Cheng / Qi / Li Analysis and Control of Boolean Networks

A Semi-tensor Product Approach
1. Auflage 2010
ISBN: 978-0-85729-097-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 472 Seiten

Reihe: Communications and Control Engineering

Analysis and Control of Boolean Networks presents a systematic new approach to the investigation of Boolean control networks. The fundamental tool in this approach is a novel matrix product called the semi-tensor product (STP). Using the STP, a logical function can be expressed as a conventional discrete-time linear system. In the light of this linear expression, certain major issues concerning Boolean network topology - fixed points, cycles, transient times and basins of attractors - can be easily revealed by a set of formulae. This framework renders the state-space approach to dynamic control systems applicable to Boolean control networks. The bilinear-systemic representation of a Boolean control network makes it possible to investigate basic control problems including controllability, observability, stabilization, disturbance decoupling etc.

Daizhan Cheng received the Ph.D. degree in systems science from Washington University, St. Louis, in 1985. Currently, he is a Professor with the Institute of Systems Science, Chinese Academy of Sciences, Beijing, China. His research interests include nonlinear systems, numerical method, complex systems, etc. Dr. Cheng is Chairman of the Technical Committee on Control Theory (since 2003), Chinese Association of Automation, a Fellow of the IEEE, and a Fellow of the International Federation of Automatic Control. Hongsheng Qi received the Ph.D. degree in systems theory from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 2008. He is currently a post-doctoral fellow at the Key Laboratory of Systems and Control, Chinese Academy of Sciences. His research interests include nonlinear systems control, complex systems, etc. Zhiqiang Li received the M.S. degree from Zhengzhou University in 2007. He is currently a Ph.D. student in the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. His research interests include nonlinear systems control, complex systems, etc.

Autoren/Hrsg.

Cheng, Daizhan

Qi, Hongsheng

Li, Zhiqiang

Inhaltsverzeichnis

1;Preface;6
1.1;References;10
2;Contents;11
3;Notation;15
4;Propositional Logic;17
4.1;Statements;17
4.2;Implication and Equivalence;21
4.3;Adequate Sets of Connectives;24
4.4;Normal Form;27
4.5;Multivalued Logic;30
4.6;References;34
5;Semi-tensor Product of Matrices;35
5.1;Multiple-Dimensional Data;35
5.2;Semi-tensor Product of Matrices;45
5.3;Swap Matrix;53
5.4;Properties of the Semi-tensor Product;57
5.5;General Semi-tensor Product;65
5.6;References;69
6;Matrix Expression of Logic;70
6.1;Structure Matrix of a Logical Operator;70
6.2;Structure Matrix for k-valued Logic;74
6.3;Logical Matrices;78
6.4;References;80
7;Logical Equations;81
7.1;Solution of a Logical Equation;81
7.2;Equivalent Algebraic Equations;82
7.3;Logical Inference;92
7.4;Substitution;98
7.5;k-valued Logical Equations;99
7.6;Failure Location: An Application;103
7.6.1;Matrix Expression of Route Logic;103
7.6.2;Failure Location;106
7.6.3;Cascading Inference;111
7.7;References;114
8;Topological Structure of a Boolean Network;116
8.1;Introduction to Boolean Networks;116
8.2;Dynamics of Boolean Networks;117
8.3;Fixed Points and Cycles;121
8.4;Some Classical Examples;132
8.5;Serial Boolean Networks;137
8.6;Higher Order Boolean Networks;139
8.6.1;First Algebraic Form of Higher Order Boolean Networks;141
8.6.2;Second Algebraic Form of Higher Order Boolean Networks;150
8.7;References;152
9;Input-State Approach to Boolean Control Networks;154
9.1;Boolean Control Networks;154
9.2;Semi-tensor Product Vector Space vs. Semi-tensor Product Space;156
9.3;Cycles in Input-State Space;159
9.4;Cascaded Boolean Networks;164
9.5;Two Illustrative Examples;167
9.6;References;174
10;Model Construction via Observed Data;175
10.1;Reconstructing Networks;175
10.2;Model Construction for General Networks;183
10.3;Construction with Known Network Graph;188
10.4;Least In-degree Model;189
10.5;Construction of Uniform Boolean Network;193
10.6;Modeling via Data with Errors;196
10.7;References;199
11;State Space and Subspaces;200
11.1;State Spaces of Boolean Networks;200
11.2;Coordinate Transformation;202
11.3;Regular Subspaces;207
11.4;Invariant Subspaces;215
11.5;Indistinct Rolling Gear Structure;218
11.6;References;223
12;Controllability and Observability of Boolean Control Networks;224
12.1;Control via Input Boolean Network;224
12.2;Subnetworks;231
12.3;Controllability via Free Boolean Sequence;233
12.4;Observability;238
12.5;References;242
13;Realization of Boolean Control Networks;243
13.1;What Is a Realization?;243
13.2;Controllable Normal Form;245
13.3;Observable Normal Form;249
13.4;Kalman Decomposition;252
13.5;Realization;256
13.6;References;258
14;Stability and Stabilization;259
14.1;Boolean Matrices;259
14.2;Global Stability;263
14.3;Stabilization of Boolean Control Networks;271
14.4;References;283
15;Disturbance Decoupling;284
15.1;Problem Formulation;284
15.2;Y-friendly Subspace;285
15.3;Control Design;292
15.4;Canalizing Boolean Mapping;298
15.5;Solving DDPs via Constant Controls;301
15.6;References;304
16;Feedback Decomposition of Boolean Control Networks;306
16.1;Decomposition of Control Systems;306
16.2;The Cascading State-space Decomposition Problem;307
16.3;Comparable Regular Subspaces;312
16.4;The Parallel State-space Decomposition Problem;314
16.5;Input-Output Decomposition;317
16.6;References;320
17;k-valued Networks;321
17.1;A Review of k-valued Logic;321
17.2;Dynamics of k-valued Networks;324
17.3;State Space and Coordinate Transformations;328
17.4;Cycles and Transient Period;332
17.5;Network Reconstruction;333
17.6;k-valued Control Networks;338
17.7;Mix-valued Logic;348
17.8;References;353
18;Optimal Control;354
18.1;Input-State Transfer Graphs;354
18.2;Topological Structure of Logical Control Networks;358
18.3;Optimal Control of Logical Control Networks;363
18.4;Optimal Control of Higher-Order Logical Control Networks;368
18.5;References;376
19;Input-State Incidence Matrices;377
19.1;The Input-State Incidence Matrix;377
19.2;Controllability;380
19.3;Trajectory Tracking and Control Design;384
19.4;Observability;385
19.5;Fixed Points and Cycles;388
19.6;Mix-valued Logical Systems;389
19.7;References;394
20;Identification of Boolean Control Networks;395
20.1;What Is Identification?;395
20.2;Identification via Input-State Data;396
20.3;Identification via Input-Output Data;399
20.4;Numerical Solutions;402
20.4.1;General Algorithm;402
20.4.2;Numerical Solution Based on Network Graph;406
20.4.3;Identification of Higher-Order Systems;409
20.5;Approximate Identification;410
20.6;References;413
21;Applications to Game Theory;414
21.1;Strategies with Finite Memory;414
21.2;Cycle Strategy;417
21.3;Compounded Games;420
21.4;Sub-Nash Solution for Zero-Memory Strategies;422
21.5;Nash Equilibrium for µ-Memory Strategies;424
21.6;Common Nash (Sub-Nash) Solutions for µ-Memory Strategies;426
21.7;References;434
22;Random Boolean Networks;435
22.1;Markov Chains;435
22.2;Vector Form of Random Boolean Variables;443
22.3;Matrix Expression of a Random Boolean Network;446
22.4;Some Topological Properties;451
22.5;References;454
23;Appendix A Numerical Algorithms;455
23.1;Computation of Logical Matrices;455
23.2;Basic Functions;457
23.3;Some Examples;462
24;Appendix B Proofs of Some Theorems Concerning the Semi-tensor Product;467
24.1;References;470
25;Index;471