1、Dynamic ExperimentsMaximizing the Information Content for Control ApplicationsCHEE825/435-Fall 20051Outline types of input signals characteristics of input signals pseudo-random binary sequence(PRBS)inputs other input signals inputs for multivariable identification input signals for closed-loop iden
2、tification CHEE825/435-Fall 20052Types of Input Signals deterministic signals steps pulses sinusoids stochastic signals white noise correlated noise what are the important characteristics?CHEE825/435-Fall 20053Outline types of input signals characteristics of input signals pseudo-random binary seque
3、nce(PRBS)inputs other input signals inputs for multivariable identification input signals for closed-loop identification CHEE825/435-Fall 20054Important Characteristics signal-to-noise ratio duration frequency content optimum input(deterministic/random)depends on intended end-use control predictionC
4、HEE825/435-Fall 20055Signal-to-Noise Ratio improves precision of model parameters predictions avoid modeling noise vs.process trade-off short-term pain vs.long-term gain process disruption vs.expensive retesting/poor controller performance note-excessively large inputs can take process into region o
5、f nonlinear behaviourCHEE825/435-Fall 20056Example-Estimating 1st Order Process Model with RBS InputTrue modely tqqu ta t().()()+=-+-106107511051015202530354000.511.522.533.54TimeStep Responseconfidenceintervals aretighter with increasing SNR1:110:1less preciseestimate ofsteady stategainmore precise
6、estimateof transientCHEE825/435-Fall 20057Example-Estimating First-Order Model with Step Input0510152025303540-2-10123456TimeStep Response1:110:1more preciseestimate ofgain vs.RBS inputless precise estimateof transientresponse99%confidenceintervalCHEE825/435-Fall 20058Test Duration how much data sho
7、uld we collect?want to capture complete process dynamic response duration should be at least as long as the settling time for the process(time to 95%of step change)failure to allow sufficient time can lead to misleading estimates of process gain,poor precisionCHEE825/435-Fall 20059Test DurationPreci
8、sion of a dynamic model improves as number of data points increases additional information for estimation0510152025303540-1-0.500.511.522.533.54TimeStep Responseas test duration increases,bias decreasesand precision increasesresponse99%confidenceinterval10 time steps30 time steps50 time stepsCHEE825
9、/435-Fall 200510“Dynamic Content”what types of transients should be present in input signal?excite process over range of interest model is to be used in controller for:setpoint tracking disturbance rejection need orderly way to assess dynamic content high frequency components-fast dynamics low frequ
10、ency components-slow dynamics/steady-state gainCHEE825/435-Fall 200511Frequency Content-Guiding PrincipleThe input signal should have a frequency content matching that for end-use.CHEE825/435-Fall 200512Looking at Frequency Content ideal-match dynamic behaviour of true process as closely as possible
11、 goal-match the frequency behaviour of the true process as closely as possible practical goal-match frequency behaviour of the true process as closely as possible,where it is most important CHEE825/435-Fall 200513Experimental Design ObjectiveDesign input sequence to minimize the following:designcost
12、error inpredicted frequency responseimportancefunction=our designobjectivesdifference in predicted vs.true behaviour-function of frequency,andthe input signal usedCHEE825/435-Fall 200514Accounting for Model Error-InterpretationOptimal solution in terms of frequency content:spectral densityfrequencye
13、rror in model vs.true processspectral densityfrequencyimportance to ourapplicationlowhighvery importantnot important*J=CHEE825/435-Fall 200515Accounting for Model Error Consider frequency content matchingGoal-best model for final application is obtained by minimizing JJG eG eC jdjTjTfrequencyrange=-
14、$()()()wwww2bias in frequencycontent modelingimportanceof matching-weightingfunctionCHEE825/435-Fall 200516Example-Importance Function for Model Predictive Control spectral densityfrequencyhigh frequency disturbance rejectionperformed by base-levelcontrollers-accuracy not importantin this rangerequi
15、re good estimateof steady state gain,slower dynamicsCHEE825/435-Fall 200517Desired Input Signal for Model Predictive Control sequence with frequency content concentrated in low frequency range PRBS(or random binary sequence-RBS)step input will provide for good estimate of gain,but not of transient d
16、ynamics CHEE825/435-Fall 200518Control ApplicationsFor best results,input signal should have frequency content in range of closed-loop process bandwidth recursive requirement!closed-loop bandwidth will depend in part on controller tuning,which we will do with identified modelCHEE825/435-Fall 200519C
17、ontrol ApplicationsOne Approach:Design input frequency content to include:frequency band near bandwidth of open-loop plant(1/time constant)frequency band near desired closed-loop bandwidth lower frequencies to obtain good estimate of steady state gainCHEE825/435-Fall 200520Frequency Content of Some
18、Standard Test Inputsfrequencypowerlow frequency-like a series of long stepshigh frequency-like a series of short stepsCHEE825/435-Fall 200521Frequency Content of Some Standard Test InputsStep Inputpowerfrequency0power is concentrated at low frequency -provides good information about steady state gai
19、n,more limited infoabout higher frequency behaviourCHEE825/435-Fall 200522Example-Estimating First-Order Model with Step Input0510152025303540-2-10123456TimeStep Response1:110:1more preciseestimate ofgain vs.RBS inputless precise estimateof transientresponse99%confidenceintervalCHEE825/435-Fall 2005
20、23Frequency Content of Some Standard Test InputsWhite Noise approximated by pseudo-random or random binary sequencespowerfrequencypower is distributed uniformlyover all frequencies-broader information,but poorerinformation about steady state gainideal curveCHEE825/435-Fall 200524Example-Estimating 1
21、st Order Process Model with RBS Input051015202530354000.511.522.533.54TimeStep Responseless preciseestimate ofsteady stategainmore preciseestimateof transient1:110:1response99%confidenceintervalCHEE825/435-Fall 200525Frequency Content of Some Standard Test InputsSinusoid at one frequencypowerfrequen
22、cypower concentrated at onefrequency correspondingto input signal-poor information aboutsteady state gain,otherfrequenciesCHEE825/435-Fall 200526Frequency Content of Some Standard Test InputsCorrelated noise consider uqucorrwhite=-011091.powerfrequencyvariability is concentrated at lowerfrequencies-
23、will lead to improved estimate ofsteady state gain,poorer estimate ofhigher frequency behaviourCHEE825/435-Fall 200527Persistent ExcitationIn order to obtain a consistent estimate of the process model,the input should excite all modes of the process refers to the ability to uniquely identify all par
24、ts of the process modelCHEE825/435-Fall 200528Persistent ExcitationPersistent excitation implies a richness in the structure of the input input shouldnt be too correlatedExamples constant step input highly correlated signal provides unique info about process gain random binary sequence low correlati
25、on signal provides unique info about additional model parametersCHEE825/435-Fall 200529Persistent Excitation-Detailed Discussion Example-consider an impulse response process representation formulate estimation problem in terms of the covariances of u(t)can we obtain the impulse weights?consider esti
26、mation matrix persistently exciting of order n-definition spectral interpretation CHEE825/435-Fall 200530Persistence of Excitation Add in defn in terms of covariance-CHEE825/435-Fall 200531Outline types of input signals characteristics of input signals pseudo-random binary sequence(PRBS)inputs other
27、 types of input signals inputs for multivariable identification input signals for closed-loop identification CHEE825/435-Fall 200532Pseudo-Random Binary Sequences(PRBS Testing)CHEE825/435-Fall 200533What is a PRBS?approximation to white noise input white noise Gaussian noise uncorrelated constant va
28、riance zero mean PRBS is a means of approximating using two levels(high/low)CHEE825/435-Fall 200534PRBS traditionally generated using a set of shift registers can be generated using random numbers switch to high/low values generation by finite representation introduces periodicity try to get period
29、large relative to data lengthCHEE825/435-Fall 200535PRBS SignalAlternates in a random fashion between two values:020406080100-2-1.5-1-0.500.511.52prbs inputtime stepvalueinput magnitudeminimumswitchingtimetest durationCHEE825/435-Fall 200536How well does PRBS approximate white noise?Compare spectra:
30、10-210-110010110210-1100101frequencypowerspectrum for 100 point PRBS signaltheoretical spectrumfor white noisenote concentrationof PRBS signalin lower frequencyrange 1 .minimum switch timeCHEE825/435-Fall 200537PRBS Design ParametersAmplitude determines signal-to-noise ratio precision vs.process ups
31、ets large magnitudes may bring in process nonlinearity as more of the operating region is covered could result in poor model because of estimation difficulties-e.g.,gains,time constants not constant over range model selection difficulties-lack of clear indication of process structureCHEE825/435-Fall
32、 200538PRBS Design ParametersMinimum switch time shortest interval in which value is held constant value is sampling period for process rule of thumb-20-30%of process time constant influences frequency content of signal small-more high frequency content large-more low frequency contentCHEE825/435-Fa
33、ll 200539PRBS Design Procedure select amplitude two levels decide on desired frequency content high/low shape frequency content by adjusting minimum switching time OR by filtering PRBS with first-order filterOR by modifying PRBS to make probability of switching 0.5 CHEE825/435-Fall 200540Other PRBS
34、Design Parameters-Switching Probability another method of adjusting frequency content given a two-level white noise input e(t),define input to process as as increases,input signal switches less frequently-lower frequencies are emphasized u tu twith probabilitye twith probability()()()=-11aaCHEE825/4
35、35-Fall 200541Switching Probability.as increases to 1,starts to approach a step this approach shapes frequency content by introducing correlation same correlation structure can be introduced using first-order filter CHEE825/435-Fall 200542Manual vs.Automatic PRBS Generation PRBS inputs can be genera
36、ted automatically using custom software using Excel,Matlab,MatrixX,Numerical Recipes routine,.shaping frequency content is usually an iterative procedure select design parameters(e.g.,switching time)and assess results,modify as required select filter parametersCHEE825/435-Fall 200543Manual Generatio
37、n sequence of step moves determined manually can resemble PRBS with appropriate design parameters gain additional benefits beyond single step test recommended procedure decide on a step sequence with desired frequency content BEFORE experimentation modify on-line as required,but assess impact of mod
38、ifications on input frequency content and thus information content of data setCHEE825/435-Fall 200544A final comment on frequency content.Increasing low frequency content typically introduces slower steps up/down brings potential benefit of being able to see initial process transient provides an ind
39、ication of time delay magnitudeCHEE825/435-Fall 200545Outline types of input signals characteristics of input signals pseudo-random binary sequence(PRBS)inputs other types of input signals inputs for multivariable identification input signals for closed-loop identification CHEE825/435-Fall 200546Wha
40、t other signals are available&when should they be used?Sinusoids particularly for direct estimation of frequency response introduce combination of sinusoids and reconstruct frequency spectrum a sequence of steps of the same duration has same properties danger-difficult to“eyeball”delay because no sh
41、arp transientsCHEE825/435-Fall 200547What other signals are available,and when should they be used?Steps and Impulses represent low frequency inputs useful for direct transient analysis indication of gain,time constants,time delays,type of process(1st/2nd order,over/underdamped)step inputs good esti
42、mate of gain less precise estimate of transientsCHEE825/435-Fall 200548Outline types of input signals characteristics of input signals pseudo-random binary sequence(PRBS)inputs other types of input signals inputs for multivariable identification input signals for closed-loop identification CHEE825/4
43、35-Fall 200549Dealing with Multivariable ProcessesApproaches Perturb inputs sequentially and estimate models for each input-output pair(SISO)Perturb all inputs simultaneously and estimate models for a given output(MISO)using independent input test sequences using correlated input test sequences Pert
44、urb all inputs simultaneously and estimate models for all outputs simultaneously(MIMO)CHEE825/435-Fall 200550SISO Approach introduce sequence of independent signals for each input estimate SISO transfer functions individually for each input/output pair advantage easier to identify model structure di
45、sadvantage reconciling disturbance models for each output difficult to guarantee all other inputs are constant residual effects of input test sequences?CHEE825/435-Fall 200551MISO Approach introduce independent signals for all inputs,use data for a single output estimate transfer functions simultane
46、ously advantage easier to identify model structure disadvantage no information about directionality of process may not identify most compact representation of processCHEE825/435-Fall 200552Why do we use a MISO approach?because of the model form used:process transfer +disturbance function modelApproa
47、ch estimate transfer functions fit disturbance to remaining residual errorCHEE825/435-Fall 200553Independent Inputs are independent when the sequence for one input does not depend on the sequence for another inputCHEE825/435-Fall 200554MIMO Approach with Correlated Inputs perturb all inputs simultan
48、eously,but with cross-correlated inputs input 1 has linear association with input 2 chances are when input 1 moves,input 2 also movesindependent inputscorrelated inputsCHEE825/435-Fall 200555MIMO Approach with Correlated Inputs advantages indication of process directionality improved model estimates
49、 disadvantages complexity of model more difficulty recognizing model structureCHEE825/435-Fall 200556Outline types of input signals characteristics of input signals pseudo-random binary sequence(PRBS)inputs other types of input signals inputs for multivariable identification input signals for closed
50、-loop identification CHEE825/435-Fall 200557Input Signals for Closed-Loop IdentificationIdentification experiments can be conducted with the controllers on automatic.Scenarios unstable processes avoiding disruption of operation quality targets highly integrated processesCHEE825/435-Fall 200558Identi
