ACS PRF | ACS
All e-Annual Reports
43229-G9
Determining Optimal Sensor Locations for State and Parameter Estimation for Nonlinear Dynamic Systems
The purpose of the funded activity was to develop a novel sensor network design strategy for monitoring chemical processes described by nonlinear dynamic systems and which can contain uncertainties in the model parameters. The activities that are required to reach this goal are
• to investigate a technique for placing a single sensor for either state or parameter estimation for a nonlinear system
o this technique for placing a single sensor can then be applied to sequentially place additional sensors, assuming some states are already measured
• to extend the developed methodology for simultaneously placing multiple sensors such that an entire sensor network can be designed
• to make use of all the information that can be obtained from a system which includes a model of the process and simulation and sensor data for determining the state of the process
• to take into account the effect that disturbances have on the sensor network design not just for reconstructing the disturbance but also for monitoring other process variables
The main idea behind establishing a technique for placing single sensors is to combine the computation of the observability covariance matrix with established measures for locating sensors for linear systems. This combination of directly using the nonlinear model and applying established measures for linear systems allows more accurate prediction of optimal sensor locations while at the same time results in a computationally tractable procedure. Additionally, by augmenting the system with parameters to be estimated, it can be shown that a submatrix of the observability covariance matrix, i.e. the part of the matrix corresponding to the augmented states, is closely related to the Fisher Information Matrix (FIM) commonly used for sensor location for parameter estimation. This portion of the work was performed by Abhay Singh who was a PhD student in the PIs group during the funded period. The developed technique resulted in a journal paper [1] which was published in the ACS journal Industrial & Engineering Chemistry Research.
The extension of the developed methodology to placing multiple sensors has been performed by formulating the sensor network design as an optimization problem. Sensors are placed such that a trade-off between process information, sensor cost, and information redundancy is taken into account. A mixed integer nonlinear optimization problem has been formulated to design a sensor network. The resulting problem is in a form that makes it very suitable for solution by a genetic algorithm (GA) as the optimization variables are given in binary form indicating if a certain state is measured. As such a GA can be directly applied without having to discretize the solution space. The result of applying a GA to solve this sensor network design problem is that processes of significant size can be analyzed with relatively small computational effort. This second part of the work has also been performed by Abhay Singh and resulted in a conference paper presented at the Chemical Process Control 7 conference [5] and won the Best Contributed Paper award. Additionally, the results have been published in Industrial & Engineering Chemistry Research [2].
The third aspect that has been considered is to include principal component analysis (PCA) as part of the measure for determining redundant information among sensors. PCA allows to more accurately use the covariance information in addition to variances when compared to traditional measures for sensor placement. As these traditional measures had been extended in the work shown in [2, 5] the investigated sensor network design procedure will also benefit from this work. This portion of the work was performed over the summer of 2006 and involved Abhay Singh as well as Manish Misra, who was supported by the summer research fellowship. Dr. Misra's expertise on PCA was crucial for this portion of the work and a paper presented at the 2006 AIChE Annual Meeting [7] and a paper currently under review at the ACS journal Industrial & Engineering Chemistry Research [3].
The last portion of the work supported by this grant involved sensor network design under the influence of disturbances [6], the effect that observability analysis has on sensor placement for distributed systems [8], and the choice of selecting parameters based upon observability analysis [9]. This work involved Yunfei Chu, who is another PhD student in the PIs group, as Abhay Singh graduated with a PhD in August 2006 and started a job as process engineer for Shell Global Solutions in Houston.
Summarizing, the grant PRF# 43229-G9 partially supported two PhD students, one visiting professor via a summer research fellowship and resulted in 4 journal papers [1-4] and 5 conference papers [5-9] that acknowledge ACS-PRF support.
