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: SteadyState 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: SteadyState 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: SteadyState 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: SteadyState 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 Openloop vs. Closedloop
8 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance
9 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance Transient response, steadystate error, stability
10 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance Transient response, steadystate error, stability The Design Process
11 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance Transient response, steadystate error, stability The Design Process specifications
12 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance Transient response, steadystate error, stability The Design Process specifications required performance measures
13 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance Transient response, steadystate error, stability The Design Process specifications required performance measures schematic
14 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance Transient response, steadystate error, stability The Design Process specifications required performance measures schematic transfer function for components
15 Unit 1: Introduction Introduction System Configurations Openloop vs. Closedloop Measuring Performance Transient response, steadystate 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 Openloop vs. Closedloop Measuring Performance Transient response, steadystate 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 partialfraction expansion to decompose ratio of polynomials
23 Transfer Functions
24 Transfer Functions Can be developed for LTI systems with zeroinitial conditions
25 Transfer Functions Can be developed for LTI systems with zeroinitial 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 zeroinitial 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 zeroinitial 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 zeroinitial 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 zeroinitial 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 zeroinitial 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! OpAmps: Utilize ideal opamp 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 momentofinertia 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 momentofinertia 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 momentofinertia 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 momentofinertia 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 momentofinertia 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 momentofinertia 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 timedomain 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 momentofinertia 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 timedomain 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 momentofinertia 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 timedomain 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 Firstorder 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 Firstorder systems yield exponential responses characterized by time constant Secondorder 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 Firstorder systems yield exponential responses characterized by time constant Secondorder 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 Firstorder systems yield exponential responses characterized by time constant Secondorder 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 Firstorder systems yield exponential responses characterized by time constant Secondorder 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 Firstorder systems yield exponential responses characterized by time constant Secondorder 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 Realaxis 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 Firstorder systems yield exponential responses characterized by time constant Secondorder 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 Realaxis 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 Signalflow 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 Signalflow 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) RouthHurwitz
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) RouthHurwitz 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) RouthHurwitz 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) RouthHurwitz 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) RouthHurwitz 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: SteadyState Error For any control system: E(s) = R(s) C(s)
66 Unit 6: SteadyState Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( )
67 Unit 6: SteadyState Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( ) Unity feedback systems
68 Unit 6: SteadyState Error For any control system: E(s) = R(s) C(s) Apply final value theorem to determine e( ) Unity feedback systems
69 Unit 6: SteadyState 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: SteadyState 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 closedloop system poles as K is varied
73 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closedloop 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 closedloop 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 closedloop 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 closedloop 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 breakin points, jω crossings, angles of departure and arrival
77 Unit 7: Root Locus Techniques Vector representation of complex numbers Root locus: locations of closedloop 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 breakin 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 closedloop 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 breakin 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 opamp
86 PID: Design for transient response, then e( ) Analog PID implemented via opamp Digital PID can be implemented in software (or digital hardware)
87 PID: Design for transient response, then e( ) Analog PID implemented via opamp 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 opamp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown ZieglerNichols (rules of thumb)
89 PID: Design for transient response, then e( ) Analog PID implemented via opamp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown ZieglerNichols (rules of thumb) Method 1: Unitstep response is Sshaped
90 PID: Design for transient response, then e( ) Analog PID implemented via opamp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown ZieglerNichols (rules of thumb) Method 1: Unitstep response is Sshaped Method 2: System appears to involve integration and/or underdamped poles
91 PID: Design for transient response, then e( ) Analog PID implemented via opamp Digital PID can be implemented in software (or digital hardware) PID tuning: Strategies to apply when system model is unknown ZieglerNichols (rules of thumb) Method 1: Unitstep response is Sshaped Method 2: System appears to involve integration and/or underdamped poles Computational search