Relation between bandwidth, response time and disturbance rejection for a control system
Every system has a time and frequency spectrum of inputs and outputs determined mass &amp; energy storage in the system, choice of inputs, sensors and tolerance vs resolution vs time response.  The ideal system matches the signal bandwidth to the desired spectrum to get the desired response with many choices of criteria;    


  Maximum SNR, minimum group delay, minimum overshoot, minimum step error, maximum disturbance rejection, linear phase, min. static error, dynamic error, min. Integrated squared error,  optimal anticipated min. error, maximal interference jitter rejection, min BER, max fault tolerance, max efficiency, max climatic stability, min. FIT rate or max MTBF with redundancy, optimal fault detection/correction/recovery.  


You must define the mission, control system with a hierarchical-input-process-output or HIPO design with full design specs. Then the spectral and time response are just necessary parameters to specify for the solution.


  The best simple explanation is the step response rise time from 10 to 90% is inversely proportional to the loop bandwidth at half signal power output as a function of frequency.
  Tr=0.35/f
  for f = max freq. BW @ -3dB. But the dampening factor or it’s inverse , Q must be low to reduce ringing or overshoot with compensation to ensure proper stability. 


Road Analogy; 

too much delay in feedback might look like a drunk driver wandering off centre or , hyperactive might kill someone in the next lane to avoid a sitting duck, or a tired or lazy driver might have more risk to accidents sampling the road traffic every few seconds rather than being attentive with a visual 25 frames per second.