Poles of a State Space Model

In this section, we show that the poles of a state-space model are given by the eigenvalues of the state-transition matrix $A$.

Beginning again with the transfer function of the general state-space model,

$$H(z) = D + C \left(zI - A\right)^{-1}B,$$

we may first observe that the poles of $H(z)$ are either the same as or some subset of the poles of

$$H_p(z) \isdef \left(zI - A\right)^{-1}$$

(They are the same when all modes are controllable and observable [37].) By Cramer's rule for matrix inversion, the denominator polynomial for $\left(zI - A\right)^{-1}$ is given by the determinant

$$D(z) \isdef \det(zI - A)$$

where $\det(Q)$ denotes the determinant of the square matrix $Q$. (The determinant of $Q$ is also often written $\left\vert Q\right\vert$.) In linear algebra, the polynomial $D(z) = \left\vert zI-A\right\vert$ is called the characteristic polynomial for the matrix $A$. The roots of the characteristic polynomial are called the eigenvalues of $A$.

Thus, the eigenvalues of the state transition matrix $A$ are the poles of the corresponding linear time-invariant system. In particular, note that the poles of the system do not depend on the matrices $B,C,D$, although these matrices, by placing system zeros, can cause pole-zero cancellations (unobservable or uncontrollable modes).