Mathematical Modeling for Industrial Processes: (19 Lecture Notes in Economics and Mathematical Systems)

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.

Table of Contents:
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.- 2.4. 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.- 3.3. Two-dimensional Search (Hill-Climbing).- 4. Design of Experiments.- 4.1. Replication.- 4.2. Blocking of Experiments.- 4.3. Randomization.- 4.4. Factorial Design.- 4.5. Orthogonality.- 4.6. Confounding.- 4.7. Fractional Factorial Design.- 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.- 5.5. 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.- 6.9. 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.- 7.6. Non-linear Estimation.- References.