Appendix 4.2

Discrete time models can be obtained directly from continuous time models, as summarized in this appendix. The MATLAB Control Toolbox can be used for these conversions, as shown in Module 4.

Consider a continuous state space model of the following form (where we have assumed D = 0)

Equation A4.1

A similar discrete state space model has the following form

Equation A4.2

Assuming that the sample time is Dt, and that the input u is held constant between time t_k and t_k+1, (A4.1) can be integrated to yield

which has the solution (although the matrix exponential is not an intuitive concept, the computation is readily performed by MATLAB)

which is usually written in the following form

where

The stability of this discrete state space model is determined by the eigenvalues of F, which must have a magnitude less that 1.0 for stability.

The discrete transfer function is, similar to the continuous time case