Applications of the Cayley-Hamilton Theorem in Linear Systems
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Keywords:
Приложение, теорема Гамильтона-Кэли, линейная системаAbstract
The classical Cayley-Hamilton theorem says that every square matrix satisfies its own characteristic equation. Cayley Hamilton theorem is widely applicable in many fields not only related to mathematics, but in other scientific fields too. This theorem is used all over in linear algebra. It also is quite useful in modern control theory, especially in the linear systems. This paper introduced the applications of the Cayley-Hamilton theorem in linear systems, mainly from seven aspects: 1.The transfer function matrix is derived from the state-space description. 2.Equivalent representation of the uncontrollable subspace of the continuous system is presented. 3. The controllability canonical form and observability canonical form of the single input – single output system is obtained. 4. Controllable subspace of the discrete system is obtained. 5. The controllability of the linear timeinvariant continuous systems after time discretization is presented. 6. The equivalent representation of the unobservable subspaces of a continuous system is obtained. 7. The observability of the linear etime-invariant discrete system is derived.References
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3. Kaczorek T. Linear Control Systems. - Tauton: Research Studies Press,1992. – 300 p.
4. Kaczorek T. Extensions of the Cayley-Hamilton theorem for 2D continuous-discrete linear systems // Appl. Math.Comput. Sci. – 1994. – Vol. 4, №4. – P. 507–515.
5. Kaczorek T. An existence of the Cayley-Hamilton theorem for nonsquare block matrices and computation of the left and right inverses of matrices // Bull. Pol. Acad. Techn. Sci. – 1995. – Vol. 43, №1. – P. 49–56.
6. Zhaolin C., Shuping M. Linear Systems Theory. – Beijing: Science Press.,2006. – 204 p.
7. Lin H. Linear Algebra in Systems and Control Theory. – Beijing: Science Press.,1984. – 770 p.
8. Xiuling Z. Principles of Automatic Control. – Beijing: Tsinghua University Press.,1984. – 345 p.9. Liang S., Jianjun Y., Daoxiong G. Theoretical Basis of Linear Systems. – Beijing: Beijing Industrial University Press.,2006. – 259 p.
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Zhumahan, K. (2015). Applications of the Cayley-Hamilton Theorem in Linear Systems. Journal of Mathematics, Mechanics and Computer Science, 87(4), 56–66. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/295
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Mathematics
