Editing Discretization (section)

=== Approximations ===
Exact discretization may sometimes be intractable due to the heavy matrix exponential and integral operations involved. It is much easier to calculate an approximate discrete model, based on that for small timesteps <math>e^{\mathbf{A}T} \approx \mathbf I + \mathbf A T</math>. The approximate solution then becomes:
<math display=block>\mathbf x[k+1] \approx (\mathbf I + \mathbf{A}T) \mathbf x[k] + T \mathbf{Bu}[k] </math>

This is also known as the [[Euler method]], which is also known as the forward Euler method. Other possible approximations are <math>e^{\mathbf{A}T} \approx (\mathbf I - \mathbf{A}T)^{-1}</math>, otherwise known as the backward Euler method and <math>e^{\mathbf{A}T} \approx (\mathbf I +\tfrac{1}{2} \mathbf{A}T) (\mathbf I - \tfrac{1}{2} \mathbf{A}T)^{-1}</math>, which is known as the [[bilinear transform]], or Tustin transform. Each of these approximations has different stability properties. The bilinear transform preserves the instability of the continuous-time system.