The concepts of controllability and observability are traditionally defined in terms of state-space models. Here, the corresponding concepts of input decoupling zeros, output decoupling zeros, and input-output decoupling zeros for systems described by a system matrix in polynomial, or state-space, form are considered. For a system described by the polynomial matrices T(s), U(s), V(s), and W(s), the corresponding transfer-function matrix model is given by
If T(s) and U(s) have a left matrix greatest common divisor L(s), then T(s) and U(s) can be expressed as
Assuming that L(s) is nonsingular, forming G(s) yields
Unless T(s) and U(s) are relatively left prime, the determinant of L(s) is a nonzero polynomial in s. The roots of this polynomial {i} are called the input decoupling zeros of the system. These input decoupling zeros cancel out in the formulation of G(s) and result in a lower-order representation. Similarly, if T(s) and V(s) have a right matrix greatest common divisor R(s), the output decoupling zeros {i} are defined as the roots of the determinant of R(s). Again, any output decoupling zeros vanish from the resulting transfer-function matrix giving a lower-order representation. If any input decoupling zeros have been removed from the system {T(s), U(s), V(s)}, and if the output decoupling zeros are then determined as the set {i}, then the input-output decoupling zeros {i} are given by the set
If G(s) is constructed from a system having no decoupling zeros, it is said to have arisen from a least-order system. These concepts also apply to a system matrix in state-space form, where the input decoupling zeros are the uncontrollable modes of the system, the output decoupling zeros are the unobservable modes of the system, and the input-output decoupling zeros are the uncontrollable and unobservable modes of the system.
Consider the single-input, single-output system described by the system matrix
The internal modes {i} of the system P(s) are given by the roots of the determinant |T(s)| = 0 as
Since the matrices T(s) and U(s) of this system matrix are not relatively left prime, and have a left matrix greatest common divisor L(s), that is, T(s) and U(s) can be expressed as
the input decoupling zeros of the system are given by the roots of the determinant |L(s)| = 0 as
Equally, since the matrices T(s) and V(s) of this system matrix are not relatively right prime, and have a right matrix greatest common divisor R(s), that is, T(s) and V(s) can be expressed as
the output decoupling zeros of the system are given by the roots of the determinant |R(s)| = 0 as
After the input decoupling zero has been removed, the resulting matrices (s) and V(s) of this system matrix are still not relatively right prime, and have a right matrix greatest common divisor R(s), that is, (s) and V(s) can be expressed as
Thus, the output decoupling zeros of the system, after the input decoupling zero has been removed, are
Therefore, the input-output decoupling zeros are
and the complete set of decoupling zeros detected are
Finally, the poles {i}, of the corresponding transfer function g(s), are given by the difference between the internal modes and the complete set of decoupling zeros as
and so the corresponding single-input, single-output transfer function g(s) calculated from the system matrix P(s) is
i} are called the input decoupling zeros of the system. These input decoupling zeros cancel out in the formulation of G(s) and result in a lower-order representation.
Similarly, if T(s) and V(s) have a right matrix greatest common divisor R(s), the output decoupling zeros {
i} are defined as the roots of the determinant of R(s). Again, any output decoupling zeros vanish from the resulting transfer-function matrix giving a lower-order representation.
If any input decoupling zeros have been removed from the system {T(s), U(s), V(s)}, and if the output decoupling zeros are then determined as the set {
i}, then the input-output decoupling zeros {
i} are given by the set
i} of the system P(s) are given by the roots of the determinant |T(s)| = 0 as
(s) and V(s) of this system matrix are still not relatively right prime, and have a right matrix greatest common divisor R(s), that is, 
(s) and V(s) can be expressed as
i}, of the corresponding transfer function g(s), are given by the difference between the internal modes and the complete set of decoupling zeros as
