But overall you are over-thinking the importance of the GND. It's important, don't get me wrong. It's just that there are other things that are as important, and getting the GND correct is relatively easy.
You specified the voltages, you didn't specify the current. Without knowing the current, we don't know the heat generated by the LDO's. And the heat will greatly influence the way the PCB is laid out. I am going to assume that the heat generated is non-trivial.
Here's what I would do...
- Rotate the caps 90 degrees (sometimes clockwise, sometimes counter-clockwise). What you are doing is putting the caps GND pins together and shortening the distance between the LDO's GND and the caps.
- Make all of your traces wider. At least as wide as the pad it's connecting to. Use multiple VIA's if you can.
- Put the +6v traces "somewhere else". Either on the back side of the PCB or on the right of the LDO's. This will make sense shortly.
- Put a copper plane on the top layer, under and around the whole thing. Connect this to the GND layer using multiple VIAs. I would use about 10 vias per LDO, mostly around the huge GND pin. The GND pin of both the LDO's and caps should be connected to this plane DIRECTLY, without any "thermal relief". This plane should be reasonably large, although the exact size depends on the space available and how much heat the LDO's will be giving off. 1 or 2 square inches per LDO is a good start.
There are two reasons for the copper plane. 1. It gives the heat from the LDO's someplace to go to be dissipated. 2. It provides a low impedance path between the caps and the LDO.
The reason for all of the vias are: 1. It allows some of the heat to be transferred to the GND layer. 2. It provides a low-impedance path from the LDO to the GND layer.
And the reason for the fatter traces and multiple vias is simply for a lower impedance path.
I will warn you, however: Doing this will make hand-soldering of the LDO's difficult. The copper planes + vias will want to suck the heat away from the soldering iron and the solder won't stay melted for very long (if at all). You can get around this somewhat by using a hotter soldering iron, or better yet pre-heat things by using a heat gun to warm up the entire PCB first. Don't get it hot enough to melt solder (use your normal iron for that). By preheating the whole board the demands placed on your iron will be less. IMHO, this isn't a big deal but it is something to be aware of and plan for.
This method will also give you a nice connection to GND, way better than anything you've told us from the datasheets.
Update, based on new information from the original poster:
Your 5v regulator is dropping 6v to 5v (a 1 volt drop) at 400 mA. This is going to produce 0.4 watts of heat. 6v to 3.3v at 150 mA = 0.4 watts. 6v to 1.8v at 200 mA = 0.84 watts. Total 1.64 watts for all three LDO's. While this isn't crazy, it is a fair amount of heat. Meaning that you must pay attention to how this is going to get cooled otherwise it will overheat. You're well on your way to getting that done properly.
You want a single plane, not three. And the plane should extend out as far as possible, I recommend at least double the area of the LDO's themselves. The larger the plane, the better the cooling effect. If the plane is really large then you'll want to put at least four vias for every square inch. By sharing the plane, the three regulators are sharing the cooling. If you didn't do this then one regulator could get really hot while the other two are just warm.
Another optimization that you can do is with how the +6v comes in to each LDO. At the moment it goes around the cap, to the LDO. Just have it go straight into the cap, without wrapping around. This will allow you to use thicker traces and keep things a little shorter. That small amount of GND plane that wraps around the cap isn't helping much anyway.
You'll want several vias from the output of the LDO to wherever that power is going. Not just the single via that you have now.
Disturbances considered in state-space systems are not constrained to be of any particular type. Step, sinusoidal, stochastic, impulse, disturbances are all described in the literature. Whether the system under consideration is continuous time or discrete doesn't matter; there is no distinction regarding the type of disturbance that can be / is analysed.
Sometimes one type of disturbance is more relevant to the problem at hand because they model real world phenomena; e.g. a step in a control system or a stochastic in a communications channel.
Step disturbances are popular for control system analysis because you usually require a zero steady-state error.
Stochastic disturbances are popularly analysed in Communications channels; but their application to control systems is also a well studied field; e.g. "Discrete Time Stochastic Systems", T.Soderstrom, Springer, 2002
It is true that discrete time controllers have become popular in the same era as stochastic approaches to control systems. This is partly coincidental but may also be due to easier analysis in discrete time; e.g. Soderstrom states "discrete time stochastic processes are much easier to handle than their continuous time counterparts, which have certain mathematical subtleties that are far from trivial to handle in a stringent way".
Best Answer
It can be hard to identify each type, but I will try to provide a guide.
Many newer devices have specifically addressed the loop stability problem with different manufacturers taking different approaches:
Analog devices has their range of AnyCap devices, and you can read about how they achieve operation insensitive to output ESR at this page.
Linear technology has a new range of regulators (typical part linked) where a current source is used for the internal reference, which permits the loop gain of the device to be constant regardless of \$Z{in}\$, and \$V_{in}\$ to \$V_{out}\$ ratio making loop stabilisation much more straightforward; in fact LT claim this part requires no output capacitance, although the dropout voltage is a little high compared to some competing parts.
Generally, most parts that have a lineage of more than 10 years age will exhibit loop stability issues without the proper amount of output ESR (the venerable LM1117 for instance is a relatively low dropout part with similar operation to the even more venerable LM117 but can be troublesome in this regard).
The lineage is important; many older devices were very simple to use and then the drop-out became an issue so manufacturers made parts that operated in a virtually identical manner but with lower drop-out (the operation is not identical without proper ESR, though but at the time of part release many capacitors had the right amount of ESR).
A quick glance at the datasheet original issue date is usually a good first step; if it predates 2006, be suspicious. That said, the easiest way is to simply search for "ESR" in the datasheet which may (or may not) uncover a stability requirement on ESR.
Some regulators require an ESR with an upper bound but no low bound.
It can be difficult to navigate this sea of parts, and experience is a key element; that said, LDO manufacturers that have devices without ESR requirements advertise it far and wide, so looking for parts that are insensitive to output ESR can be as simple as a quick search with your favourite search engine.