(222c) Sensor Placement for Process Networks Based on the Sensitivity Analysis

Sensor placement or selection is one fundamental problem in process control and arises in various applications including target tracking, agro-hydrological systems, power systems, and wireless networks. Sensor placement has been a computationally challenging problem when the number of potential sensors is large. In this work, we propose a sensitivity based approach to perform sensor placement or selection and the proposed approach is much more computationally efficient than the existing approaches.

In sensor placement, observability plays an important role. The observability Gramian for linear system or the empirical observability Gramian has been often used in sensor placement. Different approaches based on the observability Gramian have been developed such as approach based on maximizing the trace of the Gramian [1], maximizing the determinant of the Gramian [2]. When using the trace of an observability Gramian matrix as the measure of observability, it in many cases may lead to meaningless results since a system can be unobservable if it has an eigenvalue of zero, despite a potentially large trace. When the determinant of a Gramian matrix is used as a measure of the observability, it is possible to use the smallest eigenvalue as an additional measure in order to avoid the situation that the eigenvalue of the Gramian matrix is almost zero and the system is not fully observable. However, one major limitation of the above methods is that the computational complexity is very high when applied to nonlinear systems, because these methods would require simulating the system many times for each perturbed initial condition in order to compute the ensemble average.

Graph-based methods are another class of popular methods used for the sensor selection in networked systems based on the structural observability theory [3]. By examining the strongly connected components (SCCs) in a graph describing node connections, graph-based methods can determine the number and locations of sensor nodes for ensuring structural observability [4,5]. The minimal set includes one sensor in each root SCC (an SCC with no incoming edges) of the observability inference diagram. Graph-based methods has become an influential method and has been used to provide insights into the relation between network topology and observability. However, since these approaches do not explicitly take into account model parameters, in some cases, the estimator designed based on the sensors selected by the graph-based approaches may lead to less accurate state estimates [6]. A workaround is to simultaneously deal with the sensor selection and state estimation, which has the potential to improve the overall estimation performance [7].

Based on the above observations, in this work, we propose a sensitivity-based approach to determine the minimum number of sensors and their optimal locations for state estimation. The proposed approach is much more computationally efficient compared with existing methods. In the proposed approach, the local sensitivity matrix of the measured outputs to initial states is used as a measure of the degree of observability. The minimum number of sensors is determined in a way such that the local sensitivity matrix is full column rank. The subset of sensors that satisfies the full-rank condition and provides the maximum degree of observability is considered as the optimal sensor placement. Successive orthogonalization of the sensitivity matrix is conducted in the proposed approach to significantly reduce the computational complexity in selecting the sensors. To validate the effectiveness of the proposed method, it is applied to two processes including a chemical process consisting of four continuous stirred-tank reactors and a wastewater treatment plant. In both cases, the proposed approach can obtain the optimal sensor subsets.

References

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