Optimal least-square filtering and interpolation in distributed parameter systems

Hwang, Ming‐Jiu; Seinfeld, John H.; Gavalas, George R.

doi:10.1016/0022-247x(72)90224-7

Cited by 34 publications

(6 citation statements)

References 17 publications

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“…The development of these algorithms is summarized in chronological order as follows: After Kalman ( Sakawa (1972) to the linear distributed filtering theory followed. In the late 1960s and early 1970s, nonlinear distributed filtering theory was developed (Seinfeld, 1969;Sherry and Shen, 1971;Seinfeld et al, 1971;Hwang et al, 1972;Padmanabhan and Colantuoni, 1974). The developments and applications of linear and nonlinear distributed-parameter filters during the last decade have been summarized in the works of Tzafestas (1978), Ray (1978), and Bencala and Seinfeld (1969).…”

Section: Dlstrlbuted-parameter Fllterlngmentioning

confidence: 98%

Real‐time estimation of dynamic particle‐size distributions in a fluidized bed: Theoretical foundation

Cooper

Clough

1985

AIChE Journal

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Significant noise (random error) is common in an on-line signal when measuring the particle size distribution in a fluidized bed. The application of optimal estimation theory to obtain a filtering algorithm that can rectify such measurements is detailed. Off-line simulation studies demonstrate that the filtering algorithm yields a measure of particle size distribution that is well-behaved, accurate, and able to track in real time as conditions change in the bed. D. J. Cooper and D. E. Clough Department of Chemical EngineeringUniversity of Colorado Boulder, CO 80309Significant noise (random, unmodeled error) is common in an on-line signal when measuring the particle size distribution in a fluidized bed. The application of optimal estimation theory, which provides a mathematical basis for rectifying such measurements, is detailed. An optimal filtering algorithm is derived that tempers the noisy measurements with predictions from an idealized dynamic model to yield an accurate, well-behaved, real-time measure of particle size distribution. These qualities are desirable in a measure that is being monitored, and are necessary if the measure is to be used in computer control.The theoretical development includes formulation of an idealized model that describes the mixing dynamics (including elutriation and attrition) in a fluidized bed, and the derivation of a filtering algorithm that combines predictions from this model with noisy measurements to yield the optimal estimates. Several off-line simulation studies are presented to illustrate the performance and capabilities of the filtering algorithm.A general theoretical development is presented in this work. Although substantial detail has been included, the work is not intended to stand alone for those interested in implementation of this theory. Rather, for those versed in optimal estimation theory it is intended to provide a solid platform upon which they can build given the specifics of their application. For those unfamiliar with the theory it is intended to illustrate the capabilities of optimal estimation, and to introduce the methodology of application.The general theoretical development enables others to adapt or extend the method. One example of a straightforward adaptation is to obtain an accurate, well-behaved measure of the elutriate stream particle size distribution. An extension of interest includes using a different idealized dynamic model that is specific to a particular application. CONCLUSIONS AND SIGNIFICANCEAn optimal filtering algorithm is effective in producing accurate, well-behaved, real-time estimates of particle size distribution from measurements that are corrupted with random, unmodeled error (noise). Optimal estimation theory provides the mathematical basis for computing these estimates by combining the measurement data with predictions from an idealized dynamic model. Such estimates of particle size distribution are highly desirable if this parameter is being monitored during fluidized bed operation, and are necessary if it is being used in...

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Section: Dlstrlbuted-parameter Fllterlngmentioning

confidence: 98%

Real‐time estimation of dynamic particle‐size distributions in a fluidized bed: Theoretical foundation

Cooper

Clough

1985

AIChE Journal

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“…Recently derived distributed parameter filters can be readily adapted to handle virtually any catalyst decay problem of interest; however, to illustrate how this might be done we shall adapt the filtering equations derived by Hwang, et al (1972), to a particularly common class of catalyst decay problems in a packed-bed reactor.…”

Section: The Optimal Filtermentioning

confidence: 99%

“…The optimal estimation problem corresponds to determining x(z,t), and a(t) 0 < z < 1, 0 < t < it such that the objective given in (4) is minimized. When this estimate is obtained in a sequential fashion, the resulting filter, which minimizes (4), is a special case of the distributed parameter filter derived by Hwang, et al (1972). In the case where only the states, x(z,t) are to be estimated and there is no error in the boundary conditions, the best estimates can be found from the solution of dx(z,t)/dt = f(x,xz, z, t) + ff…”

Section: The Optimal Filtermentioning

confidence: 99%

“…Thus it is useful to apply recently derived distributed parameter filters to the on-line estimation of catalyst activity profiles. In this paper we shall adapt the distributed filter of Hwang, et al (1972), to a particular catalyst decay problem and then apply the resulting filter to the data and model reported by Lambrecht, et al (1972). To our knowledge, this is the first reported application of optimal filtering to actual experimental data from a reactor having catalyst deactivation.…”

Section: Introductionmentioning

confidence: 99%

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On-Line Estimation of Catalyst Activity Profiles in Packed-Bed Reactors Having Catalyst Decay

Ajinkya¹,

Ray²,

Froment³

1974

Ind. Eng. Chem. Proc. Des. Dev.

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A very general nonlinear distributed parameter filter is applied to laboratory data obtained from a fixedbed tubular catalytic reactor. The isomerization of -pentane on a platinum reforming catalyst which decayed in time was the system under investigation. A filtering algorithm to obtain the best on-line estimates of the coke content of the catalyst is outlined. The filter provided coke profile estimates which were more consistent than the model or the data alone for short catalyst lifetimes.

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“…These are to be provided by a nonlinear filter. A general nonlinear filter which provides estimates x (r, t ) of x ( r , t ) based on the noisy observations tj(ri, t ) has been derived by Seinfeld et al (1971) and Hwang et al ( 1972). The filter consists of the following equations: The weighting matrices Qij ( t ) , R ( r , s, t ) , Ro ( t ) and R1 ( t )…”

Section: Nonlinear Filtermentioning

confidence: 99%

Suboptimal control of stochastic distributed parameter systems

Seinfeld

1973

AIChE Journal

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R = gas constant T = temperature U to, we X u XT x Yo Y B Z Z , E = adsorption potential 4 = volume of adsorption p = number of theoretical units = partial pressure of adsorbate = vapor pressure of adsorbate = solute adsorbed in adsorption zone = quantity of effluent from adsorber between break-=I quantity of effluent from adsorber at breakthrough =: adsorbent composition at exhaustion =: adsorbent composition at total saturation =I volume of HF adsorbed =I adsorbate composition of entering gas =I adsorbate composition of gas at breakthrough = height of adsorbent bed = height of exchange zone through and exhaustion = density at boiling point

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Optimal least-square filtering and interpolation in distributed parameter systems

Cited by 34 publications

References 17 publications

Real‐time estimation of dynamic particle‐size distributions in a fluidized bed: Theoretical foundation

Real‐time estimation of dynamic particle‐size distributions in a fluidized bed: Theoretical foundation

On-Line Estimation of Catalyst Activity Profiles in Packed-Bed Reactors Having Catalyst Decay

Suboptimal control of stochastic distributed parameter systems

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