Transfer function computation and factorisation of 3-dimensional systems in state-space

The paper considers a general state-space model of three-dimensional (3-D) systems, i.e. dynamic systems varying with respect to three independent variables. Two problems are studied and solved. The first problem is that of determining and computing the system transfer function matrix by using directly the state transition matrix of the given state-space model along with the system characteristic polynomial. This problem is solved through an extension of the well-known Fadeeva-Leverier algorithm to the 3-D case, and a transfer function computation formula is derived which can be implemented in an iterative way. The second problem is that of factorising the transfer function of single-input single-output systems via state-variable feedback. This problem is solvable under certain conditions only if the numerator of the open-loop transfer function is by itself factorisable in three 1-D polynomials. Otherwise the factorisation is restricted to the denominator of the closed-loop transfer function. The factorisation algorithm derived provides the state-feedback gain vector through the solution of a set of algebraic equations. Two nontrivial examples are worked out, and an interpretation of the 3-D model of the paper for the case of an elliptic distributed-parameter system in 3-dimensional spatial domain is given.