(518a) Development of Constrained Estimation Algorithms for Distributed Parameter Systems for Satisfying Mass and Energy Conservation

Most real-life industrial systems can be accurately modeled as distributed parameter systems having temporal and spatial variation of both input and state variables. However, the development of dynamic models for analyzing complete spatial domain for such systems can be challenging and computationally expensive. The use of an optimal quantitative estimation algorithm utilizing a dynamic model and plant measurement data for state and parameter estimation of non-linear dynamic distributed process systems can be very helpful for process monitoring and other applications. The development of estimation techniques for distributed parameter systems has been an active research field for years which can be categorized into two major types - one is based on modifications of recursive Bayesian estimators like Kalman filter (KF), extended Kalman filter (EKF), unscented Kalman filter(UKF) or particle filters, and the other is based on optimization-based methods like the moving horizon estimator (MHE) ^[1]. Dynamic models used for estimation are typically based on partial differential equations (PDEs) or differential-algebraic equations (DAEs). However, due to the complex relationship among state variables in spatial and time domain and limited availability of spatio-temporal measurement data, estimates especially from recursive Bayesian estimators can lack physics constraints. For example, the estimates may violate mass and atom conservation in a distributed reactive system or may fail to exactly satisfy energy conservation in a heat exchanger or boiler system along the spatial domain for each location. The prediction step of an Bayesian estimator algorithm where the dynamic model includes such physics constraints would satisfy these constraints but in the update step of the estimator where real-time operating measurement data is utilized may lead to violations of mass and energy balances. While such constraints can be imposed for optimization-based estimation methods, they can be difficult to solve real-time for complex distributed systems. This talk will focus on constrained state estimation that satisfies mass and energy balance for each spatial location for distributed systems given by partial differential algebraic equation (PDAE) or differential algebraic equation (DAE) systems where the state and parameter estimation is undertaken by a modified extended Kalman filter (EKF).

Existing approaches for constrained estimation while using Bayesian estimators differ whether the process system is linear or nonlinear ^[2]. Ad-hoc clipping methods have been proposed for linear systems using standard KF for satisfying linear and nonlinear constraints using projection-based approaches where the unconstrained estimates are projected onto the constraint space ^[3]. Furthermore, for handling nonlinear constraints in linear systems, local linearization is done for the nonlinear constraints and then used in the KF to project the obtained estimate towards the nearest constrained value ^[4]. Although these methods are suitable for handling linear or nonlinear equality constraints in linear systems, they tend to give suboptimal results for highly nonlinear systems with equality and inequality constraints. Recursive optimization-based techniques such as recursive nonlinear dynamic data reconciliation (RNDDR) for nonlinear systems governed by ODEs have been proposed for state estimation ^[5] . An extension of this RNDRR approach for DAE systems based on similar methodology for handling constraints has also been investigated ^[6]. In both these approaches, a constrained optimization problem is solved at each step for enforcing the constraints on the system. A one-step correction method for enforcing equality constraints on distributed DAE model using a modified EKF framework has been developed where the constraint is applied to a minimization formulation like the RNDDR approach ^[7]. However, the optimization problem was solved initially only to enforce the constraints, and then the EKF update step was propagated ahead in time already bounded by the constraints. Recently, the use of reduced order models for constrained state estimation in distributed systems using a DAE-EKF framework has been proposed for enforcing bound constraints on the distributed system ^[8].

The modified estimation techniques discussed above are computationally not expensive and can be utilized for specific applications to impose bound or simple equality constraints. However, satisfying mass and energy balances as equality constraints for a complex distributed model requires significant modification to the EKF framework. Furthermore, there is also a need to keep the computation time low for such an algorithm. Optimization-based recursive approaches like RNDRR, with their generic constraint formulation, have the potential to solve such problems. However, they become intractable because of their high computation burden for distributed models where measurements of spatial accumulation of mass and energy are not easily available, and available information about accumulation depends on the spatial discretization used for model development. The current literature lacks any relevant work on estimation using recursive Bayesian estimators that can enforce mass and energy constraints for state estimation of distributed PDAE/DAE systems. In this work, we have developed an algorithm for constrained estimation of distributed system by modifying the update step of the EKF for satisfying mass and energy balances as well as any other equality constraint that must be satisfied. In this approach, the Kalman gain is modified to satisfy the constraint for each spatial location by solving a reduced order optimization problem instead of a generic optimization formulation without using the entire dynamic model for projection onto the constrained space. Performances of the proposed algorithm have been evaluated by applying it to a distributed tubular reactor system and to a superheater system of a power plant boiler. The algorithm developed in this work has considerably reduced computation time compared to optimization-based estimators and the approach is generic and can be readily applied to other distributed systems.

References

[1] N. Amor, G. Rasool, and N. C. Bouaynaya, “Constrained State Estimation -- A Review.” arXiv, Mar. 11, 2022. doi: 10.48550/arXiv.1807.03463.

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[5] P. Vachhani, R. Rengaswamy, V. Gangwal, and S. Narasimhan, “Recursive estimation in constrained nonlinear dynamical systems,” AIChE Journal, vol. 51, no. 3, pp. 946–959, 2005, doi: 10.1002/aic.10355.

[6] R. Kumar Mandela, R. Rengaswamy, S. Narasimhan, and L. N. Sridhar, “Recursive state estimation techniques for nonlinear differential algebraic systems,” Chemical Engineering Science, vol. 65, no. 16, pp. 4548–4556, Aug. 2010, doi: 10.1016/j.ces.2010.04.020.

[7] P. Mobed, S. Munusamy, D. Bhattacharyya, and R. Rengaswamy, “State and Parameter Estimation in Distributed Constrained Systems. 1. Extended Kalman Filtering of a Special Class of Differential-Algebraic Equation Systems,” Ind. Eng. Chem. Res., vol. 56, no. 1, pp. 206–215, Jan. 2017, doi: 10.1021/acs.iecr.6b02796.

[8] G. Seth, S. C. Patwardhan, and M. Bhushan, “Constrained profile estimation for distributed parameter system in one dimension using orthogonal collocation,” Journal of Process Control, vol. 128, p. 103011, Aug. 2023, doi: 10.1016/j.jprocont.2023.103011.