Frequency Domain Total Least Squares Identification of Linear, Periodically Time-Varying Systems from noisy input-output data

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1 Frequency Domain Total Least Squares Identification of Linear, Periodically Time-Varying Systems from noisy input-output data E. Louarroudi*, J. Lataire* and R. Pintelon* *Department of Fundamental Electricity and Instrumentation (ELEC), Vrije Universiteit Brussel, Pleinlaan 2, B-15 Brussels, Belgium, (Tel: +32 () ; Fax: +32 () ; vub.ac.be). Abstract: This paper presents an extension of the well known linear time invariant identification theory to Linear, Periodically Time-Varying (LPTV) systems. The considered class of systems is described by ordinary differential equations with coefficients that vary periodically over time, making use of multisines both for excitations as well as for the time-varying system parameters. To solve the model equation, an efficient frequency domain simulator is built and is compared with the classically time integration solvers. Further, a frequency domain identification algorithm is proposed within an errors-in-variables stochastic framework. This approach determines a parametric model for the LPTV-system from noisy input-output data. The developed estimation theory is also verified on a simulation example. Keywords: periodically time-varying systems, frequency domain, multisine excitations, total least squares, errors-in-variables 1. INTRODUCTION In the last century, the Linear Time Invariant (LTI) assumption and the corresponding system identification techniques have been utilized successfully in practice (Ljung, 1999; Pintelon and Schoukens, 21). However, in some applications the measured real-life systems do not satisfy the time invariant conditions (Niedzwiecki, 2; Fujimori and Ljung, 25; Shin et al., 25; Allen 29). Due to this reason it is necessary to extend the framework of LTIsystems to systems whereby system properties might evolve over time. According to the nature of the time-variation, two different classes of time-varying systems can be distinguished. The first one contains systems that change arbitrarily with time. In this case, the time-varying part is dictated by the random nature of the system. Hence, the system properties change randomly over time. A few concrete examples where randomly time-varying systems could be found are: pitting corrosion in metal structures, (bio)chemical processes, the vibration modes of a bridge that change due to the increasing damage, the rotor dynamics of an helicopter and so on. The second class of time-varying systems contains what we call artificially time-varying systems. The time variation in such systems is established by external time-varying parameters, the so-called scheduling parameters. Applications are for instance: flight flutter analyses with the flight speed and flight altitude as external parameters, a moving robot with the length of the robot axis as scheduling parameter, an electronic circuit whose electric properties depend on an external voltage source, to name a few. All those applications cannot be treated in the classical sense. We therefore require a more systematic approach that is able to capture such time-varying processes at once. In this paper a stochastic approach is followed for the identification. One attempts to find a model that approximates the deterministic part of the system as close as possible, by selecting a criterion of fit that eliminates the disturbing noise as much as possible. When the time variation is forced experimentally, one has to select between periodic, arbitrary (non-periodic) or hold (piecewise constant time-variation) configurations for the scheduling parameter(s). In this article we will only focus on Single-Input Single-Output (SISO) system with imposed, periodically varying scheduling parameters, as is depicted schematically in Fig. 1. max min Time SCHEDULING PARAMETERS SYSTEM Fig. 1. Block schematic representation of a periodically time-varying system. The time-variation of the system is caused by changing the scheduling parameters periodically over time. The time signals & denote respectively the (undisturbed) input and output. It could be noticed from (De Caigny, 29) that the advantage of making use of the hold configuration setup is the reduction of the modeling complexity. In those circumstances only LTI-models should be identified, at the cost of an increase in experimental time compared with the other two approaches. A set of LTI-models are created by estimating a transfer function model at every point on the scheduling parameter grid. The poly-topic model might then be obtained via an interpolation scheme that describes the estimated LTI-parameters as a function of fixed operation points (De Caigny, 29). This methodology has the shortcoming that the estimated poly-topic model strongly depends on the numerical algorithm that has been employed for the interpolation. u(t) y(t) Copyright by the International Federation of Automatic Control (IFAC) 13115

2 On the other hand, the aim of imposing non-constant timevariation (periodically or non-periodically) is to reduce the experimental time significantly, with the drawback that a more complex model (time-varying parameters) must be identified with respect to LTI-systems. As elaborated in (Fujimori and Ljung, 25), the above-mentioned drawbacks can be circumvented by building one discrete Linear Parameter Varying (LPV) model. It is assumed that the LPV system consists of scheduling parameters that are constant for each sampling interval. The continuous LPV-model is then achieved via a weighted sum of the discrete LPV-model. The difficulty in this approach is that the piecewise constant progress of the scheduling parameters may be hard to realize in practice, so that modeling errors could occur when the ideal situation is not met. For example in the case of flight flutter analyses, the flight velocity should be varied in a continuous way (e.g. linear variation), otherwise abrupt accelerations will take place. Nevertheless, from an identification point of view, all the previously discussed (periodically) time-varying systems (randomly and artificially time-varying) belong to the same group. The system is seen as a black box with measurable input and output channels. In practice, the scheduling parameters are assumed to be measurable quantities over time (Fujimori and Ljung, 25, Bamieh and Giarre, 22). As soon as the time-varying model parameters are available from an identification scheme, and if, in addition, measurements of the scheduling parameters are available; the parameters are also known as a non-parametric function of the scheduling parameters. Consequently LPV systems could also be treated in this framework without any loss of generality. In contrast to the classical LPV-approaches, the black-box framework has the supplement that it is able to account for the presence of the possible dynamics among the scheduling parameters and the varying system properties. The purpose of this work is to extract the dynamics by scanning the whole range of scheduling parameters in only one well-designed experiment within an errors-in-variables framework. So that the linear time-varying framework would be providing us a great benefit, as we do not require running multiple (expensive) experiments anymore. There exist a variety of model structures in the literature to describe Linear Periodically Time-Varying (LPTV) systems. Three common model representations either in continuoustime or discrete-time are exploited in practice. We first have the state-space approach with the corresponding Harmonic Transfer Function (HTF) operator in the frequency domain as described thoroughly in (Wereley and Hall, 2). Furthermore, the input-output formulation via the concept of impulse response or kernel has also been utilized frequently for the identification both in time and frequency domain (Sams and Marmarelis, 1988; Yin and Mehr, 29-21). Another elegant description for LPTV systems is an ordinary differential (difference) equation in either time or frequency domain with periodically time-varying parameters (Mehr and Chen, 21). In this paragraph we briefly sum up some existing identification methodologies for diverse LPTV models. In (Yin and Mehr, 29) a non-parametric estimate of the finite impulse response (FIR) of discrete LPTV-systems is proposed in the frequency domain. This non-parametric algorithm is extended from the FIR case to parametric modeling (infinite impulse response) of discrete time LPTV systems (Yin and Mehr, 21). Moreover, a non-parametric estimate of the Harmonic Transfer Function (HTF) has been derived in (Shin, 25) by handling the continuous LPTVsystem as an input-modulated Multi-Input Single-Output (MISO) LTI system. While in (Sams and Marmarelis, 1988), the basis expansion technique of the time-varying kernel is worked out as a tool for identifying continuous LPTVsystems using white noise inputs. Ultimately, a frequency domain Multi-Input Multi-Output LTI identification method making use of lifting is given in (Allen, 29) for continuous state space LPTV-systems with free response. All the identification schemes referenced above and the references therein have the limitation that the input must be known exactly. In this paper we opt for a parametric continuous-time band-limited identification setup that provides consistent estimates within an errors-in-variables stochastic framework in the frequency domain (both input and output signals are disturbed by noise). An added value of using this proposed framework is its polyvalence. The identification algorithm works for discrete-time systems as well (see section 2.2). The rest of this paper is organized as follows. Section 2 describes the class of systems that will be considered. The assumptions that will be made during the transformation and the usefulness of a multisine excitation will also be briefly mentioned. The problem of calculating the system response from a given input spectrum and system parameters is treated in this section as well. The theoretical background on frequency domain modeling is explicitly studied in section 3. This technique will be applied on simulations in section 4 before we draw our general conclusions in section 5. Some notational convention will be made to make the article easy reading. Continuous time domain scalars will be denoted by lowercase letters (e.g. ). While their corresponding frequency domain representations are written by uppercase letters. The k th -bin of the sampled Fourier transform of is denoted by Ω, with Ω = the complex discrete frequency variable, and the length of the time window. The k th Fourier coefficient of a periodic signal = + is designated by. Whereas the Discrete Fourier Transform (DFT) of a sampled (periodic) signal will be denoted by, with the sampling period. The differences in frequency domain representations for periodic signals are thus indicated by their arguments (e.g. Ω, or ). Vector and matrix quantities will be written down as boldface symbols (e.g., ). From the context it will be clear whether a vector or matrix is meant. 2.1 Considered Model Class 2. MODEL DESCRIPTION The category of systems that will be discussed here are linear, SISO, periodically time-varying systems, which can be well

3 described by ordinary differential equations with periodically time dependent parameters & : = with & R respectively the undisturbed input and output signals. The periodically time-varying parameters & R in (1) are approximated by means of a finite Fourier series (, are replaced by respectively, or, ): (1) =, (2) with Ω = C the fundamental frequency of the system parameters. Constraints are enforced on the complex periodic coefficients, &, C by the constraint matrix R to guarantee that the estimated time-dependent parameters & are real (see section 3 for a detailed analysis). 2.2 Frequency Domain Solver and Related Assumptions Till now, most of the identification techniques developed, have been carried out either in the time or frequency domain starting from a control perspective (Allen, 29; Shin, 25; Yin and Mehr, 29-21; Verhaegen and Yu, 1995; Fujimori and Ljung, 25), to name a few: the input is known exactly and only the output observations are noisy. Contrary to these methods, we will handle the more general case where all the observations (input and output signals) are corrupted by noise in the frequency domain. There are three major benefits linked with frequency domain identification. First, the frequency band of interest can be chosen such that it covers the dominant dynamics of the system important for the intended application. Second, a non-parametric noise model, obtained from the successive periods of the noisy data as will be explained in section 3, can easily be handled in the frequency domain. Finally, continuous-time models can be implemented as easy as discrete-time models. Before (1) is transformed to the frequency domain, we mention some assumptions being made on the excitation. First of all, we suppose that the input signal is a band-limited, periodic random phase multisine with harmonics: of the multisine (3) is synchronized with the system parameters (2) and is measured in steady state, such that no transient terms arise when transforming (1). If the abovementioned assumptions are valid, (1) is converted without any difficulty to the frequency domain by applying the Fourier transform properties (see Appendix A). We get:, Ω Ω Ω Ω 1 5 =, Ω Ω Ω Ω where Ω & Ω denotes respectively the windowed input-output spectra, sampled at the discrete frequencies Ω =. After evaluation of model (4) at the sampled frequencies Ω = Ω,, Ω and with some rearrangements of the equations, (4) can be rewritten after grouping the equations together in a shorthand matrix equation as: = with & C band diagonal matrices whose band widths equal the respective length of the finite Fourier series of & parameters (2). The discrete Fourier coefficients of the input-output spectra that cover the entire frequency band of interest are stored respectively in the input-output DFT vector & C. It is obvious from (5) that the system response follows straightforwardly by solving a linear set of equations. The discrete-time counter part of model (5) derived in (Mehr and Chen, 21) is developed in the same way as here. The versatility of (5) is that we could easily shift from a continuous model to a discrete-time one by only setting Ω = in our computer program. Therefore, we are able to deal with both continuous as discrete-time LPTV systems within this proposed framework. The reader should notice that (5) is a special description of the truncated HTF C model of LPTV-systems. Indeed, if we plug in = in in (Wereley and Hall, 2) then the following relation should hold =, assuming that the inverse exists. output FDS output ode45 output FDS - output ode45 (5) (4) = 1 (3) where the phases in (3) are uniformly distributed in the interval, (Pintelon and Schoukens, 21). Frequency lines Ω that correspond to in (3) are called excited lines. is a scaling factor which is equal to the number of excited lines and ensures that the power of the excitation signal would be independent of the number of excitation lines. Multisine (3) has a discrete spectrum due to its periodicity and has the benefit that the user can design an optimal signal for the intended application by proper choice of the amplitude spectrum in (3). In this paper, the excitation and the system parameters have been chosen such that frequency domain model becomes as simple as possible. The output in (1) will be periodic if the base frequency Magnitude (db) Frequency (Hz) Fig. 2. Output spectrum in decibel as a function of frequency for FDS ( o ) and ode45 solver (., coincides with o ). The difference between FDS and ode45 solver is almost 2dB below the output spectrum (solid black line). Due to the band diagonal structure of &, the output spectrum for a given multi-sine input (3) is obtained in an efficient manner, since only elements different from zero with their corresponding indices have to be stored in the 13117

4 computer s memory. None of the papers cited above compare the solvers in either domain to show the powerfulness of the Frequency Domain Simulator (FDS). To give an idea about the time-efficiency of FDS (5) a simulation has been made and is being compared with the ode45 integration solver from MATLAB. Although the output spectrum of both solvers in Fig. 2 coincides very well for the same numerical precision (relative tolerance of 1 ), the computation time of FDS is almost 1 times smaller. FDS has also the advantage that no transients are included in (5), while the time integration solver suffers from transient effects which provides an increase in calculation burden. Therefore, we are obliged to wait till the influences of the initial conditions are vanished completely. 3. IDENTIFICATION SCHEME So far we have a model (4) and a method to solve it (5); in addition we need a noise model before we can set up an identification scheme. As mentioned in the previous section, a non-parametric noise model will be used in this paper because of its simplicity in the frequency domain. The noise model is assumed to be of the additive type: = + with = and = with the transpose of., are respectively the DFT vectors of the input and the output noise where its entries are zero mean, circular complex normally distributed and uncorrelated over the frequency, such that: E = = diag Ω,, Ω E = = diag Ω,, Ω E = = diag Ω,, Ω where E, denote the expected value and the complex conjugate transpose. Operator diag in (7) stands for a diagonal matrix formed by the elements of its argument. The aim of this paper is to estimate the parameter vector =,, from measurements (6) in such a way that the residue vector, the difference between the left and the right hand side of (5), is as close as possible in least square sense to the zero vector. Because model (4) is linearin-the-parameters, and keeping in mind that the measurements are disturbed by noise, the residue vector might be written in the following matrix notation: with = the Jacobian of the residue. To impose the constraints on, matrix equation (8) ought to be first moulded in its real version. We have: with = Re Im and = Re Im (9) Im Re, where Re & Im denote respectively the real and imaginary part of quantity (Pintelon and Schoukens, 21). Further, the following constraints: (6) (7) (8) Re, = Re, & Re, = Re, Im, = Im, & Im, = Im, Im, = Im, = (1) have to be enforced on, to make sure that the estimated parameters in appear perfectly in complex conjugate pairs. Constraints (1) are linear-in-the-parameters and, therefore, they can be represented in compact form by a real matrix : =, (11) with R and the modified parameters which should be estimated from measurements. Applying the constraints on the Jacobian, the modified Jacobian turns out to be: such that (9) becomes: = = =,. (12) (13) The estimated parameters can be obtained in a numerical stable manner through the Singular Value Decomposition (SVD) of the matrix. It can be shown that this total least square estimator is inconsistent and behaves poorly at low frequencies. To get consistency and to improve the lowfrequency fit at the same time, a left and right weighting matrix should be applied (Pintelon and Schoukens, 21): / / (14) with = EΔ Δ the column covariance of the weighted modified Jacobian and Δ =. This weighted generalized total least square estimate of (14) is calculated via the Generalized SVD of the matrix pair, /. The solution is then the right singular vector corresponding to the smallest generalized singular value (see chapter 7 in (Pintelon and Schoukens, 21) for LTI case). The optimal choice for the weighting in (14) is the inverse of the residue s variance: = diag 1,,, where the variance of the residue vector at frequency bin k could be calculated as the diagonal terms of : = + 2 Herm with Herm = (15) (see Appendix B). Note that variances and covariances of the data, & are needed to construct &. A nonparametric estimate of, & can be found explicitly in (chapter 2 eq. (2-31) of Pintelon and Schoukens, 21). This information is accessible if multiple periods of the input and output signals are measured in steady state. Since the value of in (15) is not known apriori, it should be constructed iteratively. The estimator (expression (14) without left weighting) could then provide the solution to initialize the iteration procedure. The just discussed identification scheme will be tested on simulations in the next section

5 Imaginary axis SIMULATION EXAMPLE Via a simple example the performance of the estimator, developed in section III is illustrated. The proposed system had a mean transfer function of order 1 in the numerator and order 2 in the denominator: Ω = Ω + 1/Ω + Ω + 1. s-plane: Time-Varying Poles and Zeros Real Axis (16) The transfer function parameters in (16) correspond to the DC-components of the Fourier series (2) of the simulated LPTV system. The length of the Fourier series in (2) was set respectively to = = 3. To get more feeling about the time variation that is selected, a pole-zero map of the instantaneous poles and zeroes is shown in Fig. 3. The instantaneous poles and zeroes are defined as the roots of (1) in the Laplace domain keeping the system parameters in (1) fixed at some time instant =. As is clear from this figure, the time-varying poles form closed loop curves because the system parameters in (1) are T-periodic functions, while the zeroes in Fig. 3 vary periodically on the real axis. The reader should keep in mind that the concept of instantaneous poles and zeroes are used here to represent in a simple graphical way the LPTV system. However, no conclusions concerning the stability may be drawn from this graph. If the system is slowly time-varying then the idea of instantaneous poles and zeroes could give an intuitive insight in the time evolution of the system s dynamics. The simulated system contains 34 free parameters and 243 frequency lines are used for the identification process. The LPTV-system itself was excited with a multisine consisting of 24 frequency lines with a base frequency that is chosen 1 times higher than that of the scheduling parameters. Moreover, the correlated input and output were disturbed respectively by colored noise of a mean SNR of 5dB and 3dB. The mean SNR is defined as: SNR = = E. (17) Fig. 3. The true time-varying poles and zeroes are given respectively by the black crosses x and circles o. While the estimated poles and zeros are depicted respectively by the grey crosses x and stars *. The results of the identification are illustrated by its predicted output spectrum in Fig. 4. As seen in this figure, the output comprises skirt-like signals around each excited line. This energy between the adjacent lines originates from the frequency domain convolution of the multisine (3) with the spectra of the parameters (2). Note that skirts in the LTI-case are not expected anymore. There are sharply concentrated around the excited lines. The width of the skirts is therefore a kind of measure for the speed of time variation (Lataire and Pintelon, 28). Magnitude (db) Fig. 4. The figure compares the standard deviation of the output spectrum (grey line) with the difference between the true and the estimated response (black line). An improvement of almost 15dB is obtained from the parametric identification scheme of part 3. Imaginary Axis Magnitude (db) frequency (Hz) Time-Varying Poles Real Axis Frequency (Hz) Fig. 6. The ratio of the normalized residue at each frequency / is given by the black crosses ( x ). They are randomly located about the grey db line ( - ). Y o Y o - Y est σ Y excited lines The quality of the estimation process is shown in Fig. 4 by comparing the standard deviation of the output spectrum with the difference between the true and the estimated response = at frequency line. An improvement of about 15dB can be observed from this picture. From Fig. 3 it can be seen that both the estimated instantaneous poles and zeroes match very well with the true ones. A zoom of the (estimated) poles from Fig. 3 are depicted in Fig. 5. Fig. 5. A zoom of the (estimated) time-varying poles from Fig. 3. To check the quality of the identified model, a whiteness test on the residue vector is performed. To show this, the residue vector and the standard deviation of the residue vector should lie close to each other. In Fig. 6 the ratio of both quantities at each frequency / are located around the db line. A probability test shows that 37 % of the magnitude of the normalized residue components / should lie outside the db line if no modeling errors are present (see formula (15) in Pintelon et al., 21). In our case 33% of the data in Fig. 6 is found outside this confidence bound, which confirms that no modeling errors are present. 1

6 5. CONCLUSIONS In this paper a frequency domain weighted generalized total least square estimator for LPTV systems has been developed within a stochastic errors-in-variables framework. We have seen that the use of multisine excitations simplifies the model equations drastically and furthermore it delivers us more insight into the behavior and the speed of time-variations. A time efficient method for computing the system s response in the frequency domain has been generated as well. It is much faster compared with the classical numerical time domain solvers. The identification scheme has been illustrated on a simulation example. ACKNOWLEDGMENT This work is sponsored by the Fund for Scientific Research (FWO-Vlaanderen), the Flemish Government (Methusalem METH1) and the Belgian Federal Government (IUAP VI/4). E. Louarroudi is on a Ph. D. fellowship from the Methusalem project. J. Lataire s work was supported by the Research Foundation-Flanders (FWO) under a Ph.D. fellowschip. REFERENCES Allen M.S. (29), Frequency-Domain Identification of Linear Time-Periodic Systems Using LTI Techniques, Journal of Computational and Nonlinear Dynamics, Vol. 4, Iss. 4, Article Number: 414-(1-6). Bamieh B., and L. Giarre (22). Identification of linear parameter varying models, Int. J. Robust Nonlinear Control, 12: De Caigny J., J.F. Camino and J. Swevers (29), Interpolating model identification for SISO linear parameter-varying systems, Mechanical Systems and Signal Processing, vol. 23, no. 8, pp Fujimori A., and L. Ljung (25), A Polytopic Modeling of Aircraft by Using System Identification, International Conference on Control and Automation (ICCA), Budapest (Hungary), June 27-29, pp Lataire J., and R. Pintelon (28). Frequency Domain Identification of Linear Deterministically Time-Varying Systems, Proceedings 17th IFAC World Congress, Seoul (South-Korea), July 6-11, pp Ljung L. (1999). System Identification: Theory for the User. Prentice-Hall, Upper Saddle River. Mehr A.S. and T. Chen (21), On Alias-Component Matrices of Discrete-Time Linear Periodically Time- Varying Systems, IEEE Signal Processing, vol. 8, pp Niedzwiecki M. (2), Identification of Time-Varying Processes. John Wiley & Sons, Chichester. Pintelon R., and J. Schoukens (21), System Identification: a Frequency Domain Approach. IEEE Press, Piscataway (USA). Pintelon R., Y. Rolain and W. Van Moer (23). Probability density function for frequency response function measurements using periodic signals, IEEE Trans. Instrum. and Meas., vol. 52, no. 1, pp Sams A.D., and V.Z. Marmarelis (1988), Identification of Linear Periodically Time-Varying Systems Using White- Noise Test Inputs, Automatica, vol. 24, no. 4, pp Shin S.J., C.E.S. Cesnik and S.R. Hall (25), System identification technique for active helicopter Rotors. Journal of Intelligent Material Systems and Structures, vol. 16, issue: pp Verhaegen M., and X. YU (1995), A Class of Subspace Model Identification Algorithms to Identifiy Periodically and Arbitrarily Time-Varying Systems, Automatica, vol. 31, no. 2, pp N. Wereley, and S.Hall, Frequency response of linear timeperiodic systems, 29th IEEE Conference on Decision and Control, Honolulu (Hawaii), 199, pp Yin W. and A.S. Mehr (29), Identification of Linear Periodically Time-Varying Systems Using Periodic Sequences, 18th IEEE International Conference on Control Applications, Saint Petersburg (Russia), July 8-1, pp Yin W. and A.S. Mehr (21), Least Square Identification of alias components of Linear Periodically Time-Varying Systems and Optimal Training Signal Design, IET Signal Processing, vol. 4, Iss. 2, pp Appendix A. FREQUENCY DOMAIN MODEL The continuous Fourier transform for rectangle windowed signals is given by the following expression (see Appendix 5.B.2 of Pintelon and Schoukens, 21): Ω = Ω Ω + Ω Ω - - where denotes the time derivatve of order. Making the assumption that the input and output signal are T-periodic, it is easy to show from the equation above that Ω Ω = Ω Ω Ω Ω. (18) By means of the linearity of the Fourier transform and by using property (18), the model equation (1) can readily be converted to the frequency domain. We therefore get (4) after evaluating (18) at the discrete frequencies Ω =, s.t., Ω Ω Ω Ω =, Ω Ω Ω Ω. Appendix B. VARIANCE OF THE RESIDUE The covariance matrix of the residue vector is defined as: cov = E E E = E. (19) By working out of the terms between the brackets in (19): cov = E + E 2E. Substitution of (7) in the expression above gives: = + 2 Herm. (2) The variance of the residue vector is finally found as the diagonal terms of cov in (2). 1312