43321-AC9
Improved Chemical Process Operations through Data-Based Disturbance Models
1. Estimating disturbance structures from data.
We consider the following linear, discrete time state space model:
x+ = Ax + B u + Gw
y = Cx + v
in which x is the state, u is the manipulated control input, and y is
the measurement. The state at the next sample time is x+. The random
variables w and v are the process and measurement noises,
respectively. The noises w and v are assumed to be identically and
independently distributed as normals with mean zero and covariances Q
and R, respectively. The goal of state estimation is to determine the
most likely value of the state x given the model (A, B, C, G, Q, R)
and past and current measurements of y.
There are two challenges. For typical chemical processes, we do not know Q and R and must determine them from process operating data. Secondly, we often do not know the G matrix, and must determine that also from process operating data. The G matrix shapes the disturbance w entering the state. It is unlikely to have more than a few independent disturbances that affect the states. This would imply a tall G matrix with more rows than columns. In previous work in our group, the estimation of the covariances Q, R was presented. The estimation technique was based on the correlations between the measurements at different times. This technique and other previous techniques in the literature assume that the disturbance structure as given by the G matrix is known. In the absence of any knowledge about G the incorrect assumption that G is an identity matrix is often made, which implies that an independent disturbance enters each of the states.
Knowledge about the G matrix then helps improve the state estimator tuning and as a result the closed loop control performance. We developed a technique to find the structure of the G matrix from data and ensure that it represents the minimum number of independent disturbances entering the state. In addition we formulated the problem as a convex semidefinite optimization to enforce positive definite constraints on the estimated covariances. The estimated G matrix then provides both the smallest number of independent disturbances required to explain the operating data, and the structure of the disturbance. This work has appeared in 2009 in the journal Automatica and AIChE Journal [1-2].
2. Industrial Implementation
Industrial collaborations with ExxonMobil Chemicals and Eastman
Chemicals have been established to implement the techniques developed
in this grant.
ExxonMobil Chemicals provided industrial operating data on a nonlinear
blending drum that was being controlled for level and concentrations.
The techniques in Section 1 was extended to nonlinear models and
implemented on the blending tank data. Using a first principles based
model for the blending tank, the covariances of the noises affecting
the state and measurements were estimated. The estimated covariances
were used to improve the state estimation part of the controller of
this application and to obtain information about the structure of the
G matrix. This work has been submitted for publication.
3. Nonlinear State Estimation
Nonlinear state estimation is a challenging systems theory problem.
One approach to this problem receiving attention recently is particle
filtering, which is a nonlinear state estimation technique based on
Monte-Carlo type sampling. The drawback of particle filtering is the
lack of robustness in the presence of unmodelled disturbances or poor
initial guesses. Another option is moving horizon estimation, which
is a robust optimization based state estimation technique. The
nonlinear optimization may be prohibitively expensive, however, for
online use in state estimation. As part of this research, the
combination of a particle filter with moving horizon estimation was
evaluated as a better alternative than either method alone. This
combination of methods showed good results for both unmodelled
disturbances and for poor guesses for the initial state.
This work has been presented at the 2008 annual AIChE meeting and 2008
Sandia Workshop on Large-Scale Inverse Problems. It will be submitted
this year for publication [3-4].
[1] Rajamani, M. R., 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.
[2] Rajamani, M. R. and J. B. Rawlings. Estimation of the disturbance
structure from data using semidefinite programming and optimal
weighting. Automatica, 45:142-148, 2009.
[3] Rawlings, J. B. and M. R. Rajamani. A hybrid approach for state
estimation: Combining moving horizon estimation and particle
filtering. In Sandia CSRI Workshop, Large-Scale Inverse Problems
and Quantification of Uncertainty, Santa Fe, New Mexico, September
2007.
[4] Lima, F. V. and J. B. Rawlings. Nonlinear stochastic modeling for
state estimation of an industrial polymerization reactor. In AIChE
Annual Meeting, Philadelphia, November 2008.
