of Yi , respectively. The symbol k·k2 denotes the `2 norm. In this work, the model fit is computed based on a distinct dataset, extracted from an experiment without measurement noise or process disturbances.
100 90
80 60
88
40
86
20
84
0 82 0.2 0.95 0.45 0.45
0.7
1 The
process input delays from the original model reported in [15] were disregarded, to limit the simulated model order so that the influence of α(q) can be more clearly investigated. Anyway, it is important to mention that the proposed method might be applied to systems with dead-time. Nevertheless, this issue is not investigated in this work.
∆2
80
0.7
0.95
78
0.2
∆1
Fig. 1. Mean fit of the models estimated employing different roots (∆1 , ∆2 ) when the data is corrupted by measurement noise (white noise).
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Considering the results presented in the previous Subsection, the roots of α(q) are chosen to be 0.9 and 0.1, for the white and colored noise disturbance case, respectively. The comparison is accomplished based on 250 datasets of 500 samples (for each output) generated from Monte Carlo simulations. The mean F¯i and the fit standard deviation for the white noise disturbance scenario are presented in Table I. In this case, the models estimated using the ARX structure resulted in the worst fits. However, such result was already expected since the white noise is added directly to the output should be described by an output-error structure. The results produced by subspace and prediction error methods are quite similar, but they are outperformed by the proposed algorithm. As reported in Table I, the best mean fits to the validation dataset, as well as the lowest standard deviations are presented by the output-injection algorithm. To provide a better result visualization, histograms containing the fits obtained from the Monte Carlo simulations are shown2 in Fig. 3. Note that occurrences of fits greater than 95% are much more frequent for models estimated by the proposed algorithm.
The results for the simulations in which the datasets are disturbed by the colored noise vk defined in (20) is shown in Fig. 2. The fit behavior is almost the opposite to the one verified in the measurement noise case, i.e., the closer to 0 the roots of α(q) are, the better the estimated model is. It is important to remember that the process disturbance spectrum is higher in the low frequency range. Therefore, the design of α(q) can be interpreted in the following manner: α(q) plays the role of a state filter characteristic polynomial and it should be tuned with the purpose of removing the disturbance effect of the observed data. The overall fit values for the colored noise disturbance are lower than the ones provided by the white noise case. Once the SNR is maintained, there is a clear indication that identifying systems with dynamic behavior similar to the distillation column (19) in the former situation is more difficult than from data disturbed by white noise. Such a difficulty may be associated with the fact that the disturbance spectrum and the frequency response of the actual system modes are concentrated in similar frequency ranges. B. Accuracy analysis The accuracy of the proposed parameter estimation algorithm is compared to well-established techniques [8]: ARX (Auto Regressive with eXternal input) structure, subspace method (Canonical Variate Analysis algorithm [6]) and PEM (Prediction Error Method). The model estimates were computed using functions from the MATLABr System Identification Toolbox [9]. The prediction error method is employed to estimate fully-parameterized and observer canonical statespace models. Each element of the ARX transfer function matrix is a 4th order model, which implies in a multivariable model of order up to 16. The state space models estimated using the subspace and the prediction error methods have order 4, which is the order of the actual system. In order to produce a fair comparison, disturbance models are considered for prediction error methods only when the output data are corrupted by colored noise.
2 The ARX structure histograms are omitted since they are unsuitable for white noise disturbance.
TABLE I M ONTE -C ARLO SIMULATION RESULTS SUMMARY FOR THE WHITE NOISE DISTURBANCE SCENARIO .
60
82
40
80
20
78
0
76
Subspace
74
0.2 0.95 0.45
0.7 0.45
0.7
∆2
0.95
0.2
10
72 70
∆1
Fig. 2. Fit of the models estimated employing different roots (∆1 , ∆2 ) considering data corrupted by process disturbance vk .
85
90
95
100
0 80
Observer canonical
84
20
10
30
Fully−parametrized
86
80
30
20
30
Output−injection
Fit to the actual outputs (%)
88
std(F1 ) 2.23 2.09 2.03 2.10 1.54
30
0 80
90
100
F¯1 86.36 93.15 92.89 92.51 95.13
Identification method Multi-output ARX Subspace (CVA) Observer canonical Fully-parameterized Output-injection
30
30
20
20
10
10
30
20
20
10
10
0 80
85
90
95
100
0 80 30
20
20
10
10
0 80
0 80
85
85
90
95
90 95 Fit to output 1
100
100
0 80
0 80
F¯2 89.64 93.08 92.78 92.21 94.73
std(F2 ) 1.30 2.71 2.07 2.09 1.78
85
90
95
100
85
90
95
100
85
90
95
100
90 95 Fit to output 2
100
85
Fig. 3. Histogram obtained from results of the Monte Carlo simulations when the disturbance is white noise.
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MonA2.2 TABLE II S UMMARY OF THE M ONTE -C ARLO SIMULATION RESULTS FOR THE PROCESS DISTURBANCE ( COLORED NOISE ) CASE . F¯1 89.95 87.82 83.12 82.47 90.33
Identification method Multi-output ARX Subspace (CVA) Observer canonical Fully-parameterized Output-injection
std(F1 ) 3.82 4.92 7.64 6.55 3.84
F¯2 90.01 87.83 82.18 81.01 89.67
in spite of being high, this number was kept practically constant, that is, from 15 (white noise) to 16 (colored noise). But the high amount of unnecessary parameters in these models may be the cause for their low performance.
std(F2 ) 3.75 4.79 8.01 7.28 3.41
C. Overmodeling issues Some multivariable identification methods (e.g., [1], [16]) accomplish the estimation task in two steps. In the first one, an approximate overparameterized model is estimated. Then, any available model reduction technique is applied to achieve a minimal order realization. The Hankel singular values provide a measure of energy for each state in a system, hence, it is possible to identify the most significant states and perform a model reduction procedure in such a way that high energy states are retained while low energy states are eliminated. In the previous analysis, it is supposed that the model order is known and the same order of the actual system is employed in the model parameterization. Next, the Hankel singular values behavior from models estimated considering an order n larger than 4 is evaluated. In other words, the aim is to analyze how the considered estimation algorithms react when an overdimensioned structure is adopted. Such analysis is performed with the same datasets of the previous Subsection. The state space and the ARX models were estimated using a 8th order structure. So the order of the transfer function relating each input-output pair for the ARX model is 2. The Hankel singular values of overdimensioned estimated models are reported in Table III. It is important to mention that, since the energy of an unstable mode is estimated as an infinite value, the models with unstable modes are discarded, so that they are not considered in the average energy of the states. The Hankel singular values shown in III are closer to the real ones when applying the Output-injection algorithm than using any other method. Another point to be stressed is that the energy present in the 5th and 6th states (which actually do not exist) are better reproduced by the proposed method, either with with noise or with colored noise. The
ARX
The results for the process disturbance (colored noise) are summarized in Table II. The ARX and the outputinjection structures provided the best results, followed by the subspace method. As already commented, in the colored noise situation, the disturbance spectrum is concentrated in the same range in which the frequency response of the system modes are prominent, in such a way that the simulated system approaches an ARX structure. This explains why the ARX structure provided models as accurate as the ones estimated by the proposed algorithm. Compared to the others, the prediction error method provided more inaccurate models. This is clear from the histograms in Fig. 4. Note that the fits lower than 80% are frequent for both canonical observer and fully parameterized model structures. Moreover, the standard deviations of the fits produced by such models are higher. This may be explained based on the amount of estimated models with unstable modes in the observer canonical form, which increased from 4 (white noise) to 10 (colored noise). On the other hand, for the fully parameterized models,
30
30
20
20
10
10
Subspace
0
Fully−parametrized Observer canonical
80
90
100
0
30
30
20
20
10
10
0
Output−injection
70
70
80
90
100
0
30
30
20
20
10
10
0
70
80
90
100
0
30
30
20
20
10
10
0
70
80
90
100
0
30
30
20
20
10
10
0
70
80 90 Fit to output 1
100
0
70
80
90
100
70
80
90
100
70
80
90
100
70
70
80
90
80 90 Fit to output 2
TABLE III H ANKEL SINGULAR VALUES OF OVERDIMENSIONED ESTIMATED MODELS .
Method ARX Subspace Observer can. Fully-param. Output-injection
100
ARX Subspace Observer can. Fully-param. Output-injection Actual system
100
Fig. 4. Histogram obtained from results of the Monte Carlo simulations when the disturbance is colored noise.
57
Hankel singular values - white noise 2nd 3rd 4th 5th 6th 2.19 1.66 1.25 0.60 0.33 4.79 3.33 2.05 1.19 0.66 20.2 6.80 3.42 2.26 1.13 31.8 9.99 4.98 3.59 1.63 3.02 1.04 0.53 0.26 0.10 Hankel singular values - colored noise 15.96 2.88 0.45 0.29 0.14 0.064 16.65 3.94 1.40 0.86 0.55 0.31 19.43 7.37 2.97 1.51 0.90 0.48 25.22 9.36 4.40 2.37 1.40 1.73 16.00 2.78 0.48 0.30 0.15 0.07 16.03 2.72 0.45 0.078 − − 1st 14.20 16.54 28.75 41.37 16.09
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ARX method provided low singular values for the 5th and 6th states just for the colored noise case. Actually, it was expected since the ARX structure is quite similar to the one of the simulated process. In other words, if the Hankel singular values was to be applied to estimate the model order, the Output-injection algorithm provides a more reliable estimation, since the singular values relating to the additional states are closer to zero. In fact, the analysis with Hankel singular values indicates some robustness against overestimated model order. The same behavior verified for overdimensioned models of order 6 was also depicted when other orders were used (5, 7, 8 e 9). A remarkable observation is the increasing amount of unstable models when observer canonical and fully parameterized structures are estimated using PEM methods as the model order increases. Table IV presents this fact, where the labels W and C denote the white and colored disturbance scenario, respectively. The explanation for the increasing number of models with unstable modes is that such methods employ nonlinear iterative optimization algorithms.
its performance in other situations and to be able to draw more general conclusions, the authors are working on other testing platforms, such as, systems operating in closed-loop and processes subject to dead-time. There are some points to be further studied: at the same time that α(q) is an element to be determined and which requires an intervention of the user, it provides flexibility to deal with disturbances. So, a desirable feature of the proposed algorithm would be to automatically estimate such a design variable. Another point to be improved is to test other validation methods, for instance, frequency response, in order to better evaluate the identification methods. VI. ACKNOWLEDGMENTS The authors gratefully acknowledge the support of Instituto Mau´a de Tecnologia (IMT) and Fundac¸a˜ o de Amparo a` Pesquisa do Estado de S˜ao Paulo (FAPESP). They also thank the support provided by Petrobr´as. R EFERENCES [1] N. F. Al-Muthairi, S. Bingulac, and M. Zribi, “Identification of discrete-time MIMO systems using a class of observable canonicalform,” IEE Proceedings on Control Theory Applications, vol. 149, no. 2, pp. 125–130, Mar 2002. ˚ om and P. Eykhoff, “System identification - a survey,” [2] K. J. Astr¨ Automatica, vol. 7, pp. 123–162, 1971. [3] H. El-Sherief, “Multivariable system structure and parameter identification using the correlation method,” Automatica, vol. 17, no. 3, pp. 541–544, 1981. [4] R. Guidorzi, “Canonical structures in the identification of multivariable systems,” Automatica, vol. 11, pp. 361–374, 1975. [5] ——, “Invariants and canonical forms for systems structural and parametric identification,” Automatica, vol. 17, no. 1, pp. 17–133, 1981. [6] W. E. Larimore, “Canonical variate analysis in identification, filtering and adaptive control,” in Proceedings of the 29th IEEE conference on decision and control, Honolulu, 1990, p. 1990. [7] D. Liberzon, Switching in systems and control. Boston: Birkh¨auser, 2003. [8] L. Ljung, System Identification: theory for the user, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1999. [9] ——, The system identification toolbox: The manual, 7th ed., The MathWorks, Inc., MA, USA: Natick, 2007. [10] D. G. Luenberger, “Canonical forms for linear multivariable systems,” IEEE Transactions on Automatic Control, vol. AC-12, pp. 290–293, 1967. [11] A. S. Morse, “Representations and parameter identification of multioutput linear systems,” in Proceedings of the 13th IEEE Conference on Decision and Control, Dec 1974, pp. 301–306. [12] A. S. Morse and F. M. Pait, “MIMO design models and internal regulators for cyclicly switched parameter-adaptive control systems,” IEEE Transactions on Automatic Control, vol. 39, no. 9, pp. 1809– 1818, 1994. [13] F. M. Pait and F. Kassab, “On a class of switched, robustly stable, adaptive systems,” International Journal of Adaptive Control and Signal Processing, vol. 15, no. 3, pp. 213–238, 2001. [14] E. Tse and H. L. Weinert, “Structure determination and parameter identification for multivariable stochastic linear systems,” IEEE Transactions on Automatic Control, vol. 20, pp. 603–613, 1975. [15] R. K. Wood and M. W. Berry, “Terminal composition control of a binary distillation column,” Chemical Engineering Science, vol. 28, pp. 1707–1717, 1973. [16] Y. Zhu, Multivariable System Identification for Process Control. Oxford: Elsevier Science, 2001.
TABLE IV N UMBER OF MODELS WITH UNSTABLE MODES .
Identification method Observer canonical Fully-parameterized
Disturbance W C W C
n=4 4 10 15 16
Order n=6 7 9 30 19
n=8 16 10 65 21
V. CONCLUSIONS AND FUTURE WORKS The idea of the algorithm here proposed arise from the multivariable design model parameterization introduced in [12]. Nevertheless, as far as the authors know, it was never implemented to identify systems. So this paper presents an innovative application of a technique based on state space applied to system identification. As a pioneering identification method, it still has some improvements to be made. Anyway, the comparison of the proposed method based on an output-injection structure with well-established identification techniques indicated that the proposed algorithm is promising. It is important to emphasize that the superior performance of the method proposed is due to the fact that the number of parameters is kept to a minimum, that no undesired polezero cancellations can appear, and that a garden-variety leastsquares method was feasible without the need for nonlinear or iterative estimation methods. As pointed out in Section I, these are essential properties of the identification models employed. For future applications, it is worth noting that switching among a finite number of candidate models can be used effectively to deal with unknown observability indexes, as well as with unknown system order [7], [13]. In this work, the proposed algorithm was evaluated using a 2 × 2 distillation column linear model. In order to investigate 58
