Abstract
This paper proposes a new matrix product, namely, semi-tensor product. It is a generalization of the conventional matrix product. Meanwhile, it is also closely related to Kronecker (tensor) product of matrices. The purpose of introducing this product is twofold: (i) treat multi-dimensional data; (ii) treat nonlinear problems in a linear way. Then the computer and numerical methods can be easily used for solving nonlinear problems. Properties and formulas are deduced. As an application, the Morgen’s problem for control systems is formulated as a numerically solvable problem.
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References
Huang L., Linear Algebra in Systems and Control Theory (in Chinese), Beijing: Science Press, 1984.
Zhang, F., Matrix Theory, Basic Results and Techniques, New York: Springer-Verlag, 1999.
Sokolnikoff, I. S., Tensor Analysis, Theory and Applications to Geometry and Mechanics of Continua, 2nd ed., New York: John Wiley & Sons, Inc., 1964.
Boothby, W. M., An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., New York: Academic Press, 1986.
Willems, J. L., Stability Theory of Dynamical Systems, New York: John Wiley & Sons, Inc., 1970.
Ooba, T., Funahashi, Y., Two conditions concerning common quadratic Lyapunov functions for linear systems, IEEE Trans. Automat. Contr., 1997, 42(5): 719–721.
Ooba, T., Funahashi, Y., Stability robustness for linear state space models—a Lyapunov mapping approach, Sys. Contr. Lett, 1997, 29. 191–196.
Cheng, D., Xue, W., Huang, J., On general Hamiltonian Systems, Proc. of ICARCV’98, Singapore, 1998, 185–189.
Morgan, B. S., The synthesis of linear multivariable systems by state feedback, JACC, 1964, 64: 468–472.
Falb, P. L., Wolovich, W. A., Decoupling and synthesis of multivariable control systems, IEEE Trans, Aut. Contr., 1967, 12: 651–668.
Wolovich, W. A., Linear Multivariable System, New York: Springer-Verlag, 1974, 296.
Suda, M., Unahashi, K., Decoupling of nonsquare systems—a necessary and sufficient condition in terms of infinite zeros, Proc. 9th IFAC World Conf., Kyoto, 1984, Vol. 8, 88–93.
Descusse, J., Lafay, J. F., Malabre, M., Solution to Morgan’s problem, IEEE Trans. Aut. Contr., 1988, 33: 732–739.
Chen, S., Cao, L., On constrained Morgen’s problem, Science in China (in Chinese), Ser. A, 1996, 26(6): 513–523.
Xu, K., On a class of non-square Morgen’s problem, Science in China (in Chinese), Ser. A, 1996, 26(4): 694–703.
Wu, W., On Mechanization of Mathematics (Partly in Chinese), Jinan: Shandong Educational Press, 1995.
Wonham, W. M., Linear Multivariable Control, A Geometric Approach, Berlin: Springer-Verlag, 1974.
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Cheng, D. Semi-tensor product of matrices and its application to Morgen’s problem. Sci China Ser F 44, 195–212 (2001). https://doi.org/10.1007/BF02714570
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DOI: https://doi.org/10.1007/BF02714570