Time Domain Analysis Study notes Part-2

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:

• 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, t_s for second order system is:

• 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:

• 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)

2. If input is unit ramp i.e R(t) = tu(t)

3. If input is unit parabolic i.e R(t) = 0.5tu(t)

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