DynamicExperiments-ChemicalEngineering动态实验化学工程

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1、CHEE825/435 - Fall 2005 J. McLellan 1Dynamic ExperimentsMaximizing the Information Content for Control Applications CHEE825/435 - Fall 2005 J. McLellan 2 Outline types of input signals characteristics of input signals pseudo-random binary sequence (PRBS) inputs other input signals inputs for multiva

2、riable identification input signals for closed-loop identification CHEE825/435 - Fall 2005 J. McLellan 3 Types of Input Signals deterministic signals steps pulses sinusoids stochastic signals white noise correlated noise what are the important characteristics? CHEE825/435 - Fall 2005 J. McLellan 4 O

3、utline 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 identification CHEE825/435 - Fall 2005 J. McLellan 5 Important Characteristics signal-to-noise ratio du

4、ration frequency content optimum input (deterministic / random) depends on intended end-use control prediction CHEE825/435 - Fall 2005 J. McLellan 6 Signal-to-Noise Ratio improves precision of model parameters predictions avoid modeling noise vs. process trade-off short-term pain vs. long-term gain

5、process disruption vs.expensive retesting / poor controller performance note - excessively large inputs can take process into region of nonlinear behaviour CHEE825/435 - Fall 2005 J. McLellan 7 Example - Estimating 1st Order Process Model with RBS InputTrue model y t q q u t a t( ) . ( ) ( )+ = - +-

6、 -1 061 0751 1 0 5 10 15 20 25 30 35 4000.511.522.53 3.54 Time Step Response confidenceintervals aretighter with increasing SNR1:110:1 less preciseestimate ofsteady stategainmore preciseestimateof transient CHEE825/435 - Fall 2005 J. McLellan 8 Example - Estimating First-Order Model with Step Input

7、0 5 10 15 20 25 30 35 40-2-101234 56 Time Step Response1:110:1 more preciseestimate ofgain vs.RBS inputless precise estimateof transientresponse99% confidence interval CHEE825/435 - Fall 2005 J. McLellan 9 Test Duration how much data should we collect? want to capture complete process dynamic respon

8、se 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 precision CHEE825/435 - Fall 2005 J. McLellan 10 Test DurationPrecision of a dynamic model improves as numb

9、er of data points increases additional information for estimation 0 5 10 15 20 25 30 35 40-1-0.500.511.522.5 33.54 Time Step Response as test duration increases,bias decreasesand precision increasesresponse99% confidenceinterval10 time steps30 time steps50 time steps CHEE825/435 - Fall 2005 J. McLel

10、lan 11 “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 frequency

11、 components - slow dynamics / steady-state gain CHEE825/435 - Fall 2005 J. McLellan 12 Frequency Content - Guiding PrincipleThe input signal should have a frequency content matching that for end-use. CHEE825/435 - Fall 2005 J. McLellan 13 Looking at Frequency Content ideal - match dynamic behaviour

12、of true process as closely as possible 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 2005 J. McLellan 14 Experimental Design Objectiv

13、eDesign input sequence to minimize the following:designcost error inpredicted frequency response importancefunction = our designobjectivesdifference in predicted vs.true behaviour- function of frequency, andthe input signal used CHEE825/435 - Fall 2005 J. McLellan 15 Accounting for Model Error - Int

14、erpretationOptimal solution in terms of frequency content: spectral density frequencyerror in model vs. true process spectral density frequencyimportance to ourapplicationlow high very important not important*J= CHEE825/435 - Fall 2005 J. McLellan 16 Accounting for Model Error Consider frequency con

15、tent matchingGoal - best model for final application is obtained by minimizing JJ G e G e C j dj T j Tfrequencyrange= - - -$( ) ( ) ( )w w w w2bias in frequencycontent modeling importanceof matching- weightingfunction CHEE825/435 - Fall 2005 J. McLellan 17 Example - Importance Function for Model Pre

16、dictive Control spectral density frequency high frequency disturbance rejectionperformed by base-levelcontrollers- accuracy not importantin this rangerequire good estimateof steady state gain,slower dynamics CHEE825/435 - Fall 2005 J. McLellan 18 Desired Input Signal for Model Predictive Control seq

17、uence 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 dynamics CHEE825/435 - Fall 2005 J. McLellan 19 Control ApplicationsFor best results, input signal should have frequency con

18、tent 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 model CHEE825/435 - Fall 2005 J. McLellan 20 Control ApplicationsOne Approach:Design input frequency content to include: frequency ba

19、nd 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 gain CHEE825/435 - Fall 2005 J. McLellan 21 Frequency Content of Some Standard Test Inputsfrequencypower low frequency - like a series of

20、 long steps high frequency - like a series of short steps CHEE825/435 - Fall 2005 J. McLellan 22 Frequency Content of Some Standard Test InputsStep Inputpower frequency0 power is concentrated at low frequency - provides good information about steady state gain, more limited infoabout higher frequenc

21、y behaviour CHEE825/435 - Fall 2005 J. McLellan 23 Example - Estimating First-Order Model with Step Input 0 5 10 15 20 25 30 35 40-2-101234 56 Time Step Response1:110:1 more preciseestimate ofgain vs.RBS inputless precise estimateof transientresponse99% confidenceinterval CHEE825/435 - Fall 2005 J.

22、McLellan 24 Frequency Content of Some Standard Test InputsWhite Noise approximated by pseudo-random or random binary sequencespower frequencypower is distributed uniformlyover all frequencies- broader information, but poorerinformation about steady state gainideal curve CHEE825/435 - Fall 2005 J. Mc

23、Lellan 25 Example - Estimating 1st Order Process Model with RBS Input 0 5 10 15 20 25 30 35 4000.511.522.53 3.54 Time Step Response less preciseestimate ofsteady stategainmore preciseestimateof transient 1:110:1response99% confidenceinterval CHEE825/435 - Fall 2005 J. McLellan 26 Frequency Content o

24、f Some Standard Test InputsSinusoid at one frequencypower frequencypower concentrated at onefrequency correspondingto input signal- poor information aboutsteady state gain, otherfrequencies CHEE825/435 - Fall 2005 J. McLellan 27 Frequency Content of Some Standard Test InputsCorrelated noise consider

25、 u q ucorr white= - -011 09 1.power frequencyvariability is concentrated at lowerfrequencies- will lead to improved estimate ofsteady state gain, poorer estimate ofhigher frequency behaviour CHEE825/435 - Fall 2005 J. McLellan 28 Persistent ExcitationIn order to obtain a consistent estimate of the p

26、rocess model, the input should excite all modes of the process refers to the ability to uniquely identify all parts of the process model CHEE825/435 - Fall 2005 J. McLellan 29 Persistent ExcitationPersistent excitation implies a richness in the structure of the input input shouldnt be too correlated

27、Examples constant step input highly correlated signal provides unique info about process gain random binary sequence low correlation signal provides unique info about additional model parameters CHEE825/435 - Fall 2005 J. McLellan 30 Persistent Excitation - Detailed Discussion Example - consider an

28、impulse response process representation formulate estimation problem in terms of the covariances of u(t) can we obtain the impulse weights? consider estimation matrix persistently exciting of order n - definition spectral interpretation CHEE825/435 - Fall 2005 J. McLellan 31 Persistence of Excitatio

29、n Add in defn in terms of covariance - CHEE825/435 - Fall 2005 J. McLellan 32 Outline 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

30、CHEE825/435 - Fall 2005 J. McLellan 33Pseudo-Random Binary Sequences(PRBS Testing) CHEE825/435 - Fall 2005 J. McLellan 34 What is a PRBS? approximation to white noise input white noise Gaussian noise uncorrelated constant variance zero mean PRBS is a means of approximating using two levels (high/low

31、) CHEE825/435 - Fall 2005 J. McLellan 35 PRBS 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 large relative to data length CHEE825/435 - Fall 2005 J. M

32、cLellan 36 PRBS SignalAlternates in a random fashion between two values: 0 20 40 60 80 100-2-1.5-1-0.500.51 1.52 prbs input time stepvalue input magnitudeminimumswitchingtime test duration CHEE825/435 - Fall 2005 J. McLellan 37 How well does PRBS approximate white noise?Compare spectra: 10-2 10-1 10

33、0 101 10210-1100 101 frequencypower spectrum for 100 point PRBS signal theoretical spectrumfor white noisenote concentrationof PRBS signalin lower frequencyrange 1 .minimum switch time CHEE825/435 - Fall 2005 J. McLellan 38 PRBS Design ParametersAmplitude determines signal-to-noise ratio precision v

34、s. process upsets 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 struc

35、ture CHEE825/435 - Fall 2005 J. McLellan 39 PRBS 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 larg

36、e - more low frequency content CHEE825/435 - Fall 2005 J. McLellan 40 PRBS 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 pr

37、obability of switching 0.5 CHEE825/435 - Fall 2005 J. McLellan 41 Other PRBS 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 freque

38、ncies are emphasized u t u t with probabilitye t with probability( ) ( )( )= - - 1 1 aaa CHEE825/435 - Fall 2005 J. McLellan 42 Switching Probability . as increases to 1, starts to approach a step this approach shapes frequency content by introducing correlation same correlation structure can be int

39、roduced using first-order filter a CHEE825/435 - Fall 2005 J. McLellan 43 Manual vs. Automatic PRBS Generation PRBS inputs can be generated automatically using custom software using Excel, Matlab, MatrixX, Numerical Recipes routine, . shaping frequency content is usually an iterative procedure selec

40、t design parameters (e.g., switching time) and assess results, modify as required select filter parameters CHEE825/435 - Fall 2005 J. McLellan 44 Manual Generation sequence of step moves determined manually can resemble PRBS with appropriate design parameters gain additional benefits beyond single s

41、tep test recommended procedure decide on a step sequence with desired frequency content BEFORE experimentation modify on-line as required, but assess impact of modifications on input frequency content and thus information content of data set CHEE825/435 - Fall 2005 J. McLellan 45 A final comment on

42、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 indication of time delay magnitude CHEE825/435 - Fall 2005 J. McLellan 46 Outline types of input signals characteristics of

43、 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 2005 J. McLellan 47 What other signals are available & when should they be used?Sinusoids particularly for d

44、irect 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 sharp transients CHEE825/435 - Fall 2005 J. McLellan 48 What other signals are av

45、ailable, 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 estimate of gain less precise estimate of transients CHEE82

46、5/435 - Fall 2005 J. McLellan 49 Outline 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 2005 J. McLellan 50 Dealin

47、g 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 Perturb all inputs s

48、imultaneously and estimate models for all outputs simultaneously (MIMO) CHEE825/435 - Fall 2005 J. McLellan 51 SISO 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

49、disadvantage reconciling disturbance models for each output difficult to guarantee all other inputs are constant residual effects of input test sequences? CHEE825/435 - Fall 2005 J. McLellan 52 MISO Approach introduce independent signals for all inputs, use data for a single output estimate transfer

50、 functions simultaneously advantage easier to identify model structure disadvantage no information about directionality of process may not identify most compact representation of process CHEE825/435 - Fall 2005 J. McLellan 53 Why do we use a MISO approach? because of the model form used:process tran

51、sfer + disturbance function modelApproach estimate transfer functions fit disturbance to remaining residual error CHEE825/435 - Fall 2005 J. McLellan 54 Independent Inputs are independent when the sequence for one input does not depend on the sequence for another input CHEE825/435 - Fall 2005 J. McL

52、ellan 55 MIMO Approach with Correlated Inputs perturb all inputs simultaneously, but with cross-correlated inputs input 1 has linear association with input 2 chances are when input 1 moves, input 2 also movesindependent inputs correlated inputs CHEE825/435 - Fall 2005 J. McLellan 56 MIMO Approach wi

53、th Correlated Inputs advantages indication of process directionality improved model estimates disadvantages complexity of model more difficulty recognizing model structure CHEE825/435 - Fall 2005 J. McLellan 57 Outline types of input signals characteristics of input signals pseudo-random binary sequ

54、ence (PRBS) inputs other types of input signals inputs for multivariable identification input signals for closed-loop identification CHEE825/435 - Fall 2005 J. McLellan 58 Input Signals for Closed-Loop IdentificationIdentification experiments can be conducted with the controllers on automatic.Scenar

55、ios unstable processes avoiding disruption of operation quality targets highly integrated processes CHEE825/435 - Fall 2005 J. McLellan 59 Identification Signals for Closed-Loop Identification YtUtSPt+ - ControllerGc ProcessGpdither signal WtX + CHEE825/435 - Fall 2005 J. McLellan 60 Where should th

56、e input signal be introduced?Options:Dither at the controller output clearer indication of process dynamics better estimation properties preferred approachPerturbations in the setpoint additional controller dynamics will be included in estimated model CHEE825/435 - Fall 2005 J. McLellan 61 What does

57、 the closed-loop data represent? dither signal case, without disturbancesOpen-loop input-output data represents Closed-loop input-output data representsY G Wt p t=Y GG G Wt pp c t= +1 CHEE825/435 - Fall 2005 J. McLellan 62 Implications for Input Signal DesignImportance of introducing some external e

58、xcitation non-parametric estimation procedures will simply identify negative inverse of controller difficult/dangerous to estimate process transfer function from closed-loop data without external signal CHEE825/435 - Fall 2005 J. McLellan 63 Implications for Input Signal Design can still use RBS, PR

59、BS, and other signals signal to noise ratio becomes more important make dither signal dominate loop under large dither signal, properties of closed-loop estimation approach those for open-loop case may be necessary to modify frequency content to accommodate closed-loop CHEE825/435 - Fall 2005 J. McLellan 64 Interesting PointWhen the dither signal large, the closed loop experiment is equivalent to filtering dither signal input by and estimating process transfer function could be optimal for disturbance rejection controllers the input to the process, U(t), is 11+G Gp c )(1 1)( tWGGtU cp+=