Transfer function, Bode canonical form

Control theory

The Bode canonical form is a widely used representation of the transfer function. One of the main reasons is that, using the parameters contained in the formula, drawing a graphical representation of the transfer function, in the frequency domain, becomes particularly easy.

K\frac{(1+T_1s)+...(1+\frac{2\zeta_1s}{\rho_{n1}}+\frac{s^2}{\rho_{n1}^2})+...}{s^n(1+\tau_1s)+...(1+\frac{2\xi_1s}{\omega_{n1}}+\frac{s^2}{\omega_{n1}^2})+...}

The parameters that appear in the formula have a physical meaning that can be used to draw vary useful information about a system behavior.

Parameter Meaning
K Static gain of the system: if n equals zero, meaning no poles or zeroes in the origin, the static gain is the ratio of the system input and output, when the input is a constant value and the system is in its steady state.
n Number of poles in the origin, if greater than zero. Number of zeroes in the origin, if lower than zero. Roughly, the parameter tells if the system acts as an integrator, when greater than zero, or as a differentiator, when the transfer function contains one or more zeroes in the origin.
T, \tau Time constants: they are linked to the speed of the various components in the system. Usually the speed of a physical system is closely related to its slower time constants.
\zeta, \xi Damping factors: they are related to how fast the system loses its energy. Usually underdamped systems, will oscillate longer than the ones where damping factors are higher.
\rho_n, \omega_n Natural pulsations: they are related to the system resonant frequencies. In systems having a zero damping factor, the oscillation frequency can be calculated from \omega_n.
 

Binomials containing T o \tau, and trinomials containing \rho_n o \omega_n, car appear more than once or never inside the transfer function. Whereas K and s^n cannot appear more than once.

In the following articles, we will make some examples, to explain how the formula may be used in a practical context.