What is the maximal acceptable delay between pilot's input and flight control surface actuation?
Excessive phase lag is a direct contributor to Type I Pilot-Induced Oscillation (PIO). Phase lag comes from:


Rigid body dynamics of the aircraft (e.g. delay between elevator surface and pitch rate response)
Actuators (finite acceleration time between input and desired surface angle)
Structural compliance (e.g. cable friction)
Transport delay in signals
Finite computational bandwidth (e.g. loop closure bandwidth)


From NASA Report 4683, PIO susceptibility can be expressed assuming the pilot is compensatory; that is, the pilot input and the aircraft response would be exactly in phase, except for a constant time delay (across frequencies). This model is expressed as:

$$G(s)=\frac{K}{s}e^{-\tau_e s}$$

where $\tau_e$ is the effective time delay, or equivalently, phase rate as a function of frequency

From its research, it found that an effective time delay larger than 0.3 sec leads to PIO issues. Given a typical pilot time delay of 0.2 sec, this would imply an upper bound aircraft effective time delay of 0.1 sec at higher frequency (around 5 rad/s), end to end.