Editing State-space representation (section)

=== Example: continuous-time LTI case ===
Stability and natural response characteristics of a continuous-time [[LTI system]] (i.e., linear with matrices that are constant with respect to time) can be studied from the [[eigenvalue]]s of the matrix <math>\mathbf{A}</math>. The stability of a time-invariant state-space model can be determined by looking at the system's [[transfer function]] in factored form. It will then look something like this:

<math display="block"> \mathbf{G}(s) = k \frac{ (s - z_{1})(s - z_{2})(s - z_{3})
}{ (s - p_{1})(s - p_{2})(s - p_{3})(s - p_{4})
}. </math>

The denominator of the transfer function is equal to the [[characteristic polynomial]] found by taking the [[determinant]] of <math>s\mathbf{I} - \mathbf{A}</math>,
<math display="block">\lambda(s) = \left|s\mathbf{I} - \mathbf{A}\right|. </math>
The roots of this polynomial (the [[eigenvalue]]s) are the system transfer function's [[complex pole|pole]]s (i.e., the [[Mathematical singularity|singularities]] where the transfer function's magnitude is unbounded). These poles can be used to analyze whether the system is [[exponential stability|asymptotically stable]] or [[marginal stability|marginally stable]]. An alternative approach to determining stability, which does not involve calculating eigenvalues, is to analyze the system's [[Lyapunov stability]].

The zeros found in the numerator of <math>\mathbf{G}(s)</math> can similarly be used to determine whether the system is [[minimum phase]].

The system may still be '''input–output stable''' (see [[BIBO stability|BIBO stable]]) even though it is not internally stable. This may be the case if unstable poles are canceled out by zeros (i.e., if those singularities in the transfer function are [[removable singularity|removable]]).