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# 04 elec3114

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### 04 elec3114

1. 1. 1 Time Response• How to find the time response from the transfer function• How to use poles and zeros to determine the response of a control system• How to describe quantitatively the transient response of first- and second order systems• How to approximate higher-order systems as first or second order• How to find the time response from the state-space representation Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
2. 2. 2 Poles, Zeros, and System Response• The output response of a system is the sum of two responses: the forced response and the natural response.• The poles of a transfer function are the values of the Laplace transform variable, s, that cause the transfer function to become infinite.• The zeros of a transfer function are the values of the Laplace transform variable, s, that cause the transfer function to become zero. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
3. 3. 3Poles and Zeros of a First-Order System (1) Unit step response is given by: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
4. 4. 4Poles and Zeros of a First-Order System (2) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
5. 5. 5Example: Given the system below, specify the natural and forcedparts of the output c(t) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
7. 7. 7 First-Order SystemsTime constant of the system: 1/a, it istime it takes for the step response to riseto 63% of its final valueRise time, Tr : time for the waveformto go from 0.1 to 0.9 of its final value Settling time, Ts : time for the response to reach, and stay within, 2% of its final value Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
8. 8. 8First-Order Transfer Functions via TestingWith a step input, we can measure the time constant and the steady-state value,from which the transfer function can be calculated. Let first order system is and the step response is 1. Final value = 0.72 2. 63 % of final value = 0.63*0.72=0.45 3. Curve reaches 0.45 at 0.13 s, hence a = 1/0.13 = 7.7 4. Steady state value K/a = 0.72, hence K = 5.54 5.54 G(s) = s + 7.7 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
9. 9. 9 Second-Order Systems: Introduction• a second-order system exhibits a wide range of responses Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
11. 11. 11 Summary of Second-Order Systems (1)There are four possible responses: 1. Overdamped responses 2. Underdamped responses 3. Undamped response 4. Critically damped responses Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
13. 13. 13Summary of Second-Order Systems (3) ωd … damped frequency of oscillation Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
16. 16. 16 The General Second-Order System (1)Natural Frequency, ωn- the frequency of oscillation of the system without damping.Damping Ratio, ζDerivation of the second order system in terms of ωn and ζ Consider Let us determine what is the term a and the term b in terms of ωn and ζ . Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
17. 17. 17Without damping (a=0) the poles are on the imaginary axis and Then Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
18. 18. 18Now, let us assume an underdamped system. The real part of the complexpole is σ = - a/2 and it determines the exponential decay frequency. Hence Then, the general second-order transfer function is and the poles of this transfer function are: s1, 2 = −ζω n ± jωn 1 − ζ 2 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
20. 20. 20 Underdamped Second-Order Systems (1)- represent a common model for physical problems Step response: Expanding by partial fraction we obtain: Then in time domain: Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
23. 23. 23 Underdamped Second-Order Systems (4)Evaluation of Tp,Tp is found by differentiating c(t) andfinding the first zero crossing after t = 0. π Tp = ωn 1 − ζ 2 Evaluation of % OS − (ζπ / 1−ζ 2 ) %OS = e × 100 − ln(%OS / 100) ζ = π 2 + ln 2 (%OS / 100) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
24. 24. 24 Underdamped Second-Order Systems (5)Evaluation of Ts 4 Ts = ζω nEvaluation of Tr- cannot be foundanalytically Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
25. 25. 25 Underdamped Second-Order Systems (6) - now we will evaluate peak time, settling time and overshoot in terms of the pole locationFrom the Pythagorean theorem: ωd … damped frequency of oscillation σd … exponential damping frequency Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
26. 26. 26 Underdamped Second-Order Systems (7)%OS = e −(ζπ / 1−ζ 2 ) × 100 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.