Equation Error Method Research Articles

A new proof for the stability of equation-error models

Some previous works have shown that under an autoregressive constraint on the input signal, least-squares equation-error methods provide stable models of the estimated transfer function. We present an alternative proof of this fact which allows one to increase the order of the autoregressive input by one, for both the monic and unit-norm approaches.

An Augmented Lagrangian Method for Identifying Discontinuous Parameters in Elliptic Systems

The identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method. The minimization problem is then proved to be equivalent to the saddle-point problem of an augmented Lagrangian. The finite element method is used to discretize the saddle-point problem, and the convergence of the discretization is also proved. Finally, an Uzawa algorithm is suggested for solving the discrete saddle-point problem and is shown to be globally convergent.

COMPARISON OF PARAMETER ESTIMATION ALGORITHMS FOR CONTINUOUS TIME MODELS APPLIED TO D.C. MOTOR THERMAL SUPERVISION

ABSTRACT This paper deals with the thermal supervision of a D.C. motor process by means of parameter estimation. Identification is realized by applying two classes of algorithm to the measurable inputs/outputs of the process: The first deals with Reinitialized Partial time Moment modeling, which can be interpreted as an equation error method. The model parameters are computed by means of Least Square or Instrumental Variable algorithms. The second is based on the output error method. The model parameters are estimated by minimizing the output error by means of a Non Linear Programming according to the Marquardt algorithm.

Error Analysis of an Equation Error Method for the Identification of the Diffusion Coefficient in a Quasi-linear Parabolic Differential Equation

We consider the inverse problem of reconstructing the diffusion coefficient in a quasi-linear parabolic differential equation in divergence form from measurements of the solution at a finite number of points in the interior of the domain. An equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diffusion coefficient and which can be solved by the conjugate gradient method in a very efficient and stable manner. A detailed error analysis relates the required number of measurements with their accuracy. Numerical results illustrate the performance of the method.

An equation error method to recover diffusion from the distributed observation

The identification problem to recover the diffusion coefficient in a general elliptic equation is considered. For this purpose a method being a variant of the so-called equation error approach is suggested. A stability estimate for the method is given and, in particular, a very fast numerical algorithm to solve the finite-dimensional problem is presented together with computational results.

New adaptive IIR filtering algorithms based on the Steiglitz-McBride method

This correspondence introduces a family of algorithms based on the Steiglitz-McBride (SM) method for adaptive IIR filtering. The main feature of the proposed algorithms is to remove the parameter bias induced by colored measurement noise. The conceptual similarities between the SM method and the equation error method are explored in order to analyze some properties of the new algorithms. The main analysis results concern the consistency of the parameter estimates and local convergence. A set of simulations in a system identification configuration is included in order to verify the improved performance of the new algorithms based on the SM method.

New schemes and theoretical analysis of the master-slave family of recursive identification algorithms

An analysis of the Master-Slave family of system identification algorithms is developed for the white noise input case. These methods try to avoid the biased parameter estimation problem of equation error methods in the presence of noisy data, while keeping the quadratic nature of the error variances to be minimized (a feature that is lost with the output error formulation). The Master-Slave techniques may be seen as off-line schemes that construct a sequence of equation error optimization problems, each one based on the solution to the previous one. Some conditions for the stationary points of the iteration are given. Examples are presented for plants with poles close to z = 1 where undesirable attractive convergent points arise. An alternative scheme based on interpolation conditions rather than minimizing error variances is proposed and shown to have a single stationary point corresponding to the plant parameters. Finally, a simplified structure and a lattice algorithm are presented for on-line implementation.

ESTIMATION OF OUTPUT ERROR MODELS IN THE PRESENCE OF UNKNOWN BUT BOUNDED DISTURBANCES

In the context of set membership identification the feasible parameter set is defined as the set of plant parameters which are consistent with the model structure, the assumptions on (unknown but bounded) disturbances and all available measurements. It appears more convenient in practice to build an outer-bounding set, typically an ellipsoid, a parallelotope or an orthotope. This paper describes methods for adjusting set membership techniques, which are only applicable to equation error models, to the case of output error models. A method for building the optimal outer-bounding orthotope is also proposed. Equation error and output error methods are evaluated on the example of the estimation of a missile state space model in the presence of measurement noise and neglected dynamics.

運動方程式の直接当てはめによるＲＰＲＶの慣性能率の推定について

A method for the estimation of the moments of inertia of small aircraft such as Remotely-Piloted Research Vehicles (RPRVs) has been developed. When both the moments acting on a vehicle pivoted at or near its C. G. and its angular accelerations are measured simultaneously, the moments of inertia can be estimated by directly fitting the equations of motion to the measured data with multiple regression. This method, which is a variation of the equation error method for parameter identification, has great advantages over the usual output error method, in the sense that the existence of the structure which holds the vehicle has nothing to do with the measurement accuracy, and that the integration of the equations of motion is not necessary. The result of our experiments shows the accuracy and ease of application of the method.

Analysis of stabilised output error methods

In the paper, analysis of stabilised output-error methods (SOEMs) for parameter estimation of unstable aircraft is presented. These methods overcome the numerical difficulties encountered in parameter estimation of unstable systems by utilising measured states. The methods, along with the output error method (OEM) and the equation error method (EEM), are briefly described for the sake of comparison. However, the main idea of the paper is to present asymptotic analysis of the SOEM. The results of application of SOEMs to simulated data of an unstable/augmented aircraft are presented.

A study on the adaptive predictive control method for multivariable bilinear processes

A predictive control method for multivariable bilinear processes is derived based on ARMA model. To identify bilinear process models, we use simple equation error method extended to multivariable system. We can obtain the adaptive predictive controller for multivariable bilinear processes by incorporation of the identification algorithm. Offset compensator is provided to correct for the effects of unmeasured disturbances and model inaccuracies. A filter with a singled parameter is used to correct for the effects of an incorrect model. Results of simulation on multivariable bilinear processes show that the proposed control method has satisfactory performance.

Parameter identification and modeling of longitudinal aerodynamics

Using a comprehensive flight test database and a parameter identification software program produced at NASA Ames Research Center, a math model of the longitudinal aerodynamics of the Harrier aircraft was formulated. The identification program employed the equation error method using multiple linear regression to estimate the nonlinear parameters. The formulated math model structure adhered closely to aerodynamic and stability/control theory, particularly with regard to compressibility and dynamic manoeuvring. Validation was accomplished by using a three degree-of-freedom nonlinear flight simulator with pilot inputs from flight test data. The simulation models agreed quite well with the measured states. It is important to note that the flight test data used for the validation of the model was not used in the model identification.

An unbiased equation error identifier and reduced-order approximations

The equation error (EE) identification technique is modified to remove the parameter bias problem induced by uncorrelated measurement errors. The modification replaces a constraint with a constraint; the asymptotic solution replaces a normal equation with an eigenequation. The resulting algorithm is simpler than previous schemes, while at the same time preserving the desirable properties of the conventional EE method: simplicity of an on-line algorithm, unimodality of the performance surface, and consistent identification in the sufficient-order case. In the more realistic undermodeled case, a robustness result shows that the mean optimal parameter values of both the monic and unit-norm EE schemes correspond to a stable transfer function for all degrees of undermodeling, and for all stationary output disturbances, provided the input sequence satisfies an autoregressive constraint; otherwise an unstable model may result. Model approximation properties for the undermodeled case are exposed in detail for the case of autoregressive inputs; although both the monic and unit-norm variants provide Pade approximation properties, the unit-norm version is capable of autocorrelation matching properties as as well, and yields the optimal solution to a first- and second-order interpolation problem. Finally, the mismodeling error for the undermodeled case is shown to be a well-behaved function of the Hankel singular values of the unknown system. This modification allows EE methods to be admitted to the class of unbiased identification and approximation techniques. >

A new IIR adaptive echo canceler: GIVE

A novel IIR adaptive gradient instrumental variable echo canceler (GIVE) is presented. Its features include adaptive controllability during double-talk periods in acoustic conference systems; guarantee of global convergence; low computational cost (the same order as the IIR LMS algorithm of the equation error method); and flexible structures (parallel or series-parallel structures). We also show a convergence analysis for gradient adaptive algorithms including GIVE. Based on this analysis, the optimum stepsize for GIVE and three suboptimum algorithms are proposed to accelerate convergence and reduce misadjustment. In addition, a simple method that guarantees the stability of IIR filters and a configuration of GIVE applicable to closed loop systems are presented. These proposals are extensively studied by computer simulations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adaptive lattice bilinear filters

Two fast least-squares lattice algorithms for adaptive nonlinear filters equipped with bilinear system models are presented. The lattice filter formulation transforms the nonlinear filtering problem into an equivalent multichannel linear filtering problem, thus using multichannel lattice filtering algorithms to solve the nonlinear filtering problem. The computational complexity of the algorithms is an order of magnitude smaller than that of previously available methods. The first of the two approaches is an equation error algorithm that uses the measured desired response signal directly to compute the adaptive filter outputs. This method is conceptually very simple, but results in biased system models in the presence of measurement noise. The second is an approximate least-squares output error solution; the past samples of the output of the adaptive system itself are used to produce the filter output at the current time. Results indicate that the output error algorithm is less sensitive to output measurement noise than the equation error method. >

飛行データからの空力特性推定（第４報） 主成分分析と垂線誤差法

The “principal component analysis (PCA)” is one of the analysis techniques in the multivariate analysis, and also is a useful analysis technique for parameter estimations. First, this report shows to applicate the PCA to multicollinearity problems and the equation error method. Next, this report disucusses the “perpendicular error method (PE method)” proposed by this paper, which is one of the three main estimation methods under the concept of the “in-flight wind tunnel test” and corresponds to the maximum likelihood method under the concept of the response time history method. The PE method estimates the aerodynamic regression plane, on which the perpendicular aerodynamic error variance between the data and the regression plane becomes minimum. The PCA is a vital analysis technique for this PE method.

The Effect of Close or Repeated Eigenvalues on the Updating of Model Parameters from FRF Data

Methods to update the parameters of finite element models using measured vibration data usually use the experimentally derived modal model, that is, the system natural frequencies, damping coefficients, and mode shapes. Alternatively the frequency response functions have been used directly to update condensed analytical models and so avoid the sometimes difficult step of deriving the modal model. Previously the authors suggested an algorithm using FRF data that is basically a weighted equation error method based on a reduced order model. This paper investigates the performance of the algorithm for systems with closely coupled or repeated natural frequencies or eigenvalues.

飛行データからの空力特性推定 （第２報） 飛行風洞試験の概念とその解析法

This is the second report of the paper which discusses the physical meaning between flight test data and analyses, by studying the equation error method. As the more inclusive concept in the equation error method, the “in-flight wind tunnel test” concept is introduced. Under this concept, two methods can be thought out in addition to the equation error method, which respectively correspond to the output error method and the maximum likelihood method. Furthermore, it is represented that the equation error method has the built-in weighting function, which makes smaller the roles of the explanatory variables having noises as these noises increase. And the modified use of the multiregression analysis, in the case that the noise variances in the explanatory variables are known, is represented.

飛行データからの空力特性推定 （第１報） 方程式誤差法における多重共線性

This report discusses about the “multicollinearity” problem in the equation error method. This problem represents the limit of the aerodynamic characteristics information given by the flight-test data. This problem is occurred not only when the strong correlations between the explanatory variables are generated by the aircraft's maneuver, but also when the number of the explanatory variables is more than the maximum number of the variables which can be essentially independent in the equation of motion. Furthermore, the physical meanings about the partial regression coefficients and the multicollinearity are schematically explained with the regression plane.

Estimation of a temporally and spatially varying diffusion coefficient in a parabolic system by an augmented Lagrangian technique

In this paper we apply a hybrid method to estimate a temporally and spatially varying diffusion coefficient in a parabolic system. This technique combines the output-least-squares- and the equation error method. The resulting optimization problem is solved by an augmented Lagrangian approach and convergence as well as rate of convergence proofs are provided. The stability of the estimated coefficient with respect to perturbations in the observation is guaranteed.