Course SummaryUnit 1: Introduction
Unit 2: Modeling in the Frequency DomainUnit 3: Time Response
Unit 4: Block Diagram ReductionUnit 5: Stability
Unit 6: Steady-State ErrorUnit 7: Root Locus TechniquesUnit 8: Design via Root Locus
Course Summary
The course cannot be summarized in one lecture.
The intent hereis to go through each unit of the course and mention some of themost important or poorly understood concepts.
This brief tour is no replacement for the course itself! Manyimportant details will be skipped.
ENGI 5821 Course Summary
Course SummaryUnit 1: Introduction
Unit 2: Modeling in the Frequency DomainUnit 3: Time Response
Unit 4: Block Diagram ReductionUnit 5: Stability
Unit 6: Steady-State ErrorUnit 7: Root Locus TechniquesUnit 8: Design via Root Locus
Course Summary
The course cannot be summarized in one lecture. The intent hereis to go through each unit of the course and mention some of themost important or poorly understood concepts.
This brief tour is no replacement for the course itself! Manyimportant details will be skipped.
ENGI 5821 Course Summary
Course SummaryUnit 1: Introduction
Unit 2: Modeling in the Frequency DomainUnit 3: Time Response
Unit 4: Block Diagram ReductionUnit 5: Stability
Unit 6: Steady-State ErrorUnit 7: Root Locus TechniquesUnit 8: Design via Root Locus
Course Summary
The course cannot be summarized in one lecture. The intent hereis to go through each unit of the course and mention some of themost important or poorly understood concepts.
This brief tour is no replacement for the course itself!
Manyimportant details will be skipped.
ENGI 5821 Course Summary
Course SummaryUnit 1: Introduction
Unit 2: Modeling in the Frequency DomainUnit 3: Time Response
Unit 4: Block Diagram ReductionUnit 5: Stability
Unit 6: Steady-State ErrorUnit 7: Root Locus TechniquesUnit 8: Design via Root Locus
Course Summary
The course cannot be summarized in one lecture. The intent hereis to go through each unit of the course and mention some of themost important or poorly understood concepts.
This brief tour is no replacement for the course itself! Manyimportant details will be skipped.
ENGI 5821 Course Summary
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for systemanalysis and design
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 measuresschematictransfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measures
schematictransfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematic
transfer function for componentstransfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for components
transfer function for systemanalysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for system
analysis and design
Unit 1: Introduction
Introduction
System Configurations
Open-loop vs. Closed-loop
Measuring Performance
Transient response, steady-state error, stability
The Design Process
specificationsrequired performance measuresschematictransfer function for componentstransfer function for systemanalysis and design
Unit 2: Modeling in the Frequency Domain
Complex Frequency: x(t) = <{Xest}We can express constants, exponentials, sinusoids, andexponentially decaying or growing sinusoids by varying s. Theactual amplitude and phase of a particular signal is expressedin X .
Laplace Transform
Transforms any signal into a combination of complexexponentialsTables of transform pairs and theoremsUse partial-fraction expansion to decompose ratio ofpolynomials
Unit 2: Modeling in the Frequency Domain
Complex Frequency: x(t) = <{Xest}We can express constants, exponentials, sinusoids, andexponentially decaying or growing sinusoids by varying s. Theactual amplitude and phase of a particular signal is expressedin X .
Laplace Transform
Transforms any signal into a combination of complexexponentialsTables of transform pairs and theoremsUse partial-fraction expansion to decompose ratio ofpolynomials
Unit 2: Modeling in the Frequency Domain
Complex Frequency: x(t) = <{Xest}We can express constants, exponentials, sinusoids, andexponentially decaying or growing sinusoids by varying s. Theactual amplitude and phase of a particular signal is expressedin X .
Laplace Transform
Transforms any signal into a combination of complexexponentialsTables of transform pairs and theoremsUse partial-fraction expansion to decompose ratio ofpolynomials
Unit 2: Modeling in the Frequency Domain
Complex Frequency: x(t) = <{Xest}We can express constants, exponentials, sinusoids, andexponentially decaying or growing sinusoids by varying s. Theactual amplitude and phase of a particular signal is expressedin X .
Laplace Transform
Transforms any signal into a combination of complexexponentials
Tables of transform pairs and theoremsUse partial-fraction expansion to decompose ratio ofpolynomials
Unit 2: Modeling in the Frequency Domain
Complex Frequency: x(t) = <{Xest}We can express constants, exponentials, sinusoids, andexponentially decaying or growing sinusoids by varying s. Theactual amplitude and phase of a particular signal is expressedin X .
Laplace Transform
Transforms any signal into a combination of complexexponentialsTables of transform pairs and theorems
Use partial-fraction expansion to decompose ratio ofpolynomials
Unit 2: Modeling in the Frequency Domain
Complex Frequency: x(t) = <{Xest}We can express constants, exponentials, sinusoids, andexponentially decaying or growing sinusoids by varying s. Theactual amplitude and phase of a particular signal is expressedin X .
Laplace Transform
Transforms any signal into a combination of complexexponentialsTables of transform pairs and theoremsUse partial-fraction expansion to decompose ratio ofpolynomials
Transfer Functions
Can be developed for LTI systems with zero-initial conditionsTransfer function G (s) gives the system output C (s) for anyinput R(s): C (s) = R(s)G (s)
Electrical Systems
Slow method: Develop DE’s, apply Laplace, form TFFast method: Define transfer functions for individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Transfer Functions
Can be developed for LTI systems with zero-initial conditions
Transfer function G (s) gives the system output C (s) for anyinput R(s): C (s) = R(s)G (s)
Electrical Systems
Slow method: Develop DE’s, apply Laplace, form TFFast method: Define transfer functions for individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Transfer Functions
Can be developed for LTI systems with zero-initial conditionsTransfer function G (s) gives the system output C (s) for anyinput R(s): C (s) = R(s)G (s)
Electrical Systems
Slow method: Develop DE’s, apply Laplace, form TFFast method: Define transfer functions for individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Transfer Functions
Can be developed for LTI systems with zero-initial conditionsTransfer function G (s) gives the system output C (s) for anyinput R(s): C (s) = R(s)G (s)
Electrical Systems
Slow method: Develop DE’s, apply Laplace, form TFFast method: Define transfer functions for individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Transfer Functions
Can be developed for LTI systems with zero-initial conditionsTransfer function G (s) gives the system output C (s) for anyinput 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 individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Transfer Functions
Can be developed for LTI systems with zero-initial conditionsTransfer function G (s) gives the system output C (s) for anyinput R(s): C (s) = R(s)G (s)
Electrical Systems
Slow method: Develop DE’s, apply Laplace, form TFFast method: Define transfer functions for individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Transfer Functions
Can be developed for LTI systems with zero-initial conditionsTransfer function G (s) gives the system output C (s) for anyinput R(s): C (s) = R(s)G (s)
Electrical Systems
Slow method: Develop DE’s, apply Laplace, form TFFast method: Define transfer functions for individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Transfer Functions
Can be developed for LTI systems with zero-initial conditionsTransfer function G (s) gives the system output C (s) for anyinput R(s): C (s) = R(s)G (s)
Electrical Systems
Slow method: Develop DE’s, apply Laplace, form TFFast method: Define transfer functions for individualcomponents and state the problem in the freq. domain
Circuit analysis techniques developed for resistive circuitsautomatically work for L’s and C’s in the freq. domain!
Op-Amps: Utilize ideal op-amp assumptions
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms
∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)
Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Mechanical Systems
Mechanical components in translational and rotational forms∑opposing forces =
∑applied forces
One equation of motion for each linearly independent motionIn 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 shaftsMechanical impedances can be reflected between shafts tosimplify the calculation of TF
Motors: Electromechanical systems
Derived θm(s)/Ea(s) = K/s(s + α)Jump back to the time-domain to run tests and evaluatemotor parameters
Linearization
Final exam will cover only the concept and application to asimple DE
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole positionAdditional poles or zeros
Real-axis poles or zeros far to the left have little effectA pole can cancel a nearby zero
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole positionAdditional poles or zeros
Real-axis poles or zeros far to the left have little effectA pole can cancel a nearby zero
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole positionAdditional poles or zeros
Real-axis poles or zeros far to the left have little effectA pole can cancel a nearby zero
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole positionAdditional poles or zeros
Real-axis poles or zeros far to the left have little effectA pole can cancel a nearby zero
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole position
Additional poles or zeros
Real-axis poles or zeros far to the left have little effectA pole can cancel a nearby zero
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole positionAdditional poles or zeros
Real-axis poles or zeros far to the left have little effectA pole can cancel a nearby zero
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole positionAdditional poles or zeros
Real-axis poles or zeros far to the left have little effect
A pole can cancel a nearby zero
Unit 3: Time Response
The poles give the form of the response, the zeros give theweights
First-order systems yield exponential responses characterizedby time constant
Second-order systems yield four different responsescharacterized by ζ and ωn
Response specifications: Tp, %OS , and Ts
Relationship between response specs. and pole positionAdditional poles or zeros
Real-axis poles or zeros far to the left have little effectA pole can cancel a nearby zero
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 theleft or right of summing junctions and pickoff points
Moving blocks = algebraic manipulation
Signal-flow graphs
nodes are signals; edges are systems
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 theleft or right of summing junctions and pickoff points
Moving blocks = algebraic manipulation
Signal-flow graphs
nodes are signals; edges are systems
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 theleft or right of summing junctions and pickoff points
Moving blocks = algebraic manipulation
Signal-flow graphs
nodes are signals; edges are systems
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 theleft or right of summing junctions and pickoff points
Moving blocks = algebraic manipulation
Signal-flow graphs
nodes are signals; edges are systems
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 theleft or right of summing junctions and pickoff points
Moving blocks = algebraic manipulation
Signal-flow graphs
nodes are signals; edges are systems
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles 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 ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
Unit 5: Stability
Definitions: Stability of the total response and the naturalresponse (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 jωpoles for marginal stability)
Characteristic coef’s missing or differing in sign ⇒ unstable(opposite is not true)
Routh-Hurwitz
Special case: zero in first columnSpecial case: ROZ ⇒ EP factor
EP has symmetric roots so system is either unstable ormarginally stable
Problem: find K such that...
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) 11+Kp
0 0
Ramp, tu(t) ∞ 1Kv
0
Para., t2u(t) ∞ ∞ 1Ka
Disturbances: e(∞) = eR(∞) + eD(∞)
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) 11+Kp
0 0
Ramp, tu(t) ∞ 1Kv
0
Para., t2u(t) ∞ ∞ 1Ka
Disturbances: e(∞) = eR(∞) + eD(∞)
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) 11+Kp
0 0
Ramp, tu(t) ∞ 1Kv
0
Para., t2u(t) ∞ ∞ 1Ka
Disturbances: e(∞) = eR(∞) + eD(∞)
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) 11+Kp
0 0
Ramp, tu(t) ∞ 1Kv
0
Para., t2u(t) ∞ ∞ 1Ka
Disturbances: e(∞) = eR(∞) + eD(∞)
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) 11+Kp
0 0
Ramp, tu(t) ∞ 1Kv
0
Para., t2u(t) ∞ ∞ 1Ka
Disturbances: e(∞) = eR(∞) + eD(∞)
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) 11+Kp
0 0
Ramp, tu(t) ∞ 1Kv
0
Para., t2u(t) ∞ ∞ 1Ka
Disturbances: e(∞) = eR(∞) + eD(∞)
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 7: Root Locus Techniques
Vector representation of complex numbers
Root locus: locations of closed-loop system poles as K isvaried
Properties of the RL
If ∠KG (s)H(s) = (2k + 1)180o 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 andsketching rules
Unit 8: Design via Root Locus
First check acceptable operating point on uncompensated RL
PI compensation: increase system type while maintainingtransient resp.
Gc(s) =K (s + zc)
sChoose zc as a small number; Requires active amplification
Lag compensation: increase static error constant (whichreduces e(∞)
Gc(s) =K (s + zc)
s + pc
Choose pc as a small number and adjust zc accordingly
PD compensation: Adjust transient response
Gc(s) = K (s + zc)
Place zc to move RL to intersect desired operating point;Requires active amplification
All of our design techniques rely on 2nd order approx.. Verifyapprox. validity and simulate
Unit 8: Design via Root Locus
First check acceptable operating point on uncompensated RL
PI compensation: increase system type while maintainingtransient resp.
Gc(s) =K (s + zc)
sChoose zc as a small number; Requires active amplification
Lag compensation: increase static error constant (whichreduces e(∞)
Gc(s) =K (s + zc)
s + pc
Choose pc as a small number and adjust zc accordingly
PD compensation: Adjust transient response
Gc(s) = K (s + zc)
Place zc to move RL to intersect desired operating point;Requires active amplification
All of our design techniques rely on 2nd order approx.. Verifyapprox. validity and simulate
Unit 8: Design via Root Locus
First check acceptable operating point on uncompensated RL
PI compensation: increase system type while maintainingtransient resp.
Gc(s) =K (s + zc)
sChoose zc as a small number; Requires active amplification
Lag compensation: increase static error constant (whichreduces e(∞)
Gc(s) =K (s + zc)
s + pc
Choose pc as a small number and adjust zc accordingly
PD compensation: Adjust transient response
Gc(s) = K (s + zc)
Place zc to move RL to intersect desired operating point;Requires active amplification
All of our design techniques rely on 2nd order approx.. Verifyapprox. validity and simulate
Unit 8: Design via Root Locus
First check acceptable operating point on uncompensated RL
PI compensation: increase system type while maintainingtransient resp.
Gc(s) =K (s + zc)
sChoose zc as a small number; Requires active amplification
Lag compensation: increase static error constant (whichreduces e(∞)
Gc(s) =K (s + zc)
s + pc
Choose pc as a small number and adjust zc accordingly
PD compensation: Adjust transient response
Gc(s) = K (s + zc)
Place zc to move RL to intersect desired operating point;Requires active amplification
All of our design techniques rely on 2nd order approx.. Verifyapprox. validity and simulate
Unit 8: Design via Root Locus
First check acceptable operating point on uncompensated RL
PI compensation: increase system type while maintainingtransient resp.
Gc(s) =K (s + zc)
sChoose zc as a small number; Requires active amplification
Lag compensation: increase static error constant (whichreduces e(∞)
Gc(s) =K (s + zc)
s + pc
Choose pc as a small number and adjust zc accordingly
PD compensation: Adjust transient response
Gc(s) = K (s + zc)
Place zc to move RL to intersect desired operating point;Requires active amplification
All of our design techniques rely on 2nd order approx.. Verifyapprox. validity and simulate
PID: Design for transient response, then e(∞)
Analog PID implemented via op-ampDigital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shapedMethod 2: System appears to involve integration and/orunderdamped poles
Computational search
PID: Design for transient response, then e(∞)
Analog PID implemented via op-amp
Digital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shapedMethod 2: System appears to involve integration and/orunderdamped poles
Computational search
PID: Design for transient response, then e(∞)
Analog PID implemented via op-ampDigital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shapedMethod 2: System appears to involve integration and/orunderdamped poles
Computational search
PID: Design for transient response, then e(∞)
Analog PID implemented via op-ampDigital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shapedMethod 2: System appears to involve integration and/orunderdamped poles
Computational search
PID: Design for transient response, then e(∞)
Analog PID implemented via op-ampDigital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shapedMethod 2: System appears to involve integration and/orunderdamped poles
Computational search
PID: Design for transient response, then e(∞)
Analog PID implemented via op-ampDigital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shaped
Method 2: System appears to involve integration and/orunderdamped poles
Computational search
PID: Design for transient response, then e(∞)
Analog PID implemented via op-ampDigital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shapedMethod 2: System appears to involve integration and/orunderdamped poles
Computational search
PID: Design for transient response, then e(∞)
Analog PID implemented via op-ampDigital PID can be implemented in software (or digitalhardware)
PID tuning: Strategies to apply when system model isunknown
Ziegler-Nichols (rules of thumb)
Method 1: Unit-step response is S-shapedMethod 2: System appears to involve integration and/orunderdamped poles
Computational search
