Feedback Control System Design 2.017 Fall 2009AnnouncementsControl SystemsTypes of Control SystemsControl System ComparisonOverview of Closed Loop Control SystemsControl System RepresentationsLaplace TransformLaplace vs. Fourier TransformSystem Modeling (1st Order System)System Modeling (2nd Order System)2nd Order System PolesSystem Identification (Time Domain)System Identification (Frequency Domain)Closed-Loop Transfer FunctionPID Controller Transfer FunctionDisturbance Rejection (Active Vibration Cancellation)Control ActionsRoot LocusMatlab SISO Design ToolState-Space RepresentationCharacteristic PolynomialControllabilityObservabilityStabilizability and DetectabilityExampleFull-State FeedbackWhere to Place The Poles?Butterworth Pole ConfigurationsState-Space Design SummaryExampleExample Matlab CodeFrequency Design MethodsFrequency Response (Gain and Phase)Frequency Response (Bode Plot )Feedback Control System Design 2.017 Fall 2009Dr. Harrison Chin10/29/2009Announcements• Milestone Presentations on Nov 5 in class– This is 15% of your total grade: 5% group grade 10% individual grade– Email your team’s PowerPoint file to Franz and Harrison by 10 am on Nov 5– Each team gets 30 minutes of presentation + 10 minutes of Q&A– Select or design your own presentation template and styleControl Systems• An integral part of any industrial society• Many applications including transportation, automation, manufacturing, home appliances,…• Helped exploration of the oceans and space• Examples:– Temperature control– Flight control– Process control–…Types of Control SystemsPlantyrSensorsActuatorsControllerudcontrol inputdisturbancecommandoutputOpen loop systemPlant+_yreSensorsActuatorsControlleruderrorcontrol inputdisturbancecommandoutputClosed loop systemControl System Comparison • Open loop:– The output variables do not affect the input variables– The system will follow the desired reference commands if no unpredictable effects occur– It can compensate for disturbances that are taken into account– It does not change the system stability• Closed loop:– The output variables do affect the input variables in order to maintain a desired system behavior– Requires measurement (controlled variables or other variables)– Requires control errors computed as the difference between the controlled variable and the reference command– Computes control inputs based on the control errors such that the control error is minimized– Able to reject the effect of disturbances– Can make the system unstable, where the controlled variables grow without boundOverview of Closed Loop Control SystemsDisturbancesComputer /MicrocontrollerPlantInputsOutputsSensorsActuatorsADCDACControl AlgorithmScopeFunctionGeneratorModelPower AmpReference Inputs(Setpoints)Control System Representations• Transfer functions (Laplace)• State-space equations (System matrices)• Block diagrams⎩⎨⎧+=+=)()()()()()(tutxtytutxtxDCBA&+_)(sY)(sR)(sE)(sG)(sD)(sGc)(sH)(sGd++Laplace Transform• Convert functions of time into functions that are algebraic in the complex variables.• Replaces differentiation & integral operations by algebraic operations all involving the complex variable.• Allows the use of graphical methods to predict system performance without solving the differential equations of the system. These include response, steady state behavior, and transient behavior.• Mainly used in control system analysis and design.Laplace vs. Fourier Transform• Laplace transform:• Fourier transform• Laplace transforms often depend on the initial value of the function• Fourier transforms are independent of the initial value. • The transforms are only the same if the function is the same both sides of the y-axis (so the unit step function is different). ∫∞−=0)()( dtetfsFst)()( ssFtf ⇒′∫∞∞−−= dtetfFtjωω)()(System Modeling (1stOrder System)Transfer function:Differential equation:Laplace transform)()()( sFsbVssmV=+)()()( tftbvtvm=+&()()1//11)()(+=+=sbmbbmssFsV0)()(=+ tbvtvm&tmbetvtv⎟⎠⎞⎜⎝⎛−= )()(0()τ1/1−=−=−=mbbms84.151=×−e5.1≈τTime constantSystem Modeling (2ndOrder System))()()()( tftkxtxbtxm=++&&&bkmvx,)()()( tftbvtvm=+&⎟⎠⎞⎜⎝⎛+=−−−−tmkmbtmkmbtmbBeAeex)/()2/()/()2/()2/(22⎪⎪⎩⎪⎪⎨⎧==⇒++⇔++kmbmkkbsmsssnnn22222ζωωζω2ndOrder System Poles100%41212×==−=⎟⎟⎠⎞⎜⎜⎝⎛−−ζζπζωζωπeOSTTnsnp0222=++nnssωζω22,11ζωζωωσ−±−=±−=nnddjjsSystem Identification (Time Domain)Step Input, Open LoopτSystem Identification (Frequency Domain)10110210310-2100102Frequency (Hz)GainZ-Axis 101102103-300-200-1000Frequency (Hz)Phase Lag (deg)MeasuredModelFitCenter Stage ModesClosed-Loop Transfer Function• The gain of a single-loop feedback system is given by the forward gain divided by 1 plus the loop gain.)()()(1)()()(sHsGsGsGsGsGcccl+=+_)(sE)()( sGsGc)(sH)(sR)(sYPID Controller Transfer Function⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛+=++=++=ssKKsKKKssKsKKsKsKKsGidipidpidipc221)(Disturbance Rejection (Active Vibration Cancellation)ControllerdisturbanceP, PI, PD, PID,…PlantControl Actions• Proportional – improves speed but with steady-state error• Integral – improves steady state error but with less stability, overshoot,longer transient, integrator windup• Derivative – improves stability but sensitive to noiseRoot Locus• Can we increase system damping with a simple proportional control ?Dominant Poles2σ1σSISO Design Tool•command: ‘sisotool’ or ‘rltool’PID Controller Transfer FunctionCourtesy of The MathWorks, Inc. Used with permission.MATLAB and Simulink are registered trademarks of The MathWorks, Inc.See www.mathworks.com/trademarks for a list of additional trademarks.Other product or brand names may be trademarks or registered trademarks of their respective holders.MATLABMATLABState-Space Representation⎩⎨⎧+=+=)()()()()()(tutxtytutxtxDCBA&⎩⎨⎧+=+=⇒)()()()()()(sUsXsYsUsXssXDCBA)()()()()()(1sUssXsUsXsBAIBAI−−=⇒=−DBAICDBAIC+−==⇒+−=−−11)()()()()()()()(ssUsYsGsUsUssYCharacteristic PolynomialResolvent)det()det()det()det()(adj)()(0 1AIAIBCAIBAIAICBAICD−−−+−=⎥⎦⎤⎢⎣⎡−−=−=⇒=−sssssssGifMATLAB ss2tf command uses this formula to compute the transfer function(s)Characteristic polynomialControllabilityx2x(tf)x(t0)x1Courtesy of Kamal Youcef-Toumi. Used with