After linearization, the system equations can naturally be represented as a mix of linear differential equations of arbitrary order and algebraic equations. The Laplace transform of these equations (with zero initial conditions) can be written in matrix-vector form as
where , u, and y are vectors of the Laplace transformed system variables, inputs, and outputs, respectively, and T, U, V, and W are polynomial matrices of dimension r × r, r × p, q × r, and q × p, respectively. You should note that the i are the system variables, and are not necessarily the system-state variables. This set of equations can be equally written as
and the system matrix in polynomial form (Rosenbrock (1970)) is defined as
The matrix P(s) contains all the information about the system needed for analysis purposes. However, it is important to note that in the formulation of the matrix P(s), the dimension r of the matrix T(s) must be adjusted so that r ≥ , the degree of the determinant |T(s)|. Since is the order of the system, this ensures that a system matrix in polynomial form can be readily transformed to a corresponding system matrix in state-space form. Also, with this form of system description, the internal modes of the system, which can be considered here as the poles of internal subsystems, are given by the roots of |T(s)| = 0. The internal modes of the system may, or may not, appear in the elements of the corresponding transfer-function matrix description due to cancellations occurring between equal factors in the various numerator and denominator polynomial elements. For a known state-space realization, the system matrix in state-space form is defined as
The system matrix readily yields the corresponding transfer-function matrix
which for the special case of P(s) in state-space form becomes the relationship
A system matrix model for the mass-spring system can be determined by applying the Laplace transformation, with zero initial conditions, directly to the original system equations to get
where the xi, y, and u are now Laplace-transformed variables. Collecting together the terms in s associated with x(s) and u(s), these equations can also be written in the form given in Eq. (3.6), where
Since |T(s)| is a polynomial in s of degree 4, the order of the system is n = 4. Thus, to construct a system matrix, the dimension of T(s) must be made greater than or equal to 4, for example, by introducing two additional dummy variables, x{3} and x4, such that the equation set can be written as
, u, and y are vectors of the Laplace transformed system variables, inputs, and outputs, respectively, and T, U, V, and W are polynomial matrices of dimension r × r, r × p, q × r, and q × p, respectively. You should note that the 
, the degree of the determinant |T(s)|. Since 
