Reglermöte 2010

Unbiased Estimation of Gramian Based Interactions with Confidence Bounds . . Contact: Miguel Castaño ( [email_address] ) Wolfgang Birk ( [email_address] ) Abstract: The value tr(PQ) of a dynamic system is used in fields like model order reduction or analysis of process interactions. An estimation of tr(PQ) for stable linear systems is obtained from time-domain data. The distribution of the estimator is known and confidence bounds on the estimation can be generated. Conclusion. The previously existing estimator is biased, and converges to the real value of tr(PQ) when the number of available samples is infinite. The proposed estimator is unbiased, and its variance reduces when the number of available data samples increases. Example 2: A 2x2 multivariable process was excited in the presence of white noise at the output. The channel corresponding to the first input and the first output was analyzed by generating 99% confidence bounds on the estimation the value tr(P 1 Q 1 ) for different lengths of the acquired data. Example 3: The same multivariable process as in Example 2 was excited , with a length of the logged data of 600 samples and 99% confidence intervals in the estimation of tr(P j Q i ) were created. The diagonal elements are strongly dominating, and the process can be considered as composed by 2 independent SISO processes, being possible to design a decentralized control structure for the process. Conclusion. Creating confidence intervals on the estimation of tr(P j Q i ) for each of the input-output channels of a multivariable process helps to take robust decisions on the control structure selection. [1] Conley, A. and Salgado, M. (200) Gramian based interaction measure. In Proc. Of the 2000 Conference on Decision and Control, Sydney. [2] Slagado, M. and Yuz, J. (2007). Midex domain analysis of mimo dynamic interactions. In Networking, Sensing and Control, 2007 International Conference on, 340-344. tr(P 1 Q 1 ) Funded by: ,[object Object],[object Object],[object Object],[object Object],[object Object],tr(PQ) and its use in control structure selection. ,[object Object],[object Object],[object Object],Example 1: A SISO system was excited with white noise, and in the presence of output white noise. The estimators in Equations (1) and (2) were used to estimate tr(PQ) for different experiments with different length of the sampled data. Result 2. The estimator in Equation (2) is distributed as a linear combination of noncentral chi-square random variables with one degree of freedom of the form: where and are the mean and the variance of the estimated regressors . Estimation of tr(PQ). ,[object Object],[object Object],Salgado and Yuz [2] used Equation (1) to estimate tr(P j Q i ) from an estimation of the impulse response obtained by linear regression. The data was obtained by exciting the process with white noise and in the presence of output noise. And therefore the estimation is: where

Reglermöte 2010

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Reglermöte 2010