Understanding Stochastic Subspace Identification

Transcription

1 Unerstaning Stohasti Subspae Ientifiation Rune Briner Department of Strutural an Environmental Engineering University of Aalborg,Sohngaarsholmsvej 57 9 Aalborg, Denmar Palle Anersen Strutural Vibration Solutions A/S Niels Jernes Vej 1,922 Aalborg East,Denmar Nomenlature y ( System response in ontinuous time M, D,K Mass, amping an stiffness matries f ( Fore vetor x ( State vetor A, B,C State spae matries A y A N M R h hf System matrix in ontinuous time System response in isrete time System matrix in isrete time System response matrix Number of ata points Number of measurement hannels Covariane matrix at time lag Blo Hanel matrix, Past an future half part of the Blo Hanel matrix O Γ s Projetion matrix Observability matrix X Kalman State matrix [ µ i ]Ψ, Poles an eigenvetors of isrete system matrix [ λ ]Φ, Poles an eigenvetors of 2 n orer ifferential equation i Abstrat he ata riven Stohasti Subspae Ientifiation tehniques is onsiere to be the most powerful lass of the nown ientifiation tehniques for natural input moal analysis in the time omain. However, the tehniques involves several steps of mysterious mathematis that is iffiult to follow an to unerstan for people with a lassial bagroun in strutural ynamis. Also the onnetion to the lassial orrelation riven time omain tehniques is not well establishe. he purpose of this paper is to explain the ifferent steps in the SSI tehniques of importane for moal ientifiation an to show that most of the elements in the ientifiation tehniques have simple ounterparts in the lassial time omain tehniques.

2 Introution Stohasti Subspae Ientifiation (SSI) moal estimation algorithms have been aroun for more than a eae by now. he real brea-through of the SSI algorithms happene in 1996 with the publishing of the boo by van Overshee an De Moor [1]. A set of MALAB files were istribute along with this boo an the reaers oul easily onvine themselves that the SSI algorithms really were a strong an effiient tool for natural input moal analysis. Beause of the immeiate aeptane of the effetiveness of the algorithms the mathematial framewor esribe in the boo where aepte as a e fato stanar for SSI algorithms. However, the mathematial framewor is not going well together with normal engineering unerstaning. he reason is that the framewor is overing both eterministi as well as stohasti estimation algorithms. o establish this in of general framewor more general mathematial onepts has to be introue. Many mehanial engineers have not been traine to aress problems with unnown loas enabling them to get use to onepts of stohasti theory, while many ivil engineers have been traine to o so to be able to eal with natural loas lie win, waves an traffi, but on the other han, ivil engineers are not use to eterministi thining. he boo of van Overshee an De Moor [1] embraes both engineering worls an as a result the general formulation presents a mathematis that is iffiult to igest for both engineering traitions. It is the view point of the present authors, that going ba to a more traitional basis of unerstaning for aressing the response of strutural systems ue to natural input (ambient loaing) maes things more easy to unerstan. In this paper, we will loo at the SSI tehnique from a ivil engineering (stohasti) point of view. We will present the most funamental steps of the SSI algorithms base on the use of stohasti theory for Gaussian istribute stohasti proesses, where everything is ompletely esribe by the orrelation funtions in time omain or by the spetral ensities in frequeny omain. Most moal people still lie to thin about vibrations in ontinuous time, an thus the isrete time formulations use in SSI are not generally aepte. herefore a short introution is given to isrete time moels an it is shown how simple it is to introue the esription of free responses in isrete time. In the SSI tehnique it seem mysterious to many people why the response ata is gathere together in a Blo Hanel matrix, orers of magnitue larger than the original amount of ata. herefore the struture of the Blo Hanel matrix is relate to traitional ovariane estimation, an it is shown how the subsequent so-alle Projetion of this Hanel matrix onto itself an be explaine in terms of ovarianes an thus results in a set of free responses for the system. hen finally it is explaine how the physis an be estimate by performing a singular value eomposition of the projetion matrix. It is avoie to get into isussions about how to estimate the statistial part of the general moel. Normally when introue to the SSI tehnique, people will start looing at the innovation state spae formulation involving mysterious Kalman states an a Kalman gain matrix that has nothing to o with the physis. his maes most engineers with a normal bagroun in ynamis fall of the train. In this formulation, the general moel is bypasse, however the mysterious Kalman states are introue an explaine as the states for the free responses estimate by the projetion. hus, this is an invitation to the people that were isappointe in the first plae to get ba on the tra, tae a another rie with the SSI train to isover that most of what you will see you an reognize as generalize proeures well establishe in lassial moal analysis. he isrete time formulation We onsier the stohasti response from a system as a funtion of time (1) y1( y2 ( y( M y ( M he system an be onsiere in lassial formulation as a multi egree of freeom strutural system (2) M && y( + Dy& ( + Ky( f (

3 Where Μ, D, Κ is the mass, amping an stiffness matrix, an where f ( is the loaing vetor. In orer to tae this lassial ontinuous time formulation to the isrete time omain the easiest way is to introue the State Spae formulation (3) y( x( y& ( Here we are using the rather onfusing terminology from systems engineering where the states are enote x( (so please on t onfuse this with the system input, the system input is still f ( ). Introuing the State Spae formulation, the original 2 n orer system equation given by eq. (2) simplifies to a first orer equation (4) x& ( A x( + Bf ( y( Cx( Where the system matrix A in ontinuous time an the loa matrix B is given by (5) A M K B M I M D he avantage of this formulation is that the general solution is iretly available, se for instane Kailath [2] (6) t x( exp( A x() + exp( A ( t τ )) Bf ( τ ) τ Where the first term is the solution to the homogenous equation an the last term is the partiular solution. o tae this solution to isrete time, we sample all variables lie y y( an thus the solution to the homogenous equation beomes (7) x exp( A A y x exp( A CA x A x Here one shoul not be onfuse by the fat that we alulate the exponential funtion of a matrix, this onstrution is simply efine by its power series, an in pratie is alulate by performing a eigen-value eomposition of the involve matrix an then taing the exponential funtion of the eigen values. Note that the system matrix in ontinuous time an in isrete is time is not the same. he Blo Hanel Matrix In isrete time, the system response is normally represente by the ata matrix (8) N [ y y L ] 1 2 y N Where is the number of ata points. o unerstan the meaning of the Blo Hanel matrix, it is useful to onsier a more simple ase where we perform the prout between two matries that are moifiations of the

4 ata matrix given by eq. (7). Let be the ata matrix where we have remove the last ata points, an ( 1: N ) similarly, let be the ata matrix where we have remove the first ata points, then ( : N ) (9) Rˆ 1 N (1: N ) ( : N ) Is an unbiase estimate of the orrelation matrix at time lag. his follows iretly from the efinition of the orrelation estimate, se for instane Benat an Piersol [4]. he Blo Hanel matrix h efine in SSI is simply a gathering of a family of matries that are reate by shifting the ata matrix (1) h M (1: N 2s) (2: N 2s+ 1) (2s: N ) hf he upper half part of this matrix is alle the past an enote an the lower half part of the matrix is alle the future an is enote. he total ata shift is 2s an is enote the number of blo rows (of hf the upper or lower part of the Blo Hanel matrix). he number of rows in the Blo Hanel matrix is 2sM, the number of olumns is N 2s. he Projetion Here omes what in many peoples opinion is one of the most mysterious operations in SSI. In van Overshee an De Moor [1] the projetion is introue as a geometrial tool an is explaine mainly in this ontext. However, ealing with stohasti responses, projetion is efine as a onitional mean. Speifially, in SSI the projetion of the future unto the past efines the matrix (11) O E( hf ) A onitional mean lie this an for Gaussian proesses be totally esribe by its ovarianes, se for instane Melsa & Sage [3]. Sine the shifte ata matries also efines ovarianes, it is not so strange that the projetion an be alulate iretly as also efine by van Overshee an De Moor [1] (12) O hf ( ) he last matrix in this prout efines the onitions, the first four matries in the prout introues the ovarianes between hannels at ifferent time lags. A onitional mean lie given by eq. (1) simply onsist of free eays of the system given by ifferent initial onitions speifie by. he matrix is an any olumn in the matrix O is a stae free eay of the system to a (so far unnown) set of initial onitions. Using eq. (7) any olumn in O an be expresse by sm sm (13) ool Γsx C CA 2 Γ s CA M s CA

5 Γ s Now, if we new the so-alle observability matrix, then we oul simply fin the initial onitions iretly from eq. (13) (it is a useful exerise to simulate a system response from the nown system matries, use the SSI stanar proeure to fin the matrix O an then try to estimate the initial onitions iretly from eq. (13)). he Kalman States he so-alle Kalman states are simply the initial onitions for all the olumns in the matrix O, thus (14) O Γ sx Where the matrix X ontains the so efine Kalman states at time lag zero. Again, if we new the matrix Γ, then we oul simply fin all the Kalman states iretly from eq.(14), however, sine we on t now the matrix, we annot o so, an thus we have to estimate the states in a ifferent way. he tri is to use the SVD on the O matrix s Γ s (15) O USV An then efine the estimate of the matrix an the Kalman state matrix states X by Γs (16) Γˆ 1/ 2 US 1/ 2 Xˆ S V he so efine proeure for estimating the matries Γˆ an ˆX is not unique. A ertain arbitrary similarity transformation an be shown to influene the iniviual matries, but an also be shown not to influene the estimation of the system matries. A note on the name Kalman states. he Kalman state matrix ˆX is the Kalman state matrix for time lag zero. If we remove one blo row of O from the top, an then one blo row of Γ s from the bottom, then similarly we an estimate the Kalman state matrix ˆX 1 at time lag one. hus by subsequently removing blo rows from O all the Kalman states an be efine. Using the Kalman states a more general formulation for estimating also the noise part of the stohasti response moeling an be establishe. However, in this paper we fous on explaining how the system matries an be foun, an in this ontext, there is no further nee for Kalman states. Estimating the system matries he system matrix A an be foun from the estimate of the matrix Γ by removing one blo from the top an one blo from the bottom yieling (17) Γˆ (2: s) Aˆ ˆ Γ(1: s) An thus, the system matrix Â an be foun by regression. he observation matrix C an be foun simply by taing the first blo of the observability matrix (18) C ˆ Γ ˆ (11: ) Moal Analysis an pratial issues Now we are finally ba to something lie what we normally o in the fiel of strutural vibrations. First step of fining the moal parameters is to perform an eigenvalue eomposition of the system matrix Â

6 (19) Aˆ [ ] Ψ Ψ µ i he ontinuous time poles λ i are foun from the isrete time poles µ i by (2) µ i exp( λ i ) Leaing to the well nown formulas (21) ln( µ i ) λi ωi λi ωi fi π Re( 2 λi ) ς i λ i he moe shape matrix is foun from (22) Φ CΨ an the job is one from a moal point of view if we are able to mae up our min about the size of the Blo Hanel matrix. As we have seen earlier, the number s efines the size of the Blo Hanel matrix, an thus also the size of the projetion matrix O. However, the number sm efines the number of eigenvalues in our moel, thus sm efines the moel orer. Normally we woul lie to vary the moel orer to establish a stabilization iagram. his an of ourse be one by establishing a series of Blo Hanel matries of ifferent size, but it is somewhat easier, instea of varying the size of the Blo Hanel matrix, to vary the number of singular values use in eq. (16). hus in pratie the size of the Blo Hanel matrix efines the maximum moel orer, an the atual moel orer is varie by varying the number of singular values taen into aount when performing the singular value eomposition of the projetion matrix. he maximum number of eigen values sm must be ajuste to a reasonable level to inorporate the neee range of moels. Referenes [1] Peter van Overshee an Bart De Moor: Subspae Ientifiation for Linear Systems. Kluwer Aaemi Publishers, 1996 [2]. Kailath: Linear Systems. Prentie Hall In., 198. [3] James L. Melsa an Anrew P. Sage: An Introution to Probability an Stohasti Proesses. Prentie- Hall In., [4] Julius S. Benat an Allan G. Piersol: Ranom Data - Analysis an Measurement Proeures. John Wiley & Sons, 1986.