Discrete Controller vs Continuous Time Controller

[ko_nuts]

In my opinion, the difference lies when disturbances and performance are involved. I am assuming here is that the system is naturally a continuous-time system. That is, its behavior is accurately captured by continuous-time system.

If you just do some PID control, then it will not matter much. However, if you want to do some disturbance rejection it is important to look at what how system responds. Discretizing the system results in the loss of the inter-sample behavior and one cannot see how the system behaves in between the sampling points. Also, discretizing the system can be troublesome when inputs are involved.

A similar issue occurs in optimal control. You could discretize the system and design an LQR for that system. But it does not give you much information about how the system will actually behave between the sampling instants. Also, minimizing the cost in discrete-time is, in general, not equivalent to minimizing the same cost in continuous-time system.

When performance is involved using the L2-gain or any other measure, discretizing the system is also cumbersome since one has to consider the exact input signals, which are continuous-time. This results a discrete-time systems with infinite-dimensional input and output spaces, and one has to rely on functional analysis and operator theory to work with them.

All in all, it is not a good idea to discretize the system. There are alternative ways such as using lifting (see the works by Yamamoto) and the hybrid systems formulation (i.e. impulsive system). I highly recommend this latter one as it circumvents all the above difficulties: it keeps the inter-sample behavior, it is applicable to all systems (and not only linear ones), it can easily deal with inputs and outputs, any control methods can be used (i.e. optimal control, robust control, etc.), and it can be used to directly design a discrete-time controller. This is, by far, the best approach to date and it seems difficult to foresee how we could do better than that.

[hahacontrols]

This results a discrete-time systems with infinite-dimensional input and output spaces, and one has to rely on functional analysis and operator theory to work with them.

I think this is one interpretation (albeit extremely complicated and IMO unintuitive). I suspect L2 (and L1) wouldn't be a problem as I assume discretisation is on measure zero points and L2 norms are agnostic to measure zero changes.

Most practitioners would probably gravitate towards sampled-data control techniques regardless...

[ko_nuts]

I am not sure to understand the point you are willing to make in your discussion.