State and Parameter Estimation in Distributed Constrained Systems. 2. GA-EKF Based Sensor Placement for a Water Gas Shift Reactor

Abstract

Growing complexity of processes necessitates the use of information from sensors along with first-principles mathematical models to ensure safe and optimal operations. Use of sensors in complex processes requires identifying optimal location of sensors that can maximize information from a process. Classical sensor placement approaches for nonlinear systems that use state estimation schemes usually incorporate linearized models around the steady-state operating point. However, such approaches face difficulties when abnormalities or disturbances drift the system away from the normal operating point. Therefore, use of models that can appropriately track the behavior of the system in the sensor placement framework are of interest. However, the computational complexity of the detailed models makes such approaches intractable. In this work, we develop a sensor placement framework that combines genetic algorithms and the extended Kalman filter to obtain optimal sensor locations. Within this framework, we have investigated the applicability of simplified models by comparing the results of sensor placement for simplified and detailed models. The effect of the simplified models on the estimation accuracy and the optimal sensor network is further evaluated by analyzing the sensitivity to different parameters. Results show that an appropriate simplified model can not only significantly reduce the computational time of the sensor placement algorithm, but also yield a senor network with similar characteristics as the sensor network obtained using the detailed model. Further, information loss in using simplified models in sensor placement may be partially compensated through tuning of the filter parameters, resulting in acceptable, optimal sensor placement solutions.

Optimal Sensor Placement Design for Profile Estimation of Distributed Parameter Systems

“…(ii) The SPD obtained by employing the following objective function taken from the existing literature. 32 This objective function is formulated based on the normalized estimation error and is defined as…”

“…The metrics discussed above are evaluated using the optimal SPD determined using the proposed objective function, denoted as ϑ 5 . These results are then contrasted with the metrics obtained from two alternative SPDs: Heuristic SPD where sensors are placed uniformly along the spatial length of the reactor. The SPD obtained by employing the following objective function taken from the existing literature . This objective function is formulated based on the normalized estimation error and is defined as J normale normalr normalr normalo normalr = ∑ k = 1 T s true( ∑ r = 1 n N P S E r false( k false) true) Here, T s refers to the selected simulation time interval, NPSE r stands for normalized profile squared error for the r th state variable, defined as normalN normalP normalS normalE r ( k ) = ∑ i = 1 N f + 1 ( x r false( z i , k false) − χ̂ r false( z i , k false| k false) x r false( z i , k false) ) 2 , r = 1 , 2 , ... , n Here, at the k th sampling instant, x r ( z i , k ) and …”

“…Further, they correspond to minimization of an objective function computed at either an arbitrarily specified time or under steady-state conditions. Solving sensor placement problem for dynamical nonlinear systems becomes intractable in many cases …”

“…Solving sensor placement problem for dynamical nonlinear systems becomes intractable in many cases. 32 Among the recent approaches, Olanrewaju and Al-Arfaj 28 investigated sensor placement in a reactive distillation column. They employed a nonlinear model with differential equations within the extended Kalman filter (EKF) framework.…”

“…However, their work did not extensively focus on the sensor placement problem. Mobed and co-workers , studied the effect of using simplified models along with EKF for sensor placement for a water gas shift reactor modeled by a set of stochastic differential algebraic equations (DAEs) obtained after discretizing the original PDE model using the finite element method. The objective function used in their work was to minimize the sum of normalized estimation errors at discrete spatial locations, with the optimal sensor locations to be chosen from a set of candidate sensor locations using a genetic algorithm (GA).…”