System identification and uncertainty domain determination: a subspace-based approach

Transcription

1 2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010 FrC073 System identification and uncertainty domain determination: a subspace-based approach Wafa Farah, Guillaume Mercère and Thierry oinot Abstract In this paper, the problem of uncertainty domain determination for multi-input multi-output systems described with linear time-invariant state-space representation is tackled The developed method combines a subspace-based identification algorithm and a bounded-error approach To estimate the system state-space matrices, the propagator method is applied This method is characterized by the use of least-squares minimization algorithms and the fact that the estimated statespace realization is described using a fixed structure with a minimal number of parameters The problem of uncertainty domain description is then solved from the analysis of particular iso-level curves in the vicinity of the optimum A hard error bounding approach is finally considered to choose a threshold required by the developed method I INTRODUCTION Generally speaking, in the automatic control framework, the goal of system identification is to provide a system model accurate for control design [1] To reach this goal, two steps are generally necessary The first one deals with the estimation of the model parameters This model is required to understand, to control or to improve the system functioning The second step consists in designing the control law from the model parameters However, because the model is only a system approximation, it is paramount to fix some constraints so that the controller designed from the identified model achieves good performance when it is applied to the real system In other words, the controller must be robust with respect to the uncertainties of the estimated model parameters Thus, the estimated model uncertainties must be well-described In system identification theory, the uncertainty domain description is mainly based on prior assumptions about noise and unmodeled dynamics [2], [1] The first works assume that the disturbances acting on the system are random variables realizations [3], [4] Mainly based on timedomain representations [5], these first developments have quickly shown their limits, particularly for robust control [6] Furthermore, in some practical cases, the assumption of random white noise is too conservative or difficult to verify a priori [7] For this reason, other approaches have been developed Several of them rely on deterministic hypotheses, ie on non-probabilistic or hard error bounding approaches [8], [9] In these techniques, the identification error is unknown but bounded This basic idea has given W Farah, G Mercère and T oinot are with the University of oitiers, Laboratoire d Automatique et Informatique Industrielle, 40, avenue du recteur ineau, oitiers, France wafafarah@etuuniv-poitiersfr, guillaumemercere@univ-poitiersfr, thierrypoinot@univ-poitiersfr rise to a number of techniques usually addressed as boundederror or set membership identification [10] Although many set membership methods have been successfully applied on real systems [11], [12], the main drawback of this approach is its dependence on the way the bound is determined a priori Notice indeed that the error comes from two sources the unmodeled dynamics and the noise affecting the data which makes the bound determination quite difficult To deal with this problem, some works [4] propose to estimate the error modeling In this case, the bounded-error is determined through the analysis of particular iso-level curves which leads to less conservative bounds The identification and uncertainty domain description methods developed in [13] are restricted to single-input single-output SISO system represented by transfer function The main goal of this communication is to extend the basic idea of this approach to multi-input multi-output MIMO system In order to deal in a similar way with SISO and MIMO systems and because it is much more convenient for control design, state-space representations will be used As far as the identification problem is concerned, a particular subspace-based method, named the propagator method [14], [15], will be used to estimate directly a state-space realization of the system from the measured input-output I/O data Contrary to the classic subspace algorithms [16], [17], [18], this technique does not give access to fully-parametrized form but leads to a state-space representation with a minimal number of parameters, even for MIMO systems It is quite obvious that the problem of uncertainty domain determination is easier when the number of estimated parameters is constant and minimal Thanks to the fixed and minimal state-space matrices structure provided by the identification algorithm, the model is written into an input/output I/O form linear in the parameters L which makes the description of uncertainty areas easier These domains are derived from the analysis of particular quadratic criteria in the optimum vicinity The final objective is to get realistic uncertainty domains that contain all kinds of stochastic disturbances Thus, a hard error bounding approach is considered to reach this goal More particularly, an easy-tuning method is proposed to fix the value of the required bound The outline of this paper is as follows In Section II the notations, the main problem and the general assumptions are stated Section III is dedicated to the system parameters estimation using the propagator method The uncertainty domain determination problem is studied in Section IV In Section V, the global technique performance is emphasized thanks to numerical simulations Section VI concludes the /10/$ AACC 6501

2 paper II ROBLEM FORMULATION Consider the following LTI state-space model xt + 1 Axt + But yt Cxt + vt 1a 1b where ut R nu, yt R ny, xt R nx and vt R ny are respectively the input, the output, the state and the outputnoise vectors A,B,C are the system matrices relatively to a certain coordinate state-space basis It is assumed that {xt}, {ut}, {yt} and {vt}, t Z, are all ergodic and weakly stationary stochastic processes [1] The system order is assumed 1 to be known a priori The considered identification problem can be stated as follows: given realizations {ut} N t1 and {yt}n t1 of the input and output processes generated by a system of the form 1 on a finite but sufficiently wide time horizon N, estimate the matrices A,B,C represented in a particular state-space form and characterize the estimated parameters precision We start by making the following assumptions: i the input process {ut} is an ergodic and weakly stationary process that is persistently exciting, ii the output noise {vt} is a zero mean noise with a finite variance statistically uncorrelated with the input {ut}, iii the system is stable and minimal To reach the goal mentioned above, two linked problems are solved Firstly, the system matrices 1 are identified using a particular subspace-based algorithm see Section III Then, the uncertainty domains of the estimated parameters are characterized through the analysis of particular iso-level curves and an error bounded approach see Section IV III THE ROAGATOR METHOD A Basic idea of the method Subspace-based model identification SMI is a particular field of experimental modeling which has reached an interesting maturity from now on [19], [16], [20], [21], [17], [18] SMI methods are attractive because state-space realizations can be estimated directly from I/O data without using nonlinear optimization but by applying robust linear algebra tools such as the RQ factorization and the singular value decomposition SVD [22] Most of the SMI algorithms lead to minimal fully-parametrized state-space realizations [16], [17], [18] Interesting from a numerical point of view, the use of fully-parametrized model structure clearly overparametrizes the model and results in identifiability loss [23, Chapter 2] Furthermore, by applying a SVD, the person using the classic subspace algorithms has difficulties to fix the basis in which the state-space models are estimated In fact, this choice is realized during the reduction step when the system order is estimated but the reproducibility of the basis is not ensured when such a tool is employed In 1 This assumption is not really strong because many algorithms, mainly based on SVD and information criteria [17], are now available to get a reliable estimate of this parameter Thus, for LTI systems, the system order can be estimated beforehand this paper, a solution is proposed to give access to models described in a particular structured form where the number of elements in the state-space matrices is n x n u + n y, ie the minimal number of parameters of any MIMO system parametrization [24] The key problem of this identification method is the consistent estimation, from measured I/O samples, of the extended observability matrix column space defined as Γ f A,C Γ f [ C CA f 1 ] where f is a user-defined integer 2 such that f n x The starting point is the following relation [18] where Y f t Γ f Xt + H f U f t + V f t 2 Xt [ xt xt + M 1 ] u f t [ u t u t + f 1 ] U f t [ u f t u f t + M 1 ] H f is a Toeplitz matrix composed of the system Markov parameters, M is defined in a way compatible with the full number of I/O measurements N and Y f t and V f t have the same structure as U f t B Identification procedure The first step 3 of the propagator method consists in applying an orthogonal projection of the row space of Y f onto the complement of the row space of U f Y f Π U f Γ f XΠ U f 3 with 4 Π U f I nu U f U fu f 1 U f in order to remove the forced response and solve the unknown matrix H f problem The next step is the observability subspace estimation It is based on the assumptions that the system is observable and n x is known Assuming that A,C is observable, Γ f has got, at least, n x linearly independent rows Knowing a n x linearly independent rows set, the propagator basic idea is to use the matrix composed of these rows as a similarity transformation in order to fix the form of the estimated statespace realization The main problem is the selection of these n x rows When MISO observable systems are considered, the first n x rows of Γ f are linearly independent and the choice is obvious In the MIMO case, the problem is slightly more difficult Indeed, when multi-output systems are considered, there is no way to ensure that the first n x rows of Γ f are linearly independent or that the entire system dynamics are observable from one particular output To get round this difficulty, the solutions developed until now were related to the Kronecker indices [25] Because the state-space matrices 2 When n x is known, f can be chosen equal to n x Hereafter, equations are described using f to tackle a more general case 3 This equation is written assuming that v 0 The problem of noise treatment will be explained at the end of this paragraph 4 This projection can be computed in a stable and efficient way by resorting to the RQ factorisation [18] 6502

3 are unknown in our case, the determination of these indices is problematic In [15], it is proposed to reorganize the system state-space representation such that a full rank matrix can be easily extracted from the extended observability matrix More particularly, it is based on the following observation see [26, Lemma 1] for a proof roposition III1 Assume that the system is observable and the state matrix A is non-derogatory [22] Let κ [ κ1 κ 2 κ ny ] R 1 n y be a vector generated randomly from a uniform distribution Then it holds with probability one that rank { Γ nx A, κ C } n x This proposition states that the system dynamics are observable from an auxiliary output y a t n y i1 κ iy i t where y i is the ith system output ractically, when κ j, j [1, n y ], are randomly selected, a particular matrix K defined as follows κ 1 κ 2 κ ny K Rny ny can be introduced to substitute, eg, y a for the first output of the system and to obtain ȳt Kyt Then, using this transformation, Eq 3 becomes with Ȳ f Π U f Γ f XΠ U f 4 Γ f Γ f A, C c 1 c 2 C c ny ny j1 κ jc j c 2 c ny where c j, j [1, n y ], are the rows of C Noting that Γ f has full column rank, a permutation matrix S see [15] for its construction can be introduced in order to reorder the rows of Γ f and ensure that its first n x rows are linearly independent Γ nx A, c 1 Γ f nx A, c 1 A nx+1 S Γ f Γ nx A,c 2 Γ nx A,cny Γnx A, c 1 Γ c n x A, c 1 By construction, rank{γ nx A, c 1 } n x Hence, Γ nx A, c 1 can be used as a similarity transformation T Furthermore, it is obvious that the complementary part Γ c n x A, c 1 can be written as a linear combination of Γ nx A, c 1 Thus a unique operator R nyf nx nx exists, named the propagator [14], such that Γ c n x Γ nx Eq 4 becomes SȲfΠ U f S Γ f XΠ U f Γ nx A, c 1 XΠ U f XΠ U f with X TX This relation shows that the observability subspace can be described in a particular basis if the propagator can be estimated from I/O signals Indeed, assuming that is known, it holds Υ f S C CA CA f 1 with C CT 1 and A TAT 1 Concerning the estimation of, let us introduce Z as follows Z SȲfΠ Z1 U f XΠ Z 2 U f Then, under the assumptions considered until now, it is straightforward to see that a consistent estimate of can be obtained by minimizing 5 Z 2 Z 1 2 F Knowing ˆ, ˆΥf can be built easily Then, by straightforward calculations and using the Cayley Hamilton theorem [22], it is possible to show that A 2 : n x + 1, : a 0 a 1 a 2 a nx 1 c 1 [ ] 7 c j j 1f + 1, : for j [2, n y ] 8 where a j, j [0, n x 1], are the coefficients of the characteristic polynomial of A Once A and C are known, the matrix B can be obtained by linear regression [18] Observing A, B and C, the number of parameters 6 is n x n u +n y, ie the minimal number of parameters of any MIMO system parametrization [24] So, thanks to the minimal state-space matrices structure provided by the propagator method, the uncertainty areas description will be easier In the noisy data case, this method can be adapted by introducing an instrumental variable, as proposed, eg, in the MOES approach [18] The choice of the instruments depends on the inputs and disturbances properties see [18] for details 5 F is the Frobenius norm 6 Notice that B is a fully-parametrized matrix 6503

4 IV UNCERTAINTY DOMAIN DETERMINATION As soon as the user has access to the estimated state-space matrices, the uncertainty domains determination problem can be considered More precisely, the uncertainty domain description of the coefficients a i, b j i and ck i, i [0, n x 1], j [1, n u ], k [2, n y ] the state-space matrices A, B and C parameters is performed The developed approach basic idea is to study particular ellipsoidal iso-level curves by adapting a bounded-error method proposed in [13] This original method has been developed assuming that the system model is described via a linear regression Because this assumption is not satisfied with the estimated state-space model, it is firstly necessary to transform the estimated statespace form into a representation linear in the parameters Notice right now that the parameters of this modified model are combinations of the initial a i, b j i and ck i, i [0, n x 1], j [1, n u ], k [2, n y ] Then, the method developed in [13] is adapted to this MIMO representation linear in the parameters Finally, the uncertainty domain of the coefficients a i, b j i and ck i are deduced from the descriptions obtained by the I/O model All these phases are presented in the following A Input-output model description 1 General case: In this paragraph, a new representation of the estimated state-space model is proposed More particularly, an input-output I/O ARMAX-like model is obtained by getting rid of the state in Eq 1 All the calculations are available in [27] Due to the lack of space, only the main proposition is given hereafter More precisely, roposition IV1 Consider the state-space representation 1 Then, it can be described by an ARMAX-like model with yt a I ny ynx t n x + Fu nx t n x + ηt Φ tθ + ηt 9 Φ t yt n x yt 1 u nx t n x I ny ] 10 a θ 11 vecf a a 0 a 1 a nx 1 12 F a I ny Hnx + C nx R ny nxnu 13 nx [ A nx 1 B AB B ] R nx nxnu 14 and ηt a noise corresponding to a filtered version of v Furthermore, the order of the I/O model 9 is equal to n x if and only if the initial system 1 is minimal and the matrix A is non-derogatory It is important to note that, in the following, condition A is non-derogatory will be satisfied Indeed, this transformation will be applied to the estimated model, ie with a matrix A in a companion form see Eq 6 2 Case study: Consider the following SISO state-space model xt xt + b1 0 b 1 1 ut 15a a 0 a 1 a 2 b 1 2 yt [ ] xt 15b Then, the I/O model satisfies yt a 2 yt 1 a 1 yt 2 a 0 yt 3+b 1 0ut 1 + a 2 b b1 1 ut 2 + a1 b a 2b b1 2 ut 3 It is obvious that this I/O model is linear in θ but nonlinear in a i, b j i and ck i Remember that the final goal is to get the uncertainty domains description of the state-space representation coefficients B Uncertainty domain description for a model linear in the parameters In this paragraph, the uncertainty domains description of the parameters θ i composing θ is considered 7 Looking closes at Eq 9, it is obvious that the least-squares estimation of this parameter vector can be obtained by minimizing the following cost function [1, Appendix II] J θ 1 ym Ψ M θ ym Ψ M θ 2 where θ is an arbitrary estimate of θ and with y1 Φ 1 y M Ψ M ym Φ M Then, assuming that R Ψ Ψ M Ψ M is invertible, it can be shown that J θ 1 2 y M I Ψ M Ψ 1 M Ψ M Ψ M y M + 1 θ θls Ψ 2 M Ψ M θ θls J min d θ R Ψ d θ 16 with θ ls Ψ M Ψ M 1 Ψ M y M and d θ the variation of the estimate θ around θ ls This relation shows that J θ has a unique minimum at the least-squares solution θ ls and the first term of the rhs of Eq 16 is the minimum value Furthermore, if we introduce V θ J θ J min, it is obvious that V θ 1 2 d θ R Ψ d θ 17 is an ellipsoid centered in θ ls whose main directions are given by R Ψ The goal of the developed method is to determine an uncertainty domain D such that θ D Here, D will be an ellipsoidal surface depending on a user-defined criterion level J D This threshold must be chosen such that the system parameters θ is surely included in D without leading to a 7 The link with the coefficients a i, b j i and ck i is postponed for IV-C 6504

5 too conservative solution This level is obtained hereafter by using an analogy with the stochastic framework More precisely, under the Gaussian case assumptions no modeling error and zero-mean white Gaussian output noise, it is well known that [1] 1 2 θ ls θ Ψ MΨ M θ ls θ n 2 σ 2 where σ 2 is the noise variance and n R + This is again an ellipsoid with the same shape as the one given in Eq 17 This analogy leads to choose the level J D as follows J D J min n 2 σ 2 Unfortunately, this relation depends on the prior knowledge of σ 2 In order to circumvent this difficulty, a boundederror approach is used 8 To estimate this bound, symbolized hereafter by l, the first idea could be to measure the upper bound of the noise acting on the system when the system is not excited ractically, this methodology cannot involve the modeling error brought out when the system is excited Then, it is proposed to use the information contained in the residuals Indeed, the following result can be proved J min 1 2 v I Ψ M Ψ M Ψ M 1 Ψ M v 1 2 ε ε where v is the system disturbance and ε y ŷ θ are the residuals Thus, a good noise effect approximation can be deduced from ε and we fix l as follows l max { εt } C Extension to a model non-linear in the parameters The final goal of the developed method is to describe the uncertainty domains of the coefficients a i, b j i and c k i, i [0, n x 1], j [1, n u ], k [2, n y ] To reach this goal, let us introduce θ ss as the parameters vector of the estimated state-space form, ie θ ss [ a 0 a nx 1 b 1 0 b nu 0 b 1 1 b nu n x 1 c 2 0 c ny 0 c 2 1 c ny n x 1] Then, we know that θ ss and θ can be related via a mapping f known a priori see Eq 11, ie θ fθ ss Using a Taylor series expansion, the following first order approximation is satisfied f dθ θ ssˆθ ss dθ ss where ˆθ ss is the estimated state-space parameters vector Combining this relation with Eq 16, it holds with f R J J min + dθ ssrdθ ss θ ssˆθ ss R Ψ f θ ssˆθ ss 8 In this method, the only assumption is to have bounded residuals The uncertainty domains are again paraboloids whose main directions are given by R, ie a modified version of R Ψ Hence, the procedure explained in the L case can be easily extended for θ ss D Case study In order to illustrate the developed approach, assume that the identified model is described by the state-space form 15 Then, θ ss a 0 a 1 a 2 b 1 0 b 1 1 b 1 2 θ [ a 0 a 1 a 2 a 1 b a 2b b1 2 a 2 b b1 1 b0] 1 Furthermore, f b 1 0 b 1 1 a 1 a b 1 0 a V SIMULATION EXAMLE In order to show the performances of the method described beforehand, the following state-space matrices are used ] ] A B C [ [ These matrices can be rewritten as » A B C The input signal is a pseudo random binary sequences RBS of length 1000 A Monte Carlo simulation of size 1000 is carried out The output noise signal v is a zero-mean white Gaussian noise such that the signal to noise ratio equals 20dB The propagator method is applied to estimate the system matrices To reduce the noise effect, past inputs are used as instruments as for the I MOES algorithm [18] The past and future horizons are chosen equal to 5 Identification results are plotted in Figure 1 Firstly, these two graphs show that the estimates obtained using the propagator method are accurate Indeed, in each plot, the black cross +, symbolizing the mean value of the estimated parameters, almost matches the red one which corresponds to the real parameters 9 As far as the uncertainty domains are concerned, the level l is estimated such that l 111 Figure 1 shows the system parameters, the mean value of the estimated parameters + and the estimated parameters calculated during the 1000 realizations of the Monte Carlo simulation To assess the quality of these uncertainty domains and to get rid of the drawing of all the uncertainty domains, a failure rate measure, defined as the percentage of realizations for which the system parameters are outside of the ellipsoid D centered in the estimated parameters vector, is used see black disc 9 Knowing the state-space matrices, it is straightforward to compute the similarity transformation Γ nx A, c 1 and to transform the system statespace form into the structure

6 in Fig 1 The failure ratio equals 17 % for the couple a 0, a 1 and 24 % for a 2, b 1 0 These values show that the way the threshold is fixed leads to reliable uncertainty domains a1 b a a 2 Fig 1 Level surfaces of the cost function J θ The real parameters are symbolized by a red cross, the estimated parameters by a blue cross and the mean value of the estimated parameters by a black cross + Black discs are finally the failure draws VI CONCLUSIONS In this paper, the problem of uncertainty domain determination for MIMO LTI state-space systems is considered To solve this problem, a particular subspace-based method is used This technique leads to an estimated state-space form with fixed structure and a minimal number of parameters The description of the uncertainty domains is then realized in three steps Firstly, the estimated model is transformed into an ARMAX-like representation The uncertainty domains of the I/O model parameters are then described using a bounded-error approach The uncertainty domains of the initial model parameters are finally deduced from those obtained in the former step by adapting particular cost functions The experimental results have emphasized the reliability of the developed method REFERENCES [1] L Ljung, System identification Theory for the user, 2nd ed Upper Saddle River: rentice Hall, 1999 [2] G Goodwin, M Gevers, and B Ninness, Quantifying the error in estimated transfer functions with application to model error selection, IEEE Transactions on Automatic Control, vol 37, pp , 1992 [3] L Ljung, Identification model validation and control, in roceedings of the 36th Conference on Decision and Control, San Diego, California, USA, December 1997 [4], Model validation and model error modeling, in roceedings of the Aström Symposium on Control, Lund, Sweden, August 1999 [5] X Bombois, Connecting prediction error identification and robust control analysis: a new framework, hd dissertation, Université Catholique de Louvain, Louvain la Neuve, Belgium, 2000 [6] R Hakvoort, System identification for robust process control - 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