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CHAPTER 3

CHAPTER 3. TIME RESPONSE. Course Outcome. Use poles and zeros of transfer function to determine the time response of a control system Describe quantitatively the transient response of first order system s Write the general response of second order system given the pole location

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CHAPTER 3

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  1. CHAPTER 3 TIME RESPONSE

  2. Course Outcome • Use poles and zeros of transfer function to determine the time response of a control system • Describe quantitatively the transient response of first order systems • Write the general response of second order system given the pole location • Find the damping ratio and natural frequency of a second order system

  3. First Order System • A first order system without zeros can be described by the transfer function • If the input is a unit step where R(s) = 1/s, the Laplace transform of the step response is C(s), where • Taking the inverse Laplace transform, the step response is

  4. First Order System Time constant, Tc • Can be described as the time for e-at to decay to 37% of its initial values. • The time constant is the time it takes for the step response to rise to 63% of its final value.

  5. First Order System Rise Time, Tr • Defined as the time for the wave form to go from 0.1 to 0.9 of its final value.

  6. First Order System Settling Time, Ts • Is defined as the time for the response to reach and stay within 2% of its final value

  7. First Order System Example 3.3 A system has a transfer function Find the time constant settling time and rise time. Solution :

  8. Second Order System • In general, second order system has two finite poles and no zeros • By assigning appropriate value of a and b, all possible transient responses are (i) Overdamped system (ii) Underdamped system (iii) Undamped system (iv) Critically damped system The unit step response can be found using C(s) = R(s) G(s), where R(s) = 1/s

  9. Second Order System Overdamped responses Poles : Two real at and Natural response : Two exponentials with time constant with equal to the reciprocal of pole location

  10. Second Order System Underdamped responses Poles : Two complex at Natural response : damped sinusoid with an exponential envelope whose time constant is equal to the reciprocal of the pole’s real part. The radian frequency of the sinusoid, the damped frequency of oscillation, is equal to the imaginary part of the poles

  11. Second Order System Undamped responses Poles : Two imaginary at Natural response : undamped sinusoid with radian frequency equal to the imaginary part of the poles

  12. Second Order System 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 is the product of time,t and an exponential with time constant equal to the reciprocal of the pole location

  13. Second Order System • The step responses for the four cases of damping

  14. General Second Order System Consider the general system of second order as Natural Frequency, ωn • Define : The frequency of oscillation of the system without damping • Without damping, • Since the poles of this system are on the jω- axis at • Hence,

  15. General Second Order System Damping ratio, ζ • Define : ζ = Exponential decay frequency Natural frequency (rad/second) = 1 Natural period (constant) 2 π Exponential time constant Hence, Where,

  16. General Second Order System Our general second order system finally is Pole location :

  17. Second order response as a function of damping ratio

  18. Underdamped Second Order System • Transfer function of underdamped second order system is • The pole position with

  19. Underdamped Second Order System Second order underdamped system responses for damping ratio values

  20. Underdamped Second Order System • Other parameters associated with the underdamped response are (i) Peak Time, Tp (ii) Percent Overshoot, %OS (iii) Settling Time, Ts Second order underdamped system responses specification

  21. Underdamped Second Order System Peak Time, Tp Define : The time required to reach the first, or maximum, peak Settling Time, Ts Define : The time required for the transient’s damped oscillations to reach and stay within ±2% of the steady state value

  22. Underdamped Second Order System Percent Overshoot, %OS Define : The amount that the waveform overshoots the steady-state or final, value at the peak, expressed as a percentage of the steady state value Where,

  23. Second Order System Example 3.4 Given the transfer function Find the and Solution :

  24. Second Order System Example 3.5 Given the transfer function Find the Tp, Ts and %OS, Solution :

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