ON TRANSIENT DYNAMICS, OFF-EQUILIBRIUM BEHAVIOUR AND IDENTIFICATION IN BLENDED MULTIPLE MODEL STRUCTURES

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1 ON TRANSIENT DYNAMICS, OFF-EQUILIBRIUM BEHAVIOUR AND IDENTIFICATION IN BLENDED MULTIPLE MODEL STRUCTURES Roderick Mrray-Smith Dept. of Compting Science, Glasgow Uniersity, Glasgow, Scotland. Tor A. Johansen Dept. of Engineering Cybernetics, Norwegian Uni. of Science and Technology, N-79 Trondheim, Norway. Robert Shorten, Dept. of Compter Science, National Uniersity of Ireland, Maynooth, Ireland. Keywords: Nonlinear system identification, mltiple models, Gassian Process priors, Bayesian statistics, nonparametric regression. Abstract The se of mltiple-model techniqes has been reported in a ariety of control and signal processing applications. Howeer, seeral theoretical analyses hae recently appeared which otline fndamental limitations of these techniqes in certain domains of application. In particlar, the identifiability and interpretability of local linear model parameters in transient operating regimes is shown to be limited. Some modifications to the basic paradigm are sggested which oercome a nmber of the problems. As an alternatie to parametric identification of blended mltiple model strctres, nonparametric Gassian process priors are sggested as a means of proiding local models, and the reslts compared to a mltiple-model approach in a Monte Carlo simlation on some simlated ehicle dynamics data. The Mltiple-Model Framework The past few years hae shown an increase in the se of local model representations of non-linear dynamic systems (see (Johansen and Mrray-Smith 997) for a reiew). This basic strctre incldes a nmber of approaches: Tagaki Sgeno fzzy systems (Takagi and Sgeno 9), local model networks, gain-schedled control (Leith and Leithead 999), and statistical mixtre models, among them. The model parameters are obtained from prior knowledge, linearisations of a physical model or identified from measred data. Adantages of this approach are prported to be its simplicity, the insight into global dynamics obtained from the local models, and the ease with which global control laws can be constrcted from local designs. Consider the nonlinear system _x = f (x; ). By a blended local model strctre we nderstand a dynamic model of the form _x = N m X i= i (x; )f i (x; ); () where state x IR N, inpt IR P, the model f i (:; :) is one of N m ector fnctions of the state and the inpt, and is alid in a region defined by the scalar alidity fnction i, which in trn is a fnction of the aboe ariables. Typically, the local models f i are chosen to be of the form f i (x; ) = A i x + B i + d i, reslting in constitent dynamic systems i gien by, i : _x = f i (x; ) =A i x+b i +d i ; () where x; d i IR N, A i IR NN,andB i IR NP.Thisreslts in a non-linear description of plant dynamics of the form, P Nm i _x = A(x; )x + B(x; ) + d(x; ); (3) P Nm where P A(x; ) = i i (x; )A i, B(x; ) = N i (x; )B i and d(x; ) = m i i (x;)d i. Limitations of the approach Seeral limitations of the mltiple model approach are reiewed in (Shorten et al. 999). These limitations can be smmarised as being either philosophical or technical in natre. By philosophical limitations we mean difficlties, or confsion, in nderstanding the meaning of the mltiple-model model. Sch an nderstanding may be ital when we come to se the model for designing a control system. In this paper we concentrate on selected technical limitations of the mltiple model framework. In particlar, we are interested in stdying the difficlties associated with modelling offeqilibrim behaior in dynamic systems. Frthermore, we also focs on the model bias that reslts from any assmptions on the model strctre and data, or from nbiasedness of the identification algorithm. Roghly speaking, these limitations, from a practical perspectie, are related to the identifiability and interpretability of the local models. The problem of identifying off-eqilibrim linear models stems from the difficlty in gathering data of sfficient qality in these regions. In order to illstrate how interpretability problems arise in mltiple model systems we present the following abstract example of model constrction sing mltiple models. Consider the abstract case of approximating the flow associated with the dynamic system _x = f (x), in the icinity of some

2 x x x x x Error dynamics x x x Dynamics of and x x Regions R, R and Error dynamics Figre : Non-niqeness of representation. The two systems and are qalitatiely different, bt in the otlined regions (Shown in detail in R and R) we see that there is little difference, as shown by the error dynamics. ector x by the local model _x = Ax + d, wheref (:) IR N, where x; A; d are as defined earlier. Clearly for any arbitrary choice of inertible A, regardlessof its natre (stable, nstable, complex, etc.), a ector d can be fond sch that f (x )=Ax + d; () where d = f (x ), Ax. Hence, at x = x a non-niqe parameterisation of the dynamics exist, and indeed the linearisation is meaningless. Frthermore, in the neighborhood of x, sbject to some approximation error, by simply arying the location of the irtal eqilibria (or the form of the i ), it is possible to obtain many (dynamically) different parameterisations of the non-linear dynamics. This is illstrated in the following example. Example Consider the behaior of the following atonomos systems, : _x = A x; () : _x = A x + d; (),:7. where A =,: :,:, A = :,:9 3:,:, d =,: The flow associated with both of these systems is depicted in Figre. These systems are qalitatiely ery different; is a stable node with an eqilibrim point centered at the origin, whereas is a stable spiral with its eqilibrim point close to, bt not centred, at the origin. Howeer, in a small region defined by R : x ; x ; as depicted, the flow of both systems is similar. The elocity ectors point in the same direction and the maximm error, defined by, k (A, A )x + d k max = max < :: (7) xr k x k We note also that conditions exist sch that two systems, which hae the same eqilibrim point, can be identical along an entire manifold; namely, when A and A share eigenector and eigenale pairs. The manifold is defined by the eigenectors common to both systems. is bonded and small. The error dynamics _x =(A,A )x+d;x R are depicted in Figre. Hence, we conclde that in R, sbject to some appropriately defined approximation error, the dynamics described by and are in some sense eqialent. In this region both and are alid representations of an appropriate non-linear system, bt otside the region they differ considerably. This rather obios obseration is of crcial importance for two reasons. First, the identifiability of the local model parameters is poor as a direct conseqence of the fact that offeqilibrim local models with singificantly different parameters may gie ery similar dynamics within their region of alidity. Secondly, it strongly sggests that the qalitatie natre of the identified local models may say ery little abot the nonlinear dynamics een locally. This is by irte of the fact that the local model is, by definition, only alid in a local region of state space, and crcially in the off-eqilibrim case, that the local model s contribtion to the global model only comes from a restricted sb-region which does not inclde the model s eqilibrim point. This obseration, in conjnction with many similar obserations in control and identification contexts, is referred to as the Paradox of Locality in Local Model Networks. Another problem we want to emphasize is the bias being introdced de to the a priori model strctre assmptions. Strctre identification and identification of the i fnctions might obiosly improe on this. Howeer, in some cases the mltiple model strctre is not ideally sited to the system strctre and a significant bias might be difficlt to aoid. 3 Reising the Off-Eqilibria Mltiple Model Framework It was recently shown in (Johansen et al. 99) that the finite set of linearizations abot a finite nmber of points (eqilibria and transient points) can be sed to accrately approximate dynamic linearization abot arbitrary trajectories sing a blended mltiple model strctre. Despite the theoretical importance of this reslt, it is clear that the identification problems otlined in Section are paramont in a practical context. In this section we describe two complementary approaches for reising the basic mltiple model framework described in Section. The first approach inoles sggesting modifications to the existing framework which alleiate, to some extent, the problems. The second approach inoles the deelopment of a complementary nonparametric framework, with the specific aim of eliminating the problems otlined. The efficacy of both approaches is compared by means of a simple example in Section. 3. Modification of Existing Framework The dynamic linearization of _x = f (x; ) abot the point (x ; ) on some arbitrary trajectory is gien by _x = f (x ; (x ; )(x, x (x ; )(, ):

3 Introdcing deiation coordinates x = x, x, =, we get the small-signal dynamics x _ (x ; )x (x ; ) () that describe the response to small pertrbation abot a point (x ; ) on the nominal trajectory (x (t); (t)). In addition, the large-signal dynamics are locally approximated by the eqation _x = f (x ; ) which approximates the flow of the state by a constant ector near the point (x ; ) along the nominal trajectory (x (t); (t)). Now sppose we seek local linear models of the form _x = A i x + B i + d i (9) to be approximately alid in a neighborhood of a point (x i ; i ). Away from eqilibrim, the representation (9) is oerparameterized (non-niqe) since only the constant ector term d i is sfficient to gie an arbitrarily good approximation locally, see also Section. The additional degrees of freedom in the parameters A i and B i can be sed in different ways: A i and B i can be selected to increase the region of alidity of the local linear approximation (9). In this case these parameters may be completely different from the smallsignal model (x i; i ) (x i; i ) and sere only the prpose of proiding a richer class of fnction approximators. Conseqently, the local linear model may not be interpretable in terms of a small/large-signal decomposition. A i and B i can be selected to accrately represent the small-signal dynamics, i.e. A (x i; i ) and B (x i; i ). As a conseqence, the offset term will approximately characterize the large-signal dynamics, i.e. d i f (x i ; i ), A i x i, B i i. This is adantageos in terms of interpretation, analysis and applicability of the model in control systems design, bt may hae the disadantage that it may lead to a smaller region of alidity of the local model. Identification of the parameters of (9) sing, for example, a standard least-sqares criterion and some experimental data will only interpret the local model as an approximator and ths not necessarily lead to local model parameters A i and B i with a alid small-signal model interpretation. The identifiability problem is amplified by the experience that typically there is ery sparse information abot small-signal dynamics in transient operating regimes aailable in the data. The reasons for this are dierse: The system typically spends little time in transient conditions compared to stationary operating conditions, and the large signals components in the data will dominate the identification criterion. Careflly planned and expensie experiments are reqired in order to get een a small amont of small-signal dynamics information in transient operating regimes. In order to get data which are informatie with respect to both eqilibrim and transient local models, the data shold consist of two different types of excitation signals: Standard small signal pertrbations (e.g. PRBS tests) abot the releant eqilibrim points of the system, and high-freqency large step signals with sperpositioned large signal pertrbations moing the system throgh the releant transient states. For the prpose of setpoint control, we often reqire that the eqilibrim local models hae significantly higher accracy than the off-eqilibrim ones. Constrained and reglarized identication is in general a sefl tool when the data are not sfficiently informatie. Robst identification can also be achieed by directly constraining the local model parameters dring identification, see e.g. (Johansen 997). Another possibility is to take adantage of the reglarizing effect of locally weighted identification methods where each local model is identified separately by weighting each data sample according to its releance for the local model (Mrray-Smith and Johansen 997). It is obsered that this sally leads to local models with a more alid small-signal interpretation than the standard global identification method. Howeer, since these identification algorithms are biased (Johansen 997, Mrray-Smith and Johansen 997) compared to the nbiased common global least sqares identification algorithm, this improement will sally be achieed at the cost of a significantly increased model bias with the reslt that the oerall prediction performance of the model may be redced. It is therefore important that the model strctre is well tned to minimize the bias. Eentally, we are facing the well-known bias/ariance tradeoff. 3. Nonparametric alternaties Nonparametric models retain the aailable data and perform inference conditional on the crrent state and local data (called smoothing in some frameworks). As the data are sed directly in prediction, nlike the parametric methods more commonly sed in control contexts, nonparametric methods hae adantages for off-eqilibrim regions. The ncertainty of model predictions can be made dependent on local data density, and the model complexity atomatically related to the amont of aailable data (more complex models need more eidence to make them likely). Both aspects are ery sefl in sparselypoplated transient regimes. Moreoer, since weaker prior assmptions are typically applied in a nonparametric model, the bias is typically less. An example of the se of a nonparametric model is a Gassian Process prior, as reiewed in (Williams 99). In the following, the fll matrix of state and control inpt ectors is denoted, and the ector of otpt points is y. The discrete data of the regression model are k =[x(t);(t)] and y k = _x(t). The gien N data pairs sed for identification are stacked in matrices ; y and the N data pairs sed for prediction are ; y. Instead of parameterising _x = f (x; ) as a mltiple model, we can place a prior directly on the space of fnctions where f is assmed to belong. A Gassian process represents the simplest form of prior oer fnctions, so for the case with partitioned data y and y we will hae the mltiariate Nor-

4 mal distribtion (we will assme zero mean), : () y y N(; ) ; = where is the fll coariance matrix, and = T. Likethe Gassian distribtion, the Gassian Process is flly specified by a mean and its coariance fnction. The Normal assmption may seem strangely restrictie initially, bt we hae a powerfl tool in that we can adapt the model s prior expectations to a gien application by altering the strctre and parameters of the coariance fnction. The coariance fnction C( i ; j )= ij (the ij-element of ) expresses the expected coariance between y i and y j we can therefore infer y s from constant ( ; y ) s rather than bilding explicit models. We will also often iew the coariance fnction as being the combination of a coariance fnction de to the nderlying model C m and one de to measrement noise C n. The entries of this matrix are then: ij = C m ( i ; j ;)+C n ( i ; j ;),where C n () cold be ij n, for Gassian noise of ariance n.inthis paper, we se a straightforward smoothness prior coariance fnction which states that otpts associated with s closer together shold hae higher coariance than points frther apart, C m ( i ; j ;) = (j i, j j;): () (d) is a distance measre, which shold be one at d =and which shold be a monotonically decreasing fnction of d. The one sed here was (j i, j j;)=e, P p k= k( ik, jk ) : () The k s determine how qickly the fnction aries in dimension k. This estimates the relatie smoothness of different inpt dimensions. The parameter ector =( ; ;::p ) can be adapted sing standard gradient-based optimisation tools. The choice of coariance fnction is only constrained in that it mst always generate a non-negatie definite coariance matrix for any, so we can represent a spectrm of systems from ery local nonlinear models, to standard linear models sing the same framework. As in the mltinormal case, we can diide the joint probability () into a marginal Gassian process and a conditional Gassian process p(y )=p(y ;y )=p(y )p(y jy ): (3) The marginal term gies s the likelihood of the training data, P (y )=(),N j j, e, yt, y : () The conditional part of the model, which best relates to a traditional regression model is therefore the Gassian process which gies s the otpt p.d.f. conditional on the training data ; y and the test points. P (y jy ) = P (y ; y ) P (y ) T, () = e, (y,) : (y,) () N j j ; () where, as in the straightforward mltinormal case, = T, y (7) =, T, ; () so, as is dependent on we can iew this as a nonlinear regression and se f ( ) = as the expected model otpt, with a ariance of ( ) =. One adantage of the Gassian process is that, for differentiable coariance fnctions, it is easy to prodce analytic linearisations (in a limit in mean sqare sense) of the model s mean prediction (which are also Gassian processes). Vehicle dynamics example As an example, consider the longitdinal dynamics of a ehicle with mass m and speed. The interesting aspect of this experiment, oer and aboe its practical releance, is that it is a st order system, the nonlinearity is fairly smooth, and we are sing noise-free data, bt as we will see, identification of the nonlinear model can be srprisingly difficlt. The ehicle is powered by an engine which generates a longitdinal force g e (; ) where is the throttle angle. The ehicle is sbject to a distrbance force g d. A simple first order model of the ehicle is gien by the force balance m _ = g e (; ), g d ; which can be written _ = f (; ) = (g e (; ), g d )=m: (9) In the example, we set g d = N, m = kg and the engine characteristic, g e (; ) =(+3)( + arctan(, : +:)) N; is shown in Figre. With this characteristic engine cre (which corresponds to a fixed gear ratio), the engine operates in a speed interal between and m/s. 3 Linearization of the engine model (9) leads to the small-signal (; ) (; ) and drift term f (; ). These parameters are illstrated in Figre 3. The experimental data sed for identification were obtained by excitation of the ehicle by an inpt signal containing both large and small amplitde changes in order to determine the large-signal and small-signal parts of the off-eqilibrim local models. To ealate the model performance and robstness we generated data points, and identified models on nonoerlapping sbsets of 7 points. In each test the models were then tested on the remaining 993 points. The prediction performance of the different models are smmarised in Table. We can ths also analytically derie the ariance of the deriatie mean an aspect of Gassian Process priors which is difficlt to reprodce in other models withot extensie simlation. 3 This example is motiated by the experimental ehicle speed control problem considered in (Johansen et al. 99). The model is simplified, bt contains the releant aspects of the experimental ehicle in order to illstrate the main ideas.

5 t (s) t (s) engine force g(,) 3 speed ().. throttle ().. throttle angle 3 speed (ms ) 3 Figre : Left: Engine force. Right hand plots are time series of a sbseqence with 3 data points. Figre 3 gies insight into the identified parameters of the mltiple model, sing locally weighted identification of model parameters, and Gassian Process approaches, sing a fixed coariance fnction. We obsere that the Gassian Process approach prodces more accrate estimates of the small-signal parameters than the mltiple model approach with locally weighted regression. It can be seen that the model bias is the main contribtion to errors in the estimated small-signal parameters with the mltiple model approach. With the Gassian process approach the bias dominates along the eqilibrim manifold, while the ariance becomes more significant far away from eqilbrim where data are sparse. Using global least sqares, the prediction performance of the blended mlti-model can be improed, mainly de to redced bias becase it is an nbiased identification algorithm, cf. Table. In this example we fond it difficlt to redce the bias of the mltiple model strctre withot decreasing the oerall accracy de to increased ariance. Note that with a less faorable experiment design we hae experienced that the differences between mltiple models and Gassian processes become een more prononced. With the crrent experiment design and becase we are dealing with a first order system, the off-eqilbrim local models can be identified fairly well, and we do not experience high ariability of these local model parameters. Althogh the Gassian Process has consistently better test reslts from the gien data, and in the region coered by data the ariance is eenly low (nlike the mltiple model reslts which increase eenly with distance from the eqilibria), bt as we moe away from that to the edge of the plots we see a great increase in ariance of the deriatie means. This is howeer, nlike with parametric methods, accompanied by a related increase in expected prediction ariance (analytic ariance estimates for the GP s (not plotted) grew accrately in sparsely poplated areas, as desired, and match well with the Monte Carlo reslts shown). Note also the large leels of bias in the mltiple model plots, which somewhat skew the test reslts in the faor of the Gassian Process. Conclsions We hae otlined theoretical problems with the mltiple model framework when representing off-eqilibrim behaior, and illstrated them in Monte Carlo simlations. Consideration of these problems leads to new approaches to experiment planning and more robst identification methods when optimising local model parameters. This shold also proide s with more interpretable models sitable for sbseqent control system design. An alternatie approach, based on nonparametric Gassian Process prior models was deeloped and fond to proide an interesting extension of the mltiple model framework, which is simple and elegant, and can model nonlinear problems in a probabilistic framework. The disadantage is its comptational complexity, as prediction of model otpts reqires a matrix inersion of the N N coariance matrix, which becomes problematic for identification data where N >>. In transient regimes, howeer, one typically has ery few data points and we wish to make robst estimates of model behaior. This sggests a heterogeneos soltion with a mltiple-model model composed of a nmber of linear sbmodels arond eqilibrim points, and Gassian process sbmodels in transient areas. Acknowledgements R. Mrray-Smith carried ot this work while at the Department of Mathematical Modelling, Technical Uniersity of Denmark. The spport of Marie Crie TMR grant FMBICT939, and in part by the Danish Research Concils throgh the Comptational Neral Network Center (CONNECT) and the THOR Center for Neroinformatics is grateflly acknowledged. The work of T. A. Johansen was spported by the Eropean Comission nder the ESPRIT Long Term Research project H C. References Johansen, T. A. (997). On Tikhono reglarization, bias and ariance in nonlinear system identification. Atomatica 33(),. Johansen, T. A. and R. Mrray-Smith (997). The operating regime approach to nonlinear modelling and control. In: Mltiple Model Approaches to Modelling and Control (R. Mrray-Smith and T. A. Johansen, Eds.). Chap., pp Taylor and Francis, London. Johansen, T. A., K. J. Hnt, P. J. Gawthrop and H. Fritz (99). Offeqilibrim linearisation and design of gain schedled control with application to ehicle speed control. Control Engineering Practice, 7. Leith, D. and W. Leithead (999). Analytic framework for blended mltiple model systems sing linear local models. International Jornal of Control. To appear. Mrray-Smith, R. and T. A. Johansen (997). Local learning in local model networks. In: Mltiple Model Approaches to Modelling and Control (R. Mrray-Smith and T. A. Johansen, Eds.). Chap. 7, pp.. Taylor and Francis, London. Shorten, R., R. Mrray-Smith, R. Bjørgan and H. Gollee (999). On the interpretation of local models in blended mltiple model strctres. International Jornal of Control. To appear. Takagi, T. and M. Sgeno (9). Fzzy identification of systems and its applications for modeling and control. IEEE Trans. on Systems, Man and Cybernetics (), 3. Williams, C. K. I. (99). Prediction with Gassian processes: From linear regression to linear prediction and beyond. In: Learning and Inference in Graphical Models (M. I. Jordan, Ed.). Klwer.

6 Table : The m.s.e. is the root mean sqare error when predicting _ based on and, e is standard deiation of this error. Fll set Off-eqil. On-eqil. Model m.s.e. e m.s.e. e m.s.e. e Mltiple models (locally weighted fit) Mltiple models (global fit) Gassian Process From left: Mltiple model weighting fnctions i, example phase plot of data (speed s. throttle angle). Right hand plots are exact system (; (; ) speed (ms ) throttle angle Mean mltiple model small-signal parameters A(; ) and B(; ), mean GP estimates of small-signal (; (; ). Linearized A(,) Linearized system gain B(,) Absolte bias in mltiple model small signal-parameters A(; ) and B(; ) (left two) and GP small-signal (; (; ) (right two) From left: times the standard deiation of the mltiple model small signal-parameters A(; ) and B(; ) (left two) and GP small-signal (; (; ) (right two) From left: Mean bias and std. de. plots for f (; ), for mltiple model (left two) and GP (right two) Figre 3: Reslts of Monte Carlo simlation of mltiple model and Gassian Process identification processes. Note that we plot the means of models here, which are sally better models than the indiidal models.