Least squares simulation of distributed systems

Orner, P. A.; Salamon, Peter; Yu, Wanyoung

doi:10.1109/tac.1975.1100853

Cited by 7 publications

(1 citation statement)

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“…Some examples of filter implementation using orthogonal collocation include the works of Seinfeld and Chen (1971), Tanner (1972), Viskanta et al (1975, Van den Bosch (1978), Clement et al (1980), and Sorensen et al (1980). Some examples of filter implementation using Galerkin's method include the works of Prabhu and McCausland (1972), Polis et al (1973), Orner et al (1975), It0 et al (1978), andYoshimura and Campo (1982.…”

Section: Introductionmentioning

confidence: 99%

Comparison of linear distributed‐parameter filters to lumped approximants

Cooper¹,

Ramirez²,

Clough³

1986

AIChE Journal

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Optimal distributed-parameter filters are commonly implemented using approximating lumped Kalman filtering theory. The effect of such an approximation is investigated. A theoretical development shows that there is a loss in the spatial noise correlation for the lumped approximants. Two numerical examples of engineering significance illustrate that one result of this loss is slower filter convergence for the lumped approximants relative to the full distributed-parameter filters. D. J. COOPER W. F. RAMIREZ and D. E. CLOUGH Department of Chemical EngineeringUniversity of Colorado Boulder, CO 80309 SCOPESimplifying approximations are often made when designing a distributed-parameter filter for real-time implementation. Early lumping is one simplifying approximation that is commonly employed. Lumping (spatial discretization) is a process that must occur at some point during filter development so that the equations can be programmed for digital computation. Lumping can occur at several points in the filter development. The earlier that lumping occurs, the more simplifying the effect. At one extreme, early lumping involves discretizing the state model partial differential equations and approximating them as a system of ordinary differential equations. Application of optimal filtering theory to the system of ordinary differential equations results in a lumped Kalman filter that approximates the distributed-parameter filter. This is in contrast to late lumping, at the other extreme, where the full distributed-parameter filter partial differential equations are derived before they are lumped for numerical computation.Advantages of early lumping include being able to use simpler theoretical and numerical methods for filter design and implementation. Also, the early lumped filtering algorithm requires less computation for filter equation solution, so less computer resources are used when on-line. The early lumping approximation does have negative consequences, however.In this work, we present a side-by-side comparison of early lumping vs. late lumping as applied to the least-squares filtering of linear distributed-parameter systems. This investigation includes a general theoretical formalization that is applicable to such lumping methods as finite differencing, eigenfunction expansion, orthogonal collocation, and Galerkin's method. The results of the theoretical investigation are illustrated by means of two numerical examples. The first example considers the optimal estimation of dynamic temperature distributions in a heated bar. The second example considers the optimal estimation of dynamic particle-size distributions (PSD) in a fluidized bed. CONCLUSIONS AND SIGNIFICANCEThe early lumping approximation results in a loss in the spatial noise correlation that is associated with distributed-parameter systems. This spatial noise correlation is preserved in the late-lumped distributed filter.In a spatially correlated system, an occurrence at one location will directly affect neighboring locations. The early lumping approximation...

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