(542b) Computing Sensor Locations for Nonlinear Systems under the Influence of Disturbances

Sensor network design is a topic which has received a large amount of attention in recent years. Most of the performed work focuses either on maximizing some norm of observability of a system by choosing a sensor network or on using the Kalman filter error covariance matrix for computing optimal sensor locations. While initial approaches focused on linear systems or linearized nonlinear systems [1, 2], more recent work has also dealt with nonlinear processes without the requirement of linearization [3, 4, 5, 6]. However, the topic of determining a sensor network for nonlinear systems under the influence of disturbances has received comparatively little attention.

This work presents a technique for designing sensor networks under the influence of disturbances. The focus of the method is not simply on determining the disturbance itself, but on designing sensor networks that allow for reliable state reconstruction in addition to computing the magnitude of the disturbance. This is achieved by balancing the empirical observability and controllability gramians of the system, where the disturbances act as the inputs to the system. The empirical controllability gramian contains the information describing the effect that the disturbances have on the states of the system, while the empirical observability gramians contains the information about the state-to-output behavior of the process. Both empirical gramians will be balanced and the sum of the resulting Hankel singular values serves as a measure for the importance of a sensor location for state and disturbance estimation: (1) directions in state space which are not uniquely affected by the disturbances will automatically be ignored for the sensor network design as this is reflected in the empirical controllability gramian; (2) directions in the state space which cannot be uniquely observed for a chosen sensor network design will also be reflected as such; (3) balancing the empirical controllability and observability gramians ensures that the measure describing the sensor network will simultaneously reflect state reconstruction and disturbance estimation.

The technical description of the procedure is as follows: In a first step the empirical controllability gramian is computed where the disturbances are the inputs to the system. Next, empirical observability gramians have to be computed for each sensor network under consideration. The presented work makes use of properties for the computation of empirical observability gramians if multiple sensors are present in a system to reduce the computational load. The Hankel singular values for each sensor network configuration are computed by balancing the computed empirical controllability gramian with the empirical observability gramian for each possible sensor network structure.

The presented technique has been applied to two examples. In the first case, the technique has been applied to distillation column described by 32 nonlinear differential equations and in the second example, to a fixed bed reactor model described by nonlinear partial differential equations. The performance of an extended Kalman filter has been compared for the case of optimal and non-optimal measurements. The presented results show improved state estimation for the case when the measurement is placed at a location which was computed to be optimal.

REFERENCES:

[1] Muske, K.R.; Georgakis, C. Optimal measurement system design for chemical processes. AIChE Journal 2003, 49, 1488.

[2] Van den Berg, F.W.J.; Hoefsloot, H.C.J.; Boelens, H.F.M.; Smilde, A.K. Selection of optimal sensor position in a tubular reactor using robust degree of observability criteria. Chemical Engineering Science 2000, 55, 827.

[3] Wouwer, A.V.; Point, N.; Porteman, S.; Remy, M. An approach to the selection of optimal sensor locations in distributed parameter systems. Journal of Process Control 2000, 10, 291.

[4] Alonso, A.A.; Kevrekidis, I.G.; Banga, J.R.; Frouzakis, C.E. Optimal sensor location and reduced order observer design for distributed process system. Computers & Chemical Engineering 2004, 28, 27.

[5] Singh, A.K.; Hahn, J. Determining Optimal Sensor Locations for State and Parameter Estimation for Stable Nonlinear Systems. Industrial Engineering Chemistry Research 2005, 44(15), 5645.

[6] Singh, A.K.; Hahn. J. Sensor Location for Stable Nonlinear Systems: Placing Multiple Sensors. Proceedings Chemical Process Control 7 2006 , Lake Louise, Canada.