(9d) Determining Sensor Locations for Stable Nonlinear Systems: the Multiple Sensor Case

This paper presents a new technique for sensor location for nonlinear systems based on the total response energy of the outputs. In this approach sensors are placed in the system so that a trade-off between the total output energy of the system, sensor cost, and measurement redundancy is achieved. The presented technique results in a mixed-integer nonlinear programming problem when multiple sensors are placed, however, the optimization problem can be reformulated to significantly reduce the computational burden required for its solution. The resulting optimization problem is ideally suited for solution by genetic algorithms (GA) as only binary variables need to be used for the optimization after reformulation has been applied. The technique has been illustrated by applying it to a distillation column where up to 6 temperature sensors are placed along the height of the column.

The output energy of a linear system corresponds to the singular values of the observability gramian and, accordingly, the total output energy is given by the trace of the observability gramian [3]. Similarly, the output energy of nonlinear systems can be computed by using observability covariance matrices. These observability covariance matrices can be viewed as extension of linear observability gramians to nonlinear systems and can, therefore, be used for the observability analysis of nonlinear system over an operating region [6]. The total output energy of the system is equal to the sum of the diagonal elements of the observability covariance matrix. Therefore the key idea behind achieving a high degree of observability is to maximize the trace of the observability covariance matrix.

Measurement redundancy is also a desirable goal for sensor placement in addition to increasing observability of a system. Redundancy in a measurement refers to the ability to be able to make statements about a state variable even if a sensor fails. The presented sensor location technique incorporates the concept of redundancy and sensors can be placed in the system so that a desirable amount of redundancy for state variables is present in the sensor network.

A mixed integer nonlinear optimization problem has been formulated to take into account the total sensor cost, process output energy, and measurement redundancy. Sensors are placed in the system where a trade-off between process information, sensor cost, and information redundancy is taken into account. The optimization problem is reformulated such that it decomposes into a problem which is significantly less computationally demanding. This is achieved by performing the main steps of the observability and redundancy analysis outside of the optimization loop. The resulting problem is in a form which makes it very suitable for solution by a genetic algorithm as optimization variables correspond to possible locations of the sensors and each vector entry contains information in a binary form. As such a GA can be directly applied without having to discretize the solution space [4].

The presented technique has been applied to a nonlinear binary distillation column. The locations of temperature sensors have been computed for monitoring the distillation column subject to an upper bound on the number of sensors. Sensor location of up to six temperature sensors has been investigated in the presented work.

References:

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(2) Chmielewski, D.J.; Palmer, T.; Manousiouthakis, V. On the theory of optimal sensor placement. AIChE Journal 2002, 48, 1001.

(3) Fairman, F.W. (1998). Linear Control Theory: The State Space Approach. West Sussex, England : John Wiley & Sons.

(4) Goldberg, D.E. (1989). Genetic algorithm in search, optimization and machine learning. MA: Addison-Wesley.

(5) Muske, K.R.; Georgakis, C. Optimal measurement system design for chemical processes. AIChE Journal 2003, 49, 1488.

(6) Singh, A.K.; Hahn, J. On the use of empirical gramians for controllability and observability analysis. Proceedings ACC 2005. Portland, Oregon.