Given a chain of four integrators like
$$ x^{(4)} = u \,. $$
Is it possible to design a PID controller that tracks constant reference trajectories? I am asking because I tried using the Matlab PID tuning toolbox to identify a suitable PID controller, but no configuration was able to stabilize this system.
Is it possible at all to design a PID controller for this system? Is there some limitation on the maximum number of integrators in series that can be controlled using PID control?
The transfer function of your system can be written as
$$ G(s) = \frac{1}{s^4}. $$
The transfer function of the PID controller can be written as
$$ K(s) = \frac{K_I + K_P s + K_D s^2}{s}. $$
In order to explain why this controller will not be able to stabilize that system one can look at the Nyquist stability criterion. For (marginally) stable systems, like your system, this basically comes down to that the openloop (so system times controller transfer function) has to have an asymptote of a minus one slope at the 0 dB crossing in its Bode plot.
But in your case the PID controller can increase the slope by at most one. Therefore the openloop can have at most an asymptotic slope of minus three. From this it can also be concluded that a PID controller can only at most stabilize a second order integrator. This can also be checked relatively easily by using the Routh-Hurwitz. Namely from this it would follow that $K_P^2<0$ for a third order integrator and $(K_P\,K_D^2<0$ for a fourth order integrator.
It can be noted that if the system with a fourth order integrator has at least two zeros as well, then one might be able to control it with a PID controller.