csl10 9 f15

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Control Systems

First and second order system performance

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Outline of the topic

• Open loop and closed loop system• Transient and Steady state response• First order system• Performance parameters of First order system• Second order system• Performance parameters of Second order

system

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Transient and steady state response

• Closed loop or feedback control systems– Ability to adjust the transient and steady state

performance– Desired performance obtained by adjusting the

system performance

Speed at 60km/hrPress accelerator

Speed desired80km/hr

Difference

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• Transient response - disappears with time• Steady state response – exists for along time

following an input signal initiation• To analyze and design a control system – Define and measure its performance– Design specifications • Time response indices for a specified input command• Desired steady state accuracy

Transient and steady state response

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First order system -Example 1

Differential equation

𝑌 (𝑠)𝑈 (𝑠 )

= 1𝑚𝑠+𝑏

Taking Laplace transform

Transfer function

1𝑚𝑠+

-

Y(s)U(s)

𝑏

1𝑚𝑠+𝑏

U(s) Y(s)

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First order system –Example2

Differential equation

Taking Laplace transform

𝑅𝐶𝑠𝑌 (𝑠 )+𝑌 (𝑠 )=𝑈 (𝑠)

Transfer function𝑌 (𝑠)𝑈 (𝑠 )

= 1𝑅𝐶𝑠+1

1𝑅𝐶𝑠+

-

Y(s)U(s)

1𝑅𝐶𝑠+1

U(s) Y(s)

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• First order system – system with one pole• General for of first order system

First order system

𝐶 (𝑠)𝑅(𝑠)

= 1𝑠𝑇+1

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First order system with unit step

• Unit step response • Output c• Steady state response • Transient response which decays to zero as t

infinity

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First order system with unit step

• Steady state error – error after the transient response has decayed and leaves a continuous response

error signal = • As t approaches infinity, the exponential

becomes 0

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First order system with step input - response

Slope is 1/T

Pole is -1/T

At t=T, c

T is called the Time constant, time taken to reach 63.2% of the final value

c(2T) =0.895 c(3T)=0.95c(4T)=0.982c(5T)=0.993For T>4T response remains within 2% of final value, this is settling time

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First order system with unit ramp

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First order system with unit ramp

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First order system with unit impulse