Volume 3, Issue 5, May 2014
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Abstract
For the past decade noise corrupted output measurements have been a fundamental research problem to be investigated. On the other hand, the estimation of the parameters for linear dynamic systems when also the input is affected by noise is recognized as more difficult problem which only recently has received increasing attention. Representations where errors or measurement noises/disturbances are present on both the inputs and outputs are usually called errors-in-variables (EIV) models. These disturbances may also have additive effects which are also considered in this paper. Parameter estimation of false EIV problem using equation error, output error and iterative prefiltering identification schemes with and without additive uncertainty, when only the output observation is corrupted by noise has been dealt in this paper The comparative study of these three schemes has also been carried out.
Keywords: Errors-in-variable (EIV), False EIV, Equation error, Output error, Iterative prefiltering, Gaussian noise.
1.1 INTRODUCTION
In the basic dynamic EIV models the input is corrupted by noise during measurement only but there may be possibility that noise gets added in the input and affects the performance of the system, which can be termed as false EIV problem [4, 7]. This is not an actual EIV problem. So this false EIV problem can be formulated in two ways: 1) where the both input and output observations are corrupted by noise and 2) where only the output is disrupted by noise. Assumptions on input and output noises can be taken as:
Assumption 1 (A1): When both input and output noises are ARMA (auto regressive moving average) models. Assumption 2 (A2): When input noise is white noise sequence and output noise is ARMA model.
Assumption 3 (A3): When both input and output noises are white noise sequence.
Apart from above assumptions another possibility can be there regarding the noise which may be called as assumption A4.
Assumption 4 (A4): When input noise is ARMA model and output noise is white noise sequence.
This paper will be concentrated on the identification of parameters of false EIV model, with and without additive uncertainty using equation error (EQ) and output error (OP) and iterative prefiltering (IP) identification methods [6, 9, 10, 11].
1.2 FALSE DYNAMIC ERRORS-IN-VARIABLES PROBLEM
Dealing with false dynamic EIV models the output measurement noise is considered as a form of process disturbance and the total effect of and output measurement noise can be modeled as a single auto-correlated disturbance on the output side as shown in Fig. 1.1.
Fig. 1.1 False Errors-In-Variables Problem
Here is the designed input but is added (due to distortions or other unavoidable reasons) before the input reach the system as which is the noise free input in EIV model.
1.3 FALSE V/S BASIC DYNAMIC EIV MODEL
PARAMETER ESTIMATION OF FALSE
DYNAMIC EIV MODEL WITH ADDITIVE
UNCERTAINTY
1
Dr (Mrs) Dalvinder Mangal, 2Dr (Mrs) Lillie Dewan
1
Electronics and Instrumentation Engineering Department, The Technological Institute of Textile and Sciences, Bhiwani, Haryana, India
2Electrical Engineering Department, National Institute of Technology,
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The dynamics between and is the same as between and as shown in Fig. 1.1. Hence the presence of is not so problematic here as was in basic dynamic EIV model [7].
effects also the output measurements in case of false dynamic EIV problem.
For the situation as shown in Fig. 1.1, it is appropriate to regard as a form of process disturbance. The total effect of and can be modeled as a single auto correlated disturbance on the output side.
From the above considered facts it has been realized that false EIV problem is not actually an EIV problem. Therefore the false EIV problem may lie in the following two categories:
Category-1 When both input-output observations are corrupted by noise. Category-2 When only the output observation is corrupted by noise.
In the paper, the false EIV model for the category-2, where only the output is corrupted by noise, will be identified by EQ, OP and IP methods with and without additive uncertainty.
1.4 MATHEMATICAL FORMULATION FOR CATEGORY-2 FALSE EIV MODEL
In this formulation only the output observation is corrupted by noise and the ARX system is given by:
(1.1)
where
,
(1.2)
Prediction of the output Eq. 1.2 using Eq. 1.1 is given by:
or
(1.3)
(1.4)
is the regressor vector
(1.5)
is the estimated parameter vector.
The mathematical description of perturbed plant parameterized by additive perturbation in terms of non-parametric
dynamic uncertainty is given as [1, 6, 11]:
(1.6)
where is the perturbed plant.
and is the perturbation bound based on a priori physical information
with for non-parametric dynamic (additive) uncertainty (1.7)
Category-2 can be implemented using EQ, OP and IP formulations [5, 8, 9, 10].
1.5 CASE STUDY-I (WITH AND WITHOUT ADDITIVE UNCERTAINTY USING EQ AND
OP SCHEMES)
False dynamic errors-in-variables problem is now implemented on the ARX model given as [10]: (1.8)
where and
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The given ARX model is identified by using EQ and OP formulations taking periodic input and noise to the system as given by Eq. 1.9 and 1.10 respectively.
(1.9)
(1.10)
where is the white Gaussian noise with variance 0.01, and is also the white noise with variance 1. is a random variable and its distribution is uniform within (0, 1).
The system is perturbed by additive uncertainty given by Eq. 1.6 as:
where (1.11)
Here the two cases for the additive uncertainty have been considered [2, 3, 11]:
The estimated parameters are shown in Fig. 1.2 to 1.7 for EQ and OP formulations without uncertainty and with uncertainties: , .
Fig. 1.2 Estimated parameters Fig. 1.3 Estimated parameters using EQ (without uncertainty). using OP (without uncertainty).
Fig.1.4 Estimated parameters Fig. 1.5 Estimated parameters using EQ with . using OP wth .
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As observed from Fig. 1.2 to 1.7 it has been found that the parameters estimated by EQ and OP formulation are converging to their true values with less oscillations even in the case of additive uncertainty as compared to the case when system is having no uncertainty. It has also been observed that the system parameters are converging rapidly to their true value in case of equation error algorithm.
1.6 CASE STUDY-II (WITH ADDITIVE UNCERTAINTY USING EQ, OP AND IP SCHEMES) False dynamic errors-in-variables problem is now implemented on the ARX model given as:
(1.12)
where and
and is the nominal model.
Theperiodic input and noise, used in the simulation, are given by Eq. 1.9 and 1.10 respectively. The additive uncertainty is perturbing the plant as per Eq. 1.6 for which the two cases have been considered taken as:
The estimated parameters are shown in Fig. 1.8 to 1.13 for EQ, OP and IP formulations with and .
0 5 10 15 20 25 30 35 40
-1.5 -1 -0.5 0 0.5 1 1.5 e s ti m a te d p a ra m e te rs a1 a2 b0 b1
Fig. 1.8 Estimated parameters using EQ with .
0 5 10 15 20 25 30 35 40 45 50
-1 -0.5 0 0.5 1 1.5 e s ti m a te d p a ra m e te rs a1 a2 b0 b1
Fig. 1.9 Estimated parameters using EQ with .
0 10 20 30 40 50 60 70
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 e s ti m a te d p a ra m e te rs a1 a2 b0 b1
Fig. 1.10 Estimated parameters using OP with .
0 10 20 30 40 50 60 70
-0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 e s ti m a te d p a ra m e tr e s a1 a2 b0 b1
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0 5 10 15 20 25 30 35
-3 -2 -1 0 1 2 3 e s ti m a te d p a ra m e te rs a n d o u tp u t e rr o r a1 a2 b0 b1 e
Fig. 1.12 Estimated parameters using IP with .
0 5 10 15 20 25 30 35
-4 -3 -2 -1 0 1 2 3 e s ti m a te d p a ra m e te rs a n d o u tp u t e rr o r a1 a2 b0 b1 e
Fig. 1.13 Estimated parameters using IP with .
It has been observed from Fig. 1.8 to 1.13 that successful convergence within 5-10 iterations has been obtained for the system with additive uncertainty using iterative prefiltering technique as compared to equation error and output error formulations.
1.7 CONCLUSION
From the analysis of the results obtained above it is concluded that on identifying the False EIV model considering only the output observation corrupted by noise, using equation error and output error schemes with and without additive uncertainty, the system parameters are converging rapidly to their true values in case of equation error algorithm. On identifying the same system using equation error, output error and iterative prefiltering identification schemes with additive uncertainty and it is found that successful convergence within 5-10 iterations has been obtained for the system using iterative prefiltering technique as compared to equation error and output error formulations.Therefore, it can be concluded that Iterative Prefiltering method is fast as compared to other two methods in presence of uncertainty.
REFERENCES:
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[2] Douma, S. G., Paul M.J., Van den Hof, Bosgra, O. H., “Controller tuning freedom under plant identification uncertainty: double Youla beats gap in robust stability”, Proceedings of the American Control Conference, Arlington, June, pp. 3153-3158, 2001.
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Volume 3, Issue 5, May 2014
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Authors
Dr (Mrs) Dalvinder Mangal Born on February 21, 1977 at (Nilokheri) Karnal, Haryana, India, the land of Karna, Ms Dalvinder Kaur Mangal received her B.E. (Hons.), M.E. and Ph.D. degrees from DCRUST, Murthal, P.U. Chandigarh and NIT, Kurukshetra respectively in the field of Electrical Engineering. As a teacher she dedicatedly contributed her services at various colleges and Institutes of Haryana. Presently she is working as an Assistant Professor, Department of Electronics & Instrumentation Engineering, at The Technological Institute of Textile and Sciences, Bhiwani, Haryana, India. She has attended various National and International seminars and conferences. She presented/published several papers in the Journals of International/National repute.
Electronics and Instrumentation Engineering Department, The Technological Institute of Textile and Sciences, Bhiwani (127021), Haryana, India
Dr (Mrs) Lillie Dewan is a professor. She has numerous research papers in international/national journals and conferences to her credit. Her current research interest areas include identification and robust control, digital signal processing, image processing, and electrical machines.
