Recursive Algebraic Frisch Scheme: a Particle-Based Approach

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1 Recursive Algebraic Frisch Schee: a Particle-Based Approach Stefano Massaroli Renato Myagusuku Federico Califano Claudio Melchiorri Atsushi Yaashita Hajie Asaa Departent of Precision Engineering, The University of Tokyo, 7-- Hongo, Bunkyo-ku, Tokyo -8656, Japan; eail: {assaroli,iyagusuku,yaashita,asaa}@robottu-tokyoacjp Departent of Electrical, Electronic and Inforation Engineering, University of Bologna, Viale Risorgiento, Bologna 4, Italy; eail: {federicocalifano,claudioelchiorri}@uniboit Abstract: This paper presents a novel identification procedure The proposed ethod consists in a recursive forulation of the algebraic Frisch schee, which is an estiation procedure based on ild a priori assuptions in the context of error-in-variables (EIV) schees Siulations have been perfored to show the validity of the new ethodology Keywords: Syste identification, Recursive estiation, Linear systes, Frisch schee INTRODUCTION This work deals with the proble of extracting linear relations fro a set of data affected by additive noise Most of the algoriths proposed in the literature (eg, least ean squares) seek a single solution for this proble This uniqueness can be achieved only by introducing a priori assuptions in the estiation procedure whose validity cannot be ascertained on data A coplete analysis of different estiation schees has been proposed by Kalan (98), Söderströ and Stoica (989), Ljung (998), Söderströ (7, ), Iserann and Münchhof (4) Further analysis of the current state of the art of syste identification can be found in (Ljung, ) while newer trs are presented in the survey paper (Pillonetto et al, 4) The ost iportant feature of the Algebraic Frisch Schee, originally introduced by Frisch (94), is to eploy loose assuptions leading to a whole faily of solutions copatible with a given set of data rather than a unique solution In this context, according to Kalan (98), this estiation schee has fewer prejudices with respect to ost other identification procedures The properties of this schee and its loci of solutions have been deeply analyzed and developed in (Guidorzi, 99); (Guidorzi, 995); (Guidorzi et al, 8) The Frisch schee has been also exted (in the dynaic case) for the identification of ultivariable EIV odels (Diversi and Guidorzi, 7) Under assuptions that will be discussed in the following sections, if only one underlying linear relation is present, the Frisch estiation schee leads to a convex set of solutions represented by a siplex in an Euclidean space, where any point of the siplex represents a feasible solution in the context of the Frisch schee This peculiarity introduces a post-identification degree of freedo in the selection of the paraeters aong all the In this paper we always deal with the algebraic Frisch schee This specification will be oitted fro now on feasible solutions copatible with the hypothesis of the Frisch schee This aspect ay represent a resource in engineering contexts since the knowledge of physical aspects of the syste to be identified can help in this selection, ixing a pure black-box estiation step with a reasonable white-box selection procedure The ain contribution of this paper is the introduction of a novel recursive estiation schee based on the Frisch schee, which is able to restrict the searching space of solutions with respect to the traditional batch approach without introducing additional assuptions The proposed ethod is based on the idea of perforing several identifications of the sae process using partial data sets at different ties to obtain a saller final solution space Using the proposed algorith there is a consistent iproveent of the estiation precision with respect to the standard Frisch schee in only few iterations This property akes the proposed algorith suitable for realtie identification The Particle-based Recursive Frisch (PRF) schee is presented and applied to a siple EIV odel to prove its effectiveness The PRF estiation schee can be successfully used in a robust control fraework as a robust controller could be designed in such a way to deal with odel uncertainties ranging in the space of adissible solutions identified with the PRF schee This paper is organized as follows: Sec introduces soe background on EIV schees, with a particular insight on the Frisch schee, enlightening the properties of its solutions spaces In Sec the proposed PRF identification procedure is presented Siulation experients and results are shown in Sec 4 In Sec 5 conclusions and future work are drawn THE FRISCH ESTIMATION SCHEME: BACKGROUND As every systeatic procedure that deduces a odel starting fro data affected by errors, the Frisch Schee

2 relies on the odification of the observed data This is ainly due to the fact that a deterinistic atheatical relation able to fit all the observations does not exist The ain difference between different estiation schees is the set of a priori assuptions that are necessary to estiate linear relations fro noisy data Different schees use different assuptions and odify observations in different ways Coon Assuptions for EIV Estiation Schees Consider the linear relation α x + α x + + α n x n = () which relates n variables x i by eans of n coefficients α i If easureents of the variables have been collected, the observation atrix can be defined as follows x x x n x x x n X =, X IR n () x x x n Then, equation () can be rewritten as XA = () where A = (α α α n ) T In general, A IR n q where q is the nuber of indepent linear relations linking the data By defining the saple covariance atrix as Σ = XT X, () is equivalent to ΣA = (4) since Σ and X share the sae null space Assue now that noise affects the variables in an additive way, ie, x i = ˆx i + x i where ˆx i is the exact value of the variable and x i is its corresponding noise Due to this noise, Σ is positive definite and thus it is ipossible to extract fro its null space any linear relation, represented by atrix A This can only be achieved by odifying the atrix Σ Under the coon assuptions of EIV schees which relies in zero ean value of both noise and noiseless saples and in orthogonality between the, see (Guidorzi et al, 8), it can be derived that Σ = ˆΣ + Σ, for ˆΣ = ˆX T ˆX and Σ = X T X, with ˆX, X both defined as in () by replacing x i with ˆx i and x i The proble of finding linear relations copatible with noisy data can be then forulated as follows: EIV Estiation Proble (Kalan, 98) Given Σ, deterine Σ such that ˆΣ = Σ Σ, det( ˆΣ) = (5) Any base of null( ˆΣ) will span a space describing a set of linear relations copatible with the data and the assuptions, ie a solution for ˆΣA = can be found Differences aong estiation schees lie in the assuption ade on the noise variables and consequently on Σ Two estiation schees of interest in this work are now described Ordinary Least Squares (OLS) This estiation schee assues only one variable affected by noise, while the reaining ones are noise-free If noise affects the i-th variable, it follows that Σ = diag (,,, σ i,,, ) (6) where σ i is the variance of x i The solution of the estiation proble is given by σ i det (Σ) = (7) det (Σ i ) where Σ i is the atrix built by deleting the i-th row and the i-th colun of Σ, see (Guidorzi, ) In fact σ i, as defined in (7), represents the axiu aount of noise copatible with the condition presented in (5) Soe iportant aspects of this estiation schee are: In general, the noisy variable is the one considered as linearly depent, whose coefficient is noralized to The OLS solutions lead to the iniization of the squared estiation error: A = arg in ( e T e ) (8) where e T e = [X] i X i A (9) being [X] i the i-th colun of X and X i obtained by deleting and the i-th colun of X The eleents of A IR n define the depence relation of the orthogonal projection of [X] i on the subspace spanned by the coluns of X i fro these vectors When no assuptions are ade about the noisy variable, the OLS schee leads to n different solutions When the noisy variable is known to be the i-th, the estiation proble can be reforulated in the ost coon non-kernel for: [X] i = X i A In this case the iniization of the squared error is siply obtained in the pseudo-inverse closed for solution A = (X T i X i) X T i [X i], which is unique The Frisch Schee This schee is based on the assuption of utual indepence of the noise saples, ie the error covariance atrix is in the for Σ = diag ( σ, σ,, σ n ) () Given a saple covariance atrix Σ, every diagonal atrix Σ satisfying (5) is a solution of the Frisch estiation schee Notice that the n OLS solutions are included This eans that the a priori assuptions ade for the Frisch schee are ilder and ore general than those of any estiator providing a single solution like the OLS On the other hand, the coputation of solutions in closedfor is ore difficult, since there are infinite possible solutions To overcoe this proble, we use soe of the results reported in (Guidorzi et al, 8), in particular, the following definition and theore Definition The axiu corank of a saple covariance atrix Σ is defined as the axial nuber of linear relations that can be extracted fro Σ under the assuptions of the Frisch schee and it is denoted by Maxcor F (Σ): Maxcor F (Σ) := ax Σ D { di ( null( ˆΣ) )} () being D the set of all the diagonal atrices satisfying (5), ie, all the solutions of the Frisch schee Theore (Guidorzi et al, 8) If Maxcor F (Σ) =, the coefficient vectors A IR n of all linear relations copatible with the Frisch schee lie (by noralizing

3 one of the coefficients to ) inside a siplex S in the paraeter space whose vertices are defined by the n OLS solutions Furtherore, all the points of the siplex in the paraeter space are linked by a one-to-one relation to all the solutions of the Frisch schee Under the hypothesis of the latter theore, the solution space (ie, the siplex) can be easily derived by coputing the n OLS solutions However, in order to select one solution, additional inforation ust be present (see eg, (Guidorzi et al, 8)) The ain contribution of this work is the developent of an algorith which ais at reducing the solution space The decrease in size is inted with respect to the siplex that would erge by coputing a single batch estiation by eans of the classical Frisch schee The ain advantage of this new ethod is the ability of shrinking the volue of the solutions set in the paraeters space without introducing further assuptions This volue can be considered indeed as a perforance index of the estiation schee In the classical batch estiation schee the volue of the siplex is proportional to the aount of noise affecting the syste If there is no noise, the siplex collapses to one point, ie the true paraeters vector PARTICLE-BASED RECURSIVE METHOD After the first easureents, let us keep observing the linear syste described by () At tie instant t k the state of the observed syste is x(t k ) = (x (t k ) x (t k ) x n (t k )) () The observation atrix at the tie instant t k is updated as x(t k ) x(t k + ) X(t k ) = IR n () x(t k ) This update rule relies on the collection of successive observations of the state that are stacked in a atrix Note that any observation saple is used in one observation atrix only, axiizing the aount of new inforation contained in it Therefore, a new estiation can be perfored with a frequency ties slower than the one of state observation since new saples are needed to build a new observation atrix The saple covariance atrix at tie t k is: Σ(t k ) = X(t k) T X(t k ) (4) According to Theore, if Maxcor F (Σ(t k )) =, the solutions of Σ(t k )A = lie in the siplex whose vertices A i (t k ) IR n are defined by the n OLS solutions in which one entry is noralized to (or ): A i (t k ) : span(a i (t k )) = null(σ(t k ) Σ i (t k )) (5) where Σ i (t k ) = diag(,,, σi (t k ),,, ) (6) and σ i (t k ) = det(σ(t k)) (7) det(σ i (t k )) being Σ i (t k ) the atrix obtained by deleting the i-th row and the i-th colun of Σ(t k ) Without any loss of generality, let us suppose to noralize the last coefficient α n to Therefore, under the hypothesis of the Frisch schee and if Maxcor F (Σ(t k )) = the solution of the identification proble belongs the siplex S(t k ) IR n whose vertices are defined as in (5) for any value of k Intuitively, if several siplices are coputed at different iterations, it sees reasonable to seek the paraeters value inside the intersection of all the siplices Although the Frisch schee allows a siple derivation of the vertices of the siplices of solutions in the paraeters space, the coputation of their intersection fro the knowledge of the vertices is not trivial However, it is rather easy to check if a point of the paraeters space belongs to a siplex, thanks to its convexity The Particle-based ethod consists in creating a set of static points in a region of the paraeters space, called particles, each of which will be considered as a possible solution of the estiation proble At each iteration, a new siplex is coputed and the particles not belonging to the intersection of this siplex with the previous one are discarded It turns out that this approach leads to a final solution space which is saller with respect to the siplex that would erge when a single batch identification is perfored with all the data The working principle of the ethod is highlighted by the results shown in Sec 4 In the following the procedure is described in detail Initialization of the Procedure Let us define X(t ) as x(t ) x(t ) X(t ) = (8) x(t ) If we calculate the corresponding saple covariance atrix Σ(t ) = X(t)T X(t ), the initial siplex S(t ) is then uniquely defined by its vertices which are coputed as in (5) Being l(t ) the initial nuber of particles, the region of the paraeter space bounded by the initial siplex is filled uniforly with the l(t ) particles p j IR n, with the procedure described in Appix A The initial particle set is defined as P(t ) := {p j : j =,,, l(t )} (9) Updating Rule of the Solutions Set Being S(t k ) the siplex corresponding to Σ(t k ) and P(t k ), l(t k ) the particles set and the nuber of particles at the k-th tie instant, in each iteration the algorith is developed according to the pseudo-code reported in Algorith S(t k ) is initially derived fro Σ(t k ) by eans of (5) Then the updated particle set P(t k ) is assebled with the eleents of P(t k ) lying inside S(t k ) Thus, P(t k ) has the following properties: P(t k ) = {p j : p j S(t h ) h =,,, k} () { } k = p j : p j S(t h ) () h= Fro this siple idea it is possible to derive interesting consequences Indeed, it represents a different way to use

4 The relative ean error of the particles er (tk ), which is coputed at each iteration as: Algorith : PRF psudo-code Data: P(tk ), l(tk ), Σ(tk ) Result: P(tk ) Copute S(tk ); j = ; while j l(tk ) do if pj S(tk ) then pj P(tk ); else discard pj j =j+ kpj (tk ) A k (6) j= T A = (α α α ) = ( 5 ) T (7) The volue Vsol of the solution space, naely, the volue of the convex polytope of the paraeters space deliiting the particles The procedure was initially tested with =, σ = σ = σ = 6 and σ 4 = In Fig the convex hull A 5 A A 4 Case of Study and Siulation Setup A Monte Carlo Siulation of runs has been perfored in which x, x and x have been generated as three indepent sequences of zero ean rando nubers with unitary variance, while x4 has been obtained as x4 = (α x + α x + α x ) (4) Due to the value of the variances of x, x, x and the coefficients αi (i =,, 4) x4 presents a standard deviation of about 77 Indepent zero ean additive Gaussian noises has been successively added to all the variables, with variances σ, σ, σ and σ 4 Therefore, the state of the syste at any instant tk can be described as follows: x(tk ) = (x (tk ) x (tk ) x (tk ) x4 (tk )) (5) and the observation atrix has been built according to () Let tot be the nuber of saples in the dataset The total nuber of PRF iterations is N = (tot /) (excluding the initialization) The paraeter tot has been set to be 7 and the initial nuber of particles 6 The initialization of X has been done using the first saples At each iteration the vertices of the siplex of solutions have been coputed as described in Section noralizing to the last entry of each OLS solution For coparison, the total siplex Stot has been coputed with the whole dataset, ie the siplex obtained with the classical Frisch estiation schee using all the tot observations of the state In order to evaluate the efficiency of the proposed schee the following two perforance indexes are defined: l(tk )ka k where A is the vector containing true values of the paraeters corresponding to the first three variables 4 SIMULATION RESULTS AND DISCUSSION To study the effectiveness of the proposed procedure the following linear odel has been analyzed α x + α x + α x + α4 x4 = () where α = 5 α = α = α4 = () X a set of data, in the contest of the Frisch schee, as an alternative to perforing a unique batch identification The advantages of this approach, with respect to the standard case, are analyzed in the next section, also on the basis of the choice, which represents the length of the partial data set used for a single identification step Furtherore, thanks to its fast convergence tie, this new schee turns out to be suitable for online identification er (tk ) = l(tk ) A Fig Coparison between the total siplex defined by the vertices Ai coputed as in (5) with the whole data set and the convex hull enclosing the particles calculated for =, σ = σ = σ = 6 and σ 4 = enclosing the particles reained at the of the PRF identification is copared with the total siplex Stot The volue difference of the two solution spaces shows the iproveent in the estiation precision given by the PRF The convergence of the PRF can be observed in Fig where the tie evolution of the relative ean error of the particles is presented It can be noticed that er rapidly decays with respect to the nuber of iterations reaching a nonzero constant value The volue Vsol of the solution space also decreases rapidly as it can be ascertained in Fig In this figure the tie evolution of Vsol is copared with the one of Vk, the volue of the siplices obtained at different iterations and with the volue of the total siplex Vtot Results show that Vsol is halved after only ten iterations and in the worst case it becoes saller than Vtot after three iterations Fro the outcoe of the siulations can be noticed that, aong all the Monte Carlo iterations, Vtot changed of about % of its value, rearking the robustness of the standard Frisch schee identification It also turns out that Vtot corresponds always to the average of the volues of the siplices obtained at different iterations The geoetrical properties of convex polytopes and useful ethods to copute their volue can be found in (Henk et al, 4) For this reason, in Fig, only the average aong the Monte Carlo runs is represented

5 Fig Tie evolution of relative ean error er for =, σ = σ = σ = 6 and σ 4 = The solid line is the average of the values obtained in different Monte Carlo runs while the colored area (er,b ) is enclosed by the absolute iniu and axiu values of er in the Monte Carlo runs Fig 4 As function of δ and, respectively the ratio between noise and signal standard deviations and the length of the observation atrix, two quantities are represented On the left, the residual relative ean error of the particles err On the right, the ratio Vrr between the volue enclosing the particles set at the of the PRF identification (Vsol ) and the volue of the total siplex (Vtot ) ual 4 relative ean error of the particles, err = er (tn ) and the residual relative volue of the solution space Vrr = Vsol (tn )/Vtot strongly dep on the utual values of δ and, ie, err = err (, δ), Vrr = Vrr (, δ) In Fig 4 err and Vrr are represented as function of δ and while in Fig 5 they are shown as function of the observation atrix length for different values of δ Residual Relative Mean Error Fig 4 shows how err increases linearly with δ, indepently fro Fig 5 shows how err reaches exponentially with a constant ξ(δ) which is the relative ean error of the initial particles, having initialized the PRF with all the tot saples If l(t ) is sufficiently large this value is very close to the relative ean error of the total siplex er,tot, hence: ξ(δ) er,tot (8) where Z kp A kdp (9) er,tot = Vtot ka k Fig Tie evolution of the volue of the convex hull enclosing the particles Vsol copared with the volue of the siplices obtained at each iteration Vk and the volue of the total siplex Vtot for =, σ = σ = σ = 6 and σ 4 = The two solid lines are the average of Vsol and Vk obtained in different Monte Carlo runs, the dashed black line the average of Vtot while the colored areas (Vk,b and Vsol,b ) are enclosed by the absolute iniu and axiu values of Vsol and Vk in the Monte Carlo runs Stot 4 Overall Perforance Analysis If we suppose tot arbitrarily big, naely tot, it yields Several estiation tests have been perfored assigning different values to the noise variances and to the length of the observation atrix This allows to underline how the utual choice of these variables affects the PRF estiation, showing up soe iportant properties of this schee Therefore, different siulation have been perfored in which the noise variances have been chosen as σ i = (δ σ i ), for i =,,, 4 being σ i the standard deviation of the variable xi The coefficient δ has been valued between and 75 with steps of 5 In addition, the estiation procedure has been tested with ultiple values of chosen between 5 and The results of the siulations tests show that the resid- err (, δ) ξ(δ)( e /λ ) er,tot ( e /λ ) In our experiental case, the value of λ turns out to be approxiatively, indepently fro δ Residual Relative Volue Results shown in Fig 5 exhibits that the PRF leads always to a solutions set whose volue Vsol is lower than the one obtained through a single identification with the whole data Vtot since Vrr is always saller than one Like err, Vrr shows an exponential tr 4 The ter residual refers to the value at the last PRF iteration

6 7 = e rr V rr 7 = 5 estiation However, in certain conditions, the PRF schee ay becoe unstable due to both nuerical and statistical issues In fact, although sall values of and δ should iprove the overall perforance of the PRF schee, too sall values ay lead to failures of the procedure A robustness analysis follows in the next part 4 Robustness Analysis of the PRF identification [log] 7 = [log] 7 = [log] [log] Fig 5 For different values of δ (the ratio between noise and signal standard deviations), two significant data are here reported as function of the observation atrix length (in log scale) With solid blue lines, the residual relative ean error of the particles e rr With green lines the ratio V rr between the volue enclosing the particles set at the of the PRF identification (V sol ) and the volue of the total siplex (V tot) with and reaches a value close to when approaches tot On the other hand, the growth rate of V rr deps on δ The depency on δ is also exponential as it can be appreciated in Fig 4 If we suppose tot arbitrarily big, we would have: V rr (, δ) e (λ+λδ) After this analysis, it coes out that the perforances of the PRF schee strongly dep on two variables: the noise affecting the syste and the nuber of subsets in which a given dataset is divided 5 However, as syste designers, we only have control on this latter variable since we cannot change the syste noise rate Experiental results also shown, indepently on the syste noise, the optial design of the identification algorith is achieved by dividing the dataset in sall partial subsets In fact, the perforance of the PRF degrades exponentially with The results show the iproveent of the perforance of the PRF copared to a single batch approach for any value of noise variance and length of the observation atrix Therefore, whenever a sufficiently large set of data is available it is always ore convenient to perfor the PRF schee rather than use the whole data set for a single 5 ie, the nuber of iterations perfored Fig 6 Robustness ap of the PRF schee as function of and δ The blue area represents the unfeasible points where the algorith failed in at least one Monte Carlo run The gray-scale plot represents the probability of having the true paraeters vector lying inside the convex hull enclosing the survived particles for different Monte Carlo runs Throughout the siulations the feasibility of the PRF estiates has been constantly checked Firstly, at each iteration the nuber of survived particles l(t k ) has been onitored: if it dropped below one thousandth of the initial nuber the estiation has been considered unreliable In fact excessive drop of particles ight be sypto of ill conditioned estiates because a too sall solution space should not be achieved considering the aount of noise affecting the syste Consequently, the siulation was aborted and the corresponding couple δ was arked as failure (the blue area) Then, at the of each estiation it has been checked whether the true paraeters was lying inside the PRF solutions space or not If the true paraeters vector was inside the hull the value of has been assigned to the corresponding couple δ while, if it was outside, the value has been set to Finally, the results of different Monte Carlo runs has been averaged This data has been then used to build a ap of to the robustness of the PRF procedure as function of δ and, as in Fig 6 When is very sall (ie less than ) the PRF schee always fails for any aount of noise In fact, if is too sall, the observation atrix (ie the regressor) is not representative of the underlying syste, leading to unreliable estiates Furtherore, when also δ is sall, this effect is accentuated by nuerical issues (eg null space extraction fro bad conditioned atrices, check of particles inclusion inside very narrow siplices) Therefore, the estiates result to be unreliable even for higher values of Note that this latter issue ight occur also in the standard version of the Frisch schee In order to find the optial trade-off between perforance

7 and reliability, has to be chosen based on soe rough estiates of δ In addition, if the syste noise in sall, the above described nuerical issues can be overcoe by adding ore indepent additive noise in the syste This would stabilize the estiations without violating the assuptions of the Frisch schee 5 CONCLUSION In this work a novel recursive identification procedure (PRF) based on the Algebraic Frisch Schee has been presented The ain difference with respect to the standard approach consists in using different subsets of data to perfor several identifications of the sae process instead of perforing a batch identification step Siulations have been conducted to prove the effectiveness of the proposed ethod Results shown the increased reliability of the developed ethodology with respect to a single identification Furtherore, this can be achieved without introducing additional assuptions on data A robustness and sensitivity analysis of the algorith, with respect to the paraeters used to ipleent it, has been perfored through Monte Carlo siulations Future works will ext the proposed ethod to tie varying systes assigning dynaic features to the particles, and will apply the developed algorith in a robust control fraework REFERENCES Boyd, S, Vandenberghe, L, 4 Convex optiization Cabridge university press Diversi, R, Guidorzi, R, 7 The frisch schee in ultivariable errors-in-variables identification European Journal of Control 7, 4 5 Frisch, R, 94 Statistical confluence analysis by eans of coplete regression systes Vol 5 Universitetets Økonoiske Instituut Guidorzi, R, 99 Certain odels fro uncertain data: the algebraic case Systes & control letters 7 (6), Guidorzi, R, 995 Identification of the axial nuber of linear relations fro noisy data Systes & Control Letters 4 (), Guidorzi, R, Multivariable syste identification: fro observations to odels Bononia University Press Guidorzi, R, Diversi, R, Soverini, U, 8 The frisch schee in algebraic and dynaic identification probles Kybernetika 44 (5), Henk, M, Richter-Gebert, J, Ziegler, G M, 4 Basic properties of convex polytopes Handbook of discrete and coputational geoetry, 55 8 Iserann, R, Münchhof, M, 4 Identification of Dynaic Systes: An Introduction with Applications Springer Publishing Copany, Incorporated Kalan, R E, 98 Identification fro real data In: Current developents in the interface: Econoics, Econoetrics, Matheatics Springer, pp 6 96 Ljung, L, 998 Syste identification Signal analysis and prediction, 6 7 Ljung, L, Perspectives on syste identification Annual Reviews in Control 4 (), Pillonetto, G, Dinuzzo, F, Chen, T, Nicolao, G D, Ljung, L, 4 Kernel ethods in syste identification, achine learning and function estiation: A survey Autoatica 5 (), Söderströ, T, 7 Errors-in-variables ethods in syste identification Autoatica 4 (6), Söderströ, T, Syste identification for the errorsin-variables proble Transactions of the Institute of Measureent and Control 4 (7), Söderströ, T, Stoica, P, 989 Syste identification Appix A INITIALIZATION OF THE PARTICLES Given a atrix M, M i,j denotes the eleent of M corresponding to the i-th row and the j-th colun Let S(t ) be the initial siplex It is possible to define a atrix S IR (n ) n whose coluns are the vertices A j (t ) of S(t ) Then, let B be coputed as in{s,j } ax{s,j } j j in{s,j } ax{s,j } j j B = (A) in{s (n ),j } ax{s (n ),j } j j A axiu nuber l i of initial particles is chosen such that l i = a n, a N It is then possible to define a atrix R IR (n ) a whose eleents R i,j are as follow: R i,j = a j a B i, + j a B i, (A) for i =,, n, j =,, a The generation of the particles can be achieved by eploying the algorith shown in the pseudo-code Algorith To carry out the particles set initialization it is needed an Algorith : Particles set initialization pseudo-code Data: R, S(t ) Result: P(t ) P(t ) = {}; for c to a do for c to a do for c n to a do p = (R,c, R,c,, R n,cn ) if p S(t ) then add p to P(t ) algorith able to deterine if a point belongs to a convex hull (ie, a siplex), see eg, (Boyd and Vandenberghe, 4) This initialization procedure leads to a nuber of particles l(t ) that constitute P(t ) which is less than l i, l(t ) < l i The diinution of l(t ) with respect to l i is only depent on the shape of the initial siplex S(t ), eg, it ay greatly decrease if soe edges of the siplex are narrowed