## Quantifying the Error in Estimated Transfer Functions with Application to Model Order Selection (1992)

Venue: | IEEE TRANSACTIONS ON AUTOMATIC CONTROL |

Citations: | 42 - 11 self |

### BibTeX

@ARTICLE{Goodwin92quantifyingthe,

author = {Graham Goodwin and Michel Gevers and Brett Ninness},

title = {Quantifying the Error in Estimated Transfer Functions with Application to Model Order Selection},

journal = {IEEE TRANSACTIONS ON AUTOMATIC CONTROL},

year = {1992},

volume = {37},

number = {7},

pages = {913--928}

}

### Years of Citing Articles

### OpenURL

### Abstract

Previous results on estimating errors or error bounds on identified transfer functions have relied upon prior assumptions about the noise and the unmodelled dynamics. This prior information took the form of parameterized bounding functions or parameterized probability density functions, in the time or frequency domain, with known parameters. Here we show that the parameters that quantify this prior information can themselves be estimated from the data using a Maximum Likelihood technique. This significantly reduces the prior information required to estimate transfer function error bounds. We illustrate the usefulness of the method with a number of simulation examples. The paper concludes by showing how the obtained error bounds can be used for intelligent model order selection that takes into account both measurement noise and undermodelling. Another simulation study compares our method to Akaike's well known FPE and AIC criteria.

### Citations

1201 |
System Identification: Theory for the User
- Ljung
- 1987
(Show Context)
Citation Context ... the Cram'erRao lower bound on the estimated parameters. In the case of exact model structure, this tool produces reasonable variance error expressions for the estimated transfer functions : see e.g. =-=[17]-=-, [8]. This variance error typically decreases like 1 N , where N is the number of data. In the case of restricted complexity model structures, the parameters of the model have essentially no meaning:... |

84 |
System identification using Laguerre models
- Wahlberg
- 1991
(Show Context)
Citation Context ...a mapping to rational transfer functions G(q \Gamma1 ; `) with fixed denominator; that is, only the numerator is parameterized by `. FIR models and the Laguerre models studied in [19], [20], [31] and =-=[27]-=- are examples of such model structures 1 . With this assumption, the nominal model can be written as G(e \Gammaj ! ; `) = (e \Gammaj ! )`; (20) where (e \Gammaj ! ) , \Thetas1 (e \Gammaj ! ); \Delta \... |

31 |
Asymptotic variance expressions for identified black-box transfer function models
- Ljung
- 1985
(Show Context)
Citation Context ...si : (4) Suppose, for ease of presentation, that we estimate ` from the data ZN via the classical least squares estimate:s` N = arg min ` 1 N N X k=1 " 2 k (`) (5) " k (`) = y k \Gammasy k (=-=`) (6) In [18] Ljun-=-g showed that, under reasonable conditions,s` N ! `s(7) where `s= arg min ` 1 N N X k=1 E \Phi " 2 k (`) \Psi : (8) With this definition of `swe can now examine the total error between the true t... |

30 |
On the value of information in system identification—Bounded noise case
- Fogel, Huang
- 1982
(Show Context)
Citation Context ... of stochastic components are also assumed and the resultant parameter space bounds are transformed to the frequency domain. Finally, in [29], the ideas of set membership estimation developed in [5], =-=[6]-=- and [23] are used to provide hard bounds in the parameter space, which are then transformed to the frequency domain. The hard bounding approaches to quantifying errors on estimated transfer functions... |

24 |
Dynamic System Identification
- Goodwin
- 1977
(Show Context)
Citation Context ...Gamma 1 2 ln det \Sigma \Gamma 1 2 W T \Sigma \Gamma1 W + constant; (59) where \Sigma = R T \PsiC j (ff; )\Psi T R+ oe 2sR T R (60) C j (ff; ) = diag fff; ff 2 ; : : : ; ff L g: (61) It is well known =-=[10] that the covariance of an unbiased estimate-=- �� �� of �� is bounded below by the Cram'er-Rao lower bound. Covf �� ��gsM \Gamma1 �� , Cov( �� ��) (62) That is, Cov( �� ��) = M \Gamma1 �� is a lower bou... |

21 |
Design variables for bias distribution in transfer function estimation
- Wahlberg, Ljung
- 1986
(Show Context)
Citation Context ...this paper is to handle the case of estimation of restricted complexity models from a finite noisy data record. The first results on a characterization of the bias error are due to Wahlberg and Ljung =-=[28]-=- who provide an implicit description of the bias error using Parseval's formula. This formula allows for an interesting qualitative discussion of the factors affecting bias, but it does not provide an... |

19 |
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- Kosut, Lau, et al.
- 1992
(Show Context)
Citation Context ...ar approach is taken save that a Kalman Filter is used to calculate the ETFE, and FIR models are then fitted to this in the frequency domain using the interpolation theory of Lagrange. In [14], [15], =-=[16]-=-, [30] and [3] bounds are calculated in the parameter space, again based on assumptions on the smoothness of the true transfer function of the unmodelled dynamics. Compact supports for the distributio... |

15 |
the Discrete Kalman Filter
- Estimation
- 1989
(Show Context)
Citation Context ...lta (G \Delta ; fi), then the parameter vector fi can be estimated from the data. There need be no conjecture about the undermodelling being random, when we know it to be deterministic since to quote =-=[4]-=- `a random variable is like the Holy Roman Empire- it wasn't holy, it wasn't Roman and it wasn't an Empire. A random variable is neither random nor variable, it is simply a function'. In fact it is a ... |

15 | A stochastic embedding approach for quantifying uncertainty in the estimation of restricted complexity models - Goodwin, Salgado - 1989 |

13 |
Quantification of uncertainty in estimation using an embedding principle
- Goodwin, M
- 1989
(Show Context)
Citation Context ...of the unmodelled dynamics is consistent with the stochastic prior model that is typically assumed for the noise. The idea of stochastic embedding was introduced in [12] and subsequently developed in =-=[11]-=-, [21], [22] and [8]. In similar spirit to the hard-bounding work it required prior specification of likely smoothness and magnitude parameters of the true system frequency response, via a parameteriz... |

10 |
Identification with nonparametric uncertainty
- Younce, Rohrs
- 1992
(Show Context)
Citation Context ...taken save that a Kalman Filter is used to calculate the ETFE, and FIR models are then fitted to this in the frequency domain using the interpolation theory of Lagrange. In [14], [15], [16], [30] and =-=[3]-=- bounds are calculated in the parameter space, again based on assumptions on the smoothness of the true transfer function of the unmodelled dynamics. Compact supports for the distributions of stochast... |

9 |
Approximation of Stable Systems by Laguerre Filters
- Makila
- 1990
(Show Context)
Citation Context ...model structure M is a mapping to rational transfer functions G(q \Gamma1 ; `) with fixed denominator; that is, only the numerator is parameterized by `. FIR models and the Laguerre models studied in =-=[19]-=-, [20], [31] and [27] are examples of such model structures 1 . With this assumption, the nominal model can be written as G(e \Gammaj ! ; `) = (e \Gammaj ! )`; (20) where (e \Gammaj ! ) , \Thetas1 (e ... |

8 |
Identification of parameter bounds for armax models from records with bounded noise
- Norton
- 1987
(Show Context)
Citation Context ...hastic components are also assumed and the resultant parameter space bounds are transformed to the frequency domain. Finally, in [29], the ideas of set membership estimation developed in [5], [6] and =-=[23]-=- are used to provide hard bounds in the parameter space, which are then transformed to the frequency domain. The hard bounding approaches to quantifying errors on estimated transfer functions suffer f... |

8 |
Adaptive frequency response identification
- Parker, Bitmead
- 1987
(Show Context)
Citation Context ... which hard error bounds are derived on assumptions of smoothness of the true frequency response and bounds on the Gibbs effect due the finite data windowing used in calculating the ETFE. In [24] and =-=[25]-=- a similar approach is taken save that a Kalman Filter is used to calculate the ETFE, and FIR models are then fitted to this in the frequency domain using the interpolation theory of Lagrange. In [14]... |

8 |
Deterministic Adaptive Control Based on Laguerre Series Representation
- Zervos, Dumont
- 1988
(Show Context)
Citation Context ...ure M is a mapping to rational transfer functions G(q \Gamma1 ; `) with fixed denominator; that is, only the numerator is parameterized by `. FIR models and the Laguerre models studied in [19], [20], =-=[31]-=- and [27] are examples of such model structures 1 . With this assumption, the nominal model can be written as G(e \Gammaj ! ; `) = (e \Gammaj ! )`; (20) where (e \Gammaj ! ) , \Thetas1 (e \Gammaj ! );... |

6 |
Adaptive control via parameter set estimation
- Kosut
- 1988
(Show Context)
Citation Context ...[25] a similar approach is taken save that a Kalman Filter is used to calculate the ETFE, and FIR models are then fitted to this in the frequency domain using the interpolation theory of Lagrange. In =-=[14]-=-, [15], [16], [30] and [3] bounds are calculated in the parameter space, again based on assumptions on the smoothness of the true transfer function of the unmodelled dynamics. Compact supports for the... |

6 |
Laguerre series approximation of infinite dimensional systems
- Makila
- 1990
(Show Context)
Citation Context ...structure M is a mapping to rational transfer functions G(q \Gamma1 ; `) with fixed denominator; that is, only the numerator is parameterized by `. FIR models and the Laguerre models studied in [19], =-=[20]-=-, [31] and [27] are examples of such model structures 1 . With this assumption, the nominal model can be written as G(e \Gammaj ! ; `) = (e \Gammaj ! )`; (20) where (e \Gammaj ! ) , \Thetas1 (e \Gamma... |

6 |
Issues in Robust Identification
- Salgado
- 1989
(Show Context)
Citation Context ...onaverage characteristics of the total error. The technical tool for doing so is to also make the bias error a random variable, by ascribing a prior distribution to it. We therefore assume, following =-=[26]-=-,[21], [11], [12], [22], [8] that the true transfer function, G T (e \Gammaj ! ), is a stochastic process indexed by the variable !. We further assume that, for the given choice of model set M p and f... |

5 |
Bias and variance distribution in transfer function estimates
- Goodwin, Gevers, et al.
- 1991
(Show Context)
Citation Context ...ram'erRao lower bound on the estimated parameters. In the case of exact model structure, this tool produces reasonable variance error expressions for the estimated transfer functions : see e.g. [17], =-=[8]-=-. This variance error typically decreases like 1 N , where N is the number of data. In the case of restricted complexity model structures, the parameters of the model have essentially no meaning: they... |

5 |
Frequency Domain Descriptions of Linear Systems
- Parker
- 1988
(Show Context)
Citation Context ...TFE), for which hard error bounds are derived on assumptions of smoothness of the true frequency response and bounds on the Gibbs effect due the finite data windowing used in calculating the ETFE. In =-=[24]-=- and [25] a similar approach is taken save that a Kalman Filter is used to calculate the ETFE, and FIR models are then fitted to this in the frequency domain using the interpolation theory of Lagrange... |

4 |
Adaptive quantification of model uncertainties by rational approximation
- Bai
- 1991
(Show Context)
Citation Context ...or between the true transfer function and the estimated nominal model, which is usually called unmodelled dynamics. One way of treating this error is to estimate it by further parameterizing it as in =-=[7]-=-, but this amounts to replacing the nominal model by a more complex one; it amounts to modeling the unmodelled dynamics. In the no noise case treated in [7] increasing the model order is not a problem... |

4 |
Model error quantification for robust control based on quasi-bayesian estimation in closed loop
- Goodwin, Ninness
- 1991
(Show Context)
Citation Context ...to illustrate that the method may be successfully applied to the ARMAX modeling case even though the case does not fit the assumptions. The theoretical basis for the ARMAX modeling case is treated in =-=[9]-=-. 6 Model Structure Selection In this section we show how the quantified error bounds in the form of the ensemble Mean Square Error E n jG T (e \Gammaj ! ) \Gamma G(e \Gammaj ! ;s` N )j 2 o of the tra... |

3 |
Adaptive robust control via transfer function uncertainty estimation
- Kosut
- 1988
(Show Context)
Citation Context ... similar approach is taken save that a Kalman Filter is used to calculate the ETFE, and FIR models are then fitted to this in the frequency domain using the interpolation theory of Lagrange. In [14], =-=[15]-=-, [16], [30] and [3] bounds are calculated in the parameter space, again based on assumptions on the smoothness of the true transfer function of the unmodelled dynamics. Compact supports for the distr... |

3 |
Uncertainty, information and estimation
- Mayne, Salgado
- 1989
(Show Context)
Citation Context ... unmodelled dynamics is consistent with the stochastic prior model that is typically assumed for the noise. The idea of stochastic embedding was introduced in [12] and subsequently developed in [11], =-=[21]-=-, [22] and [8]. In similar spirit to the hard-bounding work it required prior specification of likely smoothness and magnitude parameters of the true system frequency response, via a parameterized pri... |

3 |
Identification with parameteric and non-parametric uncertainty
- Younce, Rohrs
- 1992
(Show Context)
Citation Context ...roach is taken save that a Kalman Filter is used to calculate the ETFE, and FIR models are then fitted to this in the frequency domain using the interpolation theory of Lagrange. In [14], [15], [16], =-=[30]-=- and [3] bounds are calculated in the parameter space, again based on assumptions on the smoothness of the true transfer function of the unmodelled dynamics. Compact supports for the distributions of ... |

2 | Robust Estimation - Ninness - 1992 |

1 |
A frequency domain estimator for use in adaptive control systems
- Athans, Stein
- 1987
(Show Context)
Citation Context ...rror on the basis of assumed prior knowledge on the noise (a known distribution or a known hard bound) and of assumed prior magnitude and smoothness bounds on the unmodelled dynamics. For example, in =-=[1]-=- a nominal parametric model is fitted to the Empirical Transfer Function Estimate (ETFE), for which hard error bounds are derived on assumptions of smoothness of the true frequency response and bounds... |

1 |
identification via membership set constraints with energy constrained noise
- System
- 1979
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Citation Context ...tions of stochastic components are also assumed and the resultant parameter space bounds are transformed to the frequency domain. Finally, in [29], the ideas of set membership estimation developed in =-=[5]-=-, [6] and [23] are used to provide hard bounds in the parameter space, which are then transformed to the frequency domain. The hard bounding approaches to quantifying errors on estimated transfer func... |

1 |
On estimation of model quality in system identification. Licentiate Thesis LIU-TEK-LIC-1990:51
- Hjalmarsson
- 1990
(Show Context)
Citation Context ...exists. The classical Cram'er-Rao expression does not apply. However, recent work has produced an asymptotic procedure for the computation of variance errors on the model parameters in this situation =-=[13]-=-. We now turn to the estimation of the second component; bias errors in the case of restricted complexity models. In the case of noiseless data this is essentially a trivial problem. Indeed, when ther... |

1 |
frequency domain model error bounds from least squares like identification techniques
- Hard
- 1990
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Citation Context ...the unmodelled dynamics. Compact supports for the distributions of stochastic components are also assumed and the resultant parameter space bounds are transformed to the frequency domain. Finally, in =-=[29]-=-, the ideas of set membership estimation developed in [5], [6] and [23] are used to provide hard bounds in the parameter space, which are then transformed to the frequency domain. The hard bounding ap... |