Dynamic systems modeling (with example)
The following chapter discusses the principles of creating dynamic models in Simultus and shows the implementation of such a system on the example of a vibrating mass. It is possible to connect the model we created to a simulated or real FATEK PLC controller!
In Simultus you can model the behavior of dynamical systems. It should be taken into account that Simultus is mainly used to observe the behavior of objects in real time and therefore works on a similar principle to an industrial controller. Which means:
1) It only simulates the operation of discrete circuits.
2) The sampling time must be constant and strictly defined.
3) It is not possible to use direct feedback loops in the diagram - the feedback value should be transferred e.g. via the VRn (or Rn) register.
Ad.1)
In Simultus, we do not have blocks for simulating continuous systems, because in practice such system must be implemented as a discrete system in a computer control system. The models of continuous systems must therefore be converted to discrete models.
Ad.2)
The sampling time must be taken into account in the control system.
The operating frequency setting of the simulation cycles can be made by selecting from the top menu of the simulation settings:
After selecting this option, a simulation settings window will appear:
If we want to observe the system’s behavior in real time, the first upper option must be selected, as in the picture above.
The simulation period is given in microseconds. It should be remembered that the calculations take a specific time for the computer, depending on its technical parameters, but also on the complexity of our simulation model. The actual simulation time is displayed in the diagram window at the bottom of the screen:
According to the note written in red font in the simulation settings window, this time may differ from the set one, which results from the way the Windows operating system works.
Cycle time is crucial for accurate calculations. Therefore, it is possible to check it in every program cycle with an accuracy of 1 microsecond.
To do this, use the SYS_DATA function block available in the 'Timers and counters' tab. The output values from this block can be copied to selected registers and, based on them, we can correct our calculations accordingly:
The value on the output 'cycle_time' specifies the computation time of one program cycle. On this basis, we can estimate the maximum frequency that we can set for our simulation so that the computer can do all the calculations in real time.
'Last_loop_time' is the execution time of the last control loop.
You can read more about the program cycles here.
In the case of function blocks where the result is dependent on time, consideration of the cycle time is essential. For example, an INTEGRATOR function block is simply an adder of the values that appear at its input in subsequent program cycles. Therefore, for the integral to be correctly computed in the time domain, the signal value should be multiplied by the sampling time in appropriate units before or after the block. The example below shows how to do it for the sampling period T = 500 [microseconds] = 0.0005 [s] (the program cycle frequency in this case is 2 kHz):
Of course, we can combine the above operation with data from the SYS_DATA block as follows (multiplication of the output value from the block by 0.000001 results from the change of units from microseconds to seconds):
EXAMPLE:
Presented below is an example of the realization of the mass model M = 250 [kg], suspended on a spring with the stiffness k = 300 [N / m], to which a damper with B = 700 [Ns / m] damping is additionally attached (requires the Simultus EDU Plus version).
The system’s mathematical model is described by the formula:
Taking into account the above remarks, the modeled system can be realized as shown in the diagram below:
The VR1 register is used to store the speed dx / dt value and the VR2 register is used to store the position x of mass M.
The rescaling of the input value to the 6DOF_AXIS block results from the conversion of the units [m] used in the model in the diagram to [mm] used in the visualization.
For the sake of clarity, the block responsible for recording the waveforms has not been shown in the diagram above.
The step force F = 300 [N] is performed by the switch (SWITCH_BUTTON).
The effect of the simulation is presented below.