User`s guide

6 Program Architecture

6-30

Integration of Continuous States

The real-time program calculates the next values for the continuous states

based on the derivative vector, dx/dt, for the current values of the inputs and

the state vector.

These derivatives are then used to calculate the next value of the states using

a state-update equation. This is the state-update equation for the first order

Euler method (

ode1).

where h is the step size of the simulation, x represents the state vector, and dx/

dt is the vector of derivatives. Other algorithms may make several calls to the

output and derivative routines to produce more accurate estimates.

Note, however, that real-time programs use a fixed-step size since it is

necessary to guarantee the completion of all tasks within a given amount of

time. This means that, while you should use higher order integration methods

for models with widely varying dynamics, the higher order methods require

additional computation time. In turn, the additional computation time may

force you to use a larger step size, which can diminish the accuracy increase

initially sought from the higher order integration method.

Generally, the stiffer the equations, (i.e., the more dynamics in the system with

widely varying time constants), the higher the order of the method that you

must use.

In practice, the simulation of very stiff equations is impractical for real-time

purposes except at very low sample rates. You should test fixed-step size

integration in Simulink to check stability and accuracy before implementing

the model for use in real-time programs.

For linear systems, it is more practical to convert the model that you are

simulating to a discrete time version, for instance, using the

c2d function in the

Control System Toolbox.

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h+=