1. CHEE825/435 - Fall 2005J. McLellan1 Dynamic Experiments Maximizing the Information Content for Control Applications

2. CHEE825/435 - Fall 2005J. McLellan2 Outline 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

3. CHEE825/435 - Fall 2005J. McLellan3 Types of Input Signals deterministic signals »steps »pulses »sinusoids stochastic signals »white noise »correlated noise what are the important characteristics?

4. CHEE825/435 - Fall 2005J. McLellan4 Outline 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

5. CHEE825/435 - Fall 2005J. McLellan5 Important Characteristics signal-to-noise ratio duration frequency content optimum input (deterministic / random) depends on intended end-use –control –prediction

6. CHEE825/435 - Fall 2005J. McLellan6 Signal-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 of nonlinear behaviour

7. CHEE825/435 - Fall 2005J. McLellan7 Example - Estimating 1st Order Process Model with RBS Input True model yt q q utat().. ()()      1 06 1075 1 1 0510152025303540 0 0.5 1 1.5 2 2.5 3 3.5 4 Time Step Response confidence intervals are tighter with increasing SNR 1:1 10:1 less precise estimate of steady state gain more precise estimate of transient

8. CHEE825/435 - Fall 2005J. McLellan8 Example - Estimating First-Order Model with Step Input 0510152025303540 -2 0 1 2 3 4 5 6 Time Step Response 1:1 10:1 more precise estimate of gain vs. RBS input less precise estimate of transient response 99% confidence interval

9. CHEE825/435 - Fall 2005J. McLellan9 Test Duration how much data should 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 precision

10. CHEE825/435 - Fall 2005J. McLellan10 Test Duration Precision of a dynamic model improves as number of data points increases »additional information for estimation 0510152025303540 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 Time Step Response as test duration increases, bias decreases and precision increases response 99% confidence interval 10 time steps 30 time steps 50 time steps

11. CHEE825/435 - Fall 2005J. McLellan11 “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 components - slow dynamics / steady- state gain

12. CHEE825/435 - Fall 2005J. McLellan12 Frequency Content - Guiding Principle The input signal should have a frequency content matching that for end-use.

13. CHEE825/435 - Fall 2005J. McLellan13 Looking at Frequency Content ideal - match dynamic behaviour 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

14. CHEE825/435 - Fall 2005J. McLellan14 Experimental Design Objective Design input sequence to minimize the following: design cost errorin predictedfrequencyresponse importance function                     our design objectives difference in predicted vs. true behaviour - function of frequency, and the input signal used

15. CHEE825/435 - Fall 2005J. McLellan15 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 our application low high very important not important * J=

16. CHEE825/435 - Fall 2005J. McLellan16 Accounting for Model Error Consider frequency content matching Goal - best model for final application is obtained by minimizing J JGeGeCjd jTjT frequency range     ()()()   2 } bias in frequency content modeling } importance of matching - weighting function

17. CHEE825/435 - Fall 2005J. McLellan17 Example - Importance Function for Model Predictive Control spectral density frequency high frequency disturbance rejection performed by base-level controllers - > accuracy not important in this range require good estimate of steady state gain, slower dynamics

18. CHEE825/435 - Fall 2005J. McLellan18 Desired 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 dynamics

19. CHEE825/435 - Fall 2005J. McLellan19 Control Applications For 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 model

20. CHEE825/435 - Fall 2005J. McLellan20 Control Applications One 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 gain

21. CHEE825/435 - Fall 2005J. McLellan21 Frequency Content of Some Standard Test Inputs frequency power low frequency - like a series of long steps high frequency - like a series of short steps

22. CHEE825/435 - Fall 2005J. McLellan22 Frequency Content of Some Standard Test Inputs Step Input power frequency0 power is concentrated at low frequency - provides good information about steady state gain, more limited info about higher frequency behaviour

23. CHEE825/435 - Fall 2005J. McLellan23 Example - Estimating First-Order Model with Step Input 0510152025303540 -2 0 1 2 3 4 5 6 Time Step Response 1:1 10:1 more precise estimate of gain vs. RBS input less precise estimate of transient response 99% confidence interval

24. CHEE825/435 - Fall 2005J. McLellan24 Frequency Content of Some Standard Test Inputs White Noise –approximated by pseudo-random or random binary sequences power frequency power is distributed uniformly over all frequencies - broader information, but poorer information about steady state gain ideal curve

25. CHEE825/435 - Fall 2005J. McLellan25 Example - Estimating 1st Order Process Model with RBS Input 0510152025303540 0 0.5 1 1.5 2 2.5 3 3.5 4 Time Step Response less precise estimate of steady state gain more precise estimate of transient 1:1 10:1 response 99% confidence interval

26. CHEE825/435 - Fall 2005J. McLellan26 Frequency Content of Some Standard Test Inputs Sinusoid at one frequency power frequency power concentrated at one frequency corresponding to input signal - poor information about steady state gain, other frequencies

27. CHEE825/435 - Fall 2005J. McLellan27 Frequency Content of Some Standard Test Inputs Correlated noise –consider u q u corrwhite    01 109 1.. power frequency variability is concentrated at lower frequencies - will lead to improved estimate of steady state gain, poorer estimate of higher frequency behaviour

28. CHEE825/435 - Fall 2005J. McLellan28 Persistent Excitation In 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 parts of the process model

29. CHEE825/435 - Fall 2005J. McLellan29 Persistent Excitation Persistent excitation implies a richness in the structure of the input –input shouldn’t be too correlated 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

30. CHEE825/435 - Fall 2005J. McLellan30 Persistent 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 estimation matrix persistently exciting of order n - definition spectral interpretation

31. CHEE825/435 - Fall 2005J. McLellan31 Persistence of Excitation Add in defn in terms of covariance -

32. CHEE825/435 - Fall 2005J. McLellan32 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

33. CHEE825/435 - Fall 2005J. McLellan33 Pseudo-Random Binary Sequences (PRBS Testing)

34. CHEE825/435 - Fall 2005J. McLellan34 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)

35. CHEE825/435 - Fall 2005J. McLellan35 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

36. CHEE825/435 - Fall 2005J. McLellan36 PRBS Signal Alternates in a random fashion between two values: 020406080100 -2 -1.5 -0.5 0 0.5 1 1.5 2 prbs input time step value input magnitude minimum switching time test duration

37. CHEE825/435 - Fall 2005J. McLellan37 How well does PRBS approximate white noise? Compare spectra: 10 -2 10 10 0 1 2 10 0 1 frequency power spectrum for 100 point PRBS signal theoretical spectrum for white noise note concentration of PRBS signal in lower frequency range 1. minimum switch time

38. CHEE825/435 - Fall 2005J. McLellan38 PRBS Design Parameters Amplitude –determines signal-to-noise ratio »precision vs. 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 structure

39. CHEE825/435 - Fall 2005J. McLellan39 PRBS Design Parameters Minimum 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 content

40. CHEE825/435 - Fall 2005J. McLellan40 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 filter OR by modifying PRBS to make probability of switching  0.5

41. CHEE825/435 - Fall 2005J. McLellan41 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 frequencies are emphasized ut utwithprobability etwithprobability () () ()       1 1  

42. CHEE825/435 - Fall 2005J. McLellan42 Switching 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

43. CHEE825/435 - Fall 2005J. McLellan43 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 –select design parameters (e.g., switching time) and assess results, modify as required –select filter parameters

44. CHEE825/435 - Fall 2005J. McLellan44 Manual Generation 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 modifications on input frequency content and thus information content of data set

45. CHEE825/435 - Fall 2005J. McLellan45 A 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 indication of time delay magnitude

46. CHEE825/435 - Fall 2005J. McLellan46 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

47. CHEE825/435 - Fall 2005J. McLellan47 What 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 sharp transients

48. CHEE825/435 - Fall 2005J. McLellan48 What 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 estimate of gain »less precise estimate of transients

49. CHEE825/435 - Fall 2005J. McLellan49 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

50. CHEE825/435 - Fall 2005J. McLellan50 Dealing with Multivariable Processes Approaches 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 simultaneously and estimate models for all outputs simultaneously (MIMO)

51. CHEE825/435 - Fall 2005J. McLellan51 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 disadvantage –reconciling disturbance models for each output –difficult to guarantee all other inputs are constant –residual effects of input test sequences?

52. CHEE825/435 - Fall 2005J. McLellan52 MISO Approach introduce independent signals for all inputs, use data for a single output estimate transfer functions simultaneously advantage –easier to identify model structure disadvantage –no information about directionality of process –may not identify most compact representation of process

53. CHEE825/435 - Fall 2005J. McLellan53 Why do we use a MISO approach? … because of the model form used: process transfer + disturbance function model Approach –estimate transfer functions –fit disturbance to remaining residual error

54. CHEE825/435 - Fall 2005J. McLellan54 Independent Inputs … are independent when the sequence for one input does not depend on the sequence for another input

55. CHEE825/435 - Fall 2005J. McLellan55 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 moves independent inputscorrelated inputs

56. CHEE825/435 - Fall 2005J. McLellan56 MIMO Approach with Correlated Inputs advantages –indication of process directionality –improved model estimates disadvantages –complexity of model –more difficulty recognizing model structure

57. CHEE825/435 - Fall 2005J. McLellan57 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

58. CHEE825/435 - Fall 2005J. McLellan58 Input Signals for Closed-Loop Identification Identification experiments can be conducted with the controllers on automatic. Scenarios –unstable processes –avoiding disruption of operation »quality targets »highly integrated processes

59. CHEE825/435 - Fall 2005J. McLellan59 Identification Signals for Closed-Loop Identification YtYt UtUt SP t + - Controller Gc Process Gp dither signal W t X + +

60. CHEE825/435 - Fall 2005J. McLellan60 Where should the input signal be introduced? Options: Dither at the controller output –clearer indication of process dynamics –better estimation properties –preferred approach Perturbations in the setpoint –additional controller dynamics will be included in estimated model

61. CHEE825/435 - Fall 2005J. McLellan61 What does the closed-loop data represent? dither signal case, without disturbances Open-loop –input-output data represents Closed-loop –input-output data represents YGW tpt  Y G GG W t p pc t   1

62. CHEE825/435 - Fall 2005J. McLellan62 Implications for Input Signal Design Importance of introducing some external excitation –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

63. CHEE825/435 - Fall 2005J. McLellan63 Implications for Input Signal Design can still use RBS, PRBS, 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

64. CHEE825/435 - Fall 2005J. McLellan64 Interesting Point When 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 1 1  GG pc