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Fall 2008 1
ME451: Control SystemsME451: Control Systems
Dr. JongeunDr.Jongeun ChoiChoi Department of Mechanical EngineeringDepartment of Mechanical Engineering Michigan State UniversityMichigan State University
Lecture 13 Lecture 13 Steady-Steady-state errorstate error
Course roadmapCourse roadmap
Laplace transformLaplace transform
Transfer functionTransfer function
Models for systemsModels for systems
- • electricalelectrical
- • mechanicalmechanical
- • electromechanicalelectromechanical Block diagramsBlock diagrams LinearizationLinearization
ModelingModeling^ AnalysisAnalysis^ DesignDesign Time responseTime response
- • TransientTransient
- • Steady stateSteady state Frequency responseFrequency response
- • Bode plotBode plot StabilityStability
- • RouthRouth--HurwitzHurwitz
- • NyquistNyquist
Design specsDesign specs Root locusRoot locus Frequency domainFrequency domain PID & LeadPID & Lead--laglag Design examplesDesign examples
(Matlab (Matlab simulations &) laboratoriessimulations &) laboratories
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Performance measures (review)Performance measures (review)
Transient responseTransient response Peak valuePeak value Peak timePeak time Percent overshootPercent overshoot Delay timeDelay time Rise timeRise time Settling timeSettling time Steady state responseSteady state response Steady state errorSteady state error
Next, we will connect Next, we will connect these measuresthese measures with s-with s-domain.domain.
(Today’ (Today’s lecture)s lecture)
(From next lecture)(From next lecture)
Steady- Steady-state error:state error: unity feedbackunity feedback
Suppose that we want output y(tSuppose that we want outputy(t) to track) to track r(tr(t).). ErrorError Steady-Steady-state errorstate error
Final value theoremFinal value theorem (Suppose CL system is stable!!!)(Suppose CL system is stable!!!)
Unity feedback! Unity feedback!
We assume that theWe assume that the CL system is stable!CL system is stable!
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Steady- Steady-state error for rampstate error for ramp r(tr(t))
KvKv
Steady- Steady-state error for parabolicstate error for parabolic r(tr(t))
KaKa
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System typeSystem type
System type of GSystem type of G (^) is defined as the orderis defined as the order (number) of poles of G(s(number) of poles ofG(s) at s=0.) at s=0.
ExamplesExamples
type 1 type 1
type 2type 2
type 3type 3
Zero steady-Zero steady-state errorstate error
If error constant is infinite, we can achieve zeroIf error constant is infinite, we can achieve zero steady-steady-state error. (Accurate tracking)state error. (Accurate tracking) For step r(tFor stepr(t))
For ramp r(tFor rampr(t))
For parabolic r(tFor parabolicr(t))
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Example 3Example 3
G(s) of type 2G(s) of type 2
By RouthByRouth--Hurwitz criterion, we can show that CLHurwitz criterion, we can show that CL system is stable.system is stable.
Step r(tStepr(t))
Ramp r(tRampr(t))
Parabolic r(tParabolicr(t))
G(s) G(s)
A control exampleA control example
Closed-Closed-loop stable?loop stable?
Compute error constantsCompute error constants
Compute steady state errorsCompute steady state errors
Summary and ExercisesSummary and Exercises
Steady-Steady-state errorstate error For unity feedbackForunity feedback (STABLE!) systems, the system(STABLE!) systems, the system type of the forward-type of the forward-path system determines if thepath system determines if the steady-steady-state error is zero.state error is zero. The key tool is the final value theoremThe key tool is thefinal value theorem!!
Next, time response of 1st-Next, time response of 1st-order systemsorder systems
ExercisesExercises Read Section 5.5.Read Section 5.5. Solve Problems 5.9 and 5.14.Solve Problems 5.9 and 5.14.
