We often analyze the dynamics of linear time invariant systems. These are characterized by linearity, that is, that any linear combination of input signals leads to the same linear combination of their respective individual output signals. These are also characterized by time invariance, that is, that a time shift in the input merely results in an equivalent time shift in the response.

$$ au_1+bu_2\rarr ay_1+by_2 $$

$$ u(t)\rarr y(t)\Rarr u(t-c)\rarr y(t-c) $$

These systems are well summarized by a transfer function, which gives the Laplace transform of the output in terms of that of the input.

$$ \boxed{u}\rarr\boxed{G_{LTI}}\rarr\boxed{y}~\Rarr~Y(s)=G(s)U(s) $$

This technique is readily applied in general to an LTI system articulated in state-space form.

$$ \bold{\dot x=Ax+B}u\\ y=\bold{Cx}+du\\ G(s)=\bold C(s\bold I-\bold A)^{-1}\bold B+d $$

A transfer function may be written as a ratio of two polynomials, or with each polynomial factored, in pole-zero form. The former is most immediately obtained when applying the Laplace transform to an ODE, the the latter more readily reveals the dynamic properties of the system. In pole-zero form, each factor in the numerator corresponds to a zero, and each in the denominator corresponds to a pole.

$$ G(s)=\frac{\Sigma a_np^n}{\Sigma b_np^n}=\frac{\Pi (s-z_n)}{\Pi (s-p_n)} $$

Pole and zero locations may be plotted along the complex plane, a particularly useful technique for obtaining a graphical intuition into the effect of pole locations on dynamic response.

Note that the imaginary component of a pole location corresponds to the frequency of oscillatory behavior, while the real component corresponds to the rate of decay behavior.