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Chapter 5 Time-Domain Analysis of Control Systems

Chapter 5 Time-Domain Analysis of Control Systems. Automatic Control Systems, 9 th Edition F. Golnaraghi & B. C. Kuo. 0, p. 253. Overview. Find and discuss transient and steady state time response of a simple control system.

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Chapter 5 Time-Domain Analysis of Control Systems

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  1. Chapter 5Time-Domain Analysis of Control Systems Automatic Control Systems, 9th Edition F. Golnaraghi & B. C. Kuo

  2. 0, p. 253 Overview • Find and discuss transient and steady state time response of a simple control system. • Develop simple design criteria for manipulating the time response. • Look at the effects of addinga simple gain or poles and zerosto the system transfer function and relate them to the concept of control. • Look at simple proportional, derivative, and integral controller design concepts in time domain.

  3. 1, p. 253 Sections 5-1 and 5-2 5-1 Introduction • Time response: 5-2 Typical Test Signals for the Time Response of Control Systems • Step-Function Input: Steady-state response Transient response

  4. 2, p. 255 Ramp- and Parabolic-Function Inputs • Ramp-Function Input: • Parabolic-Function Input:

  5. 3, p. 256 5-3 The Unit-Step Response and Time-Domain Specification Steady-state error: r yssr: reference input

  6. 4, p. 258 5-4 Steady-State Error • Definition: errorsteady-state error  • Unity feedback system (H(s) = 1) 

  7. 4, p. 259 Velocity Control System • A step input is used to control the system output that contains a ramp in the steady state.System transfer function:Kt = 10 volts/rad/sec • Closed-loop transfer function: • Unit step response ( R(s)=1/s ): • Steady-state error:Kt = 10 volts/rad/sec reference signal is 0.1t, not 1 (unit step input).

  8. 4, p. 260 Systematic Study of Steady-State Error Three Types of Control Systems • System with unity feedback: H(s) = 1 e = r  y • System with nonunity feedback,but H(0) = KH = constant e = r/KH  y • System with nonunity feedbackand H(s) has zeros at s= 0 of order N.

  9. 4, p. 260 Steady-State Error of Unity Feedback • Steady-state error: • ess depends on the number of poles G(s) has at s = 0. • Forward-path transfer function:system type = j System type (the type of the control system)

  10. 4, p. 261 Unity Feedback with Step Func. Input Step-function input:

  11. 4, p. 262 Unity Feedback with Ramp Func. Input Ramp-function input:

  12. 4, p. 263 Unity Feedback with Para. Func. Input Parabolic-function input:

  13. 4, p. 264 Table 5-1

  14. 4, p. 265 Example 5-4-2

  15. 4, p. 265 Example 5-4-2 (cont.)

  16. 4, p. 266 Relationship between Steady-State Error and Closed-Loop Transfer Function State-State Error: H(0) = KH • Error signal: • M(s) does not have any poles at s = 0: Reference signal

  17. 4, p. 267 Steady-State Error: H(0)=KH • Step-function input: R(s) = R/sess = 0 or • Ramp-function input: R(s) = R/s2

  18. 4, p. 268 Steady-State Error: H(0)=KH(cont.) • Parabolic-function input: R(s) = R/s3

  19. 4, p. 268 Example 5-4-3

  20. 4, p. 269 Example 5-4-4

  21. 4, p. 269 Example 5-4-4 (cont.) • Unit-step input:y(t)  1, ess 0 • Unit-ramp input:y(t)  t0.8, ess0.8 • Unit-parabolic input:y(t)0.5t2 0.8t11.2, ess0.8t+11.2

  22. 4, p. 270 Example 5-4-5 • Unit-step input:  0.5

  23. 4, p. 270 Example 5-4-5 (cont.) • Unit-ramp input:error: • Unit-parabolic input:error:  0.4  0.4t + 2.6  

  24. 4, p. 270 Steady-State Error:H(s) Has Nth-Order Zero at s = 0 • Reference signal: R(s)/KHsN • Error signal: R(s) = R/s

  25. 4, p. 271 Example 5-4-6

  26. 4, p. 272 Steady-State Error Caused by Nonlinear System Elements

  27. 4, p. 273 Steady-State Error Caused by Nonlinear System Elements (cont.)

  28. 5, p. 274 5-5 Time-Response of a Prototype First-Order System time constant

  29. 6, p. 275 5-6 Time-Response of a Prototype Second-Order System G(s) Characteristic equation: R(s) = 1/s (unit-step input)

  30. 6, p. 276 Unit-Step Responses Figure 5-14 Unit-step response of the prototype 2nd-order system with various damping ratios.

  31. 6, p. 277 Damping Ratio and Damping Factor • Characteristic equation: • Unit-step response: •  controls the rate of rise or decay of y(t). damping factor ( 1/  time constant ) • Damping ratio:

  32. 6, p. 278 Natural Undamped Frequency • Natural undamped frequency: n • Damped (or conditional) frequency:

  33. 6, p. 279 Figure 5-16

  34. 6, p. 280 Classification of System Dynamics

  35. 6, p. 281 Step-Response Comparison (Fig. 5-18) overdamped critically overdamped underdamped

  36. 6, p. 281 Step-Response Comparison (cont.) undamped negative overdamped negative overdamped

  37. 6, p. 281 Maximum Overshoot y() is maximum when   1. t = , n = 1 The first overshoot isthe maximum overshoot

  38. 6, p. 282 Maximum Overshoot (cont.)

  39. 6, p. 283 Maximum Overshoot (cont.) The first overshoot (n = 1) is the maximum overshoot

  40. 6, p. 283 Maximum Overshoot (cont.)

  41. 6, p. 283 Delay Time The time required for the step response to reach 50% of its final value  Set y(t) = 0.5 and solve for t • Approximation:

  42. 6, p. 284 Rise Time The time for the step response to reach from 10 to 90% of its final value. • Approximation:

  43. 6, p. 285 Settling Time: 0 <  < 0.69 The time required for the step response to decrease and staywithin a specified percentage(2% or 5%) of its final value.

  44. 6, p. 287 Settling Time:  > 0.69

  45. 6, p. 287 Settling Time

  46. 7, p. 289 5-7 Speed and Position Control of a DC Motor • Open-loop response:

  47. 7, p. 290 Speed of Motor Shaft: Open-Loop • La is very small e = La/Ra (motor electric-time constant) is neglected.Keff = Km/(RaB+KmKb): motor gain constantm = RaJm/(RaB+KmKb): motor mechanical time constant

  48. 7, p. 291 Time Response: Open-Loop Superposition: • TL(s)=0 (no disturbance and B=0) and Va(s)=A/s: • TL(s)=D/s and Va(s)=A/s:  (t) A/Kb  (t)  A/KbRaD/KmKb

  49. 7, p. 291 Closed-Loop Response

  50. 7, p. 292 Speed of Motor Shaft: Closed-Loop • La= 0 • in = A/sandTL = D/s:steady-state response: 1/c c: system mechanical-time constant

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