KEC-602 Control System Unit-3 gandgfdghhg

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1. Control System (KEC-602) Unit-3

2. Objectives and Outcomes of Unit-3 Analyze the output of a control system and present it in time domain Analysis of system is done in s-domain using Laplace transform Inverse Laplace transform is applied to describe the system Able to describe different types of systems in time domain Calculate steady state error of a system Obtain the transient and steady state response of the given system Able to calculate Time domain parameters

3. Time Domain Analysis • Time domain analysis refers to the analysis of system performance in time i.e., the study of evolution of system variables (specifically output) with time • There are two common ways of analysing the response of systems: 1. Natural response and forced response 2. Transient response and steady state response • In both cases, the complete response of the system is given by the combination of both responses i.e., natural and forced responses or transient and steady state responses

4. Natural and Forced Responses •Natural response (Zero input response) : –System's response to initial conditions with all external forces set to zero –E.g. In RLC circuits, this would be the response of the circuit to initial conditions (inductor currents or capacitor voltages) with all the independent voltage and current sources set to zero •Forced response (Zero state response) : –System's response to external forces with zero initial conditions –E.g. In RLC circuits, this would be the response of the circuit to only external voltage and current source, and zero initial conditions

5. Transient and Steady State Responses •Transient response 𝒚 tr (𝒕) –Part of the time response that goes to zero as time tends to be large –Transient response can be tied to any event that affects the equilibrium of a system viz. switching, disturbance, change in input, etc. lim (𝑡→∞) 𝑦 tr (𝑡) = 0 •Steady state response 𝒚 𝒔𝒔 (𝒕): –Steady state response is the time response of a system after transient practically vanishes and as time goes to infinity 𝑦(𝑡)=𝑦 𝑡𝑟 (𝑡) + 𝑦 𝑠𝑠 (𝑡)

6. Standard Test Inputs • In most cases, the input signals to a control system are not known prior to design of control system • Hence to analyse the performance of a control system, it is excited with standard test signals • In general, control system design specifications are also based on the response of the system to such test signals • Standard test signals include: – Unit impulse, unit step (sudden change), ramp (constant velocity), parabolic ( constant acceleration) and sinusoidal – These inputs are chosen because they capture many of the possible variations that can occur in an arbitrary input signal

7. Standard Test Inputs • Unit impulse signal: – A signal which is non-zero only at 𝑡=0 and integrates to one ℒ[𝛿𝑡]=1 • Unit step signal: – A signal that switches to one at a time instant and stays there indefinitely U(t) = 1 ∀ 𝑡 > 0 0 ∀ 𝑡 < 0 ℒ[u(t)] =  

8. Ramp signal: – A signal which increases linearly with time 𝑥(r)𝑡 = 𝐴𝑡 ∀ 𝑡≥ 0 ∀ 𝑡<0 ℒ[𝑥(𝑡)] = • Parabolic signal: 𝑥(𝑡) = (𝐴/2)𝑡 2 ∀ 𝑡 ≥ 0 0 ∀ 𝑡<0 ℒ[𝑥(𝑡)] =  

9. Standard Inputs in time and s-domain Unit Impulse Signal Unit Step Signal u Unit Ramp Signal r Unit Parabolic Signal  

10. Some terminologies Define the following Type and Order of a System Poles and Zeroes of a System Open loop gain Closed loop gain Loop Gain Unity feedback system Characteristic Polynomial/Equation

11. 1 st Order Systems • Systems with only one pole are called 1 st order systems 𝝉 : System time constant – It characterizes the speed of response of a system to an input – Higher the time constant, slower the response and vice-versa Block Diagram of a 1 st order system

12. Time Response of First Order Systems Q1. Derive the expression for the output response of a 1 st order system when different standard test signals are applied at the input. Q2. Identify the transient and steady state components of the time response.

13. Impulse Response of 1 st Order Systems

14. Step Response of 1 st Order Systems

15. Ramp Response of 1 st Order Systems

16. Parabolic Response of 1 st Order Systems

17. 2 nd Order Systems

18. 2 nd Order Systems

19. Parameters of 2 nd Order Systems

20. Response of 2 nd Order Systems

21. Damping & It’s Types

22. Damping & It’s Types

23. Impulse Response of 2 nd Order Systems

24. Impulse Response of 2 nd Order Systems

25. Impulse Response of 2 nd Order Systems

26. Impulse Response of 2 nd Order Systems

27. Step Response of 2 nd Order Systems

28. Step Response of 2 nd Order Systems

29. Step Response of 2 nd Order Systems

30. Step Response of 2 nd Order Systems

31. Step Response of 2 nd Order Systems

32. Time Response Specifications

33. Time Response Specifications

34. Time Response Specifications

35. Time Response Specifications

36. Rise Time

37. Peak Time

38. Max Peak Overshoot

39. Settling Time

40. Steady State Error

41. Steady State Error for Standard Inputs

42. Steady State Error for Standard Inputs

43. Features of Steady State Error

44. Steady State Error for Different Types

45. Steady State Error for Different Types

46. Steady State Error for Different Types

47. Static Error Coefficients There are 3 types of static error coefficients. These are Position error coefficient ( K p ) 2. Velocity error coefficient ( K v ) 3. Acceleration error coefficient ( K a )  

48. Input Type Unit Step Unit Ramp Unit Parabolic Type-0 Type-1 Type-2 Input Type Unit Step Unit Ramp Unit Parabolic Type-0 Type-1 Type-2 Relation between steady state error and static error coefficients

49. Dynamic Error Coefficients Static error coefficients are limited to standard inputs like step, ramp and parabolic signals Static error coefficients do not describe the variation in error w.r.t. time To solve these issues we define dynamic error and dynamic error coefficients Dynamic error is written as: _ _ _ _ _ _ where, C , C 1 , C 2 , C 3 , and so on are the dynamic error coefficients and given as: where, is called the error transfer function.  

50. End of Unit-3