1. Intelligent Robot Lab Pusan National University Intelligent Robot Lab Chapter 7. Forced Response Errors Pusan National University Intelligent Robot Laboratory

2. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Introduction  Steady-State Error for Unity Feedback Systems  Static Error Constants and System Type  Steady-State Error Specification  Steady-State Error for Disturbances  Steady-State Error for Non-unity Feedback Systems  Sensitivity  Steady-State Error for Systems in State Space Table of Contents Table of Contents

3. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Definition and test input  Step input Constant position commands which are useful in determining the ability of the positioning control  Ramp input Constant-velocity input to a position control system by their linearly increasing amplitude.  Parabolic input Constant-acceleration input to position control systems and can be used to represent accelerating targets. Introduction Table 7.1 Test waveforms for evaluating steady-state errors of position control systems

4. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Application to stable systems  Unstable systems represent loss of control in the steady state and are not acceptable for use at all.  The engineer must check the system for stability while performing steady-state error analysis and design.  For practice you may want to test some of the systems for stability. Introduction

5. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Evaluating steady-state errors  A step input Output1: zero steady-state error Output2: finite steady state error  A ramp input Output1: zero steady-state error Output2: finite steady state error Introduction Figure 7.2 Steady-state error: a. step input b. ramp input

6. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Sources of steady-state error  Steady-state error in control systems Nonlinear source, backlash in gears or motors that will not move unless the input voltage exceeds a threshold  Configuration of system : input, : output, : error : steady-state value of output : steady-state value of error for a step input Introduction Figure 7.4 System with: a. finite steady-state error for a step input b. zero steady-state error for step input (7.1)

7. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of  To find, the error between he input,, and the output, but  Substituting Eq.(7.3) into Eq.(7.2), simplifying, and solving for yields Steady-State Error for Unity Feedback Systems (7.2) (7.3) (7.4) Figure 7.3 Closed-loop control system error: a. general representation

8. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of  Applying the final value theorem, which allows us to use the final value of, without taking the inverse Laplace transform of, and then letting approach infinity  Substituting Eq.(7.4) into Eq.(7.5) yields Steady-State Error for Unity Feedback Systems (7.6) (7.5)

9. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of  Writing from Figure 7.3(b) but  Substituting Eq.(7.9) into Eq.(7.8) and solving for yields Steady-State Error for Unity Feedback Systems (7.8) (7.9) (7.10) Figure 7.3 Closed-loop control system error: b. representation for unity feedback systems

10. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of  Apply the final value theorem, Eq.(7.5). Assume that the closed-loop system is stable and substitute Eq.(7.10) into Eq.(7.5)  Equation (7.11) allows us to calculate the steady-state error,, given the input, and the system  Let us take Table(7.1)’s input and evaluate its effect on the steady-state error by using Eq.(7.11) Steady-State Error for Unity Feedback Systems (7.11)

11. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of -step input  The term is the DC gain of the forward transfer function, since s, the frequency variable, is approaching zero. In order to have zero steady-state error  Hence, to satisfy Eq.(7.13), must take on the following form Steady-State Error for Unity Feedback Systems (7.12) (7.13) (7.14)

12. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of -step input  If there are no integrations, then n=0. Using Eq.(7.14) which is finite and yields a finite error from Eq.(7.12)  Figure 7.2(a), output 2, is an example of this case of finite steady-state error. Steady-State Error for Unity Feedback Systems (7.15) Figure 7.2 Steady-state error: a. step input

13. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of -ramp input  To have zero steady-state error for a ramp input  If only one integration exists in the forward path, assuming Eq.(7.14) Steady-State Error for Unity Feedback Systems (7.16) (7.18) (7.17)

14. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of -ramp input  If there are no integration in the forward path  The steady-state error would be infinite and lead to diverging ramp, as shown in Figure 7.2(b), output 3. Steady-State Error for Unity Feedback Systems (7.19) Figure 7.2 Steady-state error: b. ramp input

15. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Steady-state error in terms of -parabolic input  In order to have zero steady-state error for a parabolic input  If there are only two integration in the forward path  If there is only one or less integration in the forward path  The steady-state error is infinity. Steady-State Error for Unity Feedback Systems (7.20) (7.21) (7.22) (7.23)

16. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.2 (Steady-state error for system with no integrations)  Find the steady-state errors for inputs of to the system shown in Figure 7.5. the function is the unit step. Sol)The input, whose Laplace transform is, the steady-state error will be five times as large as that given by Eq.(7.12) The input, whose Laplace transform is, the steady-state error will be five times as large as that given by Eq.(7.16) Steady-State Error for Unity Feedback Systems (7.24) Figure 7.5 Feedback control system for Example 7.2 (7.25)

17. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.2 (Steady-state error for system with no integrations) The input, whose Laplace transform is, the steady-state error will be five times as large as that given by Eq.(7.20) Steady-State Error for Unity Feedback Systems (7.26)

18. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.3 (Steady-state error for system with one integration)  Find the steady-state errors for inputs of to the system shown in Figure 7.6. the function is the unit step. Sol)The input, whose Laplace transform is, the steady-state error will be five times as large as that given by Eq.(7.12) The input, whose Laplace transform is, the steady-state error will be five times as large as that given by Eq.(7.16) Steady-State Error for Unity Feedback Systems (7.27) Figure 7.6 Feedback control system for Example 7.3 (7.28)

19. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.3 (Steady-state error for system with no integrations) The input, whose Laplace transform is, the steady-state error will be five times as large as that given by Eq.(7.20) Steady-State Error for Unity Feedback Systems (7.29)

20. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Static error constants  Relationships for steady-state error, Step input,  Relationships for steady-state error, ramp input,  Relationships for steady-state error, parabolic input, Static Error Constants and System Type (7.30) (7.31) (7.32)

21. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Static error constants  Position constant,  Velocity constant,  Acceleration constant, Static Error Constants and System Type (7.33) (7.34) (7.35)

22. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.4 (Steady state error via static error constants)  For each system of figure 7.7, evaluate the static error constants and find the expected error for the standard step, ramp, and parabolic inputs. Static Error Constants and System Type Figure 7.7 Feedback control systems for Example 7.4

23. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.4 (Steady state error via static error constants)  For each system of figure 7.7, evaluate the static error constants and find the expected error for the standard step, ramp, and parabolic inputs. Sol)for Figure 7.7(a) For a step input For a ramp input For a parabolic input Static Error Constants and System Type (7.36) (7.37) (7.38) (7.39) (7.40) (7.41)

24. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.4 (Steady state error via static error constants)  For each system of figure 7.7, evaluate the static error constants and find the expected error for the standard step, ramp, and parabolic inputs. Sol)for Figure 7.7(b) For a step input For a ramp input For a parabolic input Static Error Constants and System Type (7.42) (7.43) (7.44) (7.45) (7.46) (7.47)

25. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.4 (Steady state error via static error constants)  For each system of figure 7.7, evaluate the static error constants and find the expected error for the standard step, ramp, and parabolic inputs. Sol)for Figure 7.7(c) For a step input For a ramp input For a parabolic input Static Error Constants and System Type (7.48) (7.49) (7.50) (7.51) (7.52) (7.53)

26. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  System type  Defined steady-state errors, static error constant, and system type  The specifications for a control system’s steady-state errors will be formulated followed by some examples Static Error Constants and System Type Table 7.2 Relationships between input, system type, static error constants, and steady-state errors

27. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Static error constants  Specifications for a control system’s transient response Damping ratio, Settling time, peak time, percent overshoot  Specifications for a control system’s stead-state errors Position constant, Velocity constant, acceleration constant  Ex) Control system has the specification, conclusions The system is stable. The system is of Type1, since only Type 1 systems have ’s that are finite constants Recall that =0 for Type0 systems, whereas for Type2 systems. A ramp input is the test signal. Since is specified as a finite constant, and the steady-state error for a ramp input is inversely proportional to, we know the test input is a ramp. The stead-state error between the input ramp and out ramp is per unit of input slope. Steady-State Error Specification

28. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.6 (Gain design to meet a steady-state error specification)  Given the control system in Figure 7.10, find the value of so that there is 10% error in the steady state. Sol)only a ramp yields a finite error in a type 1 system. Thus Therefore Which yields Steady-State Error Specification Figure 7.10 Feedback control system for Example 7.6

29. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Feedback control system  Compensate for disturbances or unwanted input that enter a system.  The advantage of using feedback Regardless of these disturbances, the system can be designed the follow the input with small or zero error, as we now demonstrate.  Transform of output Steady-State Error for Disturbances Figure 7.11 Feedback control system showing disturbance (7.58) (7.59)

30. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Feedback control system  Eq.(7.59) into Eq.(7.58) and solving for Transfer function relating to :  To find the steady-state value of the error, apply final value theorem to Eq.(7.60) Steady-State Error for Disturbances (7.60) (7.61)

31. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Feedback control system : The steady-state error due to : The steady-state error due to the disturbance  Assume a step disturbance  Substituting this value into the second term of Eq.(7.61),, the steady-state error component due to a step disturbance  This Eq show that the steady-state error produced by a step disturbance can be reduce by increasing the DC gain of or decreasing the DC gain of. Steady-State Error for Disturbances (7.62)

32. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.7 (Steady-state error due to step disturbance)  Find the steady-state error component due to a step disturbance for system of Figure 7.13. Sol)Using figure 7.12 and Eq.(7.62) The result shows that the steady-state error produced by the step disturbance is inversely proportional the DC gain of. The DC gain is infinite. Steady-State Error for Disturbances (7.63) Figure 7.13 Feedback control system for Example 7.7Figure 7.12

33. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Unity feedback system from a general non-unity feedback system  Figure 7.15(a) General feedback system  Figure 7.15(b) Transducer to right path the summing junction yield the general non-unity feedback system  Figure 7.15(c) Form a unity feedback system by Adding and subtracting units feedback path  Figure 7.15(d) Combine with the negative unity feedback  Figure 7.15(e) Combine the feedback system consisting of and, leaving an equivalent forward path and a unity feedback Steady-State Error for Nonunity Feedback Systems Figure 7.15 Forming an equivalent unity feedback system from a general nonunity feedback system

34. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Non-unity feedback control system with disturbance  The steady-state error for this system, , Eq.(7.69) Steady-State Error for Nonunity Feedback Systems (7.69) (7.70)

35. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Non-unity feedback control system with disturbance  For zero error  Eq. (7.71) can always be satisfied The system stable is a Type 1 system is a Type 0 system is a Type 0 system with a DC gain of unity  The steady-state actuating signal for Figure 7.15(a) Steady-State Error for Nonunity Feedback Systems (7.71) Figure 7.17 Nonunity feedback control system with disturbance (7.72)

36. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Sensitivity  The degree to which changes in system parameters affect system transfer function, and hence performance.  A system with zero sensitivity is ideal.  The greater the sensitivity, the less desirable the effect of a parameter change.  Definition of Sensitivity which reduce to Sensitivity (7.75)

37. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.10 (Sensitivity of a close-loop transfer function)  Given the system of Figure 7.19, calculating the sensitivity of the close-loop transfer function to changes in the parameter a. How would you reduce the sensitivity? Sol) The close-loop transfer function Using Eq.(7.75), the sensitivity is given Increase in reduce the sensitivity of the close-loop transfer function to changes in the parameter. Sensitivity (7.76) (7.77) Figure 7.19 Feedback control system for Examples 7.10 and 7.11

38. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Two methods for calculating the steady-state error will be covered  Analysis via final value theorem  Analysis via input substitution  Analysis via final value theorem  Consider the closed-loop system represented in state space  The Laplace transform of the error  But Steady-State Error for Systems in State Space (7.84a) (7.84b) (7.85) (7.86)

39. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Analysis via final value theorem  Substituting Eq. (7.86) into (7.85)  Using Eq. (7.73) for  Applying the final value theorem Steady-State Error for Systems in State Space (7.87) (7.88) (7.89)

40. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.13 (Steady-state error using the final value theorem)  Evaluate The steady-state error for the system described by Eqs. (7.90) for unit step and unit ramp inputs. Use the final value theorem. Sol) Substituting Eqs. (7.90) into (7.89) For a unit step, and. For a unit ramp, and. The system behaves like a Type 0 system. Steady-State Error for Systems in State Space (7.90) (7.91)

41. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Analysis via input substitution (Step input)  Given the state Eq. (7.84), if the input is a unit step where, a steady-state solution,, for where is constant. Also  Substituting, a unit step, along with Eq. (7.92) and (7.93), into Eq.(7.84) yields Steady-State Error for Systems in State Space (7.92) (7.93) (7.94a) (7.94b)

42. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Analysis via input substitution (Step input)  is the steady-state output. Solving for yields  The final result for the steady-state error for a unit step input into a system represented in state space Steady-State Error for Systems in State Space (7.95) (7.96) (7.94b)

43. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Analysis via input substitution (Ramp input)  For unit ramp input,, a steady-state solution for where and are constant. Hence Steady-State Error for Systems in State Space (7.97) (7.98)

44. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Analysis via input substitution (Ramp input)  Substituting along with Eq.(7.97) and (7.98) into Eq.(7.84) yields  In order to balance Eq.(7.99a), equate the matrix coefficients of  Equating constant terms in Eq.(7.99a) Steady-State Error for Systems in State Space (7.99a) (7.101) (7.99b) (7.100)

45. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Analysis via input substitution (Ramp input)  Substituting Eq.(7.100) and (7.101) into (7.99b) yields  The steady-state error  Notice that in order to use this method, must exist. That is, Steady-State Error for Systems in State Space (7.102) (7.103)

46. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab.  Example 7.14 (Steady-state error using input substitution)  Evaluate The steady-state error for the system described by Eqs. (7.90) for unit step and unit ramp inputs. Use input substitution. Sol) For a unit input, the steady-state error given by Eq. (7.96)is C, A, and B are as follows For a ramp input, using Eq.(7.103) Steady-State Error for Systems in State Space (7.104) (7.105) (7.106)

47. http://robotics.pusan.ac.kr Intelligent Robot Lab Intelligent Robot Lab. T T H H A A N N K K U U Y Y O O Homework: Divide by 10