Non-parametric estimate of the system function of a time-varying system

Transcription

1 Non-parametric estimate of the system function of a time-varying system John Lataire a, Rik Pintelon a, Ebrahim Louarroudi a a Vrije Universiteit Brussel, Pleinlaan 2, 1050 Elsene Abstract The task of identifying an unknown dynamic system is made easier with prior knowledge on its behaviour. Using a frequency domain approach, the non-parametric maximum likelihood estimator of the system function, associated with the timedependent impulse response of a time-varying system, is constructed. This is accomplished by use of a simple linear least squares fitting algorithm, applied to the spectral response of the system to a multisine excitation. The noise variance on the system function is estimated simultaneously, and modelling errors can be detected, as illustrated on a simulation example. Key words: time-varying system, non-parametric estimate, frequency response function, model error estimation. 1 Introduction Engineering applications involve many time-varying systems, which include i) the resonance frequency and damping of most vibrating parts of a plane (e.g. the wings), as functions of the flight speed and height [3], ii) the impedance Acknowledgement This work was supported in part by the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (Methusalem Fund, METH1), and by the Belgian Federal Government (IUAP VI/4). J. Lataire s work was supported by the Research Foundation Flanders (FWO) under a Ph. D. fellowship. Published in Automatica, vol 48, issue 4, april 2012,pp , address: jlataire@vub.ac.be (John Lataire). Preprint submitted to Elsevier Science 3 May 2013

2 of a metal subjected to pitting corrosion as a function of the progress of the underlying chemical reaction, and iii) a robot arm (a nonlinear system) that can be viewed as a linearised system around continuously evolving (and thus time-varying) set points. When linear, these systems are unambiguously described by their time-dependent impulse response functions. The Fourier transforms of the latter are called system functions. They are a frequency domain interpretation of the response of a time-varying system, as first elaborated in [17]. This paper provides a simple methodology for extracting a non-parametric estimate of the system function, provided that it can be expanded into a series expansion. Non-parametric time-frequency methods, providing a timedependent spectral representation of non-stationary signals are thoroughly discussed in the literature. Popular methods include the short-time Fourier Transform, wavelet transforms, and Cohen s class of distributions [4]. The application of these methods to vibrational signals for the characterisation of time-varying physical systems is found in [2]. However, neither an accurate estimate of the system function nor an uncertainty or bias bounds are obtained by these methods. The current study resolves this problem, assuming that the time-varying plant can be excited by a user-defined signal, a multisine in this paper. The use of multisine excitations has been shown to provide valuable nonparametric frequency-domain information on dynamic systems, using simple algorithms [10,13,5]. In this paper, it is shown how a closer scrutiny of the spectral response of the considered class of time-varying systems to multisines allows for one to ascertain whether the system is time-varying. Next, the response to the multisine is used to estimate the system function (describing the dynamics) in the measured time window, provided that the conditions of identifiability are satisfied. This estimate is non-parametric in the frequency domain and parametric in the time domain. This means that, at each userdefined excited frequency (drawn from a discrete set), the system function is given as a continuous function of time. This can be construed as being complementary to the well known recursive identification methods [7], [8]. These identify the instantaneous dynamics of the system parametrically, but at discrete time instants. Parametric identification techniques for linear time-varying systems exist, both in the time domain [14], [15], [16] and in the frequency domain [6]. The systems are usually described by differential (or difference) equations with time-dependent coefficients. W.r.t. the parametric methods, the contribution of the non-parametric approach proposed in this paper is threefold, i) choice of an appropriate model structure, ii) provision of good starting values and iii) validation of parametric models. 2

3 The remainder of this paper is organised as follows. Section 2 describes the system model and the excitation signal considered. Section 3 interprets the output spectrum of a time-varying system excited by a multisine and relates it to the system function. In Section 4 the maximum likelihood estimator is constructed for extracting the system function from noisy data. Section 5 discusses the numerical conditioning of the method, the possible trade-offs, the estimation of the noise variance, and the detection of modelling errors. Section 6 illustrates the introduced methods on a simulation example and Section 7 draws some conclusions. 2 Problem formulation 2.1 A model for linear time-varying systems By adopting the conventions in [17], the model class considered consists of the systems denoted G V, the response y(t) of which, to an arbitrary excitation signal u(t), can be computed as y(t) = G V {u(t)} t g V (t, τ)u(τ)dτ. This system s behaviour is described by a time-varying impulse response g V (t, τ), is linear w.r.t u(t) and y(t), and is causal. Definition 1 (System function) The system function G V (jω, t) associated with the causal, linear, time-varying system with impulse response g V (t, τ) is given by G V (jω, t) = 0 g V (t, t τ)e jωτ dτ. The system function can be written as a series expansion G V (jω, t) = G p (jω)b p (t). (1) p=0 with {b p (t)} a complete set of basis functions over the interval denoted [0, T ]. From [17], the system function relates the Fourier transform U(jω) of the input signal to the output signal y(t) as y(t) = 1 G V (jω, t)u(jω)e jωt dω 2π = G p {u(t)}b p (t) p=0 (2a) (2b) with the operator G p {u(t)} = 1 2π G p(jω)u(jω)e jωt dω (obtained by plugging (1) into (2a)). As such, G p {u(t)} is recognised as the response of an LTI system to the signal with spectrum U(jω). For practical reasons, the system 3

4 to be identified is assumed to be described by a truncated version of the series (2b), viz. the following definition. Definition 2 (System model) The behaviour of the system considered is described by N p y(t) = G p {u(t)}b p (t), t [0, T ], (3) p=0 with u(t) and y(t) denoting the input and output signals respectively. G p {u(t)} denotes the response of an LTI system to the signal u(t). Assumption 1 The LTI systems G 1,..., G Np are asymptotically stable. The goal of this paper is to estimate non-parametrically the frequency response functions (FRF) of the LTI systems G p (jω) (i.e. at a discrete set of frequencies) for a given set of basis functions {b p (t)} from measurement data. This yields an estimate of the system function, which is non-parametric in the frequency domain. Measurements of the input and output signals are assumed to be available in the measurement window given by t [0, T ]. The input signal u(t) is assumed to be a multisine. The validity of the estimated system function is restricted to the measured time window and to the chosen discrete set of excited frequencies of the multisine, as explained in the next section. 2.2 The multisine as an excitation signal Definition 3 The discretised angular frequency corresponding to a record length T is given by ω k = 2πk T, k Z. Note that ω k corresponds to the kth frequency of the Discrete Fourier Transform (DFT) of a discretised signal of length T (in seconds). The index number k will be referred to as the kth DFT bin. Definition 4 (Multisine) A multisine signal is the continuous-time signal given by u ms (t) = 1 π Ã(k e ) cos(ω ke t + ϕ ke ) (4) N e k e K exc where K exc ([1, T fs ] N) is the user-defined discrete set of excited frequency 2 bins, and f s is the sampling frequency of the acquisition channel used. The user-defined discrete function Ã(k) determines the amplitudes of the sines. Their phases ϕ k are randomly distributed between π and π. The constant N e is equal to the number of elements in K exc. 4

5 1 The scaling factor Ne in (4) renders the RMS of the multisine independent of the number of excited frequencies. The multisine consists of a sum of cosines, the frequencies of which are all multiples of the same fundamental angular frequency ω 0 = 2π. For further notational simplicity, define the complex numbers T A(k) Ã(k)ejϕ k Ne for k > 0 (5) A( k) Ã(k)e jϕ k Ne The frequency-domain representation of this multisine is given by discrete points. Its Fourier spectrum is given by: U ms (jω) = k e ±K exc A(k e )δ(ω ω ke ) where δ( ) is the Dirac delta function. Note that A(k) = Ā( k) (where Ā denotes the complex conjugate of A). Definition 5 (Windowing) In this paper, applying a window to a signal x(t) means that the signal is multiplied by a rectangular window of length T, viz.: x T (t) is denoted the windowed signal. x(t) for 0 t T x T (t) = 0 for t < 0 t > T The Fourier spectrum of the windowed multisine (4) evaluated at ω k is T U ms,t (jω k ) = A(k) k ±K 2π exc (7) 0 k Z \ ±K exc Assumption 2 (Band-limited excitation) The user-defined amplitudes Ã(k) are set to zero beyond a certain frequency bin, k max. As such, the signal is bandlimited, giving U ms,t (jω k ) = 0 for k Z : ω k πf s, and aliasing is avoided when sampled at the frequency f s. Besides, some additional frequencies inside the frequency band of interest are not excited (i.e. A(k ne ) = 0, k ne Z \ ±K exc ). (6) 2.3 Spectrum and assumptions on the basis functions Definition 6 The normalised Fourier transforms of the windowed basis functions are defined at the DFT frequencies ω k = 2πk as B T p(jω k ) 1 T T 0 b p(t)e jωkt dt. The normalisation (division by T ) is introduced to allow for a notational simplification later on. As a notational convention, k e (resp. k e +) denote the 5

6 first excited frequency bin left (resp. right) of k e, viz.: k e = max exc : k < k e } k (8a) k e + = min{k K exc : k > k e } k (8b) Define ω ke ω ke + ω ke as the difference between a pair of consecutive excited angular frequencies of the multisine. Analogously, define k e k e + k e. The following assumptions on the basis functions are made: Assumption 3 (Band-limitation) b p (t) are smooth functions of time, such that their spectral content is highly concentrated at the low frequencies (compared with the density of the excited frequency grid). At relatively high frequencies, i.e. in the frequency band ], 3 2 ω k e ] [ 3 2 ω k e, [, the Fourier spectra B p (jω k ) are small and smooth. Thus, they can be approximated well by polynomials in k of order denoted N tr. This must be valid k e K exc. Remark 1 In this paper, a function is approximated well by a polynomial if the approximation error can be neglected w.r.t. the disturbing noise, which will be introduced later on. Assumption 4 (Linear independence) k e K exc, the discretised functions in the set {B 0 (jω k ),..., B Np (jω k ), (9) B 0 (jω k j ω ke ),..., B Np (jω k j ω ke ), B 0 (jω k + j ω ke ),..., B Np (jω k + j ω ke ), 1, jω k,..., (jω k ) Ntr } are linearly independent in the discretised frequency band k [ 3 2 ω k e, 3 2 ω k e ] Z. Assumptions 3 and 4 are required for the system to be identifiable, as will be discussed further on. Note that these two assumptions imply the presence of a significant (dependent on N p ) number of unexcited frequencies between any pair of excited ones. P 1 intermediate unexcited frequencies are created in a robust fashion by measuring P periods of a multisine. This assumes the repeatability of the signal generator. 6

7 Y (db) frequency (Hz) Bp (db) frequency (Hz) Bp (rad) frequency (Hz) Fig. 1. Left: Example output spectrum of a time-varying system. Centre and right: Spectra of first 4 time-scaled and shifted Legendre polynomials (17), amplitude and phase. 3 Output spectrum model 3.1 Constructing the noiseless response of the time-varying system to a multisine It can be shown (Section in [10]) that the Fourier transform of the windowed response (see Definition 5) of G p excited by u ms (t) is Y Gp,T (jω k ) = G p (jω k )U ms,t (jω k ) + T Gp (jω k ) (10) where subscript T indicates that it concerns a transient response, and T Gp (jω k ) is a rational form in jω k which is equal to zero if the system G p attained a steady-state. The spectrum of the windowed pth term in (3) is computed as Y p,t (jω k ) = [ Y Gp,T (jω) ] [T B p (jω)] ω=ωk (11) = Y p (jω k ) + T Y,p (jω k ) Y p (jω k ) = [G p (jω k )U ms,t (jω k )] [T B p (jω)] ω=ωk 1 (12) T Y,p (jω k ) = T Gp (jω k ) [T B p (jω)] ω=ωk (13) T Y,p (jω k ) is also a rational form in jω k (see Appendix A). Equation (12) is elaborated as follows. Since T B p (jω) is the spectrum of a windowed signal, G p (jω k )U ms,t (jω k ) can be replaced in (12) with the spectrum of the not windowed response G p (jω)u ms (jω). This gives Y p (jω k ) = T 2π B p (jω k jω )G p (jω )U ms (jω )dω = T A(k e )G p (jω ke )B p (jω k ke ) (14) 2π k e K exc The following theorem is proven by determining Y (jω k ) = N p p=0 Y p,t (jω k ). Theorem 1 The spectrum of the windowed response (see Definition 5) of the time-varying system, described by (3), and satisfying Assumptions 1, 3 and 4 to a multisine excitation (4) satisfying Assumption 2 consists of summed, 7

8 scaled and frequency-shifted copies of the spectra of the basis functions: Y (jω k ) = N p p=0 k e K exc θ p,ke B p (jω k ke ) + T Y (jω k ), (15) θ p,ke = T/(2π)A(k e )G p (jω ke ) (16) and T Y (jω k ) = N p p=0 T Y,p (jω k ), with T Y,p (jω k ) a rational form in jω k, given by (13). Note that the spectrum model (15) is linear-in-the-parameters θ p,ke. As such, these parameters could be extracted from the output spectrum using a simple Linear Least Squares (LLS) algorithm. This will be elaborated further on. Fig. 1, left is an example spectral response of a time-varying system. It consists of adjacent spectra of the basis functions, as given by Theorem 1. The time domain basis functions were given by b p (t) = P p (2t/T 1), p = 0,..., N p (17) (with P p (t) the pth order Legendre polynomial), and illustrated in Fig. 1, centre and right, in the frequency domain. Remark 2 The signal model in Theorem 1 is valid if the multisine excitation is applied within the time window (Definition 5). The value of the excitation signal outside this time window is of no importance and may be zero. Thus, the time of the experiment is limited to the length of the measurement window. Transient effects (at the beginning and the end of the window) are captured by T Y (jω k ). 3.2 Working with sampled signals In practical situations, the signals are available as sampled data. The required Fourier transforms can be approximated by the DFT (Discrete Fourier Transform): N 1 X(jω k ) T s n=0 x(nt s )e jω knt s X(k) (18) where x and X are replaced by u ms, y, b p /T and U ms,t, Y, B p respectively. N = T f s is the number of sampled points in the record. An exact relation is given by X(k) = X(jω k k N), (19) k = 8

9 which repeats the actual spectrum around each multiple of f s. Since u ms is perfectly band-limited (Assumption 2), (19) yields that U ms (k) = U ms (jω k ) for k < N/2. From Theorem 1 and Assumptions 2 and 3, the spectrum Y (jω k k N) for k 0 is smooth inside the excited frequency band. This justifies the following assumption. Assumption 5 For ω k πf s : Y (k) = Y (jω k ) + δ a (jω k ) with δ a (jω k ) = Y (jω k k N) k = k 0 (20a) (20b) where δ a (jω k ) can be approximated by a polynomial in k of order N tr. 3.3 Relating the noiseless output spectrum model to the non-parametric model Combining (1), (2), (7) and (16), the system function is elaborated at the excited frequencies as G V (jω ke, t) = N p p=0 θ p,ke U ms,t (jω ke ) b p(t), k e K exc (21) which can be computed as soon as the θ p,ke s are estimated from the output spectrum. 4 System function estimation from noisy data 4.1 Noise assumptions It is assumed that only the spectrum of the windowed output signal is corrupted by noise, viz.: U m (jω k ) = U ms,t (jω k ) Y m (jω k ) = Y (jω k ) + N y,t (jω k ), (22a) (22b) where the subscript m denotes a measured signal. N y,t (jω k ) is the Fourier transform of a windowed, coloured noise source. It can be shown [11] that N y,t (jω k ) = N Y (jω k ) + T H (jω k ) (23) 9

10 with N Y (jω k ) H(jω k )E(jω k ), and E(jω k ) the Fourier transform of the windowed, band-limited, white noise source. The transient term T H is, just like T Y (jω k ), a rational form in jω k. H(jω k ) is the FRF of a stable LTI system. Define the set of DFT bins lying within three consecutive excited bins around k e K exc as [ K ke = k e k e 2, k e + + k ] e Z, (24) 2 and denote N ke the number of bins in K ke. Assumption 6 T Y (jω k ) and T H (jω k ) are smooth functions that can be approximated well by polynomials in k of order N tr in the frequency band F ke = {ω k k K ke }, with K ke defined in (24). The approximation error will be taken into account further on. Denote σ 2 Y (k) = E { N Y (jω k ) 2 } as the variance of the noise at the DFT bin k. Assumption 7 The disturbing noise on the output spectrum N Y is circular complex, normally distributed. It is uncorrelated over the frequency, viz.: E { N Y (k)n Y (ν) } = δ kν σ 2 Y (k) (25) (where δ kν = 1 iff k = ν and 0 otherwise), and it is white inside K ke, that is k e K exc : σ 2 Y (k) = σ 2 Y (k e ), k K ke The assumption of whiteness requires the noise power spectrum to be piecewise constant in partially overlapping frequency bands. This is approximately true for a dense excited frequency grid and a smooth noise power spectrum. The error is proportional to the first derivative of σ Y, as mentioned further on. 4.2 LLS algorithm for estimating θ p,ke θ p,ke is estimated from the limited frequency band F ke. The following procedure is performed k e K exc. The output spectrum in F ke is modelled as: N p Y ke (k, Θ ke ) = θ p,k B p (jω k k ) + I tr (k) (26) k {k e,k e,k e + } p=0 such that only the contributions of the skirt shaped spectra centred around the three excited frequencies inside F ke are taken into account explicitly. The other smooth contributions are captured by the polynomial I tr (k) = N tr n=0 tr n k n. That is, I tr (k) captures Y ke,k e + (jω k) + T H (jω k ) + T Y (jω k ) + δ a (jω k ). (27) 10

11 In this equation, T Y, T H and δ a are defined in Theorem 1, equation (23), and equation (20b) respectively, and Y ke,k (jω e + k) = θ p,k B p (jω k k ) (28) k K exc\{k e,k e,k e + } p=0 is the contribution inside F ke of the basis spectra centred around the excited frequencies outside F ke. As per Assumption 3, these contributions are approximated by a polynomial. Remark 3 Note that the spectrum model (26) includes the noiseless spectrum from Theorem 1 and the smooth leakage contribution of the noise, T H. The parameter vector Θ ke to be identified stacks θ p,k for p {0,..., N p } and k {k e, k e, k e +}, and the coefficients of I tr (k) in a column vector, and is estimated as the following linear least squares minimiser N p ˆΘ ke = argmin e ke (Θ ke ) H e ke (Θ ke ), (29) Θ ke with e ke (Θ ke ) a column vector stacking the residuals ε ke (k, Θ ke ), k K ke, with ε ke (k, Θ ke ) = Y m (k) Y ke (k, Θ ke ). (30) As such, the residuals in (30) (evaluated in the actual parameter vector Θ 0,ke ) equal the sum of i)the stochastic part of the noise, N Y (jω k ), and ii) the bias terms of the orders of magnitude of the remainders of the Taylor series expansions of the terms in (27): ε(k, Θ 0,ke ) = N Y (jω k )... + O ( Y ke,k + T ) ( ) Ntr+1 (Ntr+1) Nke f s e + H + T Y jωke N ( ) Ntr+1 + O δ a (jω ke ) (Ntr+1) Nke f s. (31) N (where x (n) is the nth derivative of x). According to assumptions 3, 5 and 6, and consistent with Remark 1, the bias contributions in (31) can be neglected. Theorem 2 (Maximum likelihood estimator) Under Assumptions 6 and 7 on the disturbing output noise, the minimiser ˆΘ ke in (29) is the maximum likelihood estimate (MLE) of Θ ke, yielding the MLE of the system function G V (jω ke, t) in (21) of the system described by (3), under the assumptions of Theorem 1. 11

12 Proof. The minimiser of (29) is equal to the maximiser of V ml (Θ ke ) = e ε(k,θ ke ) 2 σ Y 2 (k) (32) k K ke Since the bias terms in (31) can be neglected (Assumptions 3, 5 and 6), the residuals ε(k, Θ ke ) are samples of a gaussian, uncorrelated, white noise source (Assumption 7), such that (32) is the maximum likelihood cost function. Assumptions 1 and 4 are required for the system to be identifiable. Note: if the noise is not normal and not white, the estimator is no longer maximum likelihood but remains unbiased. 5 Discussion 5.1 Numerical conditioning An alternative cost function might be built, not restricting the frequency band to three adjacent excited frequencies, but by including the complete frequency band of interest at once. The advantage of limiting the frequency band to F ke is twofold: i) a significantly smaller regression matrix, and ii) smaller bias terms in (31), as these are proportional to N ke. Assumption 4 on the spectra of the basis functions ensures a unique solution of (29). In practice, this assumption can be checked by inspecting the rank of the regression matrix, associated with the linear model (26). It has been experienced that the use of Legendre polynomials as basis functions b p (t) yields a good conditioning of the regression matrix. It is significantly better (i.e. the condition number is lower) than when using simple monomials in t, as illustrated in Fig. 2, left, for different degrees of the time variation N p. Increasing the order N tr of the polynomial I tr (k) in (26) yields a slight increase of the condition number (comparing triangles with circles), but a decrease of the bias terms, as seen in Fig. 2, right. 5.2 Trading frequency resolution for noise variance and speed of variation The number of excited frequency lines, N e, is a design parameter which should be chosen while keeping in mind the following trade-offs. More excited frequencies (for fixed T ) yield a higher frequency resolution of the estimated system function, but a lower signal-to-noise-ratio (SNR) at each individual frequency 12

13 cond reg. matrix Monomials ց տ Legendre poly. Rel. RMSBE θ (%) N p N p Fig. 2. Dots: Monomial basis functions, N tr = 9. Circles: Legendre polynomial basis functions, N tr = 4. Triangles: Legendre polynomial basis functions, N tr = 9. k e = 30 for the three plots. Left: Condition number of the regression matrix for increasing N p. Right: Relative RMS bias error of the estimated parameters θ p,ke. line. This is due to the factor 1/ N e in (4). Also, a denser excited frequency grid yields a decreased N ke and, thus, limits the order N p of the series expansion of the system function (3). 5.3 Noise variance on the estimated system function Theorem 3 Under the assumptions of Theorem 2, the noise variance of the estimated system function (21) is: σ 2 G V (k e, t) = b(t)c θ LS,ke b(t) T U ms,t (jω ke ) 2 (33) where b(t) T [b 0, b 1 (t),..., b Np (t)] and C θls,ke is the noise covariance matrix of the least squares estimates of the parameters θ p,ke, p = 0,..., N p in (29) at a given excited bin k e. Proof. Follows from (21). The covariance matrix C θls,ke is obtained as the N p N p matrix, starting and ending, respectively, at the (N p + 1)th and 2N p th diagonal elements of C Θke, the covariance matrix of the linear least squares estimate ˆΘ ke. C Θke is obtained in a straightforward fashion, see Section 6.2 in [1]. An estimate for σy 2 (k) can be obtained from the least squares residuals, as in [5]. Remark 4 Since in general C θls,ke is non-diagonal, the estimated parameters are correlated. In addition, since the estimates of Θ ke at subsequent frequencies are obtained using overlapping frequency bands, the estimates of the system function are correlated over the frequency too. The correlation length is ±2 excited frequencies. 13

14 5.4 Detecting modelling errors The required order N p of the series expansion can be determined by detecting the presence of modelling errors. This is done by using a whiteness test of the residuals (as previously used for instance in [12]). The sample auto correlation of the normalised residuals is given by ˆR ke,εε(m) = k K \m ke ε ke (k, ˆΘ ke )ε ke (k + m, ˆΘ ke ) σ Y (k)σ Y (k + m) (34) (with m < N ke ) where K \m k e is the ordered set K ke excluding the last m elements for m > 0 and the first m elements for m < 0. If no model errors are present, the residuals ε ke have the following covariance matrix: P ke,[k,l] = E { εke (k, Θ 0,ke ) σ y (k) } ε ke (l, Θ 0,ke ) σ y (l) (35) which, for a linear least squares regression with i.i.d. residuals is easily computed, see Section 6.2 in [1]. The expected value of ˆR ke,εε(m) is obtained as (subscript k e omitted for notational clarity) E { ˆRεε (m) } = Its variance is (proof in Appendix B) var { ˆRεε (m) } = k K \m ke k,l K \m ke P [k,k+m]. (36) P [k,l] P [k+m,l+m] (37) Since ˆR εε (0) is real-valued and is a sum of equally distributed terms, it is (very close to) normally distributed (for N ke sufficiently large). As such, confidence bounds from (37) are easily obtained. This gives the possibility to detect modelling errors, as demonstrated further on. 6 Simulation results The identification algorithm was applied to simulation data of a system described by (3) with N p = 3. The basis functions b p (t) were the first four, time scaled, and shifted Legendre polynomials (17). The true (black thin lines) and estimated (grey crosses) G p (jω ke ) are given in Fig. 3. Ten periods of an odd multisine with a period length of T ms = 256s was applied in the measurement 14

15 0 G p (d B ) Frequency (Hz) Fig. 3. Actual and estimated FRF of G p at ω k, k K exc (thin black lines and grey crosses respectively). Thick black lines: variances on G p. RMS error on the estimated G p, for N tr = 5 (grey dots) and N tr = 8 (white full lines). ˆRsh.(ke) k e (bins) Fig. 4. Normalised correlation at lag 0 (given by (38)), grey circles for N tr = 5, black crosses for N tr = 8. Horizontal grey lines: 95% (top line) and 50%(bottom line) confidence bounds. window t [0, T ], T = 10T ms, and u(t) = 0 for t / [0, T ]. The output signal was disturbed by coloured, band-limited noise such that the time domain SNR was 40dB. In Fig. 3, the RMS error (over 100 realisations) is significantly higher than the variance on G p in the lower frequency band for N tr = 5, indicating modelling errors. For N tr = 8, the variance and the RMS error coincide. Modelling errors are rapidly detected by inspecting the shifted correlation of the residuals at lag 0, given by ˆR sh. (k e ) = ˆRke,εε(0) E { ˆRke,εε(0) }, (38) from (34) and (36), and depicted in Fig. 4. For N tr = 8, ˆR sh. satisfies the confidence bounds, while for N tr = 5 it doesn t in the lower half of the frequency band. Thus, N tr = 5 is too low in that frequency band. The order N P can be estimated in a similar fashion. 7 Conclusions 1 In this paper a method has been presented for extracting the system function of time-varying systems. These systems were approximated by a truncated series expansion, which consisted of a parallel connection of LTI systems followed by 15

16 user defined varying gains. The system is identified non-parametrically in the frequency domain. As such it provides valuable information on the evolution of the system s dynamics with time. Tools for detecting modelling errors were provided. Simulation results were in agreement with the presented theory. References [1] D. R. Brillinger. Time Series: Data Analysis and Theory. McGraw-Hill, New York, [2] S. Conforto and T. D Alessio. Spectral analysis for non-stationary signals from mechanical measurements: a parametric approach. Mechanical Systems and Signal Processing, 13(3): , May [3] A. Fujimori and L. Ljung. Model identification of linear parameter varying aircraft systems. In Proc. IMechE, volume 220 Part G: JAERO28, pages , [4] J. K. Hammond and P. R. White. The analysis of non-stationary signals using time-frequency methods. Journal of Sound and Vibration, 190(3): , [5] J. Lataire and R. Pintelon. Estimating a non-parametric, colored noise model for linear, slowly time-varying systems. IEEE Trans. on Instrumentation and Measurement, 58(5): , May [6] J. Lataire and R. Pintelon. Frequency-domain weighted non-linear least-squares estimation of continuous-time, time-varying systems. IET Control Theory & Applications, 5(7): , [7] L. Ljung and T. Söderström. Theory and Practice of Recursive Identification. MIT Press, Cambridge, [8] M. Niedzwiecki. Identification of Time-Varying Processes. J. Wiley & Sons, Chichester, [9] B. Picinbono. Random Signal and Systems. Prentice-Hall, Englewood Cliffs, [10] R. Pintelon and J. Schoukens. System Identification A Frequency Domain Approach. IEEE Press, Piscataway, [11] R. Pintelon, J. Schoukens, and P. Guillaume. Continuous-time noise modelling from sampled data. IEEE Trans. on Instrumentation and Measurement, 55(6): , Dec [12] R. Pintelon, J. Schoukens, and Y. Rolain. Uncertainty of transfer function modelling using prior estimated noise models. Automatica, 39(10): ,

17 [13] J. Schoukens, R. Pintelon, T. Dobrowiecki, and Y. Rolain. Identification of linear systems with nonlinear distortions. Automatica, 41(3): , [14] M.D. Spiridonakos and S.D. Fassois. Parametric identification of a timevarying structure based on vector vibration response measurements. Mechanical Systems and Signal Processing, 23(6): , [15] M.K. Tsatsanis and G. B. Giannakis. Time-varying system-identification and model validation using wavelets. IEEE Trans. on Signal Processing, 41(12): , Dec [16] M. Verhaegen and X. Yu. A class of subspace model identification algorithms to identify periodically and arbitrarily time-varying systems. Automatica, 31(2): , [17] L.A. Zadeh. Frequency analysis of variable networks. Proceedings of the IRE, 38(3): , March A System transient term Following the lines in Section in [10], T Gp (s) is analytically given in the complex plane by T Gp (s) = T 1,Gp (s) e st T 2,Gp (s) (A.1) with T 1,Gp (s) and T 2,Gp (s) rational functions in s. Under Assumption 3, b p (t) is a smooth function of time and, thus, can be approximated by a low order polynomial of order, say, R. Since a multiplication in the time domain by a monomial in t is a derivative w.r.t. s in the Laplace domain, T Y,p (jω k ) is analytically given by { } R T Y,p (s) = L L 1 {T Gp (s)} γ r t r r=0 R = γ r ( 1) r dr ( r=0 ds r T1,Gp (s) e st T 2,Gp (s) ). (A.2) Working out the derivatives and evaluating the result in jω k (noting that e jωkt = 1, k Z) yields a rational function in jω k. Note that, from (10), L 1 {T Gp (s)} is windowed. This allows b p (t) to be written as a non-windowed polynomial. 17

18 B Variance of the correlation of the residuals The variance of ˆR εε (m) is computed by elaborating (subscript k e omitted for notational clarity) var { ˆRεε (m) } ( ) ε(k)ε(k + m) 2 = E k σ Y (k)σ Y (k + m) P k,k+m (B.1) Since ε(k) is circular complex normally distributed, from Section 4.6 in [9], we have that E { ε σ (k)ε σ (k + m)ε σ (l)ε σ (l + m) } = E { ε σ (k)ε σ (k + m) } E { ε σ (l)ε σ (l + m) } + E { ε σ (k)ε σ (l) } E { ε σ (k + m)ε σ (l + m) } = P k,k+m P l,l+m + P k,l P k+m,l+m, (B.2) (with ε σ (k) = ε(k) ). Elaborating (B.1) by using (B.2) gives (37). σ(k) 18