Continuous Control Systems: a review

Figure 1.4. Stability and instability domains in the s-plane

4. 
   Time Response
   

The response of a dynamic system is studied and characterized for a step input. The response of a stable system is generally of the form shown in Figure 1.5.

y(t) FV
O.9 FV
Maximum overshoot (M)
Final value (FV)
(steady state)

t
t
R
t _S
Figure 1.5. Step response

The step response is characterized by a certain number of parameters:

• 
  t_R (rise time): generally defined as the time needed to reach 90% of the final value (or as the time needed for the output to pass from 10 to 90% of the final value). For systems that present an overshoot of the final value, or that have an oscillating behavior, we often define the rise time as the time
  

needed to reach for the first time the final value. Subsequently we shall generally use the first definition of t_R.

• 
  t_S (settling time): defined as the time needed for the output to reach and remain within a tolerance zone around the final value (     2%).
  
• 
  FV (final value): a fixed output value obtained for t   .
  
• 
  M (maximum overshoot): expressed as a percentage of the final value. For example, consider the first-order system
  

H (s) 
G


1  sT

The step response for a first-order system is given by

y(t)  G(1  e^{t /T} )

Since the input is a unitary step one has

FV = G (static gain); t_R = 2.2 T
t_S = 2.2 T (for   FV) ; t_S = 3 T (for ± 5%FV) ; M = 0
and the response of such a system is represented in Figure 1.6. Note that for t = T, the output reaches 63% of the final value.


FV 90%
63%


T t