TimeDomain Analysis ofControl Systems.pptx

Chapter 5 Analysis ofControl Systems

1. Chapter 5 Time-Domain Analysis of Control Systems 1

2. Outline 5.1 Time response of continuous-data system 5.2 Typical Test Signals For The Time Response Of Control Systems 5.3 The unit-step response and time-domain specifications 5.4 Time response of A Prototype First-order System 5.5 Transient Response Of A Prototype Second-order System 5.6 Steady-state Error 5.7 Speed Control of A DC Motor 2

3. 5.1Time Response of Continuous-data System Transient response vs. steady state response Definition: Time response of a control system is the output response which is a function of time, to the applied input Transient response, Present in a short period of time immediately after the system is excited Reflect dynamic behavior of the system For the asymptotically stable system ,   Steady-state response , : Remain after the transient response has died out Reflect final accuracy of the system   3

4. 5.2 Typical Test Signals For The Time Response Of Control Systems Typical test input signals 4

5. 5.2 Typical Test Signals For The Time Response Of Control Systems Relation between standard Test Signals 5

6. 5.3 The unit step response and time domain specifications 6

7. 5.4 Time Response Of A Prototype First-order System 7

8. 5.5 Transient Response of A Prototype Second-order System ζ : decide pole of characteristic equation 8 The two roots can be expressed as

9. 5.5 Transient Response Of A Prototype Second-order System 0< ζ < 1 ( underdamped ) ζ = 0 ( undamped ( 9

10. 5.5 Transient Response Of A Prototype Second-order System ζ > 1 ( overdamped ) ζ = 1 ( critically damped) 10

11. 5.5 Transient Response Of A Prototype Second-order System Maximum overshoot 11

12. Example : 12 The unit-step response of a linear control system is shown in Figure. Find the transfer function of a second-order prototype system to model the system. Solution:

13. 5.5 Transient Response Of A Prototype Second-order System Delay time 13 Rise time Settling tim e and

14. 5.6 Steady-state Error The steady state error is the difference between the reference (desired output) and the output in the steady state .       Let us assume that the objective of the system is to have the output y(t) track the input r(t) as closely as possible . 14

15. 5.6 Steady-state Error Type of Control Systems  Systems with unity feedback; H(s) = 1. Systems with nonunity feedback, but H(0) = K H  System with H(s) =   15

16. 5.6 Steady-state Error Steady state error of a system with unity feedback       depends on the characteristics of G(s). More specifically, we can show that depends on the number of poles G(s) has at s=0. This number is known as the type of the control system.   16

17. 5.6 Steady-state Error Steady-State Error of System with a Step-Function Input   The step−error constant becomes position error constant (   17

18. 5.6 Steady-state Error Steady-State Error of System with a Step-Function Input 18

19. 5.6 Steady-state Error Steady-State Error of System with a Ramp Function Input   Laplace transform of r(t) is   The steady-State Error is   velocity error constant (   19

20. 5.6 Steady-state Error Steady-State Error of System with a Ramp Function Input 20

21. 5.6 Steady-state Error Steady-State Error of System with a Parabolic-Function Input     Defining the parabolic-error constant as acceleration error constant (   21  

22. 5.6 Steady-state Error Summary of the Steady-State Errors Due to Step-, Ramp-, and Parabolic-Function Inputs for Unity-Feedback Systems 22

23. Example : a) b) 23

24. 5.6 Steady-state Error Steady-State Error of Nonunity Feedback Systems The feedback path of a control system may be a pure gain other than unity or may have some dynamic representation R(S) Control systems may not have unity feedback because of the compensation used to improve performance or because of the physical model for the system A general feedback system E(s) = R(s) — C(s) ??? 24

25. Forming an equivalent unity feedback system from a nonunity feedback system through block diagram manipulation Unity Feedback System 5.6 Steady-state Error Steady-State Error of Nonunity Feedback Systems 25

26. Example: For the system shown in the Figure, find the system type, the appropriate error constant associated with the system type, and the steady-state error for a unit step input. Assume input and output units are the same. Convert the system into an equivalent unity feedback system Solution: static error constant The steady-state error System Type ??? 5.6 Steady-state Error Steady-State Error of Nonunity Feedback Systems 26

27. T ransfer function relating E(s) to R(s) Transfer function relating E(s) to D(s) 5.6 Steady-state Error Steady-State Error for Disturbances 27

28. Let us explore the conditions on that must exist to reduce the error due to the disturbance Assume a step disturbance, D(s) = 1/s. Applying the final value theorem 5.6 Steady-state Error Steady-State Error for Disturbances 28

29. The ss error produced by a step disturbance can be reduced by: increasing the dc gain of G 1 (s) or decreasing the dc gain of G 2 (s) System rearranged to show disturbance as input and error as output, with R(s) = 0 5.6 Steady-state Error Steady-State Error for Disturbances 29

30. Example: Find the steady-state error component due to a step disturbance for the system shown in the Figure. the steady-state error produced by the step disturbance is inversely proportional to the dc gain of G 1 (s) 5.6 Steady-state Error Steady-State Error for Disturbances 30

31. 5.6 Steady-state Error Steady-State Error for Nonunity -feedback control system with disturbance 31 Assuming step inputs and step disturbances

32. 5.7 Speed Control Of A DC Motor Speed Response and the Effects of Inductance and Disturbance-Open Loop Response 32

33. 5.7 Speed Control of A DC Motor Speed Response and the Effects of Inductance and Disturbance-Open Loop Response L/ Ra is called the motor electric-time constant 33

34. 5.7 Speed Control of A DC Motor Speed Response and the Effects of Disturbance-Open Loop Response 34      

35. 5.7 Speed Control of A DC Motor Speed Control of DC Motors: Closed-Loop Response 35

36. 5.7 Speed Control of A DC Motor Speed Control of DC Motors: Closed-Loop Response 36

37. References 37 “ Automatic Control Systems ”, Edition, Farid Golnaraghi &Benjamin C . KUO COTROL SYSTENG “Control Systems Engineering” Edition, Norman S. Nise On https ://filespayouts.com/awnu5qt38t4k/_Norman_S._Nise_-_CONTROL_SYSTEMS_ENGINEERING_(2018 )_-_libgen.li_2.pdf