A typical system that can be modelled by a first-order differential equation is a room heated by a heater, but losing heat to the outside world. If the heater output CO and external temperature D are constant, the room temperature CV will gradually rise or fall until the rate of heat loss is equal to the heater output.

A first-order model has three parameters:

From the differential equation, this is programmed in ModMultiSim as:

For those that are interested, the actual differential equation is:

     Ts.dCV(t)/dt + CV(t) = Ks.CO(t) + Kd.D(t)

where t is time, and the derivative dCv(t)/dt is the rate at which CV changes. If the derivative is approximated by the difference between successive values of CV, a "discrete" form of the differential equation is obtained:

     Ts.(CV[k+1] - CV[k])/T + CV[k] = Ks.CO[k] + Kd.D[k]

where T is the simulation time interval (i.e. CycleTime).

This can be re-arranged to produce:

     CV[k+1] = CV[k] + (Ks.CO[k] + Kd.D[k] - CV[k]).T/Ts

which is the form of equation used in the simulation example.

This model builds on the previous model so for an explanation of terms such as CV, Ks, see first-order model.

The heated room example is "self-regulating": the room temperature CV stabilizes if the controller output CO and disturbance D are constant. However, some systems are not self-regulating. A typical example is a tank being filled by a pump. If the pump flow rate CO and rate of drainage D are constant, the level of fluid CV in the tank will continue to rise or fall (unless the rate of filling and drainage are exactly equal).

This can be programmed in ModMultiSim as:

This kind of system can be modelled by a first order differential equation that does not have the CV(t) term:

     Ts.dCV(t)/dt = Ks.CO(t) + Kd.D(t)

A discrete form of this equation is:

     CV[k+1] = CV[k] + (Ks.CO[k] + Kd.D[k]).T/Ts

which is the basis for the code segment above in ModMultiSim.

This model builds on the first model, so for an explanation of terms such as CV, Ks, see first-order model.

In some systems, there may be a significant delay ("dead time") before CV starts to change when CO is changed. This is typically due the "travel time" of material (or energy) from the controller to the sensor. A "First-Order Plus Dead Time" model can be used to describe such systems.

In ModMultiSim this can be programmed as set out in the table below. Notice the use of the delay line technique to simulate a dead time of 5 simulation periods - registers CO1-CO5 and the use of the value CO5, which is the value of the controller output 5 periods earlier, to compute CV

The differential equation for a FOPDT model is:

     Ts.dCV(t)/dt + CV(t) = Ks.CO(t - Θt) + Kd.D(t)

where Θt is the system dead time.

A discrete form of this equation is:

     Ts.(CV[k+1] - CV[k])/T + CV[k] = Ks.CO[k - Θt/T] + Kd.D[k]

which re-arranged produces:

     CV[k+1] = CV[k] + (Ks.CO[k - Θt/T] + Kd.D[k] - CV[k]).T/Ts

The term CO[k - Θt/T] is the value that CO had Θt/T simulation periods earlier. As mentioned, this example uses the delay line technique to simulate a dead time of 5 simulation periods (i.e. Θt/T is 5):