Identification of Linear Systems

Transcription

1 Identification of Linear Systems Johan Schoukens Vrije Universiteit Brussel Department INDI /67

2 Basic goal Built a parametric model for a linear dynamic system from sampled data ut () yt () G Initial questions - sampled data: what s in between the samples? - plant and noise model? - cost function? Noisy data Model Cost 2/67

3 Outline Introduction Data: what is going on between the samples Model: parametric models of LTI-systems Noisy data Model Cost Cost function Frequency domain formulation Noise models Time domain formulation Validation Examples Conclusions 3/67

4 Sampled data What is going on in between the samples? Two popular assumptions ZOH zero order hold: signal piece wise constant BL band limited assumption: no power above f f s 2 6 Signals BL-spectrum ZOH-spectrum Amplitude Amplitude -6 2 (a) (b) (c) Time (s) 4/67

5 Relation signal assumption / experimental setup ZOH Band limited Generator ZOH Actuator G + y Generator ZOH G Actuator + y + + ykt ( s ) + ukt ( s ) + ykt ( s ) Choice driven by the application - ZOH: discrete control design - Band Limited: other applications 5/67

6 ZOH: discrete control design Generator u d ZOH u zoh () t Actuator ut () Gj ( ) + n p () t y () t G c ( j) G y U zoh ( ) Amplitude (b) ( ) G ZOH e jt s ( ) + y AA () t m y () t ykt ( s ) - input exactly known u d - high frequency components in input ( f f s ) - absolute calibration G y - no anti-alias filter allowed - model: from generator to output (ZOH, actuator, plant, acquisition) 6/67

7 Generator u d ZOH BL: other applications u zoh () t Actuator u g () t + n g () t u () t Plant + n p () t y () t G u G y U( ) Amplitude (a) u AA () t y AA () t m u () t m y () t + + ukt ( s ) ykt ( s ) - input and output measured - band limited data: no power above f f s 2 --> anti alias filters - relative calibration G y G u - only plant modelled G u G y 7/67

8 Conclusions signal assumption ZOH-assumption - imposes experimental condition on the excitation signal - discrete time model from generator to measured plant output - possible to transform DT --> CT model (perfect ZOH) BL-assumption - imposes condition on the observation of the signals - no constraints on the applied excitation (e.g. BL-observations of ZOH-signals can be made). - continuous-time model of the plant in the observed frequency band Violating the signal assumption - still possible to get a behaviour model - model is no longer independent of the measurement environment - the inter sample behaviour becomes an intrinsic part of the model BL-Assumption --> approximate DT-models for simulation Imperfect ZOH --> model linked to the generator 8/67

9 . Choice of the model Possible combinations of continuous/discrete-time data and models. DT-model (Assuming ZOH-setup) exact DT-model Gz () z Z Gs () s standard conditions DT modelling Not studied CT-model (Assuming BL-setup) ZOHsetup BLsetup approximate DT model G () z G ( z e jt s) Gs ( j), s digital signal processing field exact CT-model Gs () standard conditions CT modelling 9/67

10 Cascading models in simulations Lj ( ) Gj ( ) Amplitude (db) - Gj ( ) Lj ( ) -2 2 (a) Amplitude (db) Phase ( ) (b) Cascade of ZOH models ZOH of cascaded models Conclusion: BL-assumption is needed /67

11 Outline Introduction Data: what is going on between the samples Model: parametric models of LTI-systems Noisy data Model Cost Cost function Frequency domain formulation Noise models Time domain formulation Validation Examples Conclusions /67

12 Parametric models of LTI systems Continuous time Diffusion Discrete time General model G( ) Gs ( ) G( s ) Gz ( ) Bs ( ) As ( ) B( s ) A( s ) n b Bz ( ) Az ( ) B( ) b r r r with A( ) n a a r r r n b b r s r r n a a r s r r n b b m s m/ 2 m n a a n s n/ 2 n n b b r z r r n a a r z r r s continuous-time s diffusion z discrete-time 2/67

13 Parametric models of LTI systems Relation between input/output DFT spectra Input/output DFT spectra Uk () N, N utt ( )z t t s k Yk () N N ytt ( )z t t s k with z k e j2k N Remark Uk ()Yk are an ON ( ) for random excitations 3/67

14 Parametric models of LTI systems Relation between input/output DFT spectra Periodic signals Yk () G( k )Uk () if - steady state response - integer number of periods are observed 4/67

15 Parametric models of LTI systems Relation between input/output DFT spectra Arbitrary signals with where and G( ) Yk () G( k )Uk () + T G ( k ) n b B( ) b r r r , T A( ) n a G ( ) a r r r z n ig max( n a n b ) s s n ig max( n a n b ) T G ( k ) I G ( ) A( ) for periodic and time-limited signals ON ( 2 / ) arbitrary signals n ig i gr r r n a a r r r 5/67

16 Parametric models of LTI systems Full equivalence time domain - frequency domain Frequency domain leakage begin and end conditions Yk () G( k )Uk () + T G ( k ) Time domain Transient effects due to initial conditions yt () Gq ( )ut () + T G ( t ) 6/67

17 Experimental illustration: octave band-pass filter Periodic multisine excitation and random noise excitation plant model n a 6 n b 4 amplitude (db) X X X X X X X X X XX X X XX XX X X XX X X X X X XX X XXX X X X XXXX X X X X X X X X X X X XX X X X X X X X X X X X XX X XX X X X XX X X XXX X X XX X XX X X XX X XX X XX X XXX X X X X XX X XXXX X X X X XX X X XX X X X X X X X X X X XXX X X X XX XX X X X XX X X X XX X X X X X X X X X X X X X X XX X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X f (khz) measured FRF (multisine excit.) error model without transient (arbitr. excit.) error model with transient n ig 6 (arbitr. excit.) error model (multisine excit.) 7/67

18 Rational form G( ) Parametric models of LTI systems Parametrizations n b B( ) b r r r with A( ) T a n a a a a r r na b b b nb r Partial fraction expansion G( ) p L r r W( w) for s s r p r q r S r r with Gz ( ) p r p r L r z r z q S r z r z + Wz ( w) r T q Re( )Im( )Re( p )Im( p ) S S q Re( L )Im( L )Re( L p )Im( L p )w w nw 8/67

19 Parametric models of LTI systems Parametrizations (cont d) State space representation for proper transfer functions ( ) Gs ( ) CsI na A B+ D n b n a with Gz ( ) z CI z A B+ D na T vec T ( A) B T C D Pole/zero representation r r G( ) K n a disadvantage: ill conditioned for multiple poles/zeroes n b r r Systems with time delay G( ) e s B( ) A( ) Bz Gz ( ) z T ( ) s Az ( ) 9/67

20 Outline Introduction Data: what is going on in between the samples Model: parametric models of LTI-systems Cost function Validation Examples Conclusions Frequency domain formulation Noise models Time domain formulation Noisy data Model Cost 2/67

21 Setup Basic problem for BL-setup N g N p U G ( ) M U M Y Measurements Model Uk () Yk () Y + N Y, k Uk () U + N U Y G ( k )U, with G( k ) F B( k ) A( k ) Yk () n b b r r r k n a a r r k r 2/67

22 Match model and data: choice of the cost function Intuitive choice V F ( Z) -- F F k G( k ) G( k ) G 2 - G( k ): measured FRF - 2 G : uncertainty Works amazingly well in many situations Problem: good measurements in the presence of input noise lim M G( k ) S YU ( k ) G S UU ( k ) ( k ) S MU M ( U k ) S UU ( k ) 22/67

23 Alternative V F ( Z) -- F -- F F k F k Uk () U p U 2 Yk () Y p Uk () U p H Yk () Y p Y 2 2 Y 2 U Yk () Y p Uk () U p under the constraint Y p G( k )U p k 2 F Parameters to be estimated: : n a + n b + real parameters U p Y p : 2F complex parameters 23/67

24 Generalized problem correlated input output noise V F ( Z) -- F F k Yk () Y p Uk () U p H 2 k 2 Y YU () 2 UY 2 U Yk () Y p Uk () U p under the constraint Y p G( k )U p k 2 F Parameters to be estimated: : n a + n b + real parameters U p Y p : 2F complex parameters 24/67

25 Generalized problem V F ( Z) -- F F k Yk () Y p Uk () U p H 2 k 2 Y YU () 2 UY 2 U Yk () Y p Uk () U p under the constraint Y p G( k )U p k 2 F This is the maximum likelihood estimator - Gaussian distributed noise - Known covariance matrix 25/67

26 Elimination of U p Y p V F ( Z) -- F F k Yk () G( k )Uk () Y + 2 U ()G k ( k ) 2 2Re( 2 YU ()G k ( k ) ) 2 Symmetric formulation G B A V F ( Z) -- F F k A( k )Yk () B( k )Uk () Y ()A k ( k ) U ()B k ( k ) 2 2Re( 2 YU ()A k ( k )B( k ) ) 2 26/67

27 Special case : identifying from the measured FRF V F ( Z) -- F F k Yk () G( k )Uk () Y + 2 U ()G k ( k ) 2 2Re( 2 YU ()G k ( k ) ) 2 Put Yk () G( k ), Uk (), and 2 U, and 2 Y 2 G V F ( Z) -- F F k G( k ) G( k ) G 2 27/67

28 Special case 2: the input is exactly known V F ( Z) -- F F k Yk () G( k )Uk () Y + 2 U ()G k ( k ) 2 2Re( 2 YU ()G k ( k ) ) 2 Put 2 k 2 U, YU () V F ( Z) -- F F k Yk () G( k )Uk () Y 2 28/67

29 Outline Introduction Data: what is going on between the samples Model: parametric models of LTI-systems Noisy data Model Cost Cost function Frequency domain formulation Noise models Time domain formulation Validation Examples Conclusions 29/67

30 Noise models Time domain vt () Hq ( )et () + T h () t and Ev()vs r R vv ( r s) Hq ()et Frequency domain Vk () Hk ()Ek + T H and EV H Vl () 2 V kl + ON ( ) Hk ()Ek cost frequency domain ON ( ) V H 2 V ON ( ) V v T cost time domain R vv R vv ( ) R vv v V H 2 V ( k) V or H e T H e 3/67

31 Noise models cost function frequency domain cost function time domain V H V 2 V H e T H e - nonparametric noise model 2 V - no interference with plant estimate - periodic excitation --> very simple extraction - arbitrary excitation --> more complicated - simultaneous identification parametric plant/noise model - Errors-in-Variables also parametric model excitation - Only noise on output classic prediction error identif. 3/67

32 Noise model frequency domain Cost function V F ( Z) -- F F k Yk () Y p Uk () U p H 2 k 2 Y YU () 2 UY 2 U Yk () Y p Uk () U p 2nd order moments of the noise needed: to be extracted from the data Prior analysis separate signals and noise extract a nonparametric noise model ) periodic excitations 2) arbitrary excitations 32/67

33 Identify 2 U, 2 Y and YU () Noise model, prior analysis periodic excitation 2 k Additional information: the signals are periodic ut () u () t u 2 () t u l t () t Û() k,, M ---- M U l Ŷ() k l M ---- M Y l l ˆ U 2 and M M U l Û() k 2 ˆ l Y 2 M M Y l Ŷ() k l 2 ˆ YU 2 M Y M l Ŷ() k U l Û() k l 33/67

34 Noise model, prior analysis periodic excitation Properties consistency: M 4 periods are enough efficiency: M 6 periods are enough loss in efficiency M 2 C SML C M 3 ML normality: M 7 is enough Recent results 2 periods + overlapping windows are enough additional loss in efficiency: about 5% (compared to M 6, no overlap) 34/67

35 Example of a nonparametric prior noise analysis The flexible robot arm Data from Jan Swevers, KULeuven, PMA 35/67

36 Raw data Output Displacement of hand Time (s).5 Input Torque Time (s) 36/67

37 Segment the record periods Output Displacement of hand Time, s (N496, experiments) Input.4 Torque Time, s (N496, experiments) 37/67

38 Variance analysis Output Output amplitudes db db Hz Input Input amplitudes Signal std.dev Signal std.dev Hz frequency frequency 38/67

39 Variance analysis FRF Magnitudes of frf and Variances 2 FRF db std. dev. Hz Frequency 39/67

40 Estimated model num den /67

41 Noise model frequency domain Cost function V F ( Z) -- F F k Yk () Y p Uk () U p H 2 k 2 Y YU () 2 UY 2 U Yk () Y p Uk () U p 2nd order moments of the noise needed: to be extracted from the data Prior analysis separate signals and noise extract a nonparametric noise model ) periodic excitations 2) arbitrary excitations 4/67

42 Nonparametric noise model, prior analysis arbitrary excitation Simplification required: only noise on the output N g N p U G ( ) M U M Y Uk () Yk () 42/67

43 Basic idea vt () u () t linear system y () t yt () Basic idea: eliminate G ()U k and T coherence S YY () f S YU () f S U U () f 2 v () f (+ leakage errors) More advanced methods solve set of equations at multiple frequencies G ()T k smooth --> Taylor Yk () G ()U k + T + Vk () with k n k k+ n 43/67

44 N 256 N 24 N Noise model, prior analysis arbitrary excitation Example 4 Estimate System G 6 noise H /67

45 Parametric noise model Simultaneous analysis - General problem Estimate plant and noise model together Extract a parametric noise model N g N p U G ( ) M U M Y Uk () Yk () Additional constraints needed - NO cross-correlation between and M U N p M Y - input: filtered white noise --> a parametric input model is also estimated Errors-in-variables problem --> out of the scope of this course 45/67

46 Parametric noise model Simultaneous analysis - Simplified problem Input is exactly known Estimate parametric plant and noise model together No signal model needed N p U G ( ) M Y Yk () Classical prediction error method --> time domain identification 46/67

47 Outline Introduction Data: what is going on between the samples Model: parametric models of LTI-systems Cost function Frequency domain formulation Noise models Time domain formulation Validation Examples Conclusions 47/67

48 Time domain identification () Actuator + y Generator ZOH Gs + - Use discrete time models: Gz ( ) ykt ( s ) 2 k - Assume that the input is exactly known: 2 U, YU () - Use a parametric noise model: 2 Y Hz ( ) 2 - The cost function becomes: V F ( Z) N Yk () Gz ( k )Uk () --, with F Hz ( k ) 2 z k k 2 e j2k N 48/67

49 Time domain identification (Cont d) Interpretation in the time domain: model yt () G ( q)ut () + Hq ( )et () cost function ŷ( t t ) H ( q )Gq ( )ut () + H ( q ) yt (), ( t ) yt () ŷ( t t ) V N ( Z) N --- ( t ) N 2 k 49/67

50 V Time domain identification (Cont d) N --- et (), or N 2 V t --- N N k Bz k Yk () U() k Az ( k ) Hz ( k ) 2 2 Bq ) Gq ( ) and Hq ( ) > Aq ( )yt () Bq ( )ut () + et () Aq ( ) Aq ( ) problem that is linear-in-the-parameters (ARX) 2) Gq ( ) Bq Cq ( ) and Hq ( ) > Aq ( )yt () Aq ( ) Aq ( ) Bq ( )ut () + Cq ( )et () problem that is linear-in-the-parameters (ARMAX) 3) Gq ( ) Bq Bq ( ) and Hq ( ) --> yt () Aq ( ) Aq ( ) () + et () problem that is nonlinear-in-the-parameters (Output Error) 4) Gq ( ) Bq Dq ( ) Bq ( ) and Hq ( ) > yt () Aq ( ) Cq ( ) Aq ( ) () + et () Dq ( ) Cq ( ) problem that is nonlinear-in-the-parameters (Box-Jenkins) 5/67

51 Outline Introduction Data: what is going on between the samples Model: parametric models of LTI-systems Cost function Frequency domain formulation Noise models Time domain formulation Validation Examples Conclusions 5/67

52 Classic Validation - Cross-correlation R ue ()? --> test ˆ ( k) - Auto-correlation residuals or prediction errors white? --> test e Nonparametric noise models V N ˆ ( k) ( N n e ) k EV N n --> check the actual value >< theoretic value 2 V 2N n Remark: in classical prediction error framework e estimated from residuals --> includes model errors Compare FRF modelled transfer function with measured FRF 52/67

53 Linear identification framework Parametric noise model Classical prediction error frame work Non-parametric noise model Frequency domain identification Preprocessing - non-parametric noise model Estimates - parametric plant model - parametric noise model (nonlinear and disturbing noise) Estimates - parametric plant model Properties - consistent - efficient - normal Properties - consistent - efficient - normal Validation - nonlinearity is NOT detected Happy but unconscious user Validation - nonlinearity is detected - alternative validation scheme Happy but conscious user 53/67

54 Time domain versus frequency domain identification - Transforming data from time to frequency domain does not create or delete information! - There exists a full equivalence between both approaches - Practical issues are decisive some information easier accessible in one domain than in the other (non causal) prefiltering in frequency domain improved SNR --> simpler generation of starting values combining different sampling frequencies --> wide frequency range - Use periodic excitations if possible --> access to a nonparametric noise model Some of these aspects will be illustrated on the examples 54/67

55 Outline Introduction Data: what is going on between the samples Model: parametric models of LTI-systems Cost function Frequency domain formulation Noise models Time domain formulation Validation Examples Conclusions 55/67

56 3x22V Example Identification d-axis synchronous machine Armature Field Thyristor rectifier Thyristor rectifier i f i a Current controller e a e f Current controller Computer 3x22V 56/67 HP445 AW-generator HP43 A/D-convertor HP43 A/D-convertor HP43 A/D-convertor HP43 A/D-convertor HP445 AW-generator MXI controller VXI measurement system

57 Identification d-axis synchronous machine I a (dba) Current Armature -4 E a (dbv) Voltage Armature f (Hz) f (Hz) 3 M 8 N 65536, Hz 23 Hz I f (dba) Current field -4 E f (dbv) Voltage field f (Hz) f (Hz) 3 57/67

58 Estimation parametric plant model with estimated nonparametric noise model Measurement example: identification d-axis synchronous machine (cont d) -4 Z ( db) Z 2 ( db) Zs ( ) B r T n b 6 B r f (Hz) n b B r s r r n b a r s r r Z 22 ( db) f (Hz) f (Hz) measured FRF noise variance difference modelled and measured FRF 58/67

59 Estimation parametric plant model with estimated nonparametric noise model Measurement example: identification d-axis synchronous machine (cont d) n b V SML ( ˆ Z) V Theoretic e e e e e e Cost function much too large --> model errors 59/67

60 Observations - a very wide frequency range is covered [. Hz, 23 Hz] - improper models can be used (more zeros than poles) - model errors are easily detected - only a small number of frequencies is excited - a high SNR on these lines --> averaging and filtering effect - generation of initial estimates 6/67

61 averaging and filtering: Elimination of non excited frequencies Signal Samples Original signal DFT spectrum Spectral line number Signal + noise (freq. domain) original averaged ( times) averaged and filtered Spectral line number Spectral line number Spectral line number 3 Additive noise (time domain) original averaged ( times) averaged and filtered Samples Samples Samples 6/67

62 Example 2 Nuclear magnetic resonance (NMR) spectroscopy Nuclear Magnetic Resonace (NMR) scanner: ~ Tesla static magnetic field, ~ MHz oscillating field perpendicular to the static field response measured in two orthogonal directions x and y complex signal x(t) + jy(t) 62/67

63 Nuclear magnetic resonance (NMR) spectroscopy (con t) 3 abs(responsie) abs(response) tijd (s) time (s) absolute value demodulated signal x(t) + jy(t) (averaged over 64 measurements) 63/67

64 Nuclear magnetic resonance (NMR) spectroscopy (con t) 4 NMR spectrum muscle Amplitude (db) 2-2 signal model sum of complex damped exponentials Tz ( ) a r b r n 9 C n b r z r r n a r z r r frequentie (Hz) measured spectrum model residual meas.-model noise variance 64/67

65 Nuclear magnetic resonance (NMR) spectroscopy (con t) Whiteness test residuals Function of Dependency analysis + : measured - bound 5 % - - bound 95% Autocorrelation residuals Lag (in samples) autocorrelation 5% uncertainty bound (fraction outside 5.6%) 95% uncertainty bound (fraction outside 5.2%) V SML ( ˆ Z) 584 V noise 52 V2 / noise 22 65/67

66 Observations Transfer functions with complex coefficients No model errors observed 66/67

67 Outline Introduction Data: what is going on between the samples Model: parametric models of LTI-systems Cost function Frequency domain formulation Noise models Time domain formulation Validation Examples Conclusions 67/67