How to define the aileron's deflection angle (in the Laplace domain)?
It looks like a transfer function for a feedback loop, which is a function of time and transformed into the frequency domain. Input is the aileron deflection $\delta(s)$, output is the roll angle $\phi(s)$. When deflecting the ailerons, the aircraft is accelerated around the X-axis. The roll movement creates a damping force proportional to span $b$, which quickly dampens out the roll acceleration into a steady roll rate.

So achieving a roll angle depends on the moment of inertia around the X-axis, and wing area geometrical data, plus the time that the aileron is deflected at a certain angle. In order to get to a specific bank angle, the aileron needs to be deflected, then returned to zero before the bank angle is achieved - damping and inertia will then continue to roll the aircraft until the target bank angle is achieved.

The transfer functions defined in the frequency domain are used for determining gain and phase shift, more info for instance here, section 6.3 onwards. Using a signal in the frequency domain gets to be a pain once the deflection is a real world time history and not a mathematical function like a sine wave or a pure step input.  

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Once the gains are determined with the Bode plots, the block diagram of the transfer function is pretty useful for determining bank angle as a function of the time history of aileron deflection: just read the actual bank angle, subtract from the desired bank angle, and set the aileron deflection angle in real time according to the difference signal.



EDIT

Have managed to have a look into the book. I'm familiar with the equations, describing stability and control of aircraft. Your question on how to define an input signal in the frequency domain is a pertinent one: it would have to be defined mathematically in the frequency domain as well. I have only seen use of the frequency domain equations in off-line analysis, not in real-time use since the inputs are never purely mathematically defined. Equations such as 5.53 for the time domain. Then any input signal can be added in real time, multiplied with the gains found from the Laplace transform, integrated and output.


  So my first question is: suppose you want to imply the aileron rotated 45degrees how does that translate to δ(s)?


It depends how fast you bring the ailerons to 45º, for how long you keep them there, and what happens next. If it is a step response, the function is $\frac{0.785}{S}$. The aircraft will respond with a steady roll rate (in aeroplane axes) and will never establish a constant bank angle.