Reglermöte 2010
1
Unbiased Estimation of Gramian Based Interactions with Confidence Bounds. . Contact: Miguel Castaño ([email protected] ) Wolfgang Birk ([email protected] ) Funded by: 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. • P and Q are the well-known controllability and observability Gramians of a system. For a multivariable system, P j denotes the controllability Gramian related to the j-th input, and Q i denotes the observability Gramian related to the i-th output. • The trace of the product PQ quantifies the abilities of the inputs and outputs to control and observe the process state, or in other words, it quantifies the connection of the input and output spaces via the state space. • In a multivariable system , composed by elementary SISO subsystems G ij , the most important input-output channels can be considered as those with a larger value of tr(P j Q i ) [1]. • Control can be based upon a reduced model containing only the most important input-output connections, and the control structure can be selected. I.e. a diagonally dominant multivariable system, can be considered as a composition of independent SISO systems, and SISO controllers can be designed for each of these subsystems. This example is known as decentralized control structure. tr(PQ) and its use in control structure selection. •Result 1. The estimator in (2) is shown to be biased but asymptotically unbiased. Being its expected value: An unbiased estimator can be therefore obtained by subtracting the term from the estimator in Equation(2), and we denote it as 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. 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. 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). • Given a multivariable discrete system G with impulse response h(k), the trace of the product of the controllability and observability Gramians of the elementary subsystem G ij can be computed as [2]: where h ij (k) is the impulse response of the elementary subsystem G ij . 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 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 )
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Transcript of Reglermöte 2010
Unbiased Estimation of Gramian Based Interactions with Confidence Bounds.
.Contact: Miguel Castaño ([email protected]) Wolfgang Birk ([email protected]) Funded by:
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.
• P and Q are the well-known controllability and observability Gramians of a system. For a multivariable system, Pj denotes the controllability Gramian related to the j-th input, and Qi denotes the observability Gramian related to the i-th output. • The trace of the product PQ quantifies the abilities of the inputs and outputs to control and observe the process state, or in other words, it quantifies the connection of the input and output spaces via the state space.• In a multivariable system , composed by elementary SISO subsystems Gij , the most important input-output channels can be considered as those with a larger value of tr(PjQi) [1]. • Control can be based upon a reduced model containing only the most important input-output connections, and the control structure can be selected.I.e. a diagonally dominant multivariable system, can be considered as a composition of independent SISO systems, and SISO controllers can be designed for each of these subsystems. This example is known as decentralized control structure.
tr(PQ) and its use in control structure selection.
•Result 1. The estimator in (2) is shown to be biased but asymptotically unbiased. Being its expected value:
An unbiased estimator can be therefore obtained by subtracting the term from the estimator in Equation(2), and we denote it as
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.
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(P1Q1) for different lengths of the acquired 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).• Given a multivariable discrete system G with impulse response h(k), the trace of the product of the controllability and observability Gramians of the elementary subsystem Gij can be computed as [2]:
where hij(k) is the impulse response of the elementary subsystem Gij.
Salgado and Yuz [2] used Equation (1) to estimate tr(PjQi) 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
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(PjQi) 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(PjQi) 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.
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