Mathematical Modeling for Industrial Processes

ISBN-13: 9783540049432 / Angielski / Miękka / 1970 / 125 str.

L.P. Hyvarinen

Opis

These notes are based on the material presented in a series of lec tures in the IBM Systems Research Institute (ESRI) in Geneva durJng 1967-1969 to systems engineers working in the design and programming of computer systems for control and monitoring of i nustrial proc esses. The purpose of the lectures and this book is to give a survey of dif ferent approaches in developing models to describe the behavior of the process in terms of controllable variables. It does not cover the theory of control, stability of control systems, nor techniques in data acquisition or problems in instrumentation and sampling. But certain aspects in the organization of data collection and design of experiments are obtained as side products, notably the concept of orthogonality. The reader is assumed to have a working knowledge of elementary prob ability theory and mathematical statistics. Therefore, the text con tains no introduction to these concepts. The author is aware of some inaccuracies in not making proper dis tinction between population parameters and their sample estimates in the text, but this should alw s be evident from the context. The same applies to the occasional replacement of number of degrees of freedom by the number of samples in the data. In practice, computer collected sets of data consist of a high number of samples and the difference between the two is inSignificant."

Spis treści

1. Basic Concepts.- 1.1. Modeling.- 1.2. Classification of Processes.- 1.3. Process Parameters and Variables and their Classification.- 1.4. Classification of Process Models.- Capter 2. Optimizing Models.- 2.1. General Considerations.- 2.2. Objective Function — an Example.- 2.3. Designing the Objective Function.- 2.3.1. Fixed Cost.- 2.3.2. Variable Cost.- 2.3.3. Gross Revenue.- 2.3.4. Quality Factors.- 2.4. Objective Function as a Function of Time.- 2.5. Constrained and Unconstrained Optima.- 2.6. Objective Function — Example Revisited.- 3. Methods of Optimum Search.- 3.1. Problem Definition.- 3.2. Single Variable Search.- 3.2.1. Two Modes of Search.- 3.2.2. Simultaneous Search.- 3.2.3. Sequential Search.- 3.3. Two-dimensional Search (Hill-Climbing).- 3.3.1. Simultaneous Methods.- 3.3.2. Sequential Methods.- 3.3.3. Termination of the Search.- 3.3.4. Choice of Units — Affine Transformations.- 4. Design of Experiments.- 4.1. Replication.- 4.2. Blocking of Experiments.- 4.3. Randomization.- 4.3.1. Analysis of Randomized Block Designs.- 4.4. Factorial Design.- 4.4.1. Two-level, Two-factor Design — Example.- 4.4.2. Effects and Objective Function.- 4.4.3. Symbolic Notation of Effects.- 4.5. Orthogonality.- 4.6. Confounding.- 4.7. Fractional Factorial Design.- 4.7.1. Simple Example.- 4.7.2. Second Example.- 5. Dynamic Covariance Analysis.- 5.1. Dynamic Models.- 5.2. Linear Dynamic Model — Single Variable.- 5.3. End Conditions.- 5.4. Identification of Linear Model.- 5.4.1. Covariance and Correlation.- 5.4.2 Covariance Function.- 5.4.3. Solving for hk.- 5.4.4. Summary of Covariance Analysis.- 5.5. Linear Dynamic Model — Multiple Variables.- 6. Principal Component Analysis.- 6.1. Reducing Number of Variables.- 6.2. Orthogonal Coordinates in Sample Space.- 6.3. Axes with Stationary Property.- 6.4. Zero-one Normalized Variables.- 6.5. Eigenvalues and Eigenvectors.- 6.6. Orthogonality.- 6.7. Mean-square Distances — Distribution of Variance.- 6.8. Numerical Example.- 6.8.1. Principal Component Analysis.- 6.8.2. Model Identification.- 6.8.3. Stability of Model.- 6.9. Performance Variables.- 7. Regression Analysis.- 7.1. Principle of Least Squares.- 7.2. Linear Regression.- 7.3. Transformation to Linear Form.- 7.4. Choosing the Form of Model.- 7.5. Stepwise Regression.- 7.5.1. F-test.- 7.6. Non-linear Estimation.- References.