Estimating models of inverse systems

Transcription

1 Estimating models of inverse systems Ylva Jng and Martin Enqvist Linköping University Post Print N.B.: When citing this work, cite the original article. Original Pblication: Ylva Jng and Martin Enqvist, Estimating models of inverse systems, 203, Proceedings of the 52nd IEEE Conference on Decision and Control, IEEE. Personal se of this material is permitted. However, permission to reprint/repblish this material for advertising or promotional prposes or for creating new collective works for resale or redistribtion to servers or lists, or to rese any copyrighted component of this work in other works mst be obtained from the IEEE. Postprint available at: Linköping University Electronic Press

2 Estimating models of inverse systems Ylva Jng and Martin Enqvist Abstract This paper considers the problem of how to estimate a model of the inverse of a system. The se of inverse systems can be fond in many applications, sch as feedforward control and power amplifier predistortion. The inverse model is here estimated with the prpose of sing it in cascade with the system itself, as an inverter. A good inverse model in this setting wold be one that, when sed in series with the original system, reconstrcts the original inpt. The goal here is to select sitable inpts, experimental conditions and loss fnctions to obtain a good inpt estimate. Both linear and nonlinear systems will be discssed. For nonlinear systems, one way to obtain a linearizing prefilter is by Hirschorn s algorithm. It is here shown how to extend this to the postdistortion case, and some formlations of how the pre- or postinverter cold be estimated are also presented. I. INTRODUCTION The behavior of a system S can be modified in a mltitde of ways. In atomatic control, the main choices are feedforward and feedback control. Feedback can handle phenomena like distrbances, model ncertainties and nstable systems, bt a bad feedback loop may case instability. Feedforward control does not need any measrements bt ideal feedforward control (sing F f = S ) reqires perfect knowledge of the system (as well as all distrbances) and that both S and S are stable. These are of corse limiting assmptions. A benefit with feedforward is that two cascaded stable systems will always be stable. Often, both feedforward and feedback control are sed to get the advantages of both approaches. In some cases it is however not possible or desirable to se feedback. One application example of this is power amplifiers in commnication devices, which are often nonlinear and/or dynamic, casing interference in adjacent transmitting channels []. To be sefl, linearization is needed. However, one does not want to work with the amplified signal, bt rather the inpt signal to the system. A prefilter that inverts the nonlinearities/dynamics, called a predistorter, is ths preferable. In sensor applications it is rather a postdistortion that is needed. If the sensor itself has dynamics or a nonlinear behavior, the sensor otpt is not the tre signal bt will also contain some sensor contamination. This wold have to be handled at the sensor otpt. In the above applications, finding the inverse is a crcial point. The inverse model is here estimated with the prpose of sing it in cascade with the system itself, see Fig.. This inverse model can either be bilt on physical knowledge, or This work was partially spported by the Excellence Center at Linköping- Lnd in Information Technology (ELLIIT), and the Center for Indstrial Information Technology at Linköping University (CENIIT). Y. Jng and M. Enqvist are with the Department of Electrical Engineering, Linköping University, SE Linköping, Sweden, phone: +46(0) , ylvj@isy.li.se, maren@isy.li.se. S y S y S y S Fig.. The intended se of the estimated inverses. The left figre shows predistortion, where the inverse S is applied before the system S, and the right shows postdistortion, where the order is reversed. it can be estimated based on measred data. Each application entails restrictions and special conditions to attend to. In system identification, one shold sally estimate the system in the setting it shold be sed in, concerning for example the choice of inpt and the experimental conditions [2, Ch. 3], [3] and [4]. It is important to choose the inpt signal to captre the desired characteristics of the system, bt what does that correspond to in this case? Another important topic in system identification is the choice of loss fnction. It shold reflect the goal of the identification, and different properties of the system can be emphasized. In this setp, we want to se these degrees of freedom and the flexibility of the model to obtain a good inpt estimate. The contribtion in this paper is two-parted. One is the investigation of the loss fnction for the estimation of the inverse, and the adjoining discssion on the choice of inverse estimation approach. This concerns the two approaches forward estimation followed by inversion, and the estimation of the inverse system directly, for linear time invariant (LTI) dynamical systems. The other concerns Hirschorn s method [5] and how it can be rewritten into a postdistortion method, as well as the formlation of the different settings nder which this pre- or postinverter cold be estimated. The disposition of the paper is as follows. Different approaches to the inversion and estimation of inverse systems are presented in Section II. In Section III the identification of LTI dynamical systems is discssed, followed by an example in Section IV. Section V shortly introdces inversion techniqes for nonlinear systems and in Section VI, Hirschorn s method (a predistortion method), its extension to postdistorter and some aspects of the estimation of this distorter are presented. Conclsions are in Section VII. II. SYSTEM INVERSE ESTIMATION In system identification, the goal is to achieve as good a model as possible to explain the behavior of y by a prediction or simlation ŷ(t ), which depends on estimated model parameters and the inpt. This is done sing measred data, sally inpt data, (t), and otpt data, y(t). Here, a model describing the system itself will be referred to as a forward model and a model describing the inverse will be called an inverse model.

3 TABLE I THE INPUTS AND OUTPUTS TO THE IDENTIFICATION PROCEDURES. Method Inpt Otpt Reqires Model A y forward B Ŝ inverse C y inverse The inverse model is estimated with the prpose of sing it in series with the system itself, as an inverter, see Fig.. The goal is to minimize the difference between the inpt and the otpt from the cascaded systems, y. A good model in this setting is one that, when sed in series with the original system, regains the original inpt, so that y =. There are three main approaches to the estimation of an inverse of a system S, described in more detail below. A) In a first step, the forward model Ŝ is estimated in the standard way (with inpt data and otpt data y). Step two is to invert the reslting model to obtain an estimated inverse Ŝ. B) In a pre-step, the forward model Ŝ is estimated in the standard way (with inpt data and otpt data y). This model is sed in series with an inverse model,ŝ, and the inverse model parameters are estimated in this setting, by minimizing the difference between the inpt and the simlated, distorted otpt y. C) The identification is done in one step, by identifying the inverse Ŝ directly, sing inpt data y and otpt data. The inpts and otpts sed are smmarized in Table I. The identification in Method A is the standard one, as described in for example [2] and [4], and the inversion is discssed in [6] in the feedforward control application. The se of feedforward control based on an inverse model of the system in the presence of plant ncertainty is discssed in [7]. A good thing is that the identification ses standard methods, bt on the other hand, an inversion is reqired, and the weighting of the model fit is not necessarily optimal for the se intended here. Method B is often sed in power amplifier predistortion [8], [9], [0], and is then also called Direct Learning Control (DLA). The qality of the inverse and the forward models are closely copled, and two choices are available. It is often preferable to obtain a simple inverse model, as in the predistorter case, and this restriction can also be applied to the forward model. Alternatively, a more complex forward model can be sed to make sre that as mch as possible of the system behavior is captred, while keeping the inverse model less complex. The choice comes down to the implementation if the forward model has to be implemented, also this model needs to have a limited complexity. A good thing with this approach is that the estimation of the inverse is done with no noise present, bt it also reqires two (possibly nonconvex) minimizations (with the risk of obtaining local minima), and the qality of the inverse clearly depends on the qality of the forward model. Method C is also called Indirect Learning Control (ILA) in power amplifier (PA) predistortion applications and is evalated for PA predistortion in [9] and [0]. This approach assmes that the predistorter and the postdistorter are interchangeable (commtativity). An advantage with this method is that the inverse is estimated in the setting it is going to be sed, and that the weighting is possibly better than for the first method, bt the noise entering at the inpt risks casing a biased estimate. In PA predistortion applications, the third approach (ILA) is more commonly sed than the second (DLA) [0]. In [0], comparisons performed indicate that the DLA performs better in the simlation setp sed, whereas in [9] the ILA seems to perform slightly better. Inverse systems entail a nmber of problems. Any nonminimm-phase zeros of the original system will become nstable poles of the inverse system. However, if a delay can be allowed, a stable approximative filter can often be fond []. If the transfer fnction is strictly proper, the amplification will approach zero at high freqencies. The inverse of a strictly proper system will be improper, that is, high freqency contents will be amplified. The amplification of high freqency noise shold be avoided, as is described in [2]. Here, only the case when the system and its inverse are both stable and casal will be investigated. III. INVERSE IDENTIFICATION OF LTI SYSTEMS To simplify the discssion, we will start by looking at LTI dynamical systems. The model estimation is done in open loop and nder the assmption that the otpt was created according to y(t) = G 0 (q)(t) + H 0 (q)e 0 (t) () where G 0 is the tre system, H 0 is the tre noise dynamics and e 0 is a white noise seqence. In system identification, the goal is often to find the minimizing argment of a fnction of the prediction error ε(t, ) ˆ = arg min = arg min N N N ε(t, ) 2 t= N [y(t) ŷ(t )] 2, (2) t= where y(t) is the measred otpt and ŷ(t ) is the predicted otpt, given the model parameters. Here, we se a fixed noise model H sch that the prediction is described by ŷ(t ) = G(q, )(t). Looking at the identification from a freqency domain point of view, the minimization criterion in (2) can asymptotically be written as [2, (8.7)] ˆ = arg min π G 0 (e iω ) G(e iω, ) 2 Φ (ω)dω (3) where G(e iω, ) is the model and Φ (ω) is the spectrm of the inpt signal. The estimation will ths be done in a way to emphasize the model fit in freqency bands where the transfer fnction and the inpt spectrm are large enogh to have a significant impact on the total criterion. The

4 minimization is done with respect to the prodct of the model fit ( G 0 G 2 ) and the inpt spectrm. If the inpt is white noise, it is ths more important to obtain a good model fit at freqencies with a large transfer fnction magnitde. If instead the goal is to estimate the inverse model to be sed as described in Section II, the minimization criterion in the time domain can be written ˆ = arg min N N t= [ (t) ] 2 G(q, ) y(t). (4) The freqency domain eqivalent to (4), when y is noisefree, is ˆ = arg min V inv () (5) where the loss fnction is 2 V inv () = π G 0 (e iω ) G(e iω, ) Φ y (ω)dω 2 = π G 0 (e iω ) G(e iω, ) G 0 (e iω ) 2 Φ (ω)dω = G 0(e iω 2 ) π G(e iω, ) Φ (ω)dω (6) = G(e iω, ) G 0 (e iω ) 2 Φ (ω) G(e iω dω (7), ) 2 π sing Φ y = G 0 (e iω ) 2 Φ with no noise present. The minimization in (7) is similar to the weighting for the inpt error case where H = G so that y(t) = G+Ge = G(+e), that is, the error enters the system with the inpt [3]. Comparing the minimization criterion for the forward estimation in (3) to the one for the inverse estimation in (6), the weighting is clearly different. In the forward case, a relative model error at a freqency where the system amplification is small, will affect the criterion mch less than a model error at a freqency where the system amplification is large. In the inverse estimation case, thogh, a relative model error will have the same effect on the criterion, regardless of the amplification at that freqency, for freqencies with the same inpt spectral density. The weighting, and the model fit, between the different freqencies will ths be shifted to better reflect the importance of a good fit also at freqencies with a small transfer fnction magnification. The time domain criterion (4) ths leads to the freqency domain description (6), and the weighting is atomatically done to match the sage of the inverse model estimate. IV. AN ILLUSTRATIVE LINEAR DYNAMIC EXAMPLE Let s look at a small example. The goal is to obtain a system inverse to be sed in series with the original system, in order to retrieve the inpt, see Fig. (left). The inpt and the noise-free otpt y are measred. The system has two resonance freqencies, at ω = rad/s and ω = 0 rad/s. The magnitdes of the two resonance peaks are very different, with the first one a hndred times larger than the second one. The tre system G 0 is described by G 0 (s) = 0 s 4 +.s s 2 + s + 00 (8) Magnitde Freqency [rad/s] Fig. 2. Bode magnitde plot of G 0 in (8) (solid line). The stars mark the amplitde of the mltisine inpt ( in (9)) components at each freqency. and the Bode magnitde diagram is shown in Fig. 2. The inpt consists of three sinsoids arond each of the two resonance freqencies sch that the inpt power is concentrated in two bands, centered arond these resonance freqencies, = 6 a k sin(ω k t + φ k ) (9) k= with a k = for k =, 2, 3, a k = 0 for k = 4, 5, 6, ω k = { 0.9,,., 9, 0, } and φ U[ π π]. The inpt amplitde is illstrated by the stars in Fig. 2. The sampling time is T s = 0.02 s and N = simlated measrements have been collected. With the goal of sing an FIR model as prefilter to recover the inpt, two models have been estimated, sing Methods A and C in Section II. First, a forward model has been estimated as an otpt error (OE) model sing System Identification ToolBox in MATLAB, with [nb nf nk] = [ 3 0]. This model has then been inverted reslting in an FIR model with 4 terms. The approximate inverse sing Method C is an FIR model with 4 terms, i.e. [nb nf nk] = [4 0 0], and will have a very different weighting. Hence, the two inverses will catch different properties of the system. As can be seen in the Bode magnitde plot in Fig. 3, the Method A model has a mch better fit arond ω = rad/s and almost perfectly models the resonance peak, bt completely misses the second resonance peak at ω = 0 rad/s. The Method C model does not manage to catch either of the resonance peaks in a satisfactory way bt catches both of the resonance freqencies. That is, the amplification at ω = and 0 rad/s is well captred, bt not the resonance peaks arond. Estimating the forward model in the standard way will clearly focs on the freqencies where the prodct of model fit ( G 0 G 2 ) and inpt spectrm is large. When this system approximation is then inverted, the errors arond ω = 0 rad/s will become prominent. In the time domain plot in Fig. 4, it is clear that the inverse model estimated directly better reconstrcts the inpt than the inverted forward model. In Fig. 5, the periodograms of the reconstrcted inpts are shown. At the lower freqency arond ω = rad/s, the Method A model captres the inpt almost perfectly, bt arond ω = 0 rad/s, the reverse is tre and the Method C model performs better.

5 Magnitde Freqency [rad/s] Fig. 3. Bode magnitde response of G 0 (black solid line) and models sing Methods A (black dashed line) and C (gray solid line). The inverted forward model perfectly catches the resonance peak at ω = rad/s, bt completely misses the second peak. The inverse estimated directly does not catch either of the resonance peaks in a satisfactory way, bt instead has an accrate modeling of both peak freqencies Time [s] Fig. 4. The inpt (black solid line), and the reconstrcted inpt y sing the Methods A (black dashed line) and C (gray solid line). The estimation of the inverse cannot perfectly reconstrct the inpt, bt is clearly better than the inverted forward model Freqency [rad/s] Fig. 5. Periodogram of the inpt (black solid line), and the reconstrcted inpt y sing the Methods A (black dashed line) and C (gray solid line) arond ω = rad/s (top) and ω = 0 rad/s (bottom). It is clear also in the freqency domain that the forward model better captres the behavior arond ω = rad/s than the inverse estimation, bt the reverse is tre at ω = 0 rad/s. As shown in this small example, there are clearly occasions when it is advantageos to estimate an approximate inverse directly as opposed to estimating the forward model and then inverting it. V. NONLINEAR INVERSION The inversion of a nonlinear system is nontrivial, and different approaches have been sed to cope with this. In [4, p 5], the nonlinear system S is divided into a linear part, L, and a nonlinear part, N. It is shown that the inverse of S can be obtained in a feedback loop with the nonlinear part N in the feedback and the linear inverse L in the forward path. It follows that the nonlinear part N does not have to be inverted, and that only the linear part L is to be inverted. Volterra series are also sed to model nonlinear dynamical systems, and it can be shown that the pth order postinverse eqals the pth order preinverse, so that the predistorter and postdistorter are interchangeable [5]. The drawback with Volterra series is that they are comptationally very heavy, with many parameters to estimate. Hammerstein systems are often sed as a forward model in PA predistortion. The nonlinear compensation can be either a Wiener system [9] or a second Hammerstein system. This strctre can also be extended with mltiple branches, called parallel Hammerstein. VI. HIRSCHORN S METHOD A feedback soltion for nonlinear systems is the exact inpt-otpt linearization that makes se of a known model of the system to obtain overall linear dynamics, determined by the ser. In exact linearization (also called inpt-otpt linearization) [6], the otpt from a nonlinear system S, ẋ = f(x) + g(x) y = h(x) (0) is differentiated w.r.t. time, to obtain a relation between the differentiated otpt y (γ) and the inpt,. The inpt = α(x) + β(x)r () can be chosen so that it leads to a direct relation between the differentiated otpt y (γ) and the reference r, y (γ) = r. This now describes a system with linear dynamics, and the nonlinear feedback loop can be combined with linear theory to obtain the desired dynamics, denoted. The overall system from r to y (the nonlinear system with the nonlinear and linear feedbacks) will ths be linear, and the dynamics described by the transfer fnction, chosen by the ser. Exact linearization reqires the knowledge of all the states, and is therefore often sed in combination with a nonlinear observer. This can lead to a complicated feedback loop. Here, it is assmed that any zero dynamics present are stable. A. Preinversion In Hirschorn s method, exact linearization is sed to constrct a linear system [5]. Given a good enogh model, it shold be possible to se the model not only in the feedback (in the constrction of in ()) bt also as a simlation

6 S r S S y r S Ŝ Feedback Fig. 6. Block diagram of Hirschorn s method, where the system S is replaced by a model Ŝ in the exact linearization feedback loop. The inpt signal calclated in this way is then also applied to the real system S. The simlation system and feedback loop that leads to an overall linear behavior between r and y s is denoted S. The inpt to S is the reference r and the otpt is the control signal. r S S y s Fig. 7. The predistortion block S obtained sing Hirschorn s method in series with the real system leads to an overall linear behavior between r and. model. If the otpt from the simlated model is fed back to the controller, see Fig. 6, the overall system from reference r to otpt y s will be linear with the dynamics. Also, the inpt calclated for this (simlated) system leads to the desired dynamics, and the same inpt signal can be sed also for the tre system. A pre feedforward controller is ths obtained, as in [5], see Fig. 7. This can also be seen as an observer with no measrement inpts. B. Postinversion Let the nonlinear system be denoted S, the precompensation S (since it is not really an inverse of S, bt rather creates a system that, in series with S will be linear) and the dynamics of the overall linear system. The goal is to obtain a linear response to by sing a postinverse on the otpt y. The above method can be seen as an inversion of the nonlinearities of the system the otpt from the overall system will be linear with dynamics, chosen by the ser (assming the model is accrate enogh, of corse). Here, preinverse and postinverse are not interchangeable; Hirschorn s method tells s only how to determine the inpt to the nonlinear system sch that the reference-to-otpt has the linear dynamics, not how to maniplate the otpt to make it a linear response to the inpt. If a postinverse is desired, a different setp is needed. It is known that S in cascade with S leads to a linear system, so that y = r with r the reference, cf. Fig. 7. Assme that was actally created by a prefilter, S, with as otpt and the fictitios signal r as inpt, see Fig. 8. An estimate of this signal can then be obtained by r = G m y (2) Fig. 8. The (possibly fictitios) reference signal r can be seen as inpt to the block S, creating the inpt to the nonlinear system S. y r ū S Fig. 9. Hirschorn s method sed as postdistortion, when the otpt can be assmed to be created according to Fig. 8. The block S cannot simply be applied at the otpt y, bt has to be maniplated to obtain a linear behavior between and. where, if no transients or noise are present, r = r. An estimate of the inpt, called ū, can then be obtained by filtering r by S. Now, to obtain the desired dynamics, ū mst be filtered by the linear fnction, see Fig. 9. The cascade of these three blocks (/, S and ), ths make p a postdistorter that leads to a linear response between (not available for maniplation) and in Figre 0. Example Hirschorn s Postinverse: Consider the nonlinear system ẋ = x 3 + x 2 + w ẋ 2 = x w 2 (3) y = x with process noise w i N(0, 0.05) and a mltisine inpt. The nonlinear feedback = 3x 5 + 3x 2 x 2 + x 2 + ũ leads to a linear system ÿ = ũ. Now, linear theory can be applied and pole placement has been sed to get an overall system response from reference r to otpt y corresponding to the one from (s) = /(s 2 + 5s + 6). The otpt from the nonlinear system (3) is plotted in Fig. together with the otpt from the desired dynamics. A preinverse S has been constrcted as in Fig. 6. S has been sed as a preinverse, as well as a postinverse for evalation prposes. The reslts are shown in Fig. 2. Here, it is clear that the desired preinverse and postinverse are not the same, and that S cannot straight away be sed as a postinverse. If instead, the otpt y is filtered by the cascaded systems /, S and, as in Fig. 9, the reslt improves considerably, as shown in Fig. 2. The remaining errors are primarily cased by the noise. For noise-free data, the preinverse performs perfectly whereas the postinverse has some minor errors. C. Inverse estimation Hirschorn s method also opens p for qestions abot how to estimate a system inverse. In the case where the S y r S ū Fig. 0. Hirschorn s method sed as postdistortion. The postinverse consists of the three blocks /, S and. Used in this way, the overall behavior between and will be linear.

7 time [s] Fig.. The otpt from the nonlinear system (3) in gray and the desired dynamics from in black time [s] Fig. 2. A Hirschorn postinverse applied to the system (3). The otpt from (solid black) and the otpt when S was sed as a preinverse (dashed black). The otpt from the system in series with the inverse S is plotted in dashed gray when S is sed as a postinverse with no extra filtering. When the postdistortion is constrcted according to Fig. 9, the reslt improves considerably. Here, the postinverse consists of three blocks, /, S and and the postdistorted otpt is plotted in solid gray. strctre of the nonlinear system is known, bt where there are nknown parameters to be estimated, the identification can be done in several ways, as described in Section II. Method A wold correspond to measring the inpt and the otpt y, and identifying the nknown parameter vales in the standard way. This estimated model cold then be sed to provide the inverse, since if a model of the forward system is available, a model of the inverse system is as well. Method B does not really have an eqivalence in this case once the forward model is known, the exact inverse to match it is also known in the exact linearization framework. Method C wold correspond to estimating the inverse S directly. In order to do this, we wold need as otpt and the reference r as inpt. Bt, as the data was collected in open loop with no pre- or postdistorter, the signal r is not available. Now, as in Section VI-B, assme that the system was actally preceded by a system S, fed by a fictitios reference signal r, and that the overall behavior from r to y is in fact linear with dynamics described by. If this is tre, then the signal r wold be obtained by filtering y with /, and the system S can be identified sing r as inpt and as otpt. So, this eqals finding the inverse by sing (a filtered version of) the otpt y as the inpt and as otpt. A benefit with Hirschorn s method is that it provides a parameterized inverse, so that the strctre of this inverse system is already known. Estimating inverse models is mch harder in the nonlinear dynamical case, and many open qestions remain. Still, the discssion on the different methods and the merits and drawbacks of sing them is an interesting topic. VII. CONCLUSIONS This paper presents insights and discssions on the estimation of inverse systems. The estimated inverse is intended to be sed as a pre- or postdistorter of the original system. A good inverse model is ths one that, when sed in series with the original system, reconstrcts the original inpt. It is shown for the LTI case that the freqency weighting of the estimated inverse model differs, compared to the inverse of an estimated forward model. For a forward model, this contribtion will be larger in freqency bands with a large amplification, so that the minimization will focs the model fit to these bands. For the estimation of the inverse, a relative model error will contribte as mch to the total loss fnction, regardless of the amplitde of the transfer fnction gain (for a white noise inpt). For nonlinear systems, one way to obtain a linearizing prefilter is by the se of Hirschorn s algorithm. It is shown how to extend this to the postdistortion case, if the inpt is navailable for maniplation. The estimation of this pre- or postinversion model is also discssed. REFERENCES [] Y. Jng, J. Fritzin, M. Enqvist, and A. Alvandpor, Least-sqares phase predistortion of a +30dbm Class-D otphasing RF PA in 65nm CMOS, IEEE Trans. Circits Syst. I, Reg. papers, vol. 60, no. 7, pp , Jly 203. [2] L. Ljng, System Identification, Theory for the ser, 2nd ed. Prentice Hall PTR, 999. [3] M. Gevers and L. 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