The advent of small, powerful computers and sensors has made flying machines both simpler and safer. Compare old-school RC helicopters with modern quadcopters. Helicopters are both mechanically complicated and difficult to control; quadcopters are simple mechanically and easy enough that almost anyone can fly one.
All of the sophistication is in the software, of course. It’s not a trivial task, but it is well-traveled control theory. Building a reliable control system would be a straightforward task for experts. I use an open-source software stack for my own quadcopter; it works very well. A similar system for a device like you describe would be a bit more complicated but still perfectly reasonable.
Incidentally, you aren’t going to get a huge amount of runtime out of this. Assuming an optimistic 30 kg for the engines+structure (2x8.8 kg + servos/tankage, etc.), a 70 kg rider, and 25 kg of fuel, we get 125 kg, or 1.225 kN of weight. The engine spec sheet gives 40.36 g/(kN-s) of fuel consumption. That’s 505 s, or 8.4 minutes (well, maybe 9 or so when accounting for reduced consumption as the fuel is consumed). You couldn’t add much more fuel since the engines only produce 1.57 kN, and you need some extra for maneuverability (1.225 kN is probably cutting it close already).
The company does have their Lynx model with twice the thrust, and slightly better power/weight and fuel consumption specs. You could probably get 20 minutes out of that one, but it would be a substantially bigger vehicle.
I’ve never used it myself, but my understanding is that the go-to package for this kind of thing is Simulink. It allows modeling, generating, and simulating sophisticated control systems.
Simulink is expensive, but there are open-source alternatives, such as OpenModelica and Xcos. I haven’t used these either, so I don’t know if they’re up to the task.
For what it’s worth, the quadcopter software I used is Ardupilot, and it has a simulator called SITL. Not quite what you want, but probably if you were developing this, you would want to build something similar.
Simulink is an add on package for Matlab. The it provides for a range of discrete event simulation modelling paradigms that integrate with Matlab’s mathematical engine. So you model your end to end system with things connected by Simulink, use Matlab to provide those models for those components that can be described mathematically, and optionally add modules coded from scratch that need more conventional coding. Matlab is the go-to system to design control systems anyway (when used just as a mathematics engine it is the most common tool for working out parameters), so it is a good all in one solution (in principle anyway.) Matlab is expensive - so much so it is not viable for hobbyists, and adding Simulink makes it worse. So there are alternatives. But I’m not aware of anything that matches the end to end integration of Matlab plus Simulink. (I use Scilab at home, but it doesn’t have the simulation capability.)
As an example of this use - car manufactures will use exactly the above system to model and debug stability control systems and engine management systems.
You know, the more I look at that water jet platform the more it intrigues me.
It seems that stability and direction is controlled more by the pilot pivoting his ankles to keep the center of thrust under the center of gravity rather than leaning to keep the center of gravity over the center of thrust. Kind of like balancing a broomstick. It seems much more responsive that way.
Also, yaw is controlled by raising a knee. This tells me that the jets are not parallel, and raising a knee changes how much each jet contributes to the yaw axis and disrupts yaw equilibrium, allowing turns and pivots.
I wonder if it would be possible to emulating this water jet platform with jet turbines?
According to this site and several press reports, an Australian helicopter pilot was developing something similar back in 2011, although configured like a motorcycle instead of a snowboard. The site and the news items make ludicrous claims for the thing ( top speed over 170 mph, reaching altitudes of over 10,000 feet), but his site does not show any full scale flight testing (only RC models), and nothing further seems to have happened in the last few years.
I don’t think this kind of flight is as simple as the OP imagines. It seems obvious to me that in the water jet device, the supply hose provides a significant element of stability to the platform that would be missing in the OP’s concept.