Unit Impulse Response of 2nd Order System | Control Systems - Electrical Engineering (EE) PDF Download

Time Domain Characteristics

In specifying the Transient-Response characteristics of a control system to a unit step input, we usually specify the following:

• Delay time (t_d): It is the time required for the response to reach 50% of the final value in first attempt.
  The expression of delay time, t_d for second order system is:
• Rise time, (t_r): It is the time required for the response to rise from 0 to 100% of the final value for the under-damped system.
  The expression of rise time, t_r for second order system is:
• Peak time, (t_p): It is the time required for the response to reach the peak of time response or the peak overshoot.
  The expression of peak time, t_p for second order system is:
  t_P = nπ / ω_d seconds
  For first peak, n = 1 (maxima)
  t_P = π / ω_d
  For first minima, n = 2
  t_P = nπ / ω_d
  For second maxima, n = 3
  t_P = 3π / ω_d
• Settling time, (t_s): It is the time required for the response to reach and stay within a specified tolerance band ( 2% or 5%) of its final value.
  The expression of settling time, ts for second order system is:
  For 2% tolerance band,
  
  For 2% tolerance band,
  
• Peak overshoot (M_p): It is the normalized difference between the time response peak and the steady output and is defined as
  
  The expression of peak overshoot, M_p for second order system is:
  
  Where tanφ =
• Steady-state error (e_ss): It indicates the error between the actual output and desired output as ‘t’ tends to infinity.
  

Effect of Adding a Zero to a System

If we add a zero at s = -z be added to a second order system. Then we have,

• The multiplication term is adjusted to make the steady-state gain of the system unity.
  Manipulation of the above equation gives,
  
• The effect of added derivative term is to produce a pronounced early peak to the system response.
• Closer the zero to the origin, the more pronounce the peaking phenomenon.
• Due to this fact, the zeros on the real axis near the origin are generally avoided in design. However, in a sluggish system the artful introduction of a zero at the proper position can improve the transient response.
  

Types of Feedback Control System
The open-loop transfer function of a system can be written as

• If n = 0, the system is called type-0 system, if n = 1, the system is called type-1 system, if n = 2, the system is called type-2 system, etc.

Steady-State Error and Error Constants

The steady-state performance of a stable control system is generally judged by its steady-state error to step, ramp and parabolic inputs. For a unity feedback system,

Where,
E(s) is error signal
R(s) is input signal
G(s) H(s) is the open loop transfer function

It is seen that steady-state error depends upon the input R(s) and the forward transfer function G(s).

1. If input is unit step i.e R(t) = u(t)
   R(s) = 1/s
   Steady state error is
   
   where,
   
   K_p is positional error constant.
2. If input is unit ramp i.e R(t) = tu(t)
   R(s)  = 1/s²
   
   e_ss = 1/K_v
   where,
   
   Kv is velocity error constant.
3. If input is unit parabolic i.e R(t) = 0.5tu(t)
   R(s) = 1/s³
   Steady state error,
   
   e_ss = 1/K_a
   where,
   
   K_a is acceleration error constant.