Applications of the Cayley-Hamilton Theorem in Linear Systems

Abstract

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.