Real time distributed parameter state estimation applied to a two dimensional heated ingot

Lausterer, G.K.; Ray, W. Harmon; Martens, H. R.

doi:10.1016/0005-1098(78)90033-x

Cited by 20 publications

(7 citation statements)

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“…These and other investigators (e.g., Koda and Seinfeld, 1978;Lausterer et al, 1978) are in general agreement that by delaying the lumping process for as long as possible, the distributed nature of the system is best preserved. There has been little published, however, that formalizes the difference between early vs. late lumping as applied to the filtering of linear DP systems, or that investigates the character and extent of such approximations on filter performance.…”

Section: Introductionmentioning

confidence: 89%

“…There are many examples in the literature of both early-and late-lumped implementation of linear DP filters using a weighted residual method. Some examples of filter implementation using eigenfunction expansion include the works of Ajinkya (1975), Ajinkya et al (1975), Lausterer et al (1978), Lausterer and Ray (1979), and Omatu and Seinfeld (1981). 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).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Comparison of linear distributed‐parameter filters to lumped approximants

Cooper¹,

Ramirez²,

Clough³

1986

AIChE Journal

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“…Ingot models can readily be described in the form of partial differential equations (Lausterer, 1978). Ray et al (1979), have indicated some potential applications of distributed parameter system theory to control of soaking pits.…”

Section: Introductionmentioning

confidence: 99%

“…description of the temperature behavior of steel ingots during solidification, cooling and heating has received the attention of numerous investigators. t. [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] They have employed various approaches to establish the mathematical models for these ingots. For the development of these ingot models, the following main approaches have been applied to date: lumped parameter linear models, distributed parameter models, on-line optimal estimation, and regression analysis.…”

Section: Introductionmentioning

confidence: 99%

“…For the development of these ingot models, the following main approaches have been applied to date: lumped parameter linear models, distributed parameter models, on-line optimal estimation, and regression analysis. King et al (1963), have detailed the basic equations involved and their application to ingot processing modelling.Ingot models can readily be described in the form of partial differential equations (Lausterer, 1978). Ray et al (1979), have indicated some potential applications of distributed parameter system theory to control of soaking pits.…”

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confidence: 99%

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Applications of a Discrete Dynamic Model of Steel Ingot Temperature Behavior to Soaking Pit Operations

Williams

1984

Chemical Engineering Communications

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A discrete, linear, time-variant, state equation model of the heat transfer processes involved has been applied to describe the behavior of steel ingots during teeming, prior to and following stripping, during soaking pit heating and on extraction. This model gives accurate representations of the solidus formation and of the transient temperature behavior of the ingot while in the molds and while in the soaking pit furnaces. A major application of the resulting model has been to the charging of ingots with partially liquid cores into the soaking pits with a resulting substantial reduction in the energy requirements for the soaking pit operation and a correspondingly large increase in soaking pit throughput or productivity. INTRODUCTIONOne of the major areas of steel plant operation today is that of ingot processing. The objective in controlling the soaking pits is to achieve suitably high and uniform ingot temperature profiles for rolling while minimizing the total energy consumption in the soaking process.During the past few decades, the mathematical. description of the temperature behavior of steel ingots during solidification, cooling and heating has received the attention of numerous investigators. t.3-20 They have employed various approaches to establish the mathematical models for these ingots. For the development of these ingot models, the following main approaches have been applied to date: lumped parameter linear models, distributed parameter models, on-line optimal estimation, and regression analysis. King et al. (1963), have detailed the basic equations involved and their application to ingot processing modelling.Ingot models can readily be described in the form of partial differential equations (Lausterer, 1978). Ray et al. (1979), have indicated some potential applications of distributed parameter system theory to control of soaking pits. In addition, Soliman (1972) has applied the finite element method to solve the distributed parameter models , Presently Associate Professor of Chemical Engineeringof ingots during the solidification process. In practice, the solution of the partial differential equations of a distributed parameter model with the necessary boundary conditions is a difficult and time consuming task. Most investigators therefore choose to transform the model into a set of linear equations using the integral profile method or a finite difference approximation.In order to simplify the problem further, the resulting models can usually be reduced from three dimensions to one or two by means of some appropriate assumptions. Typical ingot models for this purpose are the two-dimensional model (Sevrin, 1970;Massey et al., 1971), the cylindrical model (Cook et al., 1978) and the spherical model (Maeda, 1975). The cylindrical (one dimensional) form will be followed in this investigation.The objective in controlling soaking pits is to achieve a suitable internal metal temperature profile for rolling, while minimizing surface oxidation and total energy consumption. Unfortunately, in practice, thes...

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Feedback and Control II: Modern Methodologies

Tzafestas

2018

Intelligent Systems, Control and Automation: Science and Engineering

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Real time distributed parameter state estimation applied to a two dimensional heated ingot

Cited by 20 publications

References 11 publications

Comparison of linear distributed‐parameter filters to lumped approximants

Comparison of linear distributed‐parameter filters to lumped approximants

Applications of a Discrete Dynamic Model of Steel Ingot Temperature Behavior to Soaking Pit Operations

Feedback and Control II: Modern Methodologies

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