Problem Description We remark that this classication, however, does not exhaust the possibilities to assess the model quality; see e.g., Ljung and Guo

Transcription

1 Non-Stationary Stochastic Embedding for Transfer Function Graham C. Goodwin Estimation Julio H. Braslavsky Department of Electrical and Computer Engineering The University of Newcastle Callaghan, NSW 38, Australia Phone: , Fax: Technical Report EE984 y June 998 z Mar a M. Seron seron@ee.newcastle.edu.au Abstract This paper presents a consistent framework for the quantication of noise and undermodelling errors in transfer function model estimation. We use the, so-called, stochastic embedding approach, in which both noise and undermodelling errors are treated as stochastic processes. In contrast to previous applications of stochastic embedding, in this paper we represent the undermodeling as a multiplicative error characterised by random walk processes in the frequency domain. The benet of the present formulation is that it signicantly simplies the estimation of the parameters of the embedded process yielding a closed-form expression for the model error quantication. An example illustrates how the random walk eectively captures typical cases of undermodelling found in practice. Keywords: System identication, estimation theory, least-squares estimation, parameter estimation, stochastic modelling, model approximation. Introduction The problem of tting transfer function models to experimental input-output data collected from a system is typically aected by two sources of errors. Firstly, disturbances and measurement noise (present in all real data) induce, so-called, variance errors. Secondly, the impossibility of obtaining a complete description of the system within the hypothesised model structure induce, so-called, undermodelling or bias errors. Our aim in this paper is to give a consistent paradigm for quantifying the eect of both variance and undermodelling errors on the estimated model. The quantication of model errors has attracted substantial research in recent years (see e.g. the survey paper by Ninness and Goodwin, 995). The existing approaches can be broadly classied into two streams, depending upon the type of error bounds computed. On the one hand, by characterising the error sources as deterministic one can obtain hard L-type error bounds. On the other hand, by characterising the error sources as stochastic one can obtain soft variance-type error bounds. Research supported by the Australian Research Council and the Centre for Integrated Dynamics and Control (CIDAC), The University of Newcastle, Callaghan NSW 38, Australia. y To appear IFAC'99. z Last modied on March 3, 999.

2 Problem Description We remark that this classication, however, does not exhaust the possibilities to assess the model quality; see e.g., Ljung and Guo (997), for a recent approach based on model validation. The stochastic characterisation of modelling errors has been widely accepted as having a high degree of validity in practice (Goodwin and Payne, 977; Ljung, 987; S derstr m and Stoica, 989), although it has traditionally been restricted to variance errors. A rst step to a stochastic characterisation of undermodelling errors was proposed in Goodwin and Salgado (989), in which the true system model was embedded in an underlying stochastic process in the frequency domain. A further step in developing this paradigm was suggested in Goodwin et al. (99), where a maximum likelihood procedure was proposed for estimating the parameters in the distribution of the embedded process, thus providing a self-contained mechanism for going from observed data to estimated model and stochastic error bounds. However, two diculties remained in the stochastic embedding approach. Firstly, the proposed characterisation of undermodelling as an additive stationary stochastic process in the frequency domain did not match well many kinds of undermodelling experienced in practical situations. Secondly, the estimation of the parameters in the distribution of the embedded process proved to be dicult. The current paper proposes solutions to the above diculties. We argue that a more adequate characterisation of the undermodelling is as a multiplicative error which forms a non-stationary (random walk) stochastic process in the frequency domain. As we illustrate in numerical examples, this characterisation eectively captures typical cases of undermodelling found in practice. In addition, our formulation signicantly simplies the estimation of the parameters of the embedded process and, in particular, yields a closed-form expression for the model error quantication. Problem Description We consider a single-input single-output linear dynamic system having an unknown transfer function g. Our objective is to t a model of the form g () to experimental input-output data by estimating by ^. We also seek a quantication of the size of the likely errors between g (^) and g. We assume that there exists some R p such that g(j!) = g (j!; ) + g(!) ; () where g represents the model error. The core idea of stochastic embedding (Goodwin and Salgado, 989; Goodwin et al., 99) is to think of g as a particular realization of a stochastic process; i.e. g has a probabilistic interpretation similar to that which we typically ascribe to noise and disturbances. More will be said below about the specics of the frequency domain embedded process. Note that we do not assume the existence of a value R p such that g (j!; ) = g(j!). On the contrary, () implies that the model class is fundamentally inadequate. We will approach the problem of estimating the transfer function in two stages. First, in Section 3, we estimate g(j!) for! belonging to a nite set of (not necessarily equally spaced) frequencies f! ; : : : ;! m g. Then, in Section 4, we estimate and the embedded undermodelling parameters using these frequency domain estimates. Section 5 provides a quantication of the total modelling error. The procedure is illustrated in a numerical example in Section 6. 3 Transfer Function Point Estimation Suppose that we are given a set of sampled input-output data of length N with sampling period T. In going from time domain data to an estimate of g(j!), we use a periodic input comprising m

3 3 Stochastic Embedding of the Noise Process 3 sine waves of frequencies f! ; : : : ;! m g. We aim at obtaining a rst raw estimate of the transfer function at these frequencies. The designed input is where u(t) = mx `= A` cos(!`t) ;!` = k` NT ; with k` f; : : : ; Ng ; () so that the m sine waves are exactly orthogonal on any interval of length NT. Assuming that the data are collected under steady state conditions, the corresponding sampled output response is given by y(kt) = mx `= A`g R (!`) cos(!`kt) - mx `= A`g I (!`) sin(!`kt) + v(kt) ; where g(j!`) = g R (!`) + jg I (!`) and fv(kt)g N k= is a noise process. It is then straightforward to estimate g R and g I using correlation methods. Specically, we may use ^g R (!`) = A`N ^g I (!`) = - A`N NX k= y(kt) cos(!`kt) ; (3) NX k= y(kt) sin(!`kt) : (4) 3. Stochastic Embedding of the Noise Process It is a standard paradigm in system identication to assume that the noise process fv(kt)g N k= is a realization of a time-domain stochastic process. For simplicity we assume here that the noise is a realization of a stationary uncorrelated sequence of variance v, that is EfVV g = v I ; (5) where V = [v(t); v(t); : : : ; v(nt)]. Clearly other embedded processes can be dealt with similarly. 3. Quantication of Errors At this stage there is no issue of undermodelling since, irrespective of the complexity of the true system, there exists a real and imaginary component at each frequency that gives an exact description of the response to the chosen input. We can thus use standard estimation theory to quantify the eect of V on the estimated frequency response. We start by writing equations (3), (4) in vector form as? ^G = - Y ; (6) For simplicity we assume that no!` = and that the phases are all zero. The theory can be trivially extended otherwise.

4 4 Nominal Transfer Function and Undermodelling Estimation 4 where ^G = h ^g R (! ); ^g I (! ); : : : ; ^g R (! m ); ^g I (! m )i ; Y = [y(t); y(t); : : : ; y(nt)] ; 3 cos(! T) sin(! T) : : : sin(! m T) cos(! T) sin(! T) : : : sin(! m T) = cos(! NT) sin(! NT) : : : sin(! m NT) diag [A ; -A ; A ; -A ; : : : ; A m ; -A m ] : h i Note that = diag A ; A ; : : : ; A m ; A m N=, due to the choice of the input signal. Using (5), the error in the estimated transfer function can be quantied as (Goodwin and Payne, 977, Theorem.3.) where G = E ( ^G - G)( ^G - G)? = - v ; h g R (! ); g I (! ); : : : ; g R (! m ); g I (! m )i. Since is diagonal, the stochastic process " # " g R (!`) ^g R # " (!`) g R # (!`) g I = (!`) ^g I - (!`) g I (!`) is an uncorrelated process in the frequency domain having non-stationary variance "g # " # R (!`) g R (!n ) E g I (!`) g I = D` (!` -! n ) ; (8) (! n ) D`, v A ` N " Clearly (8) depends on the unknown quantity v. However, an unbiased estimate of v can be obtained as (Goodwin and Payne, 977, Theorem.4.) ^ v = N - m (Y - ^G) (Y - ^G) : (9) Hence, the covariance (8) can be approximated by replacing v by its unbiased estimate (9). # : (7) 4 Nominal Transfer Function and Undermodelling Estimation 4. Stochastic Embedding of the Transfer Function As remarked earlier, we are not at liberty to assume that g(j!`) = g (j!`; ) for any value of. The crucial question is then, how do we think of the undermodelling error. To get a feel for this problem consider the simple case of an unmodelled pole; i.e., which yields g(s) = g (s) g(s) - g (s) = g (s) s + s ; s + This, admittedly trivial, expression reveals two important qualitative features: : ()

5 4 Stochastic Embedding of the Transfer Function 5 It is reasonable to work with multiplicative errors. The multiplicative error, in this example, has a magnitude which grows with frequency (at least up until the frequency =, which we take to be beyond the frequencies of interest). Of course, in practice, we cannot expect to have such a simple description for the undermodelling as in (). However, inspired by this and other cases that we have examined, we propose to hypothesise that the undermodelled process can be embedded in a non-stationary stochastic process whose variance grows with frequency. The simplest such process is a random walk in the frequency domain. This process has the conceptual advantage of having high complexity in the realization space but relatively simple probabilistic description. Having arrived at this conclusion, it has a certain intuitive appeal that transcends the steps that led to its genesis. Indeed, one notes that the idea of a random walk is commonly exploited in the time domain to model drifting signals. The extension of this idea to the frequency domain seems, in retrospect, to be a sensible way of capturing the notion of undermodelling. To be precise, we will assume that there exists a (unknown), a (known) and a " (unknown) such that the real and imaginary parts of g(j!) have the form g R (!) = g R (!; ) + gr (!; ) R (!) ; g I (!) = g I (!; ) + gi (!; ) I (!) ; () where f R (!)g and f R (!)g are two zero-mean independent processes having uncorrelated increments, that is with Efd"(s)d"(s)g = " ds, and R (!) = Z! d"(s) ; I (!) = Z! Ef R (!`) R (! n )g = Ef I (!`) I (! n )g = Z min(!`;!n) d"(s) ; () " ds = min(!`;! n ) " : (3) The value in () could be taken, for example, as any a priori estimate of the nominal model parameters. One could alternatively take the value = as unknown. This, however, would lead to a signicant increase in the complexity of work required to estimate. Since the model is an abstraction of reality, it seems that taking known is a reasonable simplication. Note that we have taken the undermodelling processes f R (!)g and f I (!)g in (3) as uncorrelated and of equal variance. Again, this can be viewed as a simplifying assumption. One could alternatively assume that " R # " (!`) R # " # (! n ) E I (!`) I = min(!`;! n ) : (! n ) However, the description of the embedded process would then have three unknowns ( ; ; ) and it is doubtful that, in practice, the richer structure so obtained would be warranted. Finally, for purposes of illustration, we will model the nominal transfer function g (j!; ) in terms of a given set of basis functions b (j!), : : :, b p (j!). For example, we could adopt the practice advocated in a series of recent papers of using Orthonormal Bases (Wahlberg, 99; Van den Hof et al., 993; Ninness and Gustafsson, 997).

6 4 Nominal Parameter Estimation 6 We dene the vectors B R (!) = [b R (!); : : : ; br p (!)] and BI (!) = [b I (!); : : : ; bi p (!)] containing the real and imaginary parts of the basis functions. We thus assume () to be parameterised as g R (!) = B R (!) + B R (!) R (!) ; g I (!) = B I (!) + B I (!) I (!) : (4) 4. Nominal Parameter Estimation The next stage is to use ^G obtained in (6) to estimate the parameter. From (7) and (8) we have that the observed quantities ^g R (!`); ^g I (!`) satisfy " ^g R # " (!`) g R # " # (!`) g R (!`) ^g I = (!`) g I + (!`) g I (!`) ; (5) where fg R (!`); g I (!`)g m`= is an uncorrelated vector process (in the frequency domain) having covariance D` as in (8). Combining (5), (4) and (), we have ^g R (!`) = B R (!`) + B R (!`) R (!`) + g R (!`) ; ^g I (!`) = B I (!`) + B I (!`) I (!`) + g I (!`) ; where R and I are given by (). In this model, f^g R ; ^g I g is the given set of m observed quantities, fg R ; g I g are white noise processes (in the frequency domain) having variance (8), where v can be reasonably estimated as in (9), and f R ; I g are uncorrelated processes (in the frequency domain) having unknown covariance structure (3). We will add the reasonable further assumption that To write (6), () in vector form, we dene Thus, the model (6) is expressed as (6) fg R ; g I g and f R ; I g are uncorrelated. (7) = diag[ R (! ); I (! ); : : : ; R (! m ); I (! m )] ; g R 3 (! ) B R 3 (! ) g I (! ) B I (! ) G =. ; and B = 6 4g R. : 7 6 (! m ) 5 4B R 7 (! m ) 5 g I (! m ) B I (! m ) ^G = B + B + G : (8) Lemma 4. (Nominal Parameter Estimator). An unbiased estimator for is given by the least-squares estimate with error covariance ^ = (B B) - B ^G ; (9) Ef(^ - )(^ - ) g = (B B) - B ( " + A v )B(B B) - ; () where = diag[b ] 3!! : : :!!! : : :! B6 7 I 4 5 A......!! : : :! m diag[b ] () A = N diag [A ; A ; A ; A ; : : : ; A m ; A m ] - : ()

7 5 Quantication of Modelling Errors 7 Proof. From (9) and (8) we have that ^ = (B B) - B (B + B + G). Hence, from ^ - = (B B) - B (B + G) (3) and (7), we obtain Ef(^ - )(^ - ) g = (B B) - B [EfB B g + Ef G G g]b(b B) - ; which yields () on noting that EfB B g = " ; (4) which follows from (3), and Ef G G g = A v ; (5) which follows from (8). An unbiased estimate of () is obtained using the estimate of v given by (9), and the following estimate of the undermodelling parameter ". Lemma 4. (Estimation of " ). An unbiased estimate of " is ^ " = ( ^G - B^) ( ^G - B^) trace[(i - B(B B) - B )] trace[(i - B(B B) - B )A] - trace[(i - B(B B) - B )] v : (6) Proof. By introducing the shorthand notation P = I - B(B B) - B we write, from (8) and (9), Ef( ^G - B^) ( ^G - B^)g = Ef(B + G) P(B + G)g = trace[p EfB B + G G g] (using (7)) = trace[p( " + A v )] (using (4) and (5)) = trace[p] " + trace[pa] v ; and hence Ef^ " g = " as claimed. Here again, we can replace v in (6) by its unbiased estimate from (9). 5 Quantication of Modelling Errors The main topic of this paper is the quantication of the error in the estimated transfer function under non-stationary stochastic embedding. At any frequency! n, the total modelling error is given by " g R # e G e (! (! n) n ) = g I e (! n), B(! n )^ - G(! n ) ; (7) where " g R # " (! n ) B G(! n ) = g I ; and R # (! n ) B(! n ) = (! n ) B I : (! n ) The following result gives expressions for G e (! n ) and its covariance under the proposed nonstationary stochastic embedding.

8 5 Quantication of Modelling Errors 8 Theorem 5. (Quantication of Modelling Errors). At any frequency! n, the modelling error G e (! n ) in (7) has the form G e (! n ) = B(! n )Q G + [B(!n )QB - L(! n )B(! n )] ; (8) where Q = (B B) - B and L(! n ) = diag[ R (! n ); I (! n )]. Furthermore, without loss of generality assume that! n is such that! k-! n <! k, where! k- and! k are two consecutive frequencies of the test set (). Then the error covariance e (! n ), EfG e (! n )G e (! n)g is given by the expression where e (! n ) = K v (! n ) v + K "(! n ) " ; (9) K v (! n ) = B(! n )QAQ B (! n ) K " (! n ) = B(! n )QQ B (! n ) + (diag[b(! n ) ])! n - (! n ) - (!n ) (3) with A, given in (), (), respectively, and 3 diag[b(! ) ]!. (! n ) = B(! n )Q diag[b(! k- ) ]!k- diag[b(! k ) ]!n diag[b(! n ) ] : diag[b(! m ) ]!n Proof. The expression (8) is obtained by replacing ^ - from (3) in Using (7), (4) and (5), we have e = EfG e G e g G e = B(! n )^ - B(! n ) - L(! n )B(! n ) ; = B(! n )(^ - ) - L(! n )B(! n ) : = Ef[B(! n )QB - L(!n )B(! n ) ][B(!n )QB - L(!n )B(! n ) ] g + Ef[B(! n )Q G][B(!n )Q G] g = B(! n )QAQ B(! n ) v + B(! n)qq B(! n ) " + Ef[L(! n )B(! n ) ][L(!n {z )B(! n ) ] } g (?) {z } (??) - B(! n )QEfB [L(!n )B(! n ) ] g -(??) : From (4) and () evaluated at the single frequency! n we have (?) = (diag[b(! n ) ])! n ". In a similar way it can be shown that (??) = (! n ) ". Putting everything together yields (9)-(3). Theorem 5. is the main result of the paper and gives a simple expression for model error quantication. In particular, the rst term on the rhs of (9) describes the eect of noise whilst the second captures the eect of undermodelling. Of course, the utility of the result ultimately depends on its capacity to represent the kinds of real errors met in practice. The following section illustrates its excellent performance on a test example.

9 6 Example 9 6 Example 6. Transfer function Point Estimation From the system G T (s) = 3:45(s + ) (:355s + )(:383s + )(6:4689s + )(:64s + :396s + ) ; (3) we collect N = 4 input-output data points sampled each T = :8s, starting at t = 4NT to avoid initial transients. The test input signal is selected as where m = 46 ; u(t) = mx `= cos(!`t + `)!` = k`=(nt) ; k` = f; ; : : : ; 9; ; 5; : : : ; 95; ; ; : : : ; 8; 3g ; ` = random: The simulation is conducted under additive white measurement noise of intensity v =. resulting input and output signals are plotted in Figure. The u(t) y(t) Time [s] Figure : Test input and output. The estimated ^G computed using (6) corresponds to the crosses in Figure. The true transfer function (3) is also plotted in this gure in solid line. The estimate of v provided by (9) is ^ v = :5.

10 6 Nominal Transfer Function Estimation and Error Quantication Nyquist Plot.5 True Point Estimate Imag Axis Real Axis Figure : True transfer function (solid) and point estimation (crosses). 6. Nominal Transfer Function Estimation and Error Quantication To test the proposed stochastic embedding, we t to the data three models of increasing complexity, which correspond to the sets of bases and b (s) = (:5s + )(5s + ) b (s) = (:5s + ) ; b (s) = (3s + ) b (s) = b 3 (s) = (:5s + ) ; b (s) = (:5s + ) 3 ; b 4(s) = Equations (9), (6) yield the estimated parameters ; (Set ) ; (Set ) (3s + ) ; (Set 3) (3s + ) 3 : ^ = :477 " = :8 ; (Set ) ^ = [:68; :48] " = :95 ; (Set ) ^ = [:4; :6; :77; :4] " = :86 : (Set 3) The estimated transfer function for each of the three sets of bases is plotted in dashed lines in Figures 3 to 5, respectively, together with the true transfer function in solid line. Assuming Gaussian distributions for the modelling errors, condence ellipses are drawn at each point of the estimated transfer function, using the covariance matrix e (! n ) dened in (9). The ellipses, centred at B R (! n ) + jb I (! n ), are the locus of [Re x ; Im x] [ e (! n )] - [Re x ; Im x] = c : Taking c = 4 gives approximately 8% condence regions. In the computation of e (! n ), the parameter was taken to be equal to the estimate ^ obtained for each of the sets. We observe from Figures 3 to 5 that the condence regions are subsequently reduced in size when a better model is tted to the data. Thus, the non-stationary stochastic embedding of the undermodelling correctly captures the accuracy of the estimate.

11 6 Nominal Transfer Function Estimation and Error Quantication.5 Imag Axis Figure 3: Fitting Set..5 Estimated True Real Axis.5 Imag Axis Figure 4: Fitting Set..5 Estimated True Real Axis.5 Imag Axis Figure 5: Fitting Set 3..5 Estimated True Real Axis

12 7 Conclusions 7 Conclusions This paper has described an approach to transfer function model error quantication based on non-stationary stochastic embedding in the frequency domain. The end result of the proposal is a closed-form stochastic description of the model errors due to noise and undermodelling. The strength of the suggested procedure is that it is relatively straightforward and appears to describe well the dual eect of noise and undermodelling on estimated model errors. The present work needs to be complemented by parallel developments of new robust control paradigms which account for stochastic modelling errors. A rst step in this direction is reported in (Goodwin et al., 999), where a robust control problem using stochastic (i.e. variance) model errors was formulated and solved. Acknowledgements The authors gratefully acknowledge helpful discussions with Brett Ninness during the development of this research. References Goodwin, G. and M. Salgado (989). `A stochastic embedding approach for quantifying uncertainty in the estimation of restricted complexity models'. Int. J. Adaptive Contr. Signal Processing 3(4), Goodwin, G. and R. Payne (977). Dynamic System Identication. Academic Press. New York. Goodwin, G. C., L. Wang and D. Miller (999). Bias-variance tradeo issues in robust controller design using statistical condence bounds. In `Proc. of the 4th IFAC World Congress'. Beijing, China. Goodwin, G., M. Gevers and B. Ninness (99). `Quantifying the error in estimated transfer functions with application to model order selection'. IEEE Trans. on Automatic Control 37(7), Ljung, L. (987). System Identication: Theory for the User. Prentice-Hall. Englewood Clis, NJ. Ljung, L. and L. Guo (997). `The role of model validation for assessing the size of the unmodelled dynamics'. IEEE Trans. on Automatic Control 4(9), 339. Ninness, B. and F. Gustafsson (997). `A unifying construction of orthonormal bases for system identication'. IEEE Trans. on Automatic Control 4(4), 555. Ninness, B. and G. Goodwin (995). `Estimation of model quality'. Automatica 3(), S derstr m, T. and P. Stoica (989). System Identication. Prentice-Hall. Englewood Clis, NJ. Van den Hof, P., P. Heuberger and J. Bokor (993). Identication with generalized orthonormal basis functions. Statistical analysis and error bounds. In `Selected Topics in Identication, Modelling and Control'. Vol. 6. Delft University Press. pp Wahlberg, B. (99). `System identication using Laguerre models'. IEEE Trans. on Automatic Control 36, 768.