Ljung L. System Identification : theory for the user. Prentice-Hall Information and System Sciences Series, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....m files or block diagrams, that is, between the first three methods. One of the problems in most simulation languages regards handling of algebraic loops. A typical example of an algebraic loop is when two linear systems with direct feed through are connected in a loop as shown in figure 2. 1 [52]. y = v w s 2 s 1 s 3 s u Figure 2.1: Two systems with direct feed through connected in a loop For both systems the output depends directly on the input (y depends directly on u and w depends directly on v) The signal u is equal to w, which depends directly on v, ....

L. Ljung and T. Glad. Modeling of Dynamic Systems. Prentice Hall Information and System Sciences Series, 1994.

....vector # [2,1] Hence the relation (2) is inverse to relation (1) DFT is of great interest in various engineering elds. For its relationship to Fourier series of sampled signals, DFT is frequently used in signal processing [2] One of the experimental identi cation methods employs DFT as well [7]. The close relationship of DFT to interpolation is also well known and was used recently to solvesometasks of the polynomial control theory [12, 13] and to treat robustness analysis problems of certain kind [14] For numerical computation of DFT, the ecient recursive FFT algorithm was developed ....

Ljung L., System Identication: Theory for the User, Prentice-Hall Information and Systems Sciences Series. Englewood Clis, Prentice-Hall (1987).

....to obtain a model of form in Eq. 1 using input output data, but two methods are most relevant in the present context. The #rst is the prediction error method wherein the model is #rst put in a predictor form and parameters are estimated by minimizing the prediction error for the available data #Ljung, 1987#. This method requires some canonical parameterization of state space matrices,which require signi#cant prior knowledge,especially in the multivariable case. It also results in a nonconvex optimization to be solved. An alternative is the method of subspace identi#cation, which consists of #1# the ....

Ljung, L. #1987#.System Identi#cation: Theory for the User. Prentice-Hall Information and System Sciences Series, Englewood Cli#s, NJ.

....identification scheme which estimates all frequency response functions (FRFs) of the linear building blocks in a non parametric way, using measurements of ( and ( only. These non parametric FRFs can be parametrized afterwards using classical time or frequency domain identification techniques [3], 4] This paper is structured as follows. The different structural degenerations in the nonlinear model are presented in Section II. Section III provides the description of the identification scheme. The applicability of this identification scheme on a measured nonlinear system is shown in ....

....j j j j j j j 2 first C. Identification of and G ; b G G G b G G G G : G G G G G ; G G ; G G G b G G G G G ; G G G G G G S k S Y k r k n G G b G G G G G k ; k G k k G k G k : G k ; k 6 [3] 1 2 3 2 2 22 1 2 22 1 3 22 2 3 3 12 1 2 3 21 1 21 2 21 3 11 [1] 21 12 [2] 1 2 22 [3] 1 2 3 21 12 22 3 21 12 21 12 [2] 1 2 21 12 21 12 21 12 [ 0 0 21 12 2 21 12 21 12 [2] 1 0 2 0 12 1 2 0 21 1 0 21 2 0 [2] 1 0 2 0 ( 2 3 ( 3) When taking a ....

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L. Ljung, , Prentice Hall Information and System Sciences Series. Prentice Hall, Englewood Cliffs, 1987.

....and difficult problem that is related to model structure 16 or model order determination problem. Even for linear models, choosing model order is not a trivial problem. Several approaches exist, such as generalized cross validation, model complexity penalty, statistical hypothesis test (see [30] chapter 16) and minimum description length criterion [31] Usually, both generalized cross validation and minimum description length criterion lead to the form of criterions with a term measuring the modelling error and a term corresponding to some model complexity penalty. We have numerically ....

....penalty. We have numerically tested several criterions of this type for determining the number of wavelets in wavelet network. It turns out that the Akaike s final prediction error criterion (FPE) works quite well when tested on a variety of examples. This criterion is written as follows [30, 32]: J FPE ( b f ) 1 n p =N 1 Gamma n p =N 1 2N N X k=1 ( b f (x k ) Gamma y k ) 2 where n p is the number of parameters in the estimator, x k ; y k ) 2 O N 1 are training data and N is the sample length of the training data. Because scalar dilation parameters are used for ....

L. Ljung, System Identification : Theory for the User. Prentice Hall Information and System Sciences Series, Prentice Hall, 1987.

....control command u(t) input vector to the computer A = system matrix B = input matrix for continuous time model Data values for the control command and input vector are obtained from a B737 computer simulation. The parameters A and B can be determined using the least squares estimation [7], which is given by: # = X T X) 1 X T Y (4) where: Y = # ## # # X = x(t) u(t) The vector X is the regression vector, and # is the parameter vector that contains the model parameter A as its first element and B as the vector of remaining elements. The discrete model is ....

L. Ljung, ##########################################., PrenticeHall Information and System Sciences Series , T. Kailath, Ed. (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1987).

....important and difficult problem that is related to model structure or model order determination problem. Even for linear models, choosing model order is not a trivial problem. Several approaches exist, such as generalized cross validation, model complexity penalty, statistical hypothesis test (see [30] chapter 16) and minimum description length criterion [31] Usually, both generalized cross validation and minimum description length criterion lead to the form of criterions with a term measuring the modelling error and a term corresponding to some model complexity penalty. We have numerically ....

....penalty. We have numerically tested several criterions of this type for determining the number of wavelets in wavelet network. It turns out that the Akaike s final prediction error criterion (FPE) works quite well when tested on a variety of examples. This criterion is written as follows [30, 32]: J FPE ( b f) 1 n p =N 1 Gamma n p =N 1 2N N X k=1 ( b f(x k ) Gamma y k ) 2 where n p is the number of parameters in the estimator, x k ; y k ) 2 O N 1 are training data and N is the sample length of the training data. Because scalar dilation parameters are used for ....

L. Ljung, System Identification : Theory for the User. Prentice Hall Information and System Sciences Series, Prentice Hall, 1987.

....or time and cost constraints dictate that the system has met its time and cost requirements. It is important to note that in the latter problem, the measurements are not necessarily (or even often) the parametric, time domain measurements so prevalent in the on line identification literature [1, 2, 3]. More often, nonparametric frequency domain methods are used to obtain a frequency response function (FRF) from a given system input to a given system output [4, 5, 6] Often, control design is done strictly in the frequency domain without reducing the measurement to a parametric model [7, 8, 9] ....

.... function to the frequency response function [10, 11, 12, 13, 14] This process itself is imperfect and is one of the main di#culties in obtaining decent parametric models of industrial control problems [15, 16] While there is considerable parallelism between time and frequency domain methods [17, 1], the latter do have some great advantages that make them hard to ignore: Measurements can be made on both analog and digital systems. 2 Or in the words of John Cleese, Idiom . 3 implemented in SIMULINK. 4 in this case a HP 3562A. Measurements can be made without modifying the ....

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L. Ljung, System Identification: Theory for the User. Prentice-Hall Information and System Sciences Series, Englewood Cli#s, New Jersey 07632: Prentice-Hall, 1987.

.... y T (2) x T (k) y T (k) 3 7 7 7 5 a(k) b(k) 3 7 7 7 5 fl fl fl fl fl fl fl fl fl 2 fl fl 1 2 (k)y(k) Gamma 1 2 (k)A(k)w(k) fl fl 2 (6) This approximation is known in the literature as pseudo linear regression (PLR) and its convergence criteria are reviewed in [3, 2] and studied in [4] In [4] a convergence proof for the PLR is given if (for the sufficient order case) the transfer function 1 B(z) of the unknown system is strictly positive real (SPR) i.e. e 1 B(e ) Gamma 1 2 0; 8 Gamma : To achieve the minimization of (6) ....

Ljung L. System Identification: Theory for the User. Prentice-Hall Information and Systems Sciences Series. Prentice-Hall, Inc., 1987.

....or time and cost constraints dictate that the system has met its performance and cost requirements. A key difference with the textbook problem is that the measurements are not necessarily (or even often) the parametric, time domain measurements so prevalent in the on line identification literature [1, 2, 3]. More often, nonparametric frequency domain methods are used to obtain a frequency response function (FRF) from a given system input to a given system output [4, 5, 6] Often, control design is done strictly in the frequency domain without reducing the measurement to a parametric model [7, 8, 9] ....

L. Ljung, System Identification: Theory for the User. Prentice-Hall Information and System Sciences Series, Englewood Cliffs, New Jersey 07632: Prentice-Hall, 1987.

.... y T (k) 3 7 7 7 7 7 7 5 2 4 a(k) b(k) 3 5 3 7 7 7 7 7 7 5 fl fl fl fl fl fl fl fl fl fl fl fl 2 fl fl 1 2 (k)y(k) Gamma 1 2 (k)A(k)w(k) fl fl 2 (6) This approximation is known in the literature as pseudo linear regression (PLR) and its convergence criteria are reviewed in [3] [10] and studied in [7] In [7] a convergence proof for the PLR is given if (for the sufficient order case) the transfer function 1 B(z) of the unknown system is strictly positive real (SPR) i.e. e 1 B(e ) Gamma 1 2 0; 8 Gamma : This performance criterion (6) can ....

L Ljung, System Identification: Theory for the User, Prentice-Hall Information and Systems Sciences Series. Prentice-Hall, Inc., 1987. HARTENECK, STEWART: ADAPTIVE IIR FILTERING USING QR MATRIX DECOMPOSITION 11

....a(k) and b(k) One possible approximation to break this nonlinearity is to use y(k) instead of y(k)jw , i.e. to assume that the data matrix A(k) is independent of the adaptive weights. This approximation is called in the literature pseudo linear regression (PLR) and studied extensively in [5, 6]. Now (6) can be approximated as (k) fl fl fl fl fl fl fl fl fl 1 2 k 2 6 6 6 4 2 6 6 4 d(1) d(2) d(k) 3 7 7 5 Gamma 2 6 6 6 4 x T (1) y T (1) x T (2) y T (2) x T (k) y T (k) 3 7 7 7 5 a(k) b(k) 3 7 7 7 5 fl fl fl fl fl fl fl fl fl 2 = k 1 2 ....

....r r r r r r r r r 1 r r r r r x(k) e(k) y(k 2) y(k 1) x(k 2) x(k 1) x(k) d(k) Figure 2. Signal Flow Graph of the proposed Algorithm for 3 forward and 2 feedback weights. upper triangular matrix R(k) and to use this decomposition to obtain the optimal weight vector w(k) It is shown in [5, 6] that this optimal weight vector w(k) converges for k 1 to the optimum set of weights if the unknown system satisfies the strictly positive real condition, i.e. e 1 B(e ) Gamma 1 2 0; 8 Gamma ; 8) where B(z) is the feedback polynomial of the unknown system. A ....

L Ljung. System Identification: Theory for the User. Prentice-Hall Information and Systems Sciences Series. Prentice-Hall, Inc., 1987.

....of the parameter estimation algorithm) and so break the parameter dependency of A(k)j Theta on Theta. Eq. 6) can then be approximated as (7) see top of next page) This approximation is known in the literature as pseudolinear regression (PLR) and its convergence criteria are reviewed in [4, 5] and studied in [3] In [3] a convergence proof for the PLR is given for the sufficient order case if the transfer function 1 B(z) of the unknown system is strictly positive real (SPR) i.e. e 1 B(e ) Gamma 1 2 0; 8 Gamma : k) fl fl fl fl fl fl fl fl fl fl fl ....

Ljung, L. System Identification: Theory for the User. Prentice-Hall Information and Systems Sciences Series. Prentice-Hall, Inc., 1987.

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Ljung L. System Identification : theory for the user. Prentice-Hall Information and System Sciences Series, 1987.

No context found.

Ljung L. System Identification : theory for the user. Prentice-Hall Information and System Sciences Series, 1987.

No context found.

Ljung L. System Identification : theory for the user. Prentice-Hall Information and System Sciences Series, 1987.

....H1 , etc : are essentially model based. Via weightings, the stochastic part in figure 1 may be taken into account to represent dynamic disturbances that act on the system. Our model of figure 1 is fairly general: It incorporates AR, ARMA, ARMAX, Box Jenkins etc : as special cases (see e.g. [11] [15] The structure of this paper is as follows: In section 2, we present some algebraic facts for linear time invariant systems that form the basis of our subspace approach. In section 3, we show how these facts lead to two possible deterministic identification approaches. In section 4, we ....

....the consecutive states x k ; x k j Gamma1 . The following result was derived in [5] 6] Theorem 1 A matrix input output equation Y = Gamma g X H g U (3) The singular values of the block Hankel matrix U are often used as a quantization of the concept of persistancy of excitation [6] [11] [15] Theorem 1 allows to prove the following less trivial Theorem 2 The state as intersection of past and future Let W be an 2(l m)i Theta j block Hankel matrix constructed with the vectors w k where w t k = u t k y t k ) with j (l m)i. Partition W in two equal parts ( a past ....

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Ljung L. System identification; Theory for the user. Prentice-Hall Information and System Sciences Series, 1987.

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L. Ljung. System Identification: Theory for the User. Prentice-Hall Information and System Sciences Series, Englewood Cliffs, NY, 1987.

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