## Analysis and Decision Making in Uncertain SystemsProblems, 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. |

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### Contents

Introduction to Uncertain Systems | 1 |

12 Uncertain Variables | 3 |

13 Basic Deterministic Problems | 5 |

14 Structure of the Book | 7 |

Relational Systems | 11 |

22 Analysis and Decision Making for Relational Plants | 14 |

23 Relational Plant with External Disturbances | 18 |

24 Determinization | 22 |

Parametric Optimization of Decision Systems | 201 |

92 Uncertain Controller in a Closedloop System | 206 |

93 Random Controller in a Closedloop System | 210 |

94 Descriptive and Prescriptive Approaches | 212 |

95 Fuzzy Controller in a Closedloop System | 216 |

96 Quality of Decisions Based on Nonparametric Descriptions | 220 |

Stability of Uncertain Dynamical Systems | 225 |

102 Stability Conditions | 227 |

25 Discrete Case | 25 |

Application of Random Variables | 29 |

32 Functional Plants with Random Parameters Continuous Case | 32 |

33 Functional Plants with Random Parameters Discrete Case | 39 |

34 Empirical interpretations | 41 |

35 Relational Plants with Random Parameters | 44 |

36 Determinization | 48 |

37 Nonparametric Uncertainty Continuous Case | 54 |

38 Nonparametric Uncertainty Discrete Case | 58 |

Uncertain Logics and Variables | 63 |

42 Other Versions of Uncertain Logic | 67 |

43 Uncertain Variables | 71 |

44 Additional Description of Uncertain Variables | 76 |

45 Functions of Uncertain Variables | 78 |

Application of Uncertain Variables | 85 |

52 Decision Making Problem for a Functional Plant | 86 |

53 External Disturbances | 88 |

54 Analysis for Relational Plants with Uncertain Parameters | 93 |

55 Decision Making for Relational Plants with Uncertain Parameters | 98 |

56 Computational Aspects 35 | 103 |

57 Nonparametric Uncertainty | 108 |

58 Nonparametric Problems for a Plant with External Disturbances | 115 |

Fuzzy Variables Analogies and Soft Variables | 123 |

62 Application of Fuzzy Variables in Analysis and Decision Problems | 129 |

63 Plant with External Disturbances | 134 |

64 Comparison of Uncertain Variables with Random and Fuzzy Variables | 140 |

65 Comparisons and Analogies for Nonparametric Problems | 143 |

66 Introduction to Soft Variables | 149 |

67 Application of Soft Variables to Nonparametric Problems | 151 |

68 Generalized Nonparametric Problems | 153 |

Systems with Logical Knowledge Representation | 155 |

72 Analysis and Decision Making Problems | 157 |

73 Logicalgebraic Method | 159 |

74 Analysis and Decision Making for a Plant with Random Parameters | 162 |

75 Analysis and Decision Making for a Plant with Uncertain Parameters | 164 |

76 Uncertain and Random Logical Decision Algorithms | 165 |

Dynamical Systems | 169 |

82 Analysis and Decision Making for Dynamical Plants with Uncertain Parameters | 175 |

83 Analysis and Decision Making for Dynamical Plants with Random Parameters | 182 |

84 Optimization of Random and Uncertain Multistage Decision Process | 184 |

85 Applications of Uncertain Variables for a Class of Knowledgebased Assembly Systems | 189 |

851 Knowledge Representation and Decision Problem | 190 |

852 Assembly Process with Uncertain Parameters | 193 |

86 Nonparametric Problems | 196 |

103 Special Cases | 230 |

1032 Multiplicative Uncertainty | 235 |

104 Examples | 238 |

105 An Approach Based on Random Variables | 243 |

106 An Approach Based on Uncertain Variables | 251 |

107 Stabilization | 254 |

Learning Systems | 259 |

1111 Knowledge Validation and Updating | 260 |

1112 Learning Algorithm for Decision Making in a Closedloop System | 262 |

112 Learning System Based on Knowledge of Decisions | 263 |

1121 Knowledge Validation and Updating | 264 |

1122 Learning Algorithm for Decision Making in a ClosedLoop System | 266 |

113 Learning Algorithms for a Class of Dynamical Systems 36 | 269 |

1131 Knowledge Validation and Updating | 270 |

1132 Learning Control System | 273 |

1133 Example | 274 |

114 Learning Algorithms for a Class of Knowledgebased Assembly Systems | 278 |

1142 Learning Algorithm for Decision Making in a Closedloop System | 281 |

Complex Problems and Systems | 283 |

122 Other Formulations Threelevel Uncertainty | 289 |

123 Complex Systems with Distributed Knowledge | 292 |

1232 Complex System with Uncertain and Random Parameters | 295 |

124 Knowledge Validation and Updating | 297 |

1241 Validation and Updating of the Knowledge Concerning the System | 298 |

1242 Validation and Updating of the Knowledge Concerning the Decision Making | 299 |

125 Learning System | 302 |

Complex of Operations | 313 |

132 Application of Uncertain Variables 49 54 | 316 |

133 Special Cases and Examples | 320 |

134 Decomposition and Twolevel Control | 325 |

135 Application of Random Variables 131460 | 328 |

136 Application to Task Allocation in a Multiprocessor System 33 57 | 331 |

137 Learning Algorithms | 335 |

Pattern Recognition | 339 |

142 Application of the Logicalgebraic Method | 341 |

143 Application of Uncertain Variables 54 | 344 |

144 Application of Random Variables | 350 |

145 Nonparametric Problems | 353 |

146 Learning Algorithms | 355 |

Conclusions | 361 |

363 | |

369 | |

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### Common terms and phrases

Ac(n according analogous analysis and decision analysis problem apply arg max Assume C-uncertain certainty distribution certainty index chosen randomly Closed-loop System concerning condition control system Dc(n decision making KD decision problem defuzzification denotes described in Sect determine deterministic algorithm deterministic decision algorithm Du(c Example expected value expert formulated fuzzy sets fuzzy variables fy(y given KP hu(u hy(y inequality knowledge updating knowledge validation Knowledge-based learning system Let us consider logic value matrix max hx(x mean value membership function Non-parametric Problems obtain open-loop control optimal decision plant described plant KP possible outputs presented in Sect probability density probability distribution problem consists pu(u py(y Random Parameters random variables relational knowledge representation relational plant respectively result sequence soft property soft variables solution static plant symmetric matrix Theorem time-varying system uncertain logic uncertain systems uncertain variables uncertainty unknown parameters validation and updating vector

### References to this book

Geometric Control and Nonsmooth Analysis: In Honor of the 73rd Birthday of H ... Fabio Ancona No preview available - 2008 |