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I have previously done this question on math stack, and I think that maybe here is better.

I am studying the theory of discrete-time systems, comparing them with continuous ones in the control theory field.In particular, I have focused on the advantages and disadvantages to prefer sometimes the first over the latter.

Some papers say that nowadays, most control systems are implemented through digital devices (as DSP, etc.). So sometimes, it is preferable to design directly a discrete-time controller rather than a continuous one. Indeed the engineers prefer this choice.

I have also read that designing a continuous-time controller is not entirely avoided; indeed, many mathematicians prefer this choice. In that case, the idea is that a continuous-time is approximately a discrete-time system, with an arbitrarily small sampling time.

My question is: what are the disadvantages of choosing a too small sampling time above all on the control effort?

Note: the control effort is the amount of energy or power necessary for the controller to perform its duty.

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  • $\begingroup$ Why use a system capable of sampling at 1MHz when the system only runs at 10Hz? $\endgroup$
    – Solar Mike
    Aug 7 '21 at 15:44
  • $\begingroup$ Sorry I have not understood what you want to say! Sorry but I am not so expert in this field, I am a student $\endgroup$
    – pawel
    Aug 7 '21 at 15:54
  • $\begingroup$ Why feed a system with input in rapid succession when the system is a slow-to-respond system? When cooking with a thick skillet with slow temperature response, would one adjust the flame every second or one in a few minutes? $\endgroup$
    – AJN
    Aug 7 '21 at 17:07
  • $\begingroup$ How can I know that the system is slow-to-respond? I mean I have not understood the disadvantages of using a continuous time system in view of a practical implementation...can you help me? $\endgroup$
    – pawel
    Aug 7 '21 at 17:15
  • $\begingroup$ If the frequencies of interest in the system's behavior are less than 10% of the sampling rate, everything is "cookbook" - super easy, laplace-transform techniques are used nothing to worry about. This is the preferred situation. More sampling rate is generally good, although it is critical to consider aliasing, and more effort may be required at higher rates, depending on the ADC/DAC types So usually there is a sweet spot. It's possible there may also be numerical methods issues with excessive sampling rate. $\endgroup$
    – Pete W
    Aug 7 '21 at 17:42
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I see it as some sort of "degree of freedom" for information.

you could reconstruct each signal not knowing the fully set of points over time, in frequency domain it is sufficient having less or equal to Nyquist frequency.

But in discrete models you need even less information. Information that is sufficient to calculate the control reaction.

On the other hand, the main problem on digital technologies is that they has information limits, when storing and when processing it. Then the preference is use the less information as you could work with but than is sufficient to work with.

Actually when you compute a continuous system on a program like MATLAB, it actually works with discrete system with some tolerance so that you couldn't notice. No computer (discrete system) could handle the information in a continuous model since it is actually infinite (but redundant).

Also, discrete formulation allows you to make finite matrix description, numeric approximations instead of analytics solutions (some times handle with convolutions in continuous time is not easy, in contrast in discrete time is a common matrix multiplication), and there is a lot of tools in linear algebra to use with this sort of formulations.

Then main reason is the need of improve result with the limitations of the actual devices and their limits with information.

what are the disadvantages of choosing a too small sampling time above all on the control effort?

Choosing a too small sampling, means use additional resources in physics devices for "nothing", using more information than the sufficient is a waste of energy. There are exceptions when your system has external perturbations or you wish to use the "redundant information" to make an optimal guess. but in general is not needed.

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From my -limited- experience the problem when dealing with control systems decreasing the sampling time beyond the optimum has the following adverse effects:

  • The cost of ADC goes up very fast: Usually when you need to control a motor or another physical system you need to take some measurements. Many times the output is analog and needs to be converted to analog. This Analog to Digital (ADC) conversion takes time. The faster you need data, the higher the cost. An example from a well known supplier at 50 kHz sampling rate for 1 differential channels is about 300 Euro, when you go up to 1.5 MHz the cost goes up to 2500 Euro. And although you can get even faster sampling rates, with dedicated DSP/FPGA computer cards, the total cost of development is the order of 5 digits.
  • Gather more noise: again in my experience, when the control cycle duration goes down (ie. how many control cycles per second are carried out), and assuming the ADC sampling rate remains constant, you end up with only a few or maybe one data point per cycle. This has the effect that the inherent measurement noise affects the system (you get more disturbances). In most cases, that might not be a problem, but if it combines with common problems like poor grounding it can be life hell.
  • Additional computational/programming effort: Increasing the ADC sampling rate, for a given control cycle rate, means that you have more data in each cycle. That means more processing. That processing can be as cheap as just a sum or a mean of the data, or it can have adverse effects when doing more complex dsp operations (e.g. power spectrum, of filtering). That can impose a burden on the CPU, that can sometimes affect the cycle time. Additionally depending on the hardware architecture (8-bit, 16-bit, etc) some programming effort might be required to avoid overflows during calculations.
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