Course Summary. The course cannot be summarized in one lecture.

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1 Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: Steady-State Error Unit 7: Root Locus Techniques Unit 8: Design via Root Locus Course Summary The course cannot be summarized in one lecture. ENGI 5821 Course Summary

2 Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: Steady-State Error Unit 7: Root Locus Techniques Unit 8: Design via Root Locus Course Summary The course cannot be summarized in one lecture. The intent here is to go through each unit of the course and mention some of the most important or poorly understood concepts. ENGI 5821 Course Summary

3 Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: Steady-State Error Unit 7: Root Locus Techniques Unit 8: Design via Root Locus Course Summary The course cannot be summarized in one lecture. The intent here is to go through each unit of the course and mention some of the most important or poorly understood concepts. This brief tour is no replacement for the course itself! ENGI 5821 Course Summary

4 Course Summary Unit 1: Introduction Unit 2: Modeling in the Frequency Domain Unit 3: Time Response Unit 4: Block Diagram Reduction Unit 5: Stability Unit 6: Steady-State Error Unit 7: Root Locus Techniques Unit 8: Design via Root Locus Course Summary The course cannot be summarized in one lecture. The intent here is to go through each unit of the course and mention some of the most important or poorly understood concepts. This brief tour is no replacement for the course itself! Many important details will be skipped. ENGI 5821 Course Summary

5 Unit 1: Introduction Introduction

6 Unit 1: Introduction Introduction System Configurations

7 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop

8 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance

9 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability

10 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability The Design Process

11 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability The Design Process specifications

12 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability The Design Process specifications required performance measures

13 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability The Design Process specifications required performance measures schematic

14 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability The Design Process specifications required performance measures schematic transfer function for components

15 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability The Design Process specifications required performance measures schematic transfer function for components transfer function for system

16 Unit 1: Introduction Introduction System Configurations Open-loop vs. Closed-loop Measuring Performance Transient response, steady-state error, stability The Design Process specifications required performance measures schematic transfer function for components transfer function for system analysis and design

17 Unit 2: Modeling in the Frequency Domain Complex Frequency: x(t) = R{Xe st } We can express constants, exponentials, sinusoids, and exponentially decaying or growing sinusoids by varying s. The actual amplitude and phase of a particular signal is expressed in X.

18 Unit 2: Modeling in the Frequency Domain Complex Frequency: x(t) = R{Xe st } We can express constants, exponentials, sinusoids, and exponentially decaying or growing sinusoids by varying s. The actual amplitude and phase of a particular signal is expressed in X.

19 Unit 2: Modeling in the Frequency Domain Complex Frequency: x(t) = R{Xe st } We can express constants, exponentials, sinusoids, and exponentially decaying or growing sinusoids by varying s. The actual amplitude and phase of a particular signal is expressed in X. Laplace Transform

20 Unit 2: Modeling in the Frequency Domain Complex Frequency: x(t) = R{Xe st } We can express constants, exponentials, sinusoids, and exponentially decaying or growing sinusoids by varying s. The actual amplitude and phase of a particular signal is expressed in X. Laplace Transform Transforms any signal into a combination of complex exponentials

21 Unit 2: Modeling in the Frequency Domain Complex Frequency: x(t) = R{Xe st } We can express constants, exponentials, sinusoids, and exponentially decaying or growing sinusoids by varying s. The actual amplitude and phase of a particular signal is expressed in X. Laplace Transform Transforms any signal into a combination of complex exponentials Tables of transform pairs and theorems

22 Unit 2: Modeling in the Frequency Domain Complex Frequency: x(t) = R{Xe st } We can express constants, exponentials, sinusoids, and exponentially decaying or growing sinusoids by varying s. The actual amplitude and phase of a particular signal is expressed in X. Laplace Transform Transforms any signal into a combination of complex exponentials Tables of transform pairs and theorems Use partial-fraction expansion to decompose ratio of polynomials

23 Transfer Functions

24 Transfer Functions Can be developed for LTI systems with zero-initial conditions

25 Transfer Functions Can be developed for LTI systems with zero-initial conditions Transfer function G(s) gives the system output C(s) for any input R(s): C(s) = R(s)G(s)

26 Transfer Functions Can be developed for LTI systems with zero-initial conditions Transfer function G(s) gives the system output C(s) for any input R(s): C(s) = R(s)G(s) Electrical Systems

27 Transfer Functions Can be developed for LTI systems with zero-initial conditions Transfer function G(s) gives the system output C(s) for any input R(s): C(s) = R(s)G(s) Electrical Systems Slow method: Develop DE s, apply Laplace, form TF

28 Transfer Functions Can be developed for LTI systems with zero-initial conditions Transfer function G(s) gives the system output C(s) for any input R(s): C(s) = R(s)G(s) Electrical Systems Slow method: Develop DE s, apply Laplace, form TF Fast method: Define transfer functions for individual components and state the problem in the freq. domain

29 Transfer Functions Can be developed for LTI systems with zero-initial conditions Transfer function G(s) gives the system output C(s) for any input R(s): C(s) = R(s)G(s) Electrical Systems Slow method: Develop DE s, apply Laplace, form TF Fast method: Define transfer functions for individual components and state the problem in the freq. domain Circuit analysis techniques developed for resistive circuits automatically work for L s and C s in the freq. domain!

30 Transfer Functions Can be developed for LTI systems with zero-initial conditions Transfer function G(s) gives the system output C(s) for any input R(s): C(s) = R(s)G(s) Electrical Systems Slow method: Develop DE s, apply Laplace, form TF Fast method: Define transfer functions for individual components and state the problem in the freq. domain Circuit analysis techniques developed for resistive circuits automatically work for L s and C s in the freq. domain! Op-Amps: Utilize ideal op-amp assumptions

31 Mechanical Systems

32 Mechanical Systems Mechanical components in translational and rotational forms

33 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces

34 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion

35 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass

36 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass Gears linearly relate the motions of multiple shafts

37 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass Gears linearly relate the motions of multiple shafts Mechanical impedances can be reflected between shafts to simplify the calculation of TF

38 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass Gears linearly relate the motions of multiple shafts Mechanical impedances can be reflected between shafts to simplify the calculation of TF Motors: Electromechanical systems

39 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass Gears linearly relate the motions of multiple shafts Mechanical impedances can be reflected between shafts to simplify the calculation of TF Motors: Electromechanical systems Derived θ m (s)/e a (s) = K/s(s + α)

40 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass Gears linearly relate the motions of multiple shafts Mechanical impedances can be reflected between shafts to simplify the calculation of TF Motors: Electromechanical systems Derived θ m (s)/e a (s) = K/s(s + α) Jump back to the time-domain to run tests and evaluate motor parameters

41 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass Gears linearly relate the motions of multiple shafts Mechanical impedances can be reflected between shafts to simplify the calculation of TF Motors: Electromechanical systems Derived θ m (s)/e a (s) = K/s(s + α) Jump back to the time-domain to run tests and evaluate motor parameters Linearization

42 Mechanical Systems Mechanical components in translational and rotational forms opposing forces = applied forces One equation of motion for each linearly independent motion In rotational systems torque replaces force (T = FR if F axis of rot.) and moment-of-inertia replaces mass Gears linearly relate the motions of multiple shafts Mechanical impedances can be reflected between shafts to simplify the calculation of TF Motors: Electromechanical systems Derived θ m (s)/e a (s) = K/s(s + α) Jump back to the time-domain to run tests and evaluate motor parameters Linearization Final exam will cover only the concept and application to a simple DE

43 Unit 3: Time Response The poles give the form of the response, the zeros give the weights

44 Unit 3: Time Response The poles give the form of the response, the zeros give the weights First-order systems yield exponential responses characterized by time constant

45 Unit 3: Time Response The poles give the form of the response, the zeros give the weights First-order systems yield exponential responses characterized by time constant Second-order systems yield four different responses characterized by ζ and ω n

46 Unit 3: Time Response The poles give the form of the response, the zeros give the weights First-order systems yield exponential responses characterized by time constant Second-order systems yield four different responses characterized by ζ and ω n Response specifications: T p, %OS, and T s

47 Unit 3: Time Response The poles give the form of the response, the zeros give the weights First-order systems yield exponential responses characterized by time constant Second-order systems yield four different responses characterized by ζ and ω n Response specifications: T p, %OS, and T s Relationship between response specs. and pole position

48 Unit 3: Time Response The poles give the form of the response, the zeros give the weights First-order systems yield exponential responses characterized by time constant Second-order systems yield four different responses characterized by ζ and ω n Response specifications: T p, %OS, and T s Relationship between response specs. and pole position Additional poles or zeros

49 Unit 3: Time Response The poles give the form of the response, the zeros give the weights First-order systems yield exponential responses characterized by time constant Second-order systems yield four different responses characterized by ζ and ω n Response specifications: T p, %OS, and T s Relationship between response specs. and pole position Additional poles or zeros Real-axis poles or zeros far to the left have little effect

50 Unit 3: Time Response The poles give the form of the response, the zeros give the weights First-order systems yield exponential responses characterized by time constant Second-order systems yield four different responses characterized by ζ and ω n Response specifications: T p, %OS, and T s Relationship between response specs. and pole position Additional poles or zeros Real-axis poles or zeros far to the left have little effect A pole can cancel a nearby zero

51 Unit 4: Block Diagram Reduction Recognize and reduce cascade, parallel, and feedback forms

52 Unit 4: Block Diagram Reduction Recognize and reduce cascade, parallel, and feedback forms If none of the forms are apparent, blocks can be shift to the left or right of summing junctions and pickoff points

53 Unit 4: Block Diagram Reduction Recognize and reduce cascade, parallel, and feedback forms If none of the forms are apparent, blocks can be shift to the left or right of summing junctions and pickoff points Moving blocks = algebraic manipulation

54 Unit 4: Block Diagram Reduction Recognize and reduce cascade, parallel, and feedback forms If none of the forms are apparent, blocks can be shift to the left or right of summing junctions and pickoff points Moving blocks = algebraic manipulation Signal-flow graphs

55 Unit 4: Block Diagram Reduction Recognize and reduce cascade, parallel, and feedback forms If none of the forms are apparent, blocks can be shift to the left or right of summing junctions and pickoff points Moving blocks = algebraic manipulation Signal-flow graphs nodes are signals; edges are systems

56 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response)

57 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable

58 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable All it takes is one RHP pole for instability (or one pair of jω poles for marginal stability)

59 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable All it takes is one RHP pole for instability (or one pair of jω poles for marginal stability) Characteristic coef s missing or differing in sign unstable (opposite is not true)

60 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable All it takes is one RHP pole for instability (or one pair of jω poles for marginal stability) Characteristic coef s missing or differing in sign unstable (opposite is not true) Routh-Hurwitz

61 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable All it takes is one RHP pole for instability (or one pair of jω poles for marginal stability) Characteristic coef s missing or differing in sign unstable (opposite is not true) Routh-Hurwitz Special case: zero in first column

62 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable All it takes is one RHP pole for instability (or one pair of jω poles for marginal stability) Characteristic coef s missing or differing in sign unstable (opposite is not true) Routh-Hurwitz Special case: zero in first column Special case: ROZ EP factor

63 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable All it takes is one RHP pole for instability (or one pair of jω poles for marginal stability) Characteristic coef s missing or differing in sign unstable (opposite is not true) Routh-Hurwitz Special case: zero in first column Special case: ROZ EP factor EP has symmetric roots so system is either unstable or marginally stable

64 Unit 5: Stability Definitions: Stability of the total response and the natural response (we focused on the natural response) RHP: Unstable, jω: Marginally stable, LHP: Stable All it takes is one RHP pole for instability (or one pair of jω poles for marginal stability) Characteristic coef s missing or differing in sign unstable (opposite is not true) Routh-Hurwitz Special case: zero in first column Special case: ROZ EP factor EP has symmetric roots so system is either unstable or marginally stable Problem: find K such that...

65 Unit 6: Steady-State Error For any control system: E(s) = R(s) C(s)

66 Unit 6: Steady-State Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( )

67 Unit 6: Steady-State Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( ) Unity feedback systems

68 Unit 6: Steady-State Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( ) Unity feedback systems

69 Unit 6: Steady-State Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( ) Unity feedback systems Input Type 0: e( ) Type 1: e( ) Type 2: e( ) Step, u(t) 1 1+K p Ramp, tu(t) K v 0 Para., t 2 1 u(t) K a

70 Unit 6: Steady-State Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( ) Unity feedback systems Input Type 0: e( ) Type 1: e( ) Type 2: e( ) Step, u(t) 1 1+K p Ramp, tu(t) K v 0 Para., t 2 1 u(t) K a Disturbances: e( ) = e R ( ) + e D ( )

71 Unit 7: Root Locus Techniques Vector representation of complex numbers

72 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closed-loop system poles as K is varied

73 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closed-loop system poles as K is varied Properties of the RL

74 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closed-loop system poles as K is varied Properties of the RL If KG(s)H(s) = (2k + 1)180 o we can find a K to satisfy KG(s)H(s) = 1

75 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closed-loop system poles as K is varied Properties of the RL If KG(s)H(s) = (2k + 1)180 o we can find a K to satisfy KG(s)H(s) = 1 Sketching rules

76 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closed-loop system poles as K is varied Properties of the RL If KG(s)H(s) = (2k + 1)180 o we can find a K to satisfy KG(s)H(s) = 1 Sketching rules Refinements: breakaway and break-in points, jω crossings, angles of departure and arrival

77 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closed-loop system poles as K is varied Properties of the RL If KG(s)H(s) = (2k + 1)180 o we can find a K to satisfy KG(s)H(s) = 1 Sketching rules Refinements: breakaway and break-in points, jω crossings, angles of departure and arrival Search procedure required to find points with particular spec s

78 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closed-loop system poles as K is varied Properties of the RL If KG(s)H(s) = (2k + 1)180 o we can find a K to satisfy KG(s)H(s) = 1 Sketching rules Refinements: breakaway and break-in points, jω crossings, angles of departure and arrival Search procedure required to find points with particular spec s Positive feedback requires changes to the RL definition and sketching rules

79 Unit 8: Design via Root Locus First check acceptable operating point on uncompensated RL

80 Unit 8: Design via Root Locus First check acceptable operating point on uncompensated RL PI compensation: increase system type while maintaining transient resp. G c (s) = K(s + z c) s Choose z c as a small number; Requires active amplification

81 Unit 8: Design via Root Locus First check acceptable operating point on uncompensated RL PI compensation: increase system type while maintaining transient resp. G c (s) = K(s + z c) s Choose z c as a small number; Requires active amplification Lag compensation: increase static error constant (which reduces e( ) G c (s) = K(s + z c) s + p c Choose p c as a small number and adjust z c accordingly

82 Unit 8: Design via Root Locus First check acceptable operating point on uncompensated RL PI compensation: increase system type while maintaining transient resp. G c (s) = K(s + z c) s Choose z c as a small number; Requires active amplification Lag compensation: increase static error constant (which reduces e( ) G c (s) = K(s + z c) s + p c Choose p c as a small number and adjust z c accordingly PD compensation: Adjust transient response G c (s) = K(s + z c ) Place z c to move RL to intersect desired operating point; Requires active amplification

83 Unit 8: Design via Root Locus First check acceptable operating point on uncompensated RL PI compensation: increase system type while maintaining transient resp. G c (s) = K(s + z c) s Choose z c as a small number; Requires active amplification Lag compensation: increase static error constant (which reduces e( ) G c (s) = K(s + z c) s + p c Choose p c as a small number and adjust z c accordingly PD compensation: Adjust transient response G c (s) = K(s + z c ) Place z c to move RL to intersect desired operating point; Requires active amplification All of our design techniques rely on 2 nd order approx.. Verify approx. validity and simulate

84 PID: Design for transient response, then e( )

85 PID: Design for transient response, then e( ) Analog PID implemented via op-amp

86 PID: Design for transient response, then e( ) Analog PID implemented via op-amp Digital PID can be implemented in software (or digital hardware)

87 PID: Design for transient response, then e( ) Analog PID implemented via op-amp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown

88 PID: Design for transient response, then e( ) Analog PID implemented via op-amp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown Ziegler-Nichols (rules of thumb)

89 PID: Design for transient response, then e( ) Analog PID implemented via op-amp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown Ziegler-Nichols (rules of thumb) Method 1: Unit-step response is S-shaped

90 PID: Design for transient response, then e( ) Analog PID implemented via op-amp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown Ziegler-Nichols (rules of thumb) Method 1: Unit-step response is S-shaped Method 2: System appears to involve integration and/or underdamped poles

91 PID: Design for transient response, then e( ) Analog PID implemented via op-amp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown Ziegler-Nichols (rules of thumb) Method 1: Unit-step response is S-shaped Method 2: System appears to involve integration and/or underdamped poles Computational search