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

1Outline of Todays LectureReviewPID TheoryIntegrator WindupNoise ImprovementStatic Error Constants (Review)Loop ShapingLoop Shaping with the Bode PlotLead and Lag CompensatorsLead design with Bode plotLead design with root locusLag design with Bode plot

PID: A Little TheoryConsider a 1st order function where the 1st method of Ziegler Nichols appliesThe general transfer function for this system is

The term is the transport lag and delays the action for t0 seconds. ThereforeThe term Ta is the time constant for the system. T measured on the graph is an estimate of this.

PID: A Little TheoryThe method 1 PI controller applied to the loop equation is

PID: A Little TheoryIn Method 2, the gain was increased until the system was nearly a perfect oscillatory system.Since the gain changes the oscillatory patterns, the lowest order system that this could represent would by a 3rd order system.

For this system to oscillate, there must be a solution of the characteristic function for K real and positive where s=wi

PID: A Little TheoryApplying the PI Controller:

Integrator WindupWe have tacitly assumed that the controlled devices could meet the demands of the controls that we designed. However real devices have limitations that may prevent the system from responding adequately to the control signalWhen this occurs with an integrating controller, the error which is used to amplify the control signal may build up and saturate the controller.We refer to this as integrator windup:the system cant respond and the integrator signal is extremely large (often maxed out on a real controller)the result is an uncontrolled system that can not return to normal operating conditions until the controller is resetIntegrator WindupTo avoid windup, a possible solution is to provide a correcting error from the actuator by adding another loop:(the actuator has to be extracted from the plant)

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++Derivative Noise ImprovementA major problem with using the derivative part of the PID controller that the derivative has the effect of amplifying the high frequency components which, for most systems, is likely to be noise.

Without PIDWith PIDDerivative Noise ImprovementOne way to improve the noise rejection at higher frequencies is to apply a second order filter that passes low frequency and rejects high frequencyThe natural frequency of the filter should be chosen as

with N chosen to give the controller the bandwidth necessary, usually in the range of 2 to 20The controller then has the design

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

Static Error Constants

Kv(dB)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:Everything is connected and nothing is independentWhat we gain in one area may (usually?) causes loss in other areasOften times, our best controller is a compromise between demands

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

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)xix0yb1b2kMechanicalLeadCompensatorLead 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 0.4 decades to 2 decades depending on the needsLead Compensation (Root Locus)Objective: place the dominant poles near a given position to improve transient characteristics and possibly stabilityPerformance criteria for the design:Rise timeSettling timeOvershoot limitsDamping RatioNatural FrequencyDamped FrequencyDesign strategy: place a negative real valued zero to shape the curve into the desired position and refine the position by moving the a negative real valued poleExampleAn aircraft has a pitch rate control as shown. The response of the pitch control is under damped, sluggish has an objectionable transient vibration mode. Design a lead compensator that provides a damping ratio of between 0.45 to 0.50 and 5% settling time of 150 seconds which reduces the vibration mode as much as possible.C(s)1+-RYExampleThe root locus from the complex poles has very little damping and causes the vibration seen in the response. There is a pole at -0.01 on the real axis that is dominant and causes the sluggish behavior.Strategy: Use a lead compensator to bend the curves to the left and into the 0.6 damping region. The zero of the compensator should counteract the vibrational modeExampleThe initial design with the pole at -5 and the zero at -1 had the desired effectof bending the root locus to the left and removing most of the vibration. Howeverthe pole is still too close to the origin such that 0.6 damping can not be achieved. ExampleWe achieved the specifications once the pole of the compensator was moved out to -9 and we adjusted the gain for the 0.6 damping.Lead Compensator (Frequency Design)Note:the lead compensator opensup the high frequency regionwhich could cause noise problemsThe Lead compensator addsphase fwmxix0yb1b2kMechanicalLeadCompensatorExampleAn aircraft has a pitch rate control as shown. Design a lead compensator for this system for a static velocity error of 4/sec, and a phase margin of 40 degrees.C(s)-1++Aircraft Pitch Rate DynamicsCompensatorRYExample-33dBCurrent System:Reading the plot:When design a lead compensator first adjustthe gain to meet the staticerror conditionIn this case the gain needsthe be increase by 180 or20Log10180= 45.1 dB addedExampleGain Adjusted SystemThen noting where the phase currently is, that is the desired location for the peak of the lead phase= spec Pm=40-0.153+ a small safety = 55 degFinally adjust them asnecessaryExampleInitial design: Phase good but Kv notGain needs to be increased by about 20 dBFinal design:Lag 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 marginxix0b1b2kMechanicalLagCompensatorExampleA linear motor has an open loop rate transfer function ofIt is desired that the system have a static velocity error constant greater than 20/sec, a phase margin of 45 degrees plus or minus 5 degrees and gain crossover frequency of 1 radian/sec. Design a lag compensator for this system.C(s)-1++Linear MotorRate DynamicsCompensatorRYExampleCurrent System:Phase margin islow and the static velocity error constantmust be improved.Start by correcting thestatic velocity error constant4.9dBExampleGain Adjusted:Gain and PhaseMargin problemstry placing a poleat -0.01 rps and adjust the zeroNeed to move PM to hereNeed to shapethe curve like thisExampleFinal DesignSummaryStatic Error Constants (Review)Loop ShapingLoop Shaping with the Bode PlotLead and Lag CompensatorsLead design with Bode plotLead design with root locusLag design with Bode plotNext: Sensitivity Analysis