Sensor location for nonlinear dynamic systems via observability analysis and MAX-DET optimization

Abstract This paper presents an approach for determining the optimal placement of multiple sensors for processes described by a class of nonlinear dynamic systems. This approach is based upon maximizing a criterion, i.e., the determinant, applied to the empirical observability Gramian in order to optimize certain properties of the process state estimates. The determinant directly accounts for redundancy of information for placing multiple sensors via the covariance terms in the observability matrix. However, the resulting optimization problem is nontrivial to solve as it is a mixed integer nonlinear programming problem. In order to address this point, this paper also presents a decomposition of the optimization problem such that the formulated sensor placement problem can be solved quickly and accurately on a desktop PC. Properties of the presented technique are demonstrated and discussed in two case studies, one involving a binary distillation column and the other a packed bed reactor.

[1]  Stephen P. Boyd,et al.  Determinant Maximization with Linear Matrix Inequality Constraints , 1998, SIAM J. Matrix Anal. Appl..

[2]  Alain Vande Wouwer,et al.  An approach to the selection of optimal sensor locations in distributed parameter systems , 2000 .

[3]  A. Emery,et al.  Optimal experiment design , 1998 .

[4]  Age K. Smilde,et al.  Variable selection methods as a tool to find sensor locations for distributed parameter systems , 2008 .

[5]  J. Hahn,et al.  On the use of empirical gramians for controllability and observability analysis , 2005, Proceedings of the 2005, American Control Conference, 2005..

[6]  J. Kiefer Optimum Experimental Designs , 1959 .

[7]  Thomas F. Edgar,et al.  An improved method for nonlinear model reduction using balancing of empirical gramians , 2002 .

[8]  Zuyi Huang,et al.  Sensor Network Design via Observability Analysis and Principal Component Analysis , 2007 .

[9]  Eric Walter,et al.  Qualitative and quantitative experiment design for phenomenological models - A survey , 1990, Autom..

[10]  D. Ucinski Optimal measurement methods for distributed parameter system identification , 2004 .

[11]  Stephen P. Boyd,et al.  Sensor Selection via Convex Optimization , 2009, IEEE Transactions on Signal Processing.

[12]  John F. MacGregor,et al.  Optimal sensor location with an application to a packed bed tubular reactor , 1980 .

[13]  Miguel J. Bagajewicz,et al.  Design of Nonlinear Sensor Networks for Process Plants , 2008 .

[14]  Gérard Cornuéjols,et al.  An algorithmic framework for convex mixed integer nonlinear programs , 2008, Discret. Optim..

[15]  N. L. Johnson,et al.  Breakthroughs in Statistics , 1992 .

[16]  H. Wynn Results in the Theory and Construction of D‐Optimum Experimental Designs , 1972 .

[17]  Juergen Hahn,et al.  Determining Optimal Sensor Locations for State and Parameter Estimation for Stable Nonlinear Systems , 2005 .

[18]  G. Froment,et al.  Parametric sensitivity and runaway in fixed bed catalytic reactors , 1970 .

[19]  H. Weber,et al.  Analysis and optimization of certain qualities of controllability and observability for linear dynamical systems , 1972 .

[20]  J. Alvarez,et al.  On the effect of the estimation structure in the functioning of a nonlinear copolymer reactor estimator , 2004 .

[21]  David Zumoffen,et al.  A Systematic Approach for the Design of Optimal Monitoring Systems for Large Scale Processes , 2010 .

[22]  J. Hahn,et al.  Sensor Location for Stable Nonlinear Dynamic Systems: Multiple Sensor Case , 2006 .

[23]  Maciej Patan,et al.  Sensor network design for the estimation of spatially distributed processes , 2010, Int. J. Appl. Math. Comput. Sci..

[24]  Donald J. Chmielewski,et al.  Covariance-based hardware selection-Part II: equivalence results for the sensor, actuator, and simultaneous selection problems , 2006, IEEE Transactions on Control Systems Technology.

[25]  Age K. Smilde,et al.  Selection of optimal sensor position in a tubular reactor using robust degree of observability criteria , 2000 .

[26]  J. P. Babary,et al.  Observer Design and Sensor Location in Distributed Parameter Bioreactors 2 , 1992 .

[27]  Miguel J. Bagajewicz,et al.  Design and retrofit of sensor networks in process plants , 1997 .

[28]  A. Krener,et al.  Nonlinear controllability and observability , 1977 .

[29]  J. M. A. Scherpen,et al.  Balancing for nonlinear systems , 1993 .

[30]  Denis Dochain,et al.  On the use of observability measures for sensor location in tubular reactor , 1998 .

[31]  Ch. Venkateswarlu,et al.  Optimal selection of sensors for state estimation in a reactive distillation process , 2009 .

[32]  D. Georges The use of observability and controllability gramians or functions for optimal sensor and actuator location in finite-dimensional systems , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

[33]  David L. Woodruff,et al.  Pyomo — Optimization Modeling in Python , 2012, Springer Optimization and Its Applications.

[34]  W. Brogan Modern Control Theory , 1971 .