Optimal Filtering and Residual Analysis in Errors-in-Variables Model Identification

Abstract

Dynamic model identification from time series data is a critical component of process control, monitoring, and diagnosis. An important adjunct of model identification is the derivation of filtered estimates of the variables and consequent one-step-ahead prediction errors (residuals) which are very useful for model assessment and iterative model identification. In this work, we present an optimal filtering and residual generation method for the errors-in-variables (EIV) scenario, wherein both the input and output variable measurements are contaminated with errors. The main idea is to combine an EIV-identification strategy with the EIV-Kalman filter (EIV-KF) that is known to provide optimal filtered estimates and residuals of both inputs and outputs for a linear dynamical process in the EIV case. In this work, we combine the EIV-KF with the dynamic iterative principal component analysis (DIPCA) approach that has been recently developed for EIV model identification. This work assumes prominence in that the optimally generated residuals are critical to the tasks of model assessment, fault detection, and diagnosis. The use of residuals in model assessment and reidentification is illustrated in this article, while pointing out that the use of DIPCA alone leads to nonunique filtered estimates and hence nonunique residuals. We remark that the proposed method can be used with any other EIV identification technique.

Identification of Errors-in-Variables Models Using Dynamic Iterative Principal Component Analysis

“…From Table IV, It can be clearly observed that the last 5 eigenvalues are not equal to unity for d guess = 5, while for d guess = 4, it can be observed that the last 4 eigenvalues are approximately unity. This can also be confirmed from hypothesis testing for equality of the smallest d guess eigenvalues, which are expected to be unity [28]. The results for hypothesis testing for each value of d guess are reported in Table V.…”

“…The lagged data matrix Z L is constructed for L = 6, and the proposed algorithm is applied starting with the maximum possible value of d guess = 6. We apply the hypothesis test as described in [28] to test whether the smallest d guess eigenvalues are all equal. The hypothesis test results are presented in Table X for different values of d guess .…”

“…Although the system order can be correctly estimated, the model parameters cannot be directly estimated using the eigenvector corresponding to the smallest eigenvalue, as discussed in the preceding section. This is because the multiple linear equations obtained from eigenvectors corresponding to unity eigenvalues can be linear combinations of the discrete shifted model equations [26], [28]. A procedure to recover the model parameters from these eigenvectors is described.…”

Modeling and Simulation of a Packed Column Batch Still for Fruit Wine Distillations

“…The four free model parameters in (62) were regressed by minimum least-squares using a global optimization algorithm. We choose a global over a local solver given our model's strong nonlinearities and multiple free parameters, making it prone to falling into a local optimum.…”

“…States are represented by (59), implicit variables by (60), and input variables by (61), which include the manipulated variables and measured disturbance 𝑇 𝑤.𝑖𝑛 . The four model parameters in (62) were regressed from the experimental data. A consistent initial condition is obtained by solving ( 55) and ( 56) simultaneously at steady-state (zeros on the left-hand side).…”