Create a Spark Motivational Contest

Professor Walter W. OlsonDepartment of Mechanical, Industrial and Manufacturing EngineeringUniversity of ToledoSensitivity

1Outline of Todays LectureReviewStatic Error Constants (Review)Loop ShapingLoop Shaping with the Bode PlotLead and Lag CompensatorsLead design with Bode plotLead design with root locusLag design with Bode plotImportant transfer functionsGang of SixGang of FourDisturbance RejectionNoise RejectionLimitations

Static Error ConstantsIf the system is of type 0 at low frequencies will be level. A type 0 system, (that is, a system without a pole at the origin,)will have a static position error, Kp, equal to

If the system is of type 1 (a single pole at the origin) it will have a slope of -20 dB/dec at low frequenciesA type 1 system will have a static velocity error, Kv, equal to the value of the -20 dB/dec line where it crosses 1 radian per secondIf the system is of type 2 ( a double pole at the origin) it willhave a slope of -40 dB/dec at low frequenciesA type 2 system has a static acceleration error,Ka, equal to the value of the -40 dB/dec line where it crosses 1 radian per second

Loop ShapingWe have seen that the open loop transfer function, has profound influences on the closed loop response

The key concept in loop shaping designs is that there is some ideal open loop transfer (B(s)) that will provide the design specifications that we require of our closed loop system

Loop shaping is a trial and error process

To perform loop shaping we can used either the root locus plots or the Bode plots depending on the type of response that we wish to achieve

We have already considered an important form of loop shaping as the PID controller

ErrorsignalE(s)++Outputy(s)Open LoopSignalB(s)PlantP(s)ControllerC(s)Inputr(s)Sensor-1

Loop Shaping with the Bode PlotThe open loop Bode plot is the natural design tool when designing in the frequency domain. For the frequency domain, the common specifications are bandwidth, gain cross over frequency, gain margin, resonant frequency, resonant frequency gain, phase margin, static errors and high frequency roll off.

Bandwidth rps-3 dbGain cross overfrequency rpsRoll off Rate dB/decResonant peak frequency rpsResonant peak gain, dBLoop Shaping with the Bode Plot

Increase of gainalso increasesbandwidth and resonant gainBreak frequencycorresponds to thecomponent pole or zeroPoles bend the magnitude and phase down

Zeros bend the magnitude and the phase upLead and Lag CompensatorsThe compensator with a transfer function

is called a lead compensator if aaThe lead and the lag compensator can be used togetherNote: the compensator does add a steady state gain ofthat needs to be accounted for in the final designThere are analytical methods for designing these compensators (See Ogata or Franklin and Powell)Lead CompensatorThe lead compensator is used to improve stability and to improve transient characteristics.The lead compensator can be designed using either frequency response or root locus methodsUsually, the transient characteristics are better addressed using the root locus methodsAddressing excessive phase lag is better addressed using the frequency methodsThe pole of the system is usually limited by physical limitations of the components use to implement the compensatorIn the lead compensator, the zero and pole are usually separated in frequency from about .4 decades to 1.5 decades depending on the designLead Compensator (Frequency Design)Note:the lead compensator opensup the high frequency regionwhich could cause noise problemsThe Lead compensator addsphase fwmxix0yb1b2kMechanicalLeadCompensatorLag CompensatorLag compensators are used to improve steady state characteristics where the transient characteristics are adequate and to attenuate high frequency noise In order to not change the transient characteristics, the zero and pole are located near the origin on the root locus plotThe starting point for the design on a root locus is to start with a pole location at about s = -0.001 and then locate the pole as needed for the desired effectIn order to not give up too much phase, the zero and pole are located away from the phase margin frequencyLag CompensatorabNote that the lag compensator causesa drop in the magnitude and phaseThis could be useful in reducing bandwidth, and improving gain margin; however it might reduce phase marginxix0b1b2kMechanicalLagCompensatorSensitivitySensitivity is an evaluation of how the system responds to various signals compared to the design signalIn general, we want the system to respond to the reference inputWe do not want the system to respond to noises and other signals that do not contribute to the accuracy of the desired outputSeveral Transfer Functions++-1++++RYEUDNuhYControllerProcessDisturbancesMeasurementNoiseThe Model++-1++++RYEUDNuhYControllerProcessGang of SixSensitivityFunctionLoadSensitivityFunctionComplementarySensitivityFunctionNoiseSensitivityFunctionGang of FourExample++-1++++RYEUDNuhYControllerProcessDisturbance RejectionWe want our system designed such that the disturbances to the system are attenuatedHarold S. Black gave us the answer: negative feedback++-1++++RYEUDNuhYControllerProcessDisturbance RejectionDisturbances witha frequency lessthan the cross overfrequency are attenuated whilefrequencies higherwould be passedFortunately, most loaddisturbances are lowfrequency and oftencan be treated as stepinputsNoise RejectionWe would also like noise rejectionNoise is most often high frequency signals caused by the sensors used to measureNoise is presented as a result of the feedback termsWe do not have noise as defined here in an open systemIn the closed loop error, noise is multiplied by T, the complementary sensitivity function,In a system without a pre-filter, this is the transfer functionFor this reason high frequency roll-off is importantLimitationsSystems with right hand side poles and zeros are inherently hard to controlFor a system with right hand side poles, pk, Bode showed thatImprovements in one frequency region are met with deteriorations in another frequency regionSometimes called the waterbed effect SummaryImportant transfer functionsGang of SixGang of FourDisturbance RejectionNoise RejectionLimitationsNext: Robustness