Abstract
We address the problem of nonlinear dynamical system identification in state space formulation using an approximate Gaussian process (GP) regression framework where the basis for GP are learned from data. Approximate GP inference is used to address the high computational cost of exact GP frameworks, and is based on basis function expansion concept. We address the design of these basis functions using data and eigenvalue decomposition framework. The need for such design arises in practical applications where the collected data has outliers, discontinuities and other non-stationary characteristics which limit not only the applicability but also the performance of traditional GP framework with inherent stationary assumptions. Using the available data set, a kernel eigenvalue problem is formulated which is solved using Monte Carlo techniques to construct basis functions. Then, a low-rank approximation to the exact GP is obtained by an approach similar to kernel principle component analysis (k-PCA). The coefficients in the basis function expansion and other unknown parameters are learned from data using sequential Monte Carlo technique. The proposed GP framework for dynamical system identification is tested and validated against fixed basis framework using simulations and experimental data.
Highlights
In practical industry, a number of critical operations such as process control [1], [2], state estimation [3], process monitoring [4], fault diagnosis [5], gross error detection and data reconciliation [6] are mostly model based where a good model describes the underlying process dynamics with an accurate and correct mathematical formulation
In all the following examples, we separate the data into training and testing data and compare the results in terms of following two performance metrics: root mean square error (RMSE), and mean negative log probability (MNLP), obtained on testing data
We report the averaged RMSE, MNLP and execution time results in Table 1 which have been computed ans averaged after performing the simulations 20 times
Summary
A number of critical operations such as process control [1], [2], state estimation [3], process monitoring [4], fault diagnosis [5], gross error detection and data reconciliation [6] are mostly model based where a good model describes the underlying process dynamics with an accurate and correct mathematical formulation. The model in (15) is in a parametric form which allows us to make inference on a rather large set of parameters contained in unknown weight matrix along with the system state xt , and process noise covariance matrix Q This identification helps in identifying the function f(.) according to (14), provided basis functions are known. This extended formulation of SSM enables to incorporate the designed basis functions during inference and learning process and provides much more reliable state trajectory estimates. Get estimate of Xt using PGAS as described in [40] end Obtain state trajectory x0:T using systematic resampling [41] of all Xt ∀ t Sample A and Q using (31) end end
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