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1、Dynamic Experiments,Maximizing the Information Content for Control Applications,CHEE825/435 - Fall 2005,1,J. McLellan,Dynamic ExperimentsMaximizing,Outline,types of input signalscharacteristics of input signalspseudo-random binary sequence (PRBS) inputsother input signalsinputs for multivariable ide
2、ntificationinput signals for closed-loop identification,CHEE825/435 - Fall 2005,2,J. McLellan,Outlinetypes of input signalsC,Types of Input Signals,deterministic signalsstepspulsessinusoidsstochastic signalswhite noisecorrelated noisewhat are the important characteristics?,CHEE825/435 - Fall 2005,3,
3、J. McLellan,Types of Input Signalsdetermin,Outline,types of input signalscharacteristics of input signalspseudo-random binary sequence (PRBS) inputsother input signalsinputs for multivariable identificationinput signals for closed-loop identification,CHEE825/435 - Fall 2005,4,J. McLellan,Outlinetype
4、s of input signalsC,Important Characteristics,signal-to-noise ratiodurationfrequency contentoptimum input (deterministic / random) depends on intended end-usecontrolprediction,CHEE825/435 - Fall 2005,5,J. McLellan,Important Characteristicssigna,Signal-to-Noise Ratio,improves precision of modelparame
5、terspredictionsavoid modeling noise vs. processtrade-off short-term pain vs. long-term gainprocess disruption vs.expensive retesting / poor controller performancenote - excessively large inputs can take process into region of nonlinear behaviour,CHEE825/435 - Fall 2005,6,J. McLellan,Signal-to-Noise
6、Ratioimproves,Example - Estimating 1st Order Process Model with RBS Input,True model,y,t,q,q,u,t,a,t,(,),.,.,(,),(,),+,=,-,+,-,-,1,0,6,1,0,75,1,1,0,5,10,15,20,25,30,35,40,0,0.5,1,1.5,2,2.5,3,3.5,4,Time,Step Response,confidenceintervals aretighter with increasing SNR,1:1,10:1,less preciseestimate ofs
7、teady stategain,more preciseestimateof transient,CHEE825/435 - Fall 2005,7,J. McLellan,Example - Estimating 1st Order,Example - Estimating First-Order Model with Step Input,0,5,10,15,20,25,30,35,40,-2,-1,0,1,2,3,4,5,6,Time,Step Response,1:1,10:1,more preciseestimate ofgain vs.RBS input,less precise
8、estimateof transient,response,99% confidenceinterval,CHEE825/435 - Fall 2005,8,J. McLellan,Example - Estimating First-Ord,Test Duration,how much data should we collect?want to capture complete process dynamic responseduration should be at least as long as the settling time for the process (time to 9
9、5% of step change)failure to allow sufficient time can lead to misleading estimates of process gain, poor precision,CHEE825/435 - Fall 2005,9,J. McLellan,Test Durationhow much data sho,Test Duration,Precision of a dynamic model improves as number of data points increasesadditional information for es
10、timation,0,5,10,15,20,25,30,35,40,-1,-0.5,0,0.5,1,1.5,2,2.5,3,3.5,4,Time,Step Response,as test duration increases,bias decreasesand precision increases,response,99% confidenceinterval,10 time steps,30 time steps,50 time steps,CHEE825/435 - Fall 2005,10,J. McLellan,Test DurationPrecision of a dy,“Dyn
11、amic Content”,what types of transients should be present in input signal?excite process over range of interestmodel is to be used in controller for:setpoint trackingdisturbance rejectionneed orderly way to assess dynamic contenthigh frequency components - fast dynamicslow frequency components - slow
12、 dynamics / steady-state gain,CHEE825/435 - Fall 2005,11,J. McLellan,“Dynamic Content”what types of,Frequency Content - Guiding Principle,The input signal should have a frequency content matching that for end-use.,CHEE825/435 - Fall 2005,12,J. McLellan,Frequency Content - Guiding Pr,Looking at Frequ
13、ency Content,ideal - match dynamic behaviour of true process as closely as possiblegoal - match the frequency behaviour of the true process as closely as possiblepractical goal - match frequency behaviour of the true process as closely as possible, where it is most important,CHEE825/435 - Fall 2005,
14、13,J. McLellan,Looking at Frequency Contentid,Experimental Design Objective,Design input sequence to minimize the following:,design,cost,error,in,predicted,frequency,response,importance,function,=,our designobjectives,difference in predicted vs.true behaviour- function of frequency, andthe input sig
15、nal used,CHEE825/435 - Fall 2005,14,J. McLellan,Experimental Design ObjectiveD,Accounting for Model Error - Interpretation,Optimal solution in terms of frequency content:,spectral density,frequency,error in model vs.true process,spectral density,frequency,importance to ourapplication,low,high,very i
16、mportant,not important,*,J=,CHEE825/435 - Fall 2005,15,J. McLellan,Accounting for Model Error - I,Accounting for Model Error,Consider frequency content matchingGoal - best model for final application is obtained by minimizing J,J,G,e,G,e,C,j,d,j,T,j,T,frequency,range,=,-,-,-,$,(,),(,),(,),w,w,w,w,2,
17、bias in frequencycontent modeling,importanceof matching- weightingfunction,CHEE825/435 - Fall 2005,16,J. McLellan,Accounting for Model Error Con,Example - Importance Function for Model Predictive Control,spectral density,frequency,high frequency disturbance rejectionperformed by base-levelcontroller
18、s- accuracy not importantin this range,require good estimateof steady state gain,slower dynamics,CHEE825/435 - Fall 2005,17,J. McLellan,Example - Importance Function,Desired Input Signal for Model Predictive Control,sequence with frequency content concentrated in low frequency rangePRBS (or random b
19、inary sequence - RBS)step inputwill provide for good estimate of gain, but not of transient dynamics,CHEE825/435 - Fall 2005,18,J. McLellan,Desired Input Signal for Model,Control Applications,For best results, input signal should have frequency content in range of closed-loop process bandwidthrecurs
20、ive requirement!closed-loop bandwidth will depend in part on controller tuning, which we will do with identified model,CHEE825/435 - Fall 2005,19,J. McLellan,Control ApplicationsFor best r,Control Applications,One Approach:Design input frequency content to include:frequency band near bandwidth of op
21、en-loop plant (1/time constant)frequency band near desired closed-loop bandwidthlower frequencies to obtain good estimate of steady state gain,CHEE825/435 - Fall 2005,20,J. McLellan,Control ApplicationsOne Approa,Frequency Content of Some Standard Test Inputs,frequency,power,low frequency - like a s
22、eries of long steps,high frequency - like a series of short steps,CHEE825/435 - Fall 2005,21,J. McLellan,Frequency Content of Some Stan,Frequency Content of Some Standard Test Inputs,Step Input,power,frequency,0,power is concentrated at low frequency - provides good information about steady state ga
23、in, more limited infoabout higher frequency behaviour,CHEE825/435 - Fall 2005,22,J. McLellan,Frequency Content of Some Stan,Example - Estimating First-Order Model with Step Input,0,5,10,15,20,25,30,35,40,-2,-1,0,1,2,3,4,5,6,Time,Step Response,1:1,10:1,more preciseestimate ofgain vs.RBS input,less pr
24、ecise estimateof transient,response,99% confidenceinterval,CHEE825/435 - Fall 2005,23,J. McLellan,Example - Estimating First-Ord,Frequency Content of Some Standard Test Inputs,White Noise approximated by pseudo-random or random binary sequences,power,frequency,power is distributed uniformlyover all
25、frequencies- broader information, but poorerinformation about steady state gain,ideal curve,CHEE825/435 - Fall 2005,24,J. McLellan,Frequency Content of Some Stan,Example - Estimating 1st Order Process Model with RBS Input,0,5,10,15,20,25,30,35,40,0,0.5,1,1.5,2,2.5,3,3.5,4,Time,Step Response,less pre
26、ciseestimate ofsteady stategain,more preciseestimateof transient,1:1,10:1,response,99% confidenceinterval,CHEE825/435 - Fall 2005,25,J. McLellan,Example - Estimating 1st Order,Frequency Content of Some Standard Test Inputs,Sinusoid at one frequency,power,frequency,power concentrated at onefrequency
27、correspondingto input signal- poor information aboutsteady state gain, otherfrequencies,CHEE825/435 - Fall 2005,26,J. McLellan,Frequency Content of Some Stan,Frequency Content of Some Standard Test Inputs,Correlated noiseconsider,u,q,u,corr,white,=,-,-,0,1,1,0,9,1,.,.,power,frequency,variability is
28、concentrated at lowerfrequencies- will lead to improved estimate ofsteady state gain, poorer estimate ofhigher frequency behaviour,CHEE825/435 - Fall 2005,27,J. McLellan,Frequency Content of Some Stan,Persistent Excitation,In order to obtain a consistent estimate of the process model, the input shou
29、ld excite all modes of the processrefers to the ability to uniquely identify all parts of the process model,CHEE825/435 - Fall 2005,28,J. McLellan,Persistent ExcitationIn order,Persistent Excitation,Persistent excitation implies a richness in the structure of the inputinput shouldnt be too correlate
30、dExamplesconstant step input highly correlated signal provides unique info about process gainrandom binary sequence low correlation signalprovides unique info about additional model parameters,CHEE825/435 - Fall 2005,29,J. McLellan,Persistent ExcitationPersisten,Persistent Excitation - Detailed Disc
31、ussion,Example - consider an impulse response process representationformulate estimation problem in terms of the covariances of u(t)can we obtain the impulse weights?consider estimation matrix persistently exciting of order n - definitionspectral interpretation,CHEE825/435 - Fall 2005,30,J. McLellan
32、,Persistent Excitation - Detail,Persistence of Excitation,Add in defn in terms of covariance -,CHEE825/435 - Fall 2005,31,J. McLellan,Persistence of ExcitationAdd i,Outline,types of input signalscharacteristics of input signalspseudo-random binary sequence (PRBS) inputsother types of input signalsin
33、puts for multivariable identificationinput signals for closed-loop identification,CHEE825/435 - Fall 2005,32,J. McLellan,Outlinetypes of input signalsC,Pseudo-Random Binary Sequences,(PRBS Testing),CHEE825/435 - Fall 2005,33,J. McLellan,Pseudo-Random Binary Sequences,What is a PRBS?,approximation to
34、 white noise inputwhite noise Gaussian noiseuncorrelatedconstant variancezero meanPRBS is a means of approximating using two levels (high/low),CHEE825/435 - Fall 2005,34,J. McLellan,What is a PRBS?approximation t,PRBS,traditionally generated using a set of shift registerscan be generated using rando
35、m numbersswitch to high/low valuesgeneration by finite representation introduces periodicitytry to get period large relative to data length,CHEE825/435 - Fall 2005,35,J. McLellan,PRBStraditionally generated us,PRBS Signal,Alternates in a random fashion between two values:,0,20,40,60,80,100,-2,-1.5,-
36、1,-0.5,0,0.5,1,1.5,2,prbs input,time step,value,input magnitude,minimumswitchingtime,test duration,CHEE825/435 - Fall 2005,36,J. McLellan,PRBS SignalAlternates in a ran,How well does PRBS approximate white noise?,Compare spectra:,10,-2,10,-1,10,0,10,1,10,2,10,-1,10,0,10,1,frequency,power,spectrum fo
37、r 100 point PRBS signal,theoretical spectrumfor white noise,note concentrationof PRBS signalin lower frequencyrange,1 .minimum switch time,CHEE825/435 - Fall 2005,37,J. McLellan,How well does PRBS approximate,PRBS Design Parameters,Amplitudedetermines signal-to-noise ratioprecision vs. process upset
38、slarge magnitudes may bring in process nonlinearity as more of the operating region is coveredcould result in poor model because ofestimation difficulties - e.g., gains, time constants not constant over rangemodel selection difficulties - lack of clear indication of process structure,CHEE825/435 - F
39、all 2005,38,J. McLellan,PRBS Design ParametersAmplitud,PRBS Design Parameters,Minimum switch timeshortest interval in which value is held constantvalue is sampling period for processrule of thumb - 20-30% of process time constantinfluences frequency content of signalsmall - more high frequency conte
40、ntlarge - more low frequency content,CHEE825/435 - Fall 2005,39,J. McLellan,PRBS Design ParametersMinimum,PRBS Design Procedure,select amplitudetwo levelsdecide on desired frequency contenthigh/lowshape frequency content byadjusting minimum switching time OR by filtering PRBS with first-order filter
41、OR by modifying PRBS to make probability of switching 0.5,CHEE825/435 - Fall 2005,40,J. McLellan,PRBS Design Procedureselect am,Other PRBS Design Parameters - Switching Probability,another method of adjusting frequency contentgiven a two-level white noise input e(t), define input to process asas inc
42、reases, input signal switches less frequently - lower frequencies are emphasized,u,t,u,t,with,probabilit,y,e,t,with,probabilit,y,(,),(,),(,),=,-,-,1,1,a,a,CHEE825/435 - Fall 2005,41,J. McLellan,Other PRBS Design Parameters -,Switching Probability .,as increases to 1, starts to approach a stepthis ap
43、proach shapes frequency content by introducing correlationsame correlation structure can be introduced using first-order filter,CHEE825/435 - Fall 2005,42,J. McLellan,Switching Probability .as,Manual vs. Automatic PRBS Generation,PRBS inputs can be generated automatically using custom softwareusing
44、Excel, Matlab, MatrixX, Numerical Recipes routine, .shaping frequency content is usually an iterative procedureselect design parameters (e.g., switching time) and assess results, modify as requiredselect filter parameters,CHEE825/435 - Fall 2005,43,J. McLellan,Manual vs. Automatic PRBS Gene,Manual G
45、eneration,sequence of step moves determined manuallycan resemble PRBS with appropriate design parametersgain additional benefits beyond single step testrecommended proceduredecide on a step sequence with desired frequency content BEFORE experimentationmodify on-line as required, but assess impact of
46、 modifications on input frequency content and thus information content of data set,CHEE825/435 - Fall 2005,44,J. McLellan,Manual Generationsequence of s,A final comment on frequency content.,Increasing low frequency content typically introduces slower steps up/downbrings potential benefit of being a
47、ble to see initial process transientprovides an indication of time delay magnitude,CHEE825/435 - Fall 2005,45,J. McLellan,A final comment on frequency c,Outline,types of input signalscharacteristics of input signalspseudo-random binary sequence (PRBS) inputsother types of input signalsinputs for mul
48、tivariable identificationinput signals for closed-loop identification,CHEE825/435 - Fall 2005,46,J. McLellan,Outlinetypes of input signalsC,What other signals are available & when should they be used?,Sinusoidsparticularly for direct estimation of frequency responseintroduce combination of sinusoids
49、 and reconstruct frequency spectruma sequence of steps of the same duration has same propertiesdanger - difficult to “eyeball” delay because no sharp transients,CHEE825/435 - Fall 2005,47,J. McLellan,What other signals are availab,What other signals are available, and when should they be used?,Steps
50、 and Impulsesrepresent low frequency inputsuseful for direct transient analysisindication of gain, time constants, time delays, type of process (1st/2nd order, over/underdamped)step inputsgood estimate of gainless precise estimate of transients,CHEE825/435 - Fall 2005,48,J. McLellan,What other signa
