FORMULATION OF HANKEL SINGULAR VALUES AND SINGULAR VECTORS IN TIME DOMAIN. An-Pan Cherng

Transcription

1 FORMULATION OF HANKEL SINGULAR VALUES AND SINGULAR VECTORS IN TIME DOMAIN An-Pan Cherng Associate Professor, Dept. of Agricultural Machinery Engineering Nationaii-Lan Institute of Agriculture and Technology Shen-Lung Rd., -Lan, Taiwan 65, R.O.C. ABSTRACT. Singular Value Decomposition (SVD) has been widely applied in engineering problems, such as System Identification, Model Reduction, Vibration Control, and Sensor Placement, etc.. Understanding the evolution of SVD components helps one make appropriate decisions on such problems. To this end, this paper is intended to derive analytical formulae of the decomposition components, i.e., singular values and singular vectors, for a Hankel matrix so that the mechanisms of the decomposition can be unfolded. The derivations show how singular vectors are composed of natural frequency and damping ratio, and how singular values appear in pairs and how they are affected by modal parameters. The optimal size of a candidate Hankel matrix is also discussed for single mode case. Finally, numerical examples are given and compared with analytical results. Their excellent agreements confirm the results of this work. NOMENCLATURE y(t) : impulse displacement response N : number of modes r : index for mode r k : k th sample, or matrix size OJ : natural frequency, rad./s t; : viscous damping A : amplitude (real) J=~ * J : complex conjugate T J : transpose ~~ : sampling interval K( ) : condition number Bold uppercase letter : matrix Bold lowercase letter : vector INTRODUCTION Modal parameter estimation techniques in time domain have received a lot attention in recent years, especially for on-line identification applications. Under the assumption that the modal or signal information is uncorrelated to random noise induced in the process of acquiring data, noise-contaminated data of a multiple DOF system are usually grouped into a matrix. The matrix is then decomposed into signal and noise subspaces. Finally, modal information is filtered out and calculated by identification techniques. Among the decomposition methods available in the literature (8], Singular Value Decomposition (SVD) is one of the most popular methods [ OJ, due to its numerical stability and inherent property for separating non-correlated data [4]. SVD has been applied to engineering problems, such as system identification (5], model reduction [][5], vibration control [3][], sensor/actuator placement [6), as well as detection of structural changes [][7). A lot of Algorithms developed for such applications treat the subspaces as a whole, either using signal subspace or noise subspace. On the contrary, the individual vectors (singular vectors) and their corresponding singular values in the subspace matrix have drawn less attention in the literature. Formulations of controllability and observability grammians have been derived in the literature (9), but the singular vectors were not considered. In fact, understanding the evolution of SVD on Hankel matrix (or data matrix) would be very useful in explaining and evaluating algorithms based on such techniques. Taking the Hankel matrix of an impulse response as an example. What have been well-known are that, singular values appear in pairs but ill-conditioning of the data matrix occur when closely-spaced modes exist, and the spectra of the singular vectors exhibit peaks where natural frequencies reside []. Such phenomena have also been reported in numerical simulations. This paper is thus intended to develop analytical expressions for 436

2 the singular vectors and singular values of a candidate signal subspace. The results of such derivations successfully show that the singular vectors are mainly composed of natural frequency and damping ratio, while singular values contain information of all modal parameters, including amplitude. DEVELOPMENT OF THEORY (SINGLE-INPUT SINGLE-OUTPUT CASE) Assuming the structure under investigation contains N modes with viscous damping. The impulse response of displacement can be represented as y(t) =.!._ f C4,e,..l + ;c e.t;r) r=l () A X = _/' (J) + ;'~- /' (J) = (J) e±ja, r' r f:,r r- 4:tr r r where A =-JA A =JA r r ' r r a, = Cos- ( -(,) The k lh sampled data can be abbreviated as y(km) = y(tt) = yk N =-LA,(-Je-<rr~ r=l + Je.<.r~) N = L A,e -r;,oj,r~ sin(~ - (;OJ J k ) r=l The sampled data can be sequentially filled to form a Hankel matrix, H, as follows Yo Y Yn-l () Hmxn = Y Y Yn (3) Ym-l Ym Ym+n- The above formulation is straight forward, and the decomposition matrices are complex. However, the SVD of H is a real decomposition. To derive the analytical expressions of the SVD matrices, the original complex decomposition was used as a start. Consider a single mode case, r-. H can be rewritten as JA H=[v JA =[-v toi a:~j vi] r = -vlv VI ) T [ '] A V To convert the above complex decomposition to real decomposition, we may assume that there exists a transformation between the two decomposition forms. Let u - v el 8 I- I U - v e-jb - I where angle ()is unknown for the time being. The relations in equation(9) can be grouped to form a matrix and written as J[- ( u - u ~) ( u - u; ) ] = J[- r r ] = R = J[- VI or simply Similarly, J[(u - u~) (u - u;)] = J[r r ] = R (8) (9) Without loss of generality, we shall let m = n = k. By defining the following two complex column vectors, ea.,l!.r V T = [ e-<.&.. ea.(k-l)!!.tr r v: = [ e.<;& e-<;&.. e.t;(k-l)l!.t Matrix H becomes H where ~ ; A ( T T - -T = }r, - V r V r + V r V r ) = VA V r=l A =.diag(-ja 7 JA 7...,-JAN,JAN) VN V~] (4) (5) 6) m 437 or R = JV Q ) After rearranging equations ( ) and ( ), we get V =-JR Q~ V =-R Q; () where R ' R E R mx V V C mx ' E Q Q E cx - Substituting equation () back to (8), we obtain the following result

3 _ ja (- ")R Q_;Q-rRr(-.) - } II J --jar (QTQ )-IRT - I I =-jar [jsin() T I } sin() ] R = -A R RT 4sin() That is, H= -A R Rr (3) 4sin() The conversion from complex decomposition to real decomposition has completed. All we have to do is to find a suitable 8 such that real matrices R and R are orthogonal. -I Equation (6) shows that angle () is a function of modal parameters and matrix size k. Once the angle o is found, all of the orthogonal conditions in equation (5) stand. Our final step is to nonnalize the column vectors of R and R The nann square of each vector is calculated by llr,ll = I!Jrlll = l~(ul- u:)r _ {'{e -(lllf* cos(~ mt k + - a)}) = --- -cos(-a). ~(I) +(e-'lllt* _ ) {7) To this end, consider the column vectors of R and R. By assuming the sampling interval,.t, to be close to zero (relative to the natural frequencies), we may define the inner product of two vectors as (r, r ) {k- } =. ll~ t;<uli- u~;)(u ;- u;)m } ilt T = - ( u - u ) (u - u )dt,to I I To ensure the orthogonal property, we need (± rl, r) = ±j(u - u:), j(u - u;)) = ( (4) ~ By expanding the inner product tenns and dropping the subscripts of modal parameters to simplify the notations, we get eu.t.+ja +ea."t.-ja -(eja +e-ja) cos() ( _,lllf ) + e -I= ~ The shifted angle can be explicitly represented by cos() = - ~{e- 'lllf cos(~ OJik -a)- cos( a)} e-l(lllt - then (6) 438 Therefore the SVD of H can be written as H =UL:Vr - Jr~ ] 4 sin(8)m llrl 4sin()M. T Jrl llr,ll jr"{ llrl (9) The column vectors in U and V and diagonal elements in ~ correspond to singular vectors and singular values. Notice that each matrix of the above decomposition is real and the two singular values are close but slightly different. The only negative sign in one of the (right and left) singular vectors may be moved among them, but the result stands. There are some special fonnulae which are useful in evaluating the perfonnance of SVD. For example, as the matrtx size becomes large, i.e., t~c ~ oo, the above solution approaches to cos() = -~cos( a) = ~ For lightly damping structures, 7! 7! cos() ~ = cos(-) ::::::> (} ~ - 4 In addition, if ~~-~{)/k =i7! i =,,3,4,...

4 cos( B) = I + cos(a) = ( which is the same as when t k ~ oo. This result indicates that the size of the Hankel matrix is not necessarily to be too large. An appropriate matrix size is close to the multiple of the (half) period of mode interested. The condition number is simply the ratio of the two singular values, as follows llr! ~e(h) = llrt +( ::::: -- ::::: + ~' if tk ~ oo and r << -( '=', <:. () which is clearly a function of damping ratio. For lightly damped structures, this ratio is close to unity. That is, the singular values appear in pairs for the same mode. In summary, jjj(u _ u;)jl = _ + t;cos(8- a)!!j ;{J) ( ) and if tk ~ IIJ(ut -u;)f = ~ l+t;co;~b-a)(l-e-'"" ) if J - t; {J)( k = i lc () IIJ(u -u;f = l+t;cos(8+a) tlt ;{J) (3) if tk ~ lij<u -u;)f = ~ I+t;co;:B+a)(l-e-'"" ) SIMULATION RESULTS if J- ( {J)(k = i;r (4) A single-mode impulse response of displacement, with f = OOHz~ flt = -4 sec. and variable (but square) matrix size, is used for comparisons between analytical (from derivations above) and simulated solutions. Notice that the period of the mode is columns. The condition number, i.e., the ratio of the two singular values, as a function of matrix size is depicted in Fig.. It can be seen that the condition number converges to equation () as matrix size grows. The oscillating but also decaying period is half cycle of the natural frequency, as seen in Fig.. This indicates that the optimal matrix size is not necessarily to be too large; rather, the (half) multiple of the mode period is appropriate. When the matrix size expands, the growth of signal singular vaiues slows down, but the power of noise accumulates steadily. As a result, a smaller Signal to Noise Ratio (SNR) occurs, and consequently causes less accuracy in identification of modal parameters. On the other hand, damping ratio affects the converging speed of condition number. The larger the damping is, the faster the convergence is, as shown in Fig. 3 (using equation ()). Turning to singular vectors, we know that the vectors are normalized and the amplitude of the signal is stored in the singular values. The singular vector is simply an impulse response with a shifted angle () with respect to the original data sequence. The shift is a function of matrix size, damping ratio, and natural frequency. Fig. 4 shows the shifted angle as a function of matrix size by using equation (6). It is evident that the angle converges to almost 45 degrees (the offset depends on damping ratio) as the matrix size grows. Similar to the condition number, the converging speed of shifted angle is also determined by the damping ratio. CONCLUSIONS An analytical formulation of SVD for an impulse Hankel matrix in time domain has been presented in this paper. The results of this work show that singular vector can be treated as a normalized impulse response but with a shifted angle with respect to the original data used to comprise the data matrix. In addition, each singular vector is a function of matrix size, damping ratio, and natural frequency. The shifted angle converges to the vicinity of 45 degrees, depending on the damping ratio. It can also be seen that the singular values of the same mode appear in pairs. Each singular value contains all modal information, especially proportional to amplitude. The ratio of a singular value pair goes up as damping ratio increases. Finally, the optimal size of a candidate Hankel matrix is the multipre of the mode period at which the condition of the matrix is closest to unity. A large data matrix needs time-consuming calculations and may not serve as a suitable tool for identification purposes. Fig. shows the singular values versus matrix size. The derivations in this paper did not discuss the Their excellent agreements verify the formulae influence of rectangular matrix, and closed-spaced derived in previous section. It can be observed that modes to the performance of SVD. For fairly the two singular values move close to each other separated modes, the analytical solutions of this work every half cycle (5 columns) of the mode period. is adequate and useful. It is hoped that more work of 439

5 this research will be reported in the future. REFERENCES. Balmes, E., "Optimal Ritz Vectors for Component Mode Synthesis Using the Singular Value Decompositionft, AIAA Journal, Vol. 34, No.6,pp. 56-6,996.. Cherng, A.P., Identification of Pseudo Stationary Modal Parameters - Updating and Detection, Ph.D. dissertation, U. of Maryland at Baltimore County, Garg, S., "Robust Eigenspace Assignment Using Singular Value Decompositionft, AIAA J. of Guidance, Vol. 4, No., pp , Golub, G.H., and C.F. Van Loan, Matrix Computations, nd edition, Johns Hopkins University Press, Baltimore, Juang, J-N., and R.S. Pappa, "An Eigensystem Realization Algorithm for Modal Parameter Identification and Model Reduction", AIAA J. of Guidance, Vol. 8, No.5, pp. 6-67, Kammer, D.C., "Optimal Sensor Placement for Mudal Identification Using System- Realization Methods", AIAA J. of Guidance, Vol. 9, No.3, pp , Lenzen, A., and H. Waller, "Identification Using the Algorithm of SVD - An Application to the Realization of Dynamic Systems and to Fault Detection and Localization", Mechanical Systems and Signal Processing, Vol., No. 3, pp , Liang, Z., and D.J. Inman, "Matrix Decomposition Methods in Experimental Modal Analysis", Trans. ASME J. of Vibration and Acoustics, Vol., pp. 4-43, Lim, K.B., and W. Gawronski, "Hankel Singular Values of Flexible Structures in Discrete Time", AIAA J. of Guidance, Vol. 9, No 6., pp , Maia, N.M., "Fundamentals of Singular Value Decomposition", 9 h IMAC, pp. 55-5,99.. Oshman, Y., "Linear Quadratic Stochastic Control Using the Singular Value Decomposition", AIAA J. of Guidance, Vol. 5, No.4, pp ,

6 f= Hz; dt=.; zeta=. 45~------~ ~ ~------~ ~ , 4 35 sigma_ CD 3 ~ 5 ~... til ~ c Ul 5 5 sigma_ solid - simulated dotted - analytical Fig. Comparisons of singular values between analytical and simulated results. 4 ' \ \ I I I f= Hz; dt=.; zeta=. solid- simulated dotted - analytical Cii.c E 8 ::I c : +- ' : Fig. Comparisons of condition numbers between analytical and simulated results. 44

7 f= Hz; dt=. 4r ~ ~ ~ , r solid- zeta=. dashed- zeta=.5 dotted- zeta=.....d E 8 :::J :;:; : ~------~~------~ ~ ~ ~ ~ Fig. 3 Comparisons of condition numbers for different damping ratios. f= Hz; dt= ~ ~ r r ~ , 5 Ill 4 ~ ) "C ~3 Ill "C -= ~ solid- zeta=. dashed- zeta=.5 dotted- zeta=. ~------~------~ ~------~------~------~ Fig. 4 Comparisons of shifted angles for different damping ratios. 44