The identification task consists of making a model of a system from measured input and output signals. Wiener models consist of a linear dynamic system, followed by a static nonlinearity. We derive an algorithm to calculate the maximum likelihood estimate of the model for this class of systems. We describe an implementation in some detail and show simulation results where a test system is successfully identified from data.
En Wienermodell är en olinjär struktur som består av ett linjärt dynamiskt system, följt av en dynamisk olinjäritet. Vi presenterar en metod för identifiering av Wienermodeller, genom numerisk sökning efter maximum likelihoodskattningen av parametrarna. För att undvika problem med lokala minima föreslås en initialisering baserad på en minsta kvadratskattning.
A Wiener model consists of a linear dynamic block followed by with a nonlinear static block. When identifying the parameters of such a system, the Prediction Error Method (PEM) can be used. Depending on how noise enters the system, the predictor can be difficult to express, and an approximate predictor may be interesting. The estimate obtained from using this approximate predictor is however not always consistent. In this report we investigate this inconsistency.
The identification of nonlinear systems by the minimization of a prediction error criterion suffers from the problem of local minima. To get a reliable estimate we need good initial values for the parameters. In this paper we discuss the class of nonlinear Wiener models, consisting of a linear dynamic system followed by a static nonlinearity. By selecting a parameterization where the parameters enter linearly in the error, we can obtain an initial estimate of the model via linear regression. An example shows that this approach may be preferential to trying to estimate the linear system directly form input-output data, if the input is not Gaussian. We discuss some of the users choices and how the linear regression initial estimate can be converted to a desired model structure to use in the prediction error criterion minimization. The method is also applied to experimental data.
The identification of nonlinear systems by the minimization of a predictionerror criterion suffers from the problem of local minima. To get a reliableestimate we need good initial values for the parameters. In this paper wediscuss the class of nonlinear Wiener models, consisting of a linear dynamicsystem followed by a static nonlinearity. By selecting a parameterizationwhere the parameters enter linearly in the error, we can obtain an initialestimate of the model via linear regression. An example shows that thisapproach may be preferential to trying to estimate the linear system directlyform input-output data, if the input is not Gaussian. We discuss some of theusers choices and how the linear regression initial estimate can be convertedto a desired model structure to use in the prediction error criterionminimization. The method is also applied to experimental data.
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Keywords: teaching, education, problem based learning, PBL
This paper reviews and compares two methods for fault detection and isolation in a stochastic setting, assuming additive faults on input and output signals and stochastic unmeasurable disturbances. The first method is the parity space approach, analyzed in a stochastic setting. This leads to Kalman filter like residual generators, but with a FIR filter rather than an IIR filter as for the Kalman filter. The second method is to use principal component analysis (PCA). The advantage is that no model or structural information about the dynamic system is needed, in contrast to the parity space approach. We explain how PCA works in terms of parity space relations. The methods are illustrated on a simulation model of an F-16 aircraft, where six different faults are considered. The result is that PCA has similar fault detection and isolation capabilities as the stochastic parity space approach.
This paper compares two methods for fault detection and isolation in a stochastic setting. We assume additive faults on input and output signals, and stochastic unmeasurable disturbances. The first method is the parity space approach, analyzed in a stochastic setting. The stochastic parity space approach is similar to a Kalman filter, but uses an FIR fiter, while the Kalman filter is IIR. This enables faster response to changes. The second method is to use PCA, principal component analysis. In this case no model is needed, but fault isolation will be more difficult. The methods are illustrated on a simulation model of an F-16 aircraft. The fault detection probabilities can be calculated explicitly for the parity space approach, and are verified by simulations. The simulations of the PCA method suggest that the residuals have similar fault detection and isolation capabilities as for the stochastic parity space approach.
Problem Based Learning, PBL, has been used for several years, especially in the medical commmunity. At Linköping University, now also the program in Information Technology is taught using this method. We describe how PBL is used in a basic course in control theory, including linear algebra and Laplace transforms. The experience from the first three years of the course is promising. The students are in general more active and more motivated than students in traditional courses. The teachers spend roughly the same number of hours on the course, but the focus has shifted to more contact with the students.
A Wiener model consists of a linear dynamic system followed by a static nonlinearity. The input and output are measured, but not the intermediate signal. We discuss the Maximum Likelihood estimate for Gaussian measurement and process noise, and the special cases when one of the noise sources is zero.
Many parametric identification routines suffer from the problem with local minima. This is true also for the prediction-error approach to identifying Wiener models, i.e. linear models with a static non-linearity at the output. We here suggest a linear regression initialization, that secures a consistent and efficient estimate, when used in conjunction with a Gauss-Newton minimization scheme.
The Wiener model is a block oriented model, having a linear dynamic system followed by a static nonlinearity. The dominating approach to estimate the components of this model has been to minimize the error between the simulated and the measured outputs. We show that this will, in general, lead to biased estimates if there are other disturbances present than measurement noise. The implications of Bussgang's theorem in this context are also discussed. For the case with general disturbances, we derive the Maximum Likelihood method and show how it can be efficiently implemented. Comparisons between this new algorithm and the traditional approach, confirm that the new method is unbiased and also has superior accuracy.
The Wiener model is a block oriented model having a linear dynamicsystem followed by a static nonlinearity.The dominating approachto estimate the components of this model has been to minimize theerror between the simulated and the measured outputs. We show thatthis will in general lead to biased estimates if there is otherdisturbances present than measurement noise. The implications ofBussgangs theorem in this context are also discussed. For the casewith general disturbances we derive the Maximum Likelihood methodand show how it can be efficiently implemented. Comparisons betweenthis new algorithm and the traditional approach confirm that the newmethod is unbiased and also has superior accuracy.