Editing Resonance (section)

=== Generalizing resonance and antiresonance for linear systems ===

Next consider an arbitrary linear system with multiple inputs and outputs. For example, in [[state-space representation]] a third order [[linear time-invariant system]] with three inputs and two outputs might be written as
<math display="block">\begin{bmatrix} \dot{x}_1 \\ \dot{x}_2 \\ \dot{x}_3 \end{bmatrix} =
 A \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + B \begin{bmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \end{bmatrix},</math>
<math display="block">\begin{bmatrix} y_1(t) \\ y_2(t) \end{bmatrix}
= C \begin{bmatrix} x_1(t) \\ x_2(t) \\ x_3(t) \end{bmatrix} + D \begin{bmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \end{bmatrix},</math>
where ''u''<sub>''i''</sub>(''t'') are the inputs, ''x''<sub>''i''</sub>(t) are the state variables,  ''y''<sub>''i''</sub>(''t'') are the outputs, and ''A'', ''B'', ''C'', and ''D'' are matrices describing the dynamics between the variables.

This system has a [[transfer function matrix]] whose elements are the transfer functions between the various inputs and outputs. For example,
<math display="block">
\begin{bmatrix} Y_1(s) \\ Y_2(s) \end{bmatrix} = 
\begin{bmatrix} 
H_{11}(s) & H_{12}(s) & H_{13}(s) \\
H_{21}(s) & H_{22}(s) & H_{23}(s)
\end{bmatrix}
\begin{bmatrix} U_1(s) \\ U_2(s) \\ U_3(s) \end{bmatrix}.
</math>

Each ''H''<sub>''ij''</sub>(''s'') is a scalar transfer function linking one of the inputs to one of the outputs. The RLC circuit examples above had one input voltage and showed four possible output voltages–across the capacitor, across the inductor, across the resistor, and across the capacitor and inductor combined in series–each with its own transfer function. If the RLC circuit were set up to measure all four of these output voltages, that system would have a 4×1 transfer function matrix linking the single input to each of the four outputs.

Evaluated along the imaginary axis, each ''H''<sub>''ij''</sub>(''iω'') can be written as a gain and phase shift,
<math display="block">H_{ij}(i\omega) = G_{ij}(\omega)e^{i\Phi_{ij}(\omega)}.</math>

Peaks in the gain at certain frequencies correspond to resonances between that transfer function's input and output, assuming the system is [[exponential stability|stable]].

Each transfer function ''H''<sub>''ij''</sub>(''s'') can also be written as a fraction whose numerator and denominator are polynomials of ''s''.
<math display="block">H_{ij}(s) = \frac{N_{ij}(s)}{D_{ij}(s)}.</math>

The complex roots of the numerator are called zeroes, and the complex roots of the denominator are called poles. For a stable system, the positions of these poles and zeroes on the complex plane give some indication of whether the system can resonate or antiresonate and at which frequencies. In particular, any stable or [[marginal stability|marginally stable]], complex conjugate pair of poles with imaginary components can be written in terms of a natural frequency and a damping ratio as
<math display="block">s = -\zeta\omega_0 \pm i\omega_0\sqrt{1-\zeta^2},</math>
as in Equation ({{EquationNote|5}}). The natural frequency ''ω''<sub>0</sub> of that pole is the magnitude of the position of the pole on the complex plane and the damping ratio of that pole determines how quickly that oscillation decays. In general,{{sfn|Hardt|2004}}
* Complex conjugate pairs of ''poles'' near the imaginary axis correspond to a peak or resonance in the frequency response in the vicinity of the pole's natural frequency. If the pair of poles is ''on'' the imaginary axis, the gain is infinite at that frequency.
* Complex conjugate pairs of ''zeroes'' near the imaginary axis correspond to a notch or antiresonance in the frequency response in the vicinity of the zero's frequency, i.e., the frequency equal to the magnitude of the zero. If the pair of zeroes is ''on'' the imaginary axis, the gain is zero at that frequency.

In the RLC circuit example, the first generalization relating poles to resonance is observed in Equation ({{EquationNote|5}}). The second generalization relating zeroes to antiresonance is observed in Equation ({{EquationNote|7}}). In the examples of the harmonic oscillator, the RLC circuit capacitor voltage, and the RLC circuit inductor voltage, "poles near the imaginary axis" corresponds to the significantly underdamped condition ζ < 1/<math>\sqrt{2}</math>.