An introduction to semi-tensor product of matrices and its applications

An Introduction to Semi-Tensor Product of Matrices and ITS Applications

Volumn , Issue , 2012, Pages 1-587

  Cheng, Daizhan ^a   Qi, Hongsheng ^a   Zhao, Yin ^a  

^a INSTITUTE OF GEOLOGY AND GEOPHYSICS   (China)

Author keywords

[No Author keywords available]

Indexed keywords

BOOLEAN FUNCTIONS; FUZZY CONTROL;

ANALYSIS AND CONTROLS; ARBITRARY MATRICES; COMMUTATIVE PROPERTY; FUNDAMENTAL PROPERTIES; HIGHER-DIMENSIONAL; MULTI-LINEAR MAPPINGS; SEMI-TENSOR PRODUCT; SEMI-TENSOR PRODUCT OF MATRICES;

TENSORS;

EID: 84995421769     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1142/8323     Document Type: Book

References (175)

1. Abraham, R. and Marsden, J. (1978). Foundations of Mechanics, 2nd edn. (Benjamin/Cummings Pub. Com. Inc.).
2. Agaian, S., Astola, J. and Egiazarian, K. (1995). Binary Polynomial Transforms and Nonlinear Digital Filters (Marcel Dekker, New York).
3. Agaian, S., Panetta, K., Nercessian, S. and Danahy, E. (2010). Boolean derivatives with application to edge detection for imaging systems, IEEE Trans. Syst., Man., Man, Cybern. B, Cubern. 40, 2, pp. 371-382.
4. Akers, S. (1959). On a theory of Boolean functions, Journal of the Society for Industrial and Applied Mathematics 7, 4, pp. 487-498.
5. Akutsu, T., Hayashida, M., Ching, W. and Ng, M. (2007). Control of Boolean networks: Hardness results and algorithms for tree structured networks, J. Theoretical Biology 244, 4, pp. 670-679.
6. Akutsu, T., Miyano, S. and Kuhara, S. (2000). Inferring qualitative relations in genetic networks and metabolic pathways, Bioinformatics 16, pp. 727-734.
7. Albert, R. and Barabási, A. (2000). Dynamics of complex systems: Scaling laws for the period of Boolean networks, Physical Review Letters 84, 24, pp. 5660-5663.
8. Aldana, M. (2003). Boolean dynamics of networks with scale-free topology, Phys-ica D: Nonlinear Phenomena 185, 1, pp. 45-66.
9. Arnold, V. (1983). Geometrical Methods on the Theory of Ordinary Differential Equations (Springer-Verlag, New York).
10. Ashenhurst, R. (1957). The decomposition of switching functions, in Proceedings of an International Symposium on the Theory of Switching, pp. 74-116.
11. Azariadis, P. and Aspragathos, N. (2001). Computer graphics representation and transformation of geometric entities using dual unit vectors and line transformations, Computers & Graphiscs 25, pp. 195-205.
12. Barnes, D. and Mack, J. (1975). An Algebraic Introduction to Mathematical Logic (Springer-Verlag, Texas).
13. Bates, D. and Watts, D. (1980). Relative curvature measures of nonlinearity, Journal of the Royal Statistical Society. Series B (Methodological) 42, pp. 1-25.
14. Bates, D. and Watts, D. (1981). Parameter transformations for improved approximate confidence regions in nonlinear least squares, The Annals of Statistics 9, pp. 1152-1167.
15. Bochmann, D. (1978). Boolean Differential Calculus (Larl Marx Stadt, German Democratic Pepublic).
16. Boothby, W. (1986). An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. (Academic Press, Inc., Orlando).
17. Borota, N., Flores, E. and Osler, T. (2000). Spacetime numbers the easy way, Mathematics and Computer Education 34, 2, pp. 159-168.
18. Brayton, R. and Khatri, S. (1999). Multi-valued logic synthesis, in International Conference on VLSI Design (Goa, India), pp. 196-205.
19. Brown, F. (2003). Boolean Reasoning: The Logic of Boolean Equations, 2nd edn. (Dover Pub., New York).
20. Burris, S. and Sankappanavar, H. (1981). A Course in Universal Algebra, Number 78 in Graduate Texts in Mathematics (Springer-Verlag).
21. Cairo, L. and Feix, M. (1992). Families of invariants of the motion for Lotka-Volterra 1st family, J. Math. Phys. 33, 7, pp. 2240-2455.
22. Cairo, L., Feix, M. and Llibre, J. (1999). Darboux method and search of invariants for the Lotka-Volterra and complex quadratic systems, J. Math. Phys. 40, 4, pp. 2074-2091.
23. Carlet, C. (2010). Boolean Methods and Models in Mathematics, Computer Sci-ence, and Engineering, chap. Boolean functions for cryptography and errorcorrecting codes (Cambridge Univ. Press, Cambridge), pp. 257-397.
24. Carr, J. (1981). Applications of Center Manifold Theory (Springer-Verlag).
25. Chartrand, G. and Zhang, P. (2005). Introduction to Graph Theory (McGraw-Hill Inc., New York).
26. Cheng, D. (2001). On Lyapunov mapping and its applications, Commu. on In-form. Sys. 1, 3, pp. 255-272.
27. Cheng, D. (2002). Matrix and Polynomial Approach to Dynamic Control Systems (Science Press, Beijing).
28. Cheng, D. (2007). Sime-tensor product of matrices and its applications: A survey, in Proc. 4th International Congress of Chinese Mathematicians (Higher Edu. Press, Int. Press, Hangzhou), pp. 641-668.
29. Cheng, D. (2009). Input-state approach to Boolean networks, IEEE Trans. Neural Networks 20, 3, pp. 512-521.
30. Cheng, D. (2011). Disturbance decoupling of Boolean control networks, IEEE Trans. Aut. Contr. 56, 1, pp. 2-10.
31. Cheng, D., Feng, J. and Lv, H. (2011a). Solving fuzzy relational equations via semi-tensor product, IEEE Trans. Fuzzy System, (to appear).
32. Cheng, D., Guo, L. and Huang, J. (2003). On quadratic Lyapunov functions, IEEE Trans. Aut. Contr. 48, 5, pp. 885-890.
33. Cheng, D. and Qi, H. (2007). Semi-tensor Product of Matrices — Theory and Applications (Science Press, Beijing), in Chinese.
34. Cheng, D. and Qi, H. (2009). Controllability and observability of Boolean control networks, Automatica 45, 7, pp. 1659-1667.
35. Cheng, D. and Qi, H. (2010a). A lienar representation of dynamics of Boolean networks, IEEE Trans. Aut. Contr. 55, 10, pp. 2251-2258.
36. Cheng, D. and Qi, H. (2010b). State-space analysis of Boolean networks, IEEE Trans. Neural Networks 21, 4, pp. 584-594.
37. Cheng, D., Qi, H. and Li, Z. (2011b). Analysis and Control of Boolean Networks: A Semi-tensor Product Approach (Springer, London).
38. Cheng, D., Qi, H., Li, Z. and Liu, J. B. (2011c). Stability and stabilization of Boolean networks, Int. J. Robust Nonlinear Contr. 21, 2, pp. 134-156.
39. Cheng, D., Xi, Z., Lu, Q. and Mei, S. (2000). Geometric structure of generalized controlled Hamiltonian systems and its application, Science in China, Series E 43, 4, pp. 365-379.
40. Cheng, D. and Xu, X. (2011). Bi-decomposition of logical mappings via semitensor product of matrices, Preprint.
41. Cheng, D., Xue, W. and Huang, J. (1998). On general Hamiltonian systems, in Proc. ICARCV’98 (Singapore), pp. 185-189.
42. Cheng, D. and Zhao, Y. (2011). Identification of Boolean control networks, Au-tomatica 47, 4, pp. 702-710.
43. Cheng, D., Zhao, Y. and Mu, Y. (2010). Strategy optimization with its application to dynamic games, in Proc. IEEE CDC’2010, pp. 5822-5827, doi:10.1109/CDC.2010.5717060.
44. Cheng, D., Zhao, Y. and Xu, X. (2011d). Matrix approach to Boolean calculus, Preprint.
45. Cheng, D., Zhu, Y. and Qi, H. (2009). A conjecture on the norm of Lyapunov mapping, J. Control Theory & Applications 7, 1, pp. 48-50.
46. Cheng, H. and Thompson, S. (1997). Dual iterative displacement analysis of spatial mechanisms using the C programming language, Mechanism and Machine Theory 32, 2, pp. 193-207.
47. Chiang, H., Hirsch, M. and Wu, F. (1988). Stability regions of nonlinear autonomous dynamical systems, IEEE Trans. Aut. Contr. 33, 1, pp. 16-27.
48. Choudhury, M. and Mohanram, K. (2010). Bi-decomposition of large Boolean functions using blocking edge graphs, in Proc. 2010 IEEE/ACM Int. Conf. Comput.-aided Design (San Jose), pp. 586-591
49. Clifford, W. (1873). Preliminary sketch of biquaternions, Proc. London Mathe-matical Society 4, 64, pp. 381-395.
50. Cohen, D. (1989). An Introduction to Hilbert Space and Quantum Logic (Springer-Verlag, New York).
51. Curtis, H. (1962). A New Approach to the Design of Switching Circuits (Van Nostrand, Princeton, N.J.).
52. Daniell, P. (1917). The modular difference of classes, Bull. Amer. Math. Soc. 23, pp. 446-450.
53. Datta, A., Choudhary, A., Bittner, M. and Dougherty, E. (2003). External control in Markovian genetic regulatory networks, Machine Learning 52, pp. 169-191.
54. Datta, A., Choudhary, A., Bittner, M. and Dougherty, E. (2004). External control in Markovian genetic regulatory networks: The imperfect information case, Bioinformatics 20, pp. 924-930.
55. Davio, M., Deschamps, J. and Thayse, A. (1978). Discrete and Switching Functions (McGraw-Hill, New York).
56. Descusse, J. and Morg, C. (1985). Decoupling with dynamic compensation for strong invertible affine nonlinear systems, Int. J. Contr. 43, pp. 1385-1398.
57. Devanathan, R. (2001). Linearization condition through state feedback, IEEE Trans. Aut. Contr. 46, 8, pp. 1257-1260
58. Di Benedetto, M., Glumineau, A. and Moog, C. (1994). The nonlnear interactor and its application to imput-output decoupling, IEEE Trans. Aut. Contr. 39, 9, pp. 1240-1250.
59. Dixon, J. and Mortimer, B. (1996). Permutation Groups (Springer-Verlag, London).
60. Drossel, B., Mihaljev, T. and Greil, F. (2005). Number and length of attractors in a critical Kauffman model with connectivity one, Physical Review Letters 94, 8, p. 88701.
61. Dubois, D. and Prade, H. (eds.) (2000). Fundamentals of Fuzzy Sets (Kluwer Acad. Pub., Boston).
62. Duchet, P. (1995). Hypergraphs, in R. Graham, M. Grötschel and L. Lovász (eds.), Handbook of Combinatorics 1 (MIT Press, North-Holland), pp. 381-432.
63. Emmott, S. (2006). Towards 2020 Science (Microsoft Prsearch Ltd., Cambridge).
64. Falb, P. and Wolovich, W. (1967). Decoupling and synthesis of multivariable control systems, IEEE Trans. Aut. Contr. 12, 5, pp. 651-669.
65. Farrow, C., Heidel, J., Maloney, J. and Rogers, J. (2004). Scalar equations for synchronous Boolean networks with biological applications, IEEE Trans. Neural Networks 15, 2, pp. 348-354.
66. Feng, J., Lu, H. and Cheng, D. (2011). Matrix-based approach to fuzzy logic and fuzzy control, (Submitted for pub.).
67. Foster, J. and Nightngale, J. (1995). A Short Course in General Relativity (Springer-Verlag).
68. Freund, E. (1975). The structure of decoupled nonlinear systems, Int. J. Contr. 21, pp. 443-450.
69. Fudenberg, D. and Tirole, J. (1991). Game Theory (MIT Press, Cambridge).
70. Gerla, G. (2001). Fuzzy Logic, Mathematical Tools for Approximate Reasoning (Kluwer).
71. Gibbons, R. (1992). A Primer in Game Theory (Prentice Hall).
72. Glumineau, A. and Moog, C. (1992). Nonlinear Morgan’s problem: case of (p+1) inputs and p outputs, IEEE Trans. Aut. Contr. 37, 7, pp. 1067-1072.
73. Goodwin, B. (1963). Temporal Organization in Cells (Acad. Press, New York).
74. Greub, W. (1978). Multilinear Algebra, 2nd edn. (Springer-Verlag, New York).
75. Greub, W. (1981). Linear Algebra, 4Thedn. (Springer-Verlag).
76. Guckenheimer, J. and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Sys-tems and Bifurcations of Vector Fields (Springer, Berlin).
77. Hachtel, G. and Somenzi, F. (2000). Logic Synthesis and Verification (Kluwer Academic Pub.).
78. Hamilton, A. (1988). Logic for Mathematicians, revised edn. (Cambridge Univ Press, Cambridge).
79. Han, D., Wei, Q., Li, Z. and Sun, W. (2008). Control of oriented mechanical systems: A method based on dual quaternion, in Proc. 17th IFAC World Congress (Seoul), pp. 3836-3841.
80. Harris, S., Sawhill, B., Wuensche, A. and Kauffman, S. (2002). A model of transcriptional regulatory networks based on biases in the observed regulation rules, Complexity 7, 4, pp. 23-40.
81. Hartshorne, R. (1977). Algebraic Geometry (Springer-Verlag, New York).
82. Heidel, J., Maloney, J., Farrow, C. and Rogers, J. (2003). Finding cycles in synchronous Boolean networks with applications to biochemical systems, Int. J. Bifurcation & Chaos 13, 3, pp. 535-552.
83. Heymann, M. (1968). Pole assignment in multi-input linear systems, IEEE Trans. Aut. Contr. 13, 6, pp. 748-749.
84. Horn, R. and Johnson, C. (1991). Topics in Matrix Analysis, Vol. 1, 2 (Cambridge Univ. Press, Cambridge).
85. Hu, B. (2010). Foundation of Fuzzy Theory, 2nd edn. (Wuhan University Press, Wuhan), in Chinese.
86. Huang, J. (2004). Nonlinear Output Regulation, Theory and Applications (Siam, Philadelphia).
87. Hungerford, T. (1974). Algebra (Springer-Verlag).
88. Ideker, T., Galitski, T. and Hood, L. (2001). A new approach to decoding life: Systems biology, Annu. Rev. Genomics Hum. Genet. 2, pp. 343-372.
89. Isidori, A. (1995). Nonlinear Control Systems, 3rd edn. (Springer)
90. Kauffman, S. (1993). The Origins of Order: Self-organization and Selection in Evolution (Oxford University Press, New York).
91. Keren, O. (2008). Reduction of the average path length in binary decision diagrams by spectral methods, IEEE Trans. Computers 57, 4, pp. 520-531.
92. Kerre, E., Huang, C. and Ruan, D. (2004). Fuzzy Set Theory and Approximate Reasoning (Wuhan University Press, Wuhan).
93. Khalil, H. (1996). Nonlinear Systems, 3rd edn. (Prentice Hall).
94. Knobloch, H., Isidori, A. and Flockerzi, D. (1993). Topics in Control Theory (Birkhauser Verlag, Basel).
95. Krener, A. and Kang, W. (1990). Extended normal forms of quadratic systems, in Proc. 29th IEEE Conference on Decision and Control, pp. 2091-2096.
96. Lee, K. (2005). First Course on Fuzzy theory and Applications (Springer-Verlag, Berlin).
97. Lewis, F., Campos, J. and Selmic, R. (2002). Neuro-Fuzzy Control of Industrial Systems with Actuator Nonlinearities (SIAM, Philadelphia).
98. Li, D., Zhao, J. and Song, C. (2008). On the positive definiteness of the left semi-tensor product of matrices, in E. Jiang, T. Huang and C. Yang (eds.), Advances in Matrix Theory and Its Applications, Vol. 1 (World Academic Union, London), pp. 119-122.
99. Li, H. and Wang, Y. (2010). Boolean derivative calculation with application to fault detection of combinational circuits via the semi-tensor product method, Preprint.
100. Li, T., Tong, S. and Feng, G. (2010). A novel robust adaptive-fuzzy-tracking control for a class of nonlinear multi-input/multi-output systems, IEEE Trans. Fuzzy Systems 18, 1, pp. 150-160.
101. Li, Z., Zhao, Y. and Cheng, D. (2011). Structure of higher order Boolean networks, Journal of the Graduate School of the Chinese Academy of Sciences 28, 4, pp. 431-447, in Chinese.
102. Liu, X., Wang, H. and Zhao, J. (2008). The properties of bounded tridiagonal matrices, in E. Jiang, T. Huang and C. Yang (eds.), Advances in Matrix Theory and Its Applications, Vol. 2 (World Academic Union, London), pp. 104-109.
103. Liu, Z. and Liu, Y. (1996). Fuzzy Logic and Neural Network (BUAA Press, Beijing), in Chinese.
104. Ljung, L. and Söderström, T. (1982). Theory and Practice of Recursive Identification (MIT Press).
105. Mamdani, E. (1974). Applications of fuzzy algorithms for control of simple dynamic plant, Proceedings of the Institution of Electrical Engineers 121, 12, pp. 1585-1588.
106. Mei, S., Liu, F. and Xue, A. (2010). Semi-tensor Product Approach to Transient Analysis of Power Systems (Tsinghua Univ. Press, Beijing), in Chinese.
107. Mishchenko, A., Steinbach, B. and Perkowski, M. (2001). An algorithm for bidecomposition of logic functions, in Proc. 2001 IEEE/ACM 38th Design Automation Conference (Las Vegas), pp. 103-108.
108. Ooba, T. and Funahashi, Y. (1997). Two conditions concerning common quadratic Lyapunov functions for linear systems, IEEE Trans. Aut. Contr. 42, 5, pp. 719-721.
109. Passino, K. and Yurkovich, S. (1998). Fuzzy Control (Addison Wesley Longman).
110. Passino, K. and Yurkovich, S. (2002). Fuzzy Control (Tsinghua Univ. Press & Addison-Wesley).
111. Pennestri, E. and Valentini, P. (2009). Dual quaternions as a tool for rigid body motion analysis: A tutorial with an application to biomechanics, in K. Arczewski, J. Fraczek and M. Wojtyra (eds.), Multibody Dynamics 2009, EC-COMAS Thematic Conference (Warsaw), pp. 1-17
112. Posthoff, C. and Steinbach, B. (2004). Logic Functions and Equations: Binary Models for Computer Science (Springer, Dordrecht, The Netherlands).
113. Qiao, Y., Yuan, Y. and Cheng, D. (2011). Finite Fliess functional expansion and its application to ight control of missiles, Int. J. Robot. Autom. 26, 2, pp. 173-181.
114. Rade, L. and Westergren, B. (1998). Mathematics Handbook for Science and Engineering, 4Thedn. (Studentlitteratur, Sweden).
115. Reed, I. (1954). A class of multiple error-correction and the decoding scheme, IRE Trans. Inform. Theory IT-4, pp. 38-49.
116. Rohde, C. (1966). Some results on generalized inverses, SIAM Review 8, 2, pp. 201-205.
117. Roth, J. and Karp, R. (1962). Minimization over Boolean graphs, IBM Journal, pp. 227-238.
118. Runkler, T. (1996). Extended defuzzification methods and their properties, in Proc. 9th IEEE Int. Conf. Fuzzy Sys., pp. 694-700.
119. Saade, J. and Diab, H. (2000). Defuzzification techniques for fuzzy controllers, IEEE Trans. Systems, Man, and Cybernetics, Part B: Cybernetics 30, 1, pp. 223-229.
120. Sachs, R. and Wu, H. (1977). General Relativity for Mathematicians (Springer-Verlag).
121. Saha, S., Fouad, A., Kliemann, W. and Vittal, V. (1997). Stability boundary approximatin of a power system using the real normal form of vector fields, IEEE Trans. Power Systems 12, 2, pp. 797-802.
122. Samuelsson, B. and Troein, C. (2003). Superpolynomial growth in the number of attractors in Kauffman networks, Physical Review Letters 90, 9, p. 98701.
123. Sanchez, E. (1979). Inverses of fuzzy relations. Application to possibility distributions and medical diagnosis, Fuzzy Sets and Systems 2, 1, pp. 75-86.
124. Sanchez, E. (1996). Truth-qualification and fuzzy relations in matural languages, application to medical diagnosis, Fuzzy Sets and Systems 84, pp. 155-167.
125. Sanchez, E. (2002). Functional relations and fuzzy relational equations, in Fuzzy Information Processing Society, 2002. Proceedings. NAFIPS. 2002 Annual Meeting of the North American (IEEE), ISBN 0780374614, pp. 451-456.
126. Sasao, T. (1999). Switching Theory for Logic Synthesis (Kluwer Academic Publishers).
127. Sasao, T. and Butler, J. (1997). On bi-decompositions of logic functions, in Proc. Int. Worksop on Logic Synthesis, Vol. 2 (Lake Tahoe), pp. 1-6.
128. Sasao, T. and Fujita, M. (eds.) (1996). Representations of Discrete Functions (Kluwer Academic Publishers).
129. Schneeweiss, W. (1989). Boolean Functions, with engineering Applications and Computer Programs (Springer).
130. Shmulevich, I., Dougherty, E., Kim, S. and Zhang, W. (2002). Probabilistic Boolean networks: A rule-based uncertainty model for gene regulatory networks, Bioinformatics 18, 2, pp. 261-274.
131. Sidi, M. (1997). Spacecraft Dynamics and Control, A Practical Engineering Approach (Cambridge Univ. Press, Cambridge).
132. Song, C. (2009). Some Properties and Applications of Left Semi-tensor Product of Matrices, Master degree thesis, Liaocheng University.
133. Song, C., Zhao, J. and Li, D. (2008). The application of generalized permutation matrix in the commutation of tensor products of matrices, in E. Jiang, T. Huang and C. Yang (eds.), Advances in Matrix Theory and Its Applications, Vol. 1 (World Academic Union, London), pp. 16-18.
134. Soriano, J., Olarte, A. and Melgarejo, M. (2005). Fuzzy controller for MIMO systems using defuzzification based on Boolean relations (DBR), in Proc. 14th IEEE Int. Conf. Fuzzy Sys., pp. 271-275.
135. Sun, Z. and Xia, X. (1997). On nonregular feedback linearization, Automatica 33, 7, pp. 1739-1744.
136. Svozil, K. (1998). Quantum Logic (Springer, New York).
137. Thayse, A. (1984). P-functions and Boolean Matrix Factorization (Springer, Berlin).
138. Truemper, K. (2004). Design of Logic-based Intelligent Systems (Wiley & Sons, New Jersey).
139. Tsai, C. (1983). Contributions to the Design and Analysis of Nonlinear Models, Ph.D. thesis, Univ. of Minisota.
140. Tsukamoto, Y. (1979). An approach to fuzzy reasoning method, in M. Gupta, R. Ragade and R. Yager (eds.), Advances in Fuzzy Set Theory and Application (North-Holland, Amsterdam).
141. Tucker, J., Tapia, M. and Wayne Bennett, A. (1988). Boolean integral calculus, Applied Mathematics and Computation 26, 3, pp. 201-236
142. van der Schaft, A. (2000). L2-Gain and Passivity Techniques in Nonlinear Control (Springer-Verlag).
143. Varadarajan, V. (1984). Lie Groups, Lie Algebras, and Their Representations (Springer-Verlag, New York).
144. Venkatasubramanian, V. and Ji, W. (1997). Numerical approximation of (n-1)-dimensional stable manifolds in large systems such as the power system, Automatica 33, 10, pp. 1877-1883.
145. Verbruggen, H. and Babuška, R. (eds.) (1999). Fuzzy Logic Control: Advances in Applications (World Scientific Pub. Co. Inc., Singapore).
146. Vichniac, G. (1990). Boolean derivatives on cellular automata, Physica D: Non-linear Phenomena 45, 1-3, pp. 63-74.
147. Wang, G., Wei, Y. and Qiao, S. (2004). Generalized Inverses: Theory and Computations (Science Press, Beijing).
148. Wang, G. and Xu, Z. (2006). The reverse order law for the W-weighted Drazin inverse of multiple matrices product, Journal of Applied Mathematics and Computing 21, 1, pp. 239-248.
149. Wang, H. (2010). Some Advances of Left Semi-tensor Product of Matrice, Master degree thesis, Liaocheng University.
150. Wang, H., Liu, X. and Zhao, J. (2009a). The estimates for singular values of Schur complements of the left semi-tensor product of matrices, in Proc. 3rd Int. Workshop Matrix Analy. Appl., Vol. 1, pp. 286-289.
151. Wang, H., Si, C. and Niu, L. (2009b). On the metapositive of the left semi-tensor product of complex matrices, in Proc. 3rd Int. Workshop Matrix Analy. Appl., Vol. 1, pp. 290-293.
152. Wang, L. (1996). A Course in Fuzzy Systems and Control (Prentice-Hall, Inc., Upper Saddle River, NJ, USA).
153. Wang, X. (2002). Parameter Estimate of Nonlinear Models: Theory and Applications (Wuhan Univ. Press, Wuhan), in Chinese.
154. Wei, B. (1986). Second moments of LS estimate of nonlinear regressive model, Applied Math., J. Chinese Universities 1, 2, pp. 279-285, in Chinese.
155. Wen, Q., Niu, X. and Yang, Y. (2000). Boolean Functions in Modern Cryptography (Science Press, Beijing), in Chinese.
156. Willems, J. (1970). Stability Theory of Dynamical Systems (John Wiley & Sons Inc.).
157. Wilson, R. (1996). Introduction to Graph Theory, 4Thedn. (Prentice Hall, London).
158. Wolfram, S. (1986). Theory and Applications of Cellular Automata (World Scientific, Singapore).
159. Wonham, W. (1979). Linear Multivariable Control: A Geometric Aproach, 2nd edn. (Springer, Berlin).
160. Wonham, W. and Morse, A. (1970). Decoupling and pole assignment in linear multivariable systems: A geometric approach, SIAM J. Control 8, 1, pp. 1-18.
161. Wu, W. (1995). On Mechanization of Mathematics (Shandong Educational Press, Ji’nan).
162. Xue, A., Toe, K., Lu, Q. and Mei, S. (2006). Polynomial approximations for the stable and unstable manifolds of hyperbolic equilibrium point using semitensor product, Int. J. Innov. Comp. Inform. Contr. 2, 3, pp. 1-16.
163. Yanushkevich, S. (1998). Logic differential calculus in multi-valued logic design, Prace Naukowe Politechniki Szczecińskiej. Instytut Informatyki 537, 1, pp. 7-326.
164. Zaborszky, J., Huang, J., Zheng, B. and Leung, T. (1988). On the phase protraits of a class of large nonlinear dynamic systems such as the power systems, IEEE Trans. Aut. Contr. 33, 1, pp. 4-15.
165. Zadeh, L. (1965). Fuzzy Sets, Information and Control 8, pp. 338-353.
166. Zadeh, L. (1972). Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Systems, Man and Cybernetics 3, 1, pp. 28-44.
167. Zadeh, L. and Kacprzyk, J. (eds.) (1992). Fuzzy Logic for the Management of Uncertainty (John Wiley & Sons, Inc. New York, NY, USA).
168. Zhang, X. (2004). Matrix Analysis and Applications (Tsinghua Univ. Press & Springer, Beijing).
169. Zhang, Y. (1993). Theory of Multi-Edge Matrix (Chinese Statistics Press, Henan), in Chinese.
170. Zhang, Y., Liu, Z. and Wang, Y. (2009). A three-dimensional probabilistic fuzzy control systems for network queue management, J. Contr. Theory & Appl. 7, 1, pp. 29-34.
171. Zhao, Q. (2005). A remark on ‘Scalar equations for synchronous Boolean networks with biologicapplications’ by C. Farrow, J. Heidel, J. Maloney, and J. Rogers, IEEE Trans. Neural Networks 16, 6, pp. 1715-1716.
172. Zhao, Y., Gao, X. and Cheng, D. (2010a). Semi-tensor product approach to Boolean functions, Preprint.
173. Zhao, Y., Li, Z. and Cheng, D. (2010b). Optimal control of logical control networks, Preprint.
174. Zhao, Y., Qi, H. and Cheng, D. (2010c). Input-state incidence matrix of Boolean control networks and its applications, Sys. Contr. Lett. 59, 12, pp. 767-774.
175. Zhu, J. (2005). Fuzzy System and Control Theory (China Machine Press, Beijing), in Chinese.