(325c) Identifying Noise Covariance Matrices From Industrial Datasets

Performance monitoring for any controller, including model predictive controllers, requires realistic measures of optimal performance.  The achieved performance is then compared to these optimal benchmarks.  However, the calculation of optimal benchmarks requires accurate knowledge of the disturbances affecting the system.  For a linear state space model with normal, zero-mean, white noise disturbances affecting the system, autocovariance least-squares (ALS) provides a method of identifying these covariances based on input-output data alone, based on the correlations of the prediction error (using any stable estimator) [2,3].  ALS uses the model and current estimator to predict how the prediction error is correlated and forms a least-squares problem by comparing the model to the data.  This technique distinguishes the process noises affecting the states from the measurement noises which affect only the outputs, based on how the process noises propagate through the system.  The ALS method can also be used to identify the optimal covariances to be used in an integrated disturbance model to provide offset free control) [4].  

However, applying ALS in practice becomes challenging for large systems which have many more states than outputs.  In this case, the output data does not provide enough information to uniquely identify the process noises which affect the states of the system.  In addition, the quality of the ALS results is highly sensitive to the observability of the system.  When the system is unobservable, noises can affect the unobservable modes without affecting the measured outputs, and the ALS problem is guaranteed to not have a unique solution.  When the system is observable but the observability matrix is poorly conditioned, the system is considered weakly observable.  Since noises affecting the weakly observable modes have little effect on the measurements, this poor conditioning carries over to the ALS problem.

To overcome these challenges, a modified ALS method can be used by reducing the least-squares problem to a smaller and better-conditioned problem.  The modified method consists of: (1) Reducing the original model of the system (excluding the integrating disturbance model) to a smaller fully observable subsystem, (2) augmenting this subsystem with an integrating disturbance model, (3) performing ALS to find the process noise covariance matrix of lowest rank, and (4) transforming the results to apply to the original model.

The first step is performed to eliminate the ill-conditioning caused by the weakly observable model.  Since the original model likely consists of states which are a combination of fully observable and weakly observable modes, a similarity transformation is used to identify those modes which are weakly observable.  Since these states have little impact on the output, they can be removed without significantly reducing the ability of the model to predict future outputs.  The reduced model is chosen such that the observability matrix is as well conditioned as possible without significantly increasing the prediction error.  

In the MPC, the original state-space model is augmented with an integrating disturbance model to provide offset free control.  These integrating disturbances are treated as additional states in the system.  However, if these states are included in the model when it is transformed and reduced, part of the integrating states could be removed in the first step.  Because it is essential for offset free control that these extra states to behave as integrators, they are added to the model after it is reduced.  Then when ALS is preformed on the reduced and augmented system, it identifies both the covariances for these integrators as well as for the reduced states.

When there is no knowledge of the disturbance structure, even the reduced ALS problem does not have a unique solution.  Therefore, the least-squares problem is modified to find the process noise covariance matrix which has the lowest rank, i.e. the model with the fewest number of independent noises entering the system [3].  In this modification, the ALS problem is written as a weighted optimization problem to minimize both the least-squares term as well as the trace of the process noise covariance.  (The trace serves as a substitute for the rank and results in a convex optimization problem).  This problem is solved with different weights on the trace term to develop a trade-off between the  quality of fit to data (least-squares) and the number of independent disturbances (trace).  The solution is decomposed to determine the structure of the process noises.  Provided this structure meets a certain rank condition, the ALS problem has a unique solution for that noise structure.

Because the ALS problem is solved using the transformed and reduced system, the solution must be transformed back into the original coordinates.  This transformation is performed by assuming that there is no noise affecting the weakly observable states (an assumption which is reasonable as little noise from these states affects the outputs), and then using the inverse of the transformation.  The resulting process noise covariance matrix applies to the original system.

The results of ALS can then be applied to controller performance monitoring as well as estimator design.  The noise covariances can be used to provide a benchmark for the expected performance, given the noises affecting the system, and can be compared to the actual performance calculated from the input/output data [5].  For an unconstrained linear system, this theoretical performance measure is calculated analytically. When there are constraints on the system, this benchmark is estimated from simulations and Monte Carlo methods.  For nonlinear systems, Monte Carlo methods are also used to calculate the ideal benchmark, although a linearized version of the model must be used to estimate the noise covariances from ALS) [1].

One benchmark uses the current estimator and the noise covariances estimated from ALS.  The true performance of the system is expected to match this benchmark unless other factors, such as plant model mismatch or unmeasured disturbances are affecting the system.  A second benchmark is calculated to give the performance possible given the optimal estimator, whose gain is calculated using the results of ALS.  This measure indicates the degree of improvement which is possible for the system if the estimator were to be redesigned.

References

[1] F. V. Lima, M. R. Rajamani, T. A. Soderstrom, and J. B. Rawlings.  Covariance and state estimation of weakly observable systems: Application to polymerization processes.  IEEE Trans. Cont. Sys. Tech., 2012.  Accepted for publication.

[2] B. J. Odelson, M. R. Rajamani, and J. B. Rawlings.  A new autocovariance least-squares method for estimating noise covariances.  Automatica, 42(2): 303–308, February 2006.

[3] M. R. Rajamani and J. B. Rawlings.  Estimation of the disturbance structure from data using semidefinite programming and optimal weighting.  Automatica, 45:142–148, 2009.

[4] M. R. Rajamani, J. B. Rawlings, and S. J. Qin.  Achieving state estimation equivalence for misassigned disturbances in offset-free model predictive control.  AIChE J., 55(2):396–407, February 2009.

[5] M. A. Zagrobelny, L. Ji, and J. B. Rawlings.  Quis custodiet ipsos custodes?  In IFAC Conference on Nonlinear Model Predictive Control 2012, Noordwijkerhout, the Netherlands, August 2012.