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Manchester University

Manchester University. Electrical and Electronic Engineering. Control Systems Centre (CSC). A COMPARITIVE STUDY BETWEEN A DATA BASED MODEL OF LEAST SQUARES AND PARTIAL LEAST SQUARES ALGORITHMS. Prepared by AWAD R. SHAMEKH. Under supervision of Dr. BARRY LENNOX. 10-5-2006.

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Manchester University

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  1. ManchesterUniversity Electrical and ElectronicEngineering Control Systems Centre (CSC) A COMPARITIVE STUDY BETWEEN A DATA BASED MODEL OF LEAST SQUARES AND PARTIAL LEAST SQUARES ALGORITHMS Prepared by AWAD R. SHAMEKH Under supervision of Dr. BARRY LENNOX 10-5-2006

  2. The Presentation Contents • MODELLING AND IDENTIFICATION • LEAST SQUARES ALGORITHMS • MULTIVARIATE STATISTICAL METHODES • SIMULATION RESULTS AND CONCLUSIONS • FURTHER WORK

  3. Modelling and Identification Modelling is a useful way to consolidate information about a system and to explore its characteristics. a model can be constructed either theoretically or by identification. The Steps in identifying a model of a process are as follows: • Collection of data 2.Selection of identification algorithm 3.Selection of a model structure 4.Specifying of Criteria

  4. Least squares algorithms Since its invention in1795 by Gauss, the least squares technique remains the most popular tool in the identification field. The reasons for its popularity are that it does not contain 1. high-level mathematical analysis 2. it is easy to implement and 3. modification and extensions have been made to it that make it extremely robust and applicable

  5. Recursive Least Squares algorithm (RLS) In the recursive computation technique, the identification of the current parameters is performed based on the old estimated parameters and therefore the capacity of memory storage will be significantly reduced. Summary of the recursive least squares 1. It is commonly used in on-line controlling systems. It could be performed explicitly as in the self tuning regulators or implicitly as in case of model reference control. 2. Dose not required a large memory size. 3. It can provide an over view about a system behaviour, such advantage arises when the system is subjected to drastic changes in the operating conditions.

  6. Multivariate Statistical Methods The multivariate statistics is a modern data analysis technique that has been widely used in Industry with good results. By using the multivariate statistical algorithms the data can be compressed in a manner that retains the essential information in small number of factors which describe of how the variables are related to each other. Principle component analysis (PCA) and Partial least squares (PLS) are dominant techniques in multivariate statistics. Partial Least squares PLS PLS regression originated in social science by Herman Wold, 1966, and then Entered in chemometrics by his son Svante. The PLS decomposes X and Y data into orthogonal sets of scores (T,U), loadings (P,Q) and Weights (W,C) which are evaluated to maximize covariance between the scores of X and Y.

  7. Select the first column of Y as , in case of multi-output system Non-Iterative Partial Least Squares (NIPALS) The regression coefficient b for the inner relation is: the X and Y block residuals are calculated as follows: The same procedure for all Y columns should be repeated, results PLS parameters vector

  8. Recursive Partial Least Squares (RPLS) In this study the modified kernel PLS algorithm is implemented to develop a RPLS Model. As introduced by Dayal and MacGregor, the algorithm contains the following steps: The covariance matrices should update as

  9. For a >1 is computed as the eigenvectors corresponding to the largest eigenvalue of the output covariance matrix deflated as The RPLS model Coefficients are calculated by

  10. Simulation Results Three cases are under taken to demonstrate the performance of the four types of identification algorithm, OLS, PLS, U-D RLS and RPLS 1.Correlated data A set of highly correlated variables denoted by X-matrix and observations of y-vector are used to test a model of four different ways of identification, Ordinary Least Squares (OLS), NIPALS Partial Least Squares (PLS), U-D Recursive Least Squares (RLS), and modified kernel Recursive Partial Least squares (RPLS). The X-data and y-outputs are

  11. Model Actual -0.5800 0.0000 1.3330 OLS 0.8166 -1.0503 2.0335 PLS 0.8166 -1.0503 2.0335 RLS -0.7459 0.1273 1.2493 RPLS 0.8166 -1.0503 2.0335 Model Actual - 0.5800 0.0000 1.3330 OLS 0.8166 -1.0503 2.0335 PLS -0.6287 0.0362 1.3095 RLS -0.7459 0.1273 1.2493 PRLS -0.6287 0.0362 1.3095 Table (4.1.a), parameters of the estimated models from X& y. (LV=3) Table (4.1.b), parameters of the estimated models from X& y. (LV=2)

  12. , LV=3 , LV=2 + + Table (4.2.a), parameters of the estimated models from X& Table (4.2.b), parameters of the estimated models from X& 0.001 0.001 y y Model Actual - 0.5800 0.0000 1.3330 OLS 0.0007 -0.04364 1.6246 PLS -0.6284 0.0366 1.3094 RLS -0.7455 0.1277 1.2493 RPLS -0.6284 0.0366 1.3094 Model Actual - 0.5800 0.0000 1.3330 OLS 0.0007 -0.04364 1.6246 PLS 0.0007 -0.04364 1.6246 RLS -0.7455 0.1277 1.2493 RPLS 0.0007 -0.04364 1.6246

  13. 2. Artificial data The following process transfer function has been excited by GBN signal And its out put is estimated by means of OLS, PLS, U-D RLS and RPLS The objective is to identify the ARX model for the considered process as in the structure below at different signal to noise ratio. Remark: In the case of recursive identification, the output error criterion is applied

  14. Model Actual 1.0000 0.5000 -1.5000 0.7000 OLS 0.9884 0.9065 -1.1547 0.3422 PLS 0.9884 0.9065 -1.1547 0.3422 RLS 0.9816 0.4952 -1.5037 0.7050 Model RPLS 0.9828 0.4920 -1.5041 0.7053 Actual 1.0000 0.5000 -1.5000 0.7000 OLS 0.9949 0.6197 -1.4219 0.6259 PLS 0.9949 0.6197 -1.4219 0.6259 RLS 0.9934 0.5001 -1.5010 0.7015 RPLS 0.9937 0.4993 -1.5010 0.7015 Table (4.2.a), the estimated parameters compared with the actual at signal-to-noise ratio (0.5) all latent variables (4) are considered in the PLS's estimation. Table (4.2.b), the estimated parameters compared the actual .The PLS's estimation are performed with (LV=4) at signal-to-noise ratio (0.05).

  15. The process is defined as irreversible A B and the reaction is carried out perfectly in a mixed CSTR. CSTR 3. Non-isothermal Continuous Stirred Tank Reactor (CSTR) An ARX model of each output variable is identified individually where the system is driven by random walk of Arrhenius rate constant the desired model has the following structure: The prediction is carried out recursively by U-D RLS and RPLS

  16. Conclusions The study reveals some notes about the studied and applied algorithms, these are summarized as 1. The motivation behind using the ARX model is its simplicity and to ensure model accuracy number of lags should be increased. In contrast, this leads to a huge regression vector especially in the fat system data, as in the case of a distillation column. 2. A system can be identified perfectly with OLS algorithm assuming that the variables of regression matrix are independent. But such conditions in many situations are not guaranteed, which is related to an unstable model.

  17. 3. As it has been documented in the literature that the importance of the PLS appears in heavy multivariable systems. 4. From the results apparently, there is on difference in the model accuracy of U-D RLS and RPLS. However, in some studies, it has been shown that RPLS-based model is better than its competitor, U-D RLS, when they are used in control design.

  18. Further work 1. Survey and analysis of General Model Predictive control (GPC). 2. Design of Dynamic Matrix control for the CSTR case study. 3. Comparison between U-D RLS and RPLS using the error optimization. technique.

  19. Thank you

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