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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.
6. 6. 6 First-Order SystemsStep response: 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.
10. 10. 10Second-Order Systems: Introduction 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.
12. 12. 12Summary of Second-Order Systems (2) 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.
14. 14. 14Summary of Second-Order Systems (4) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
15. 15. 15Summary of Second-Order Systems (5) 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.
19. 19. 19Control 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.
21. 21. 21Underdamped Second-Order Systems (2) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
22. 22. 22 Underdamped Second-Order Systems (3)Specifications: 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.
27. 27. 27Underdamped Second-Order Systems (8) Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
28. 28. 28Problem: Given the plot, find ζ, ωn, Tp, %OS and Ts Solution: ωn = 7 2 + 32 = 7.62 Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
29. 29. 29 System Response with Additional Poles (1)• If a system such has more than two poles or has zeros, we cannot use the previously derived formulae to calculate the performance specifications• However, under certain conditions, a system with more than two poles or with zeros can be approximated as a second- order system that has just two complex dominant poles Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
30. 30. 30 System Response with Additional Poles (2)- Let us consider a system with complex poles and a real pole s3 = −α r- Then the step response can be found using partial fraction expansion Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
31. 31. 31 System Response with Additional Poles (3)• We can assume that the exponential decay is negligible after five time constants. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
32. 32. 32 System Response with Zeros (1) If a >> b, c :• zero looks like a simple gain factor and does not change the relative amplitudes of the components of the response. Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
33. 33. 33 System Response with Zeros (2)Different approach to investigate the effect of a zero:• Let C(s) be the response of a system, T(s), with unity in the numerator• If we add a zero to the transfer function, yielding (s + a)T(s), the Laplace transform of the response will be If a is very large and positive, then the response is approximately a scaled version of the original response If a is small and positive, then the derivative term has a greater effect on the response – more overshoot Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
34. 34. 34 System Response with Zeros (3) If a is negative (in the right haf-plane), then the derivative term will be of opposite sign from the scaled response termIf the response begins to turn towards the negative direction even though thefinal value is positive then the system is known as nonminimum-phasesystem - (zero is in the right half-plan). Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
35. 35. 35 Pole-zero cancellation Consider:• If the zero at –z is very close to the pole at –p3 then it can be shown using partial fraction expansion of T(s) that the residue of the pole at – p3 is much smaller than any of the other residues Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
36. 36. 36Laplace Transform Solution of State EquationsTaking the Laplace transform of After taking the Laplace transform of y(t)=L-1{Y(s)} Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.
37. 37. 37Laplace Transform Solution of State EquationsEigenvalues and Transfer Function PolesIf x(0) = 0 : Hence, we can find poles from Control Systems Engineering, Fourth Edition by Norman S. Nise Copyright © 2004 by John Wiley & Sons. All rights reserved.