Alternative Analytical Mean-Square Calculations for Dynamic Systems Subject to Random Inputs

2001 ◽  
Author(s):  
Robin C. Redfield
Keyword(s):  
Transfer Function ◽  
Frequency Response ◽  
Dynamic Systems ◽  
Analytical Techniques ◽  
System Response ◽  
Mean Square ◽  
Random Inputs ◽  
Linear Dynamic Systems ◽  
Linear Dynamic ◽  
Insight Into

Abstract Output variables of linear dynamic systems subject to random inputs are often quantified by mean square calculations. Computationally, these involve integration of the frequency response magnitude squared over all frequency. Numerically, this is an easy task and analytically, methods exist to find mean square values as functions of transfer function (frequency response) coefficients. This paper develops further analytical techniques to calculate mean-square values as functions of system pole-zero locations and as functions of eigenproperties and system matrices. These other analytical representations may provide paths to further insight into dynamic system response and optimal design/tuning of dynamic systems.

Random Response Relationships to Transfer Function and State-Space Properties

2007 ◽  
Vol 129 (5) ◽  
pp. 672-677
Author(s):  
Robin C. Redfield
Keyword(s):  
Transfer Function ◽  
State Space ◽  
Dynamic Systems ◽  
System Response ◽  
Mean Square ◽  
System Matrix ◽  
Matrix Transfer Function ◽  
Mean Square Response ◽  
The Mean ◽  
Insight Into

Output variables of dynamic systems subject to random inputs are often quantified by mean-square calculations. Computationally for linear systems, these typically involve integration of the output spectral density over frequency. Numerically, this is a straightforward task and, analytically, methods exist to find mean-square values as functions of transfer function (frequency response) coefficients. These formulations offer analytical relationships between system parameters and mean-square response. This paper develops further analytical relationships in calculating mean-square values as functions of transfer function and state-space properties. Specifically, mean-square response is formulated from (i) system pole-zero locations, (ii) as a spectral decomposition, and (iii) in terms of a system matrix transfer function. Direct, closed-form relationships between response and these properties are afforded. These new analytical representations of the mean-square calculation can provide significant insight into dynamic system response and optimal design/tuning of dynamic systems.

scholarly journals On the Uncertainty Identification for Linear Dynamic Systems Using Stochastic Embedding Approach with Gaussian Mixture Models

Sensors ◽  
2021 ◽  
Vol 21 (11) ◽  
pp. 3837
Author(s):  
Rafael Orellana ◽  
Rodrigo Carvajal ◽  
Pedro Escárate ◽  
Juan C. Agüero
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Dynamic Systems ◽  
Gaussian Mixture ◽  
Error Model ◽  
Fault Detection And Diagnosis ◽  
Deterministic Models ◽  
Linear Dynamic Systems ◽  
Linear Dynamic

In control and monitoring of manufacturing processes, it is key to understand model uncertainty in order to achieve the required levels of consistency, quality, and economy, among others. In aerospace applications, models need to be very precise and able to describe the entire dynamics of an aircraft. In addition, the complexity of modern real systems has turned deterministic models impractical, since they cannot adequately represent the behavior of disturbances in sensors and actuators, and tool and machine wear, to name a few. Thus, it is necessary to deal with model uncertainties in the dynamics of the plant by incorporating a stochastic behavior. These uncertainties could also affect the effectiveness of fault diagnosis methodologies used to increment the safety and reliability in real-world systems. Determining suitable dynamic system models of real processes is essential to obtain effective process control strategies and accurate fault detection and diagnosis methodologies that deliver good performance. In this paper, a maximum likelihood estimation algorithm for the uncertainty modeling in linear dynamic systems is developed utilizing a stochastic embedding approach. In this approach, system uncertainties are accounted for as a stochastic error term in a transfer function. In this paper, we model the error-model probability density function as a finite Gaussian mixture model. For the estimation of the nominal model and the probability density function of the parameters of the error-model, we develop an iterative algorithm based on the Expectation-Maximization algorithm using the data from independent experiments. The benefits of our proposal are illustrated via numerical simulations.

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