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Chapter: Control Systems - Time Response Analysis

System Time Response

First-order system time response v Transient v Steady-state Second-order system time response v Transient v Steady-state

System Response


First-order system time response

v    Transient

 

v    Steady-state

 

Second-order system time response

v   Transient

 

v    Steady-state

 

First Order System

Y s / R(s) = K / (1+ K+sT) = K / (1+sT)

 

Step Response of First Order System

 

Evolution of the transient response is determined by the pole of the transfer function at s=-1/t where t is the time constant

 

Also, the step response can be found:

 



Second-order systems

LTI second-order system






Second order system responses

 

Overdamped response: Poles: Two real at

- σ 1 - - σ 2

Natural response: Two exponentials with time constants equal to the reciprocal of the pole location

C( t)= k1 e-ζ1+ k2 e-ζ2

Poles: Two complex at

 

Underdamped response:

-σ1±jWd

 

Natural response: Damped sinusoid with an exponential envelope whose time constant is equal to the reciprocal of the pole‗s radian frequency of the sinusoid, the damped frequency of oscillation, is equal to the imaginary part of the poles

 

Undamped Response:

Poles: Two imaginary at

 

±jW1

Natural response: Undamped sinusoid with radian frequency equal to the imaginary part of the poles

C(t) = Acos(w1t-φ)

 

Critically damped responses:

Poles: Two real at

 

Natural response: One term is an exponential whose time constant is equal to the reciprocal of the pole location. Another term product of time and an exponential with time constant equal to the reciprocal of the pole location.

 

Second order system responses damping cases



Second- order step response

Complex poles



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