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2004 | Buch

Introduction to Uncertain Systems Kapitel lesen Erstes Kapitel lesen

Analysis and Decision Making in Uncertain Systems

verfasst von: Professor Zdzislaw Bubnicki, PhD

Verlag: Springer London

Buchreihe : Communications and Control Engineering

Enthalten in: Professional Book Archive

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Über dieses Buch

Problems, methods and algorithms of decision making based on an uncertain knowledge now create a large and intensively developing area in the field of knowledge-based decision support systems. The main aim of this book is to present a unified, systematic description of analysis and decision problems in a wide class of uncertain systems described by traditional mathematical models and by relational knowledge representations. A part of the book is devoted to new original ideas introduced and developed by the author: the concept of uncertain variables and the idea of a learning process consisting in knowledge validation and updating. In a certain sense this work may be considered as an extension of the author's monograph Uncertain Logics, Variables and Systems (Springer-Verlag, 2002). In this book it has been shown how the different descriptions of uncertainty based on random, uncertain and fuzzy variables may be treated uniformly and applied as tools for general analysis and decision problems, and for specific uncertain systems and problems (dynamical control systems, operation systems, knowledge-based pattern recognition under uncertainty, task allocation in a set of multiprocessors with uncertain execution times, and decision making in an assembly system as an example of an uncertain manufacturing system). The topics and the organization of the text are presented in Chapter 1 (Sects 1. 1 and 1. 4). The material presented in the book is self-contained.

Inhaltsverzeichnis

Frontmatter
1. Introduction to Uncertain Systems
Abstract
Uncertainty is one of the main features of complex and intelligent decision making systems. Various approaches, methods and techniques in this field have been developed for several decades, starting with such concepts and tools as adaptation, stochastic optimization and statistical decision theory (see e.g. [2, 3, 68, 79, 80]). The first period of this development was devoted to systems described by traditional mathematical models with unknown parameters. In the past two decades new ideas (such as learning, soft computing, linguistic descriptions and many others) have been developed as a part of modern foundations of knowledge-based Decision Support Systems (DSS) in which the decisions are based on uncertain knowledge. Methods and algorithms of decision making under uncertainty are especially important for design of computer control and management systems based on incomplete or imperfect knowledge of a decision plant. Consequently, problems of analysis and decision making in uncertain systems are related to the following fields:
1.
General systems theory and engineering.
 
2.
Control and management systems.
 
3.
Information technology (knowledge-based expert systems).
 
Zdzislaw Bubnicki
2. Relational Systems
Abstract
This chapter is concerned with analysis and decision making problems for a static input—output plant described by a relation which is not reduced to the function 0 considered in Sect. 1.3. Consequently, for the given relation, the output is not determined by the input. The analysis problem consists in finding the output property (or the set of possible outputs) for the given input property (or the given set of inputs), and the decision problem consists in finding the input property (or the set of possible inputs) for the given output property (or the set of acceptable outputs, required by a user). For the functional plant presented in Sect. 1.3, the input and output properties have the form “ u = u* ” and “ y = y* ”, respectively. For the relational plant the respective properties have the form “ u ∈ D u ” and “ y ∈ D y ” where D u and D y are subsets of U and Y, respectively.
Zdzislaw Bubnicki
3. Application of Random Variables
Abstract
This chapter presents an application of random variables in the analysis and decision problems for a static plant. In the parametric case, the unknown parameters in the function or in the relation describing the plant are assumed to be values of random variables with the given probability distributions. In the non-parametric case, the plant is described by the given conditional probability distribution. The foundations of random variables and probabilistic theory are presented in many books in this classical area (see e.g. [66, 73, 94]). In Sect. 1.1, a very short description of random variables is given to introduce the notation and to bring together formalisms concerning random, uncertain and fuzzy variables in a unified framework.
Zdzislaw Bubnicki
4. Uncertain Logics and Variables
Abstract
In this chapter, we present basic theoretical foundations of uncertain variables — as a comparatively new tool for analysis and decision problems in uncertain systems [44, 46, 54]. The uncertain variable is described by a certainty distribution given by an expert and characterizing his or her knowledge on approximate values of the variable. The uncertain variables are related to random and fuzzy variables, but there are also essential differences. The comparison with random and fuzzy variables will be presented in Chapter 6. The definitions of the uncertain variables are based on uncertain logics described in Sects. 4.1 and 4.2. It is worth noting that the uncertain logics and variables are defined for any metric space X (the set of values). Starting from Sect. 4.4, the considerations are concerned with a real number vector space X.
Zdzislaw Bubnicki
5. Application of Uncertain Variables
Abstract
The purpose of this chapter is to show how uncertain variables may be applied to analysis and decision problems for a static plant. In the parametric case, we assume that the unknown parameters in the function or the relation describing the plant are values of uncertain variables described by certainty distributions given by an expert. In the non-parametric case, the plant is described by the conditional certainty distribution characterizing the expert’s knowledge of the plant. The considerations are analogous to those for the random variables in Chapter 3.
Zdzislaw Bubnicki
6. Fuzzy Variables, Analogies and Soft Variables
Abstract
The first part of this chapter presents the application of fuzzy variables to non-parametric problems for a static plant, analogous to those described for random and uncertain variables. In Sect. 6.1, a very short description of fuzzy variables (see e.g. [69, 71, 74, 75, 84, 103, 104]) is given in the form needed to formulate our problems and to indicate analogies for non-parametric problems based on random, uncertain and fuzzy variables. These analogies lead to a generalization in the form of soft variables and their applications to non-parametric decision problems.
Zdzislaw Bubnicki
7. Systems with Logical Knowledge Representation
Abstract
The relations introduced in Chapter 2 may have a specific form of logical formulas concerning input, output and additional variables. In this case the so-called logic-algebraic method may be used to formulate and solve the analysis and decision problems [17, 18, 19, 21–24, 26, 29]. The main idea of this method consists in replacing individual reasoning concepts based on inference rules by unified algebraic procedures based on the rules in two-value logic algebra.
Zdzislaw Bubnicki
8. Dynamical Systems
Abstract
The aim of this chapter is to show how the approaches and methods presented in the previous chapters may be applied to discrete-time dynamical plants described by traditional functional models or by relational knowledge representations. Special attention is paid to the relational plants and the descriptions based on uncertain variables [22, 54]. The considerations are completed with the optimization of a random and uncertain multistage decision process (dynamic programming under uncertainty) and with applications to a class of assembly systems. Other considerations for dynamical systems are presented in Chapter 9 (uncertain, random and fuzzy controllers in closed-loop systems), in Chapter 10 (stability) and in Chapter 11 (dynamical learning systems).
Zdzislaw Bubnicki
9. Parametric Optimization of Decision Systems
Abstract
In the previous chapters the decision problems for non-deterministic plants with different descriptions of the uncertainty (different forms of the knowledge representation) have been considered. The typical procedure of finding the deterministic decision algorithm has been as follows:
1.
The determination of the non-deterministic decision algorithm (the knowledge of the decision making KD) using the knowledge of the plant KP and the requirement concerning the output of the plant.
 
2.
The determination of the deterministic decision algorithm by applying a determinization of KD.
 
Zdzislaw Bubnicki
10. Stability of Uncertain Dynamical Systems
Abstract
The analysis and decision making problems considered in the previous chapters may be called quantitative problems. In many cases there is a need to formulate and solve quantitative analysis problems, which consist in an investigation of some properties concerning the system under consideration, such as stability, controllability, observability, etc. Let us consider a system with two unknown vector parameters in its description: c ∈ C and b ∈ B. The uncertainties concerning c and b are formulated as follows:
Zdzislaw Bubnicki
11. Learning Systems
Abstract
This chapter is concerned with plants described by a knowledge representation in the form of relations with unknown parameters. The learning process consists here in step by step knowledge validation and updating [20, 28, 32, 34, 37, 51, 54]. At each step one should prove if the current observation “belongs” to the knowledge representation determined before this step (knowledge validation) and if not — one should modify the current estimation of the parameters in the knowledge representation (knowledge updating). The results of the successive estimation of the unknown parameters are used in the current determination of the decisions in a learning decision making system. This approach may be considered as an extension of the known idea of adaptation via identification for the plants described by traditional mathematical models (see e.g. [14]). We shall consider two versions of learning systems. In the first version the knowledge validation and updating is concerned with the knowledge of the plant (i.e. the relation R describing the plant), and in the second version — with the knowledge of the decision making (i.e. the set of decisions D u ). In both versions the learning algorithms based on the knowledge validation and updating will be presented.
Zdzislaw Bubnicki
12. Complex Problems and Systems
Abstract
In the previous chapters we have considered a single decision plant with one unified form of the uncertainty based on random, uncertain or fuzzy variables. Complex problems arise when there are different forms of uncertainty in the knowledge representation of a single plant or when there are different levels of uncertainty concerning a single plant. If the relational knowledge representation presented in Chapter 2 is considered as a basic level, then the description of unknown parameters in the relations, using random or uncertain variables, forms the second (the upper) level of uncertainty or the second-order uncertainty, as was explained in Sect. 3.6 for the relational and random levels. The first part of this chapter concerns this kind of complex problems.
Zdzislaw Bubnicki
13. Complex of Operations
Abstract
There exists a great variety of decision problems concerning allocation and scheduling in a complex of operations (a complex operation system), with many applications to manufacturing and computer systems (see e.g. [8]). This chapter is concerned with the control of the complex of parallel operations containing unknown parameters in the relational knowledge representation. The complex of parallel operations is considered here as a specific uncertain decision plant. The control consists in a proper distribution of the given size of a task or the given amount of a resource, taking into account the execution time of the whole complex. It may mean the distribution of raw material in the case of a manufacturing process or a load distribution in a group of parallel computers. In the deterministic case where the operations are described by functions determining the relationship between the execution time and the size of the task or the amount of the resource, the optimization problem consisting in the determination of the distribution that minimizes the execution time of the complex may be formulated and solved (see e.g. [13]). In the case of uncertainty, various formulations of decision problems adequate for the different descriptions of the uncertainty may be considered [15, 16, 49, 51, 54].
Zdzislaw Bubnicki
14. Pattern Recognition
Abstract
Knowledge-based recognition (or classification) under uncertainty may be considered as a good example of the general problems and methods presented in the previous chapters. In this chapter recognition problems based on the relational and logical knowledge representation are described. Classical methods based on probabilistic description and new approaches (application of uncertain variables are the learning process consisting in the knowledge validation and updating) are presented and discussed in a uniform way, as specific analysis and decision problems with the descriptions of the uncertainty considered in Chapters 3, 5, 6, 7, 11.
Zdzislaw Bubnicki
Conclusions
Abstract
This book has been concerned with a wide class of uncertain systems described by traditional mathematical models and by relational knowledge representations. For the systems with different formal models of uncertainty, a uniform description of analysis and decision making problems has been presented and discussed. A special emphasis has been put on new approaches:
1.
Uncertain variables and their applications.
 
2.
Learning concepts consisting in step by step knowledge validation and updating.
 
3.
Soft variables as a tool for unification and generalization of non-parametric problems.
 
Zdzislaw Bubnicki
Backmatter
Metadaten
Titel
Analysis and Decision Making in Uncertain Systems
verfasst von
Professor Zdzislaw Bubnicki, PhD
Copyright-Jahr
2004
Verlag
Springer London
Electronic ISBN
978-1-4471-3760-3
Print ISBN
978-1-84996-909-3
DOI
https://doi.org/10.1007/978-1-4471-3760-3