Equation Error Method Research Articles

Some Aspects on the Accuracy of System Identification

This paper studies accuracy in the usual system identification where continuous-time system is identified in the form of the corresponding discrete-time system. First we introduce condition number as a measure of identification accuracy, or identification worthiness of input signal. We show through numerical experiments that proper sampling period, system order and scaling of output signal can be determined by examining the condition number. Next we adopt the identification accuracy referred to continuous-time system coefficients, compare two identification method based on output error and equation error, and illustrate that the output error method gives more accurate results than the equation error method.

Recent developments in damage detection based on system identification methods

The output error and equation error methods of system identification are compared for their effectiveness in assessing damage in structural systems. Damage is modeled on an element-by-element basis as changes in sectional properties, which then contribute to variations in the terms of the structural stiffness matrix. Both static displacements and eigenmodes of the structure are used in the damage detection process. A rational basis for the proper selection of eigenmodes and loading conditions for the identification process is also presented. Characteristics of the unconstrained optimization design space for the two approaches are discussed in context of their ability to yield the location and extent of damage.

The Augmented Lagrangian Method for Parameter Estimation in Elliptic Systems

In this paper a new technique for the estimation of parameters in elliptic partial differential equations is developed. It is a hybrid method combining the output-least-squares and the equation error method. The new method is realized by an augmented Lagrangian formulation, and convergence as well as rate of convergence proofs are provided. Technically the critical step is the verification of a coercivity estimate of an appropriately defined Lagrangian functional. To obtain this coercivity estimate a seminorm regularization technique is used.

An equation error method for the design of digital controllers

The purpose of the proposed equation error technique is to design a digital controller such that the response of the resulting closed-loop system matches approximately a specified response. The technique is based on the minimization of the equation error in closed loop. The necessary and sufficient condition for minimizing the equation error is given, and is used in the design method. >

式誤差法によるInterconnected Systemの構造を保持した次数低減化問題

式誤差法によるInterconnected Systemの構造を保持した次数低減化問題

Turbulent passive scalar field of a small Prandtl number

The statistical-mechanics theory of the passive scalar field convected by turbulence, developed in an earlier paper [Phys. Fluids 28, 1299 (1985)], is extended to the case of a small molecular Prandtl number. The set of governing integral equations is solved by the equation-error method. The resultant scalar-variance spectrum for the inertial range is F(k)∼x−5/3/[1+1.21x1.67(1+0.353x2.32)], where x is the wavenumber scaled by Corrsin’s dissipation wavenumber. This result reduces to the − (5)/(3) law in the inertial-convective range. It also approximately reduces to the − (17)/(3) law in the inertial-diffusive range, but the proportionality constant differs from Batchelor’s by a factor of 3.6.

システム同定における式誤差規範(EEM)と出力誤差規範(OEM)の相違

システム同定における式誤差規範(EEM)と出力誤差規範(OEM)の相違

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正誤表

Universal equilibrium range of turbulence

The variational approach to the closure problem of turbulence theory, reported in an earlier paper, is applied to the study of the universal equilibrium range of turbulence. Two integral equations are derived for two unknown functions, taking account of viscous dissipation. The equation-error method of parameter estimation of control theory is used to obtain the approximate solution of the two integral equations. The resulting energy spectrum for the universal equilibrium range is E(k)=ε2/3k−5/3F(k/kd) with F(x)=1.19(1+5.3x2/3) ×exp(−5.4x4/3). Here ε is the energy dissipation rate and kd is the Kolmogorov wavenumber. The corresponding one-dimensional energy spectrum and dissipation spectrum are calculated and are in agreement with the experimental results. Since F(x) is not a monotonically decreasing function, but has a maximum near x=0.1, the usual experimental values of the Kolmogorov constant will be greater than its theoretical value F(0)≂1.2 and will depend upon the range of k/kd.

Stability of reduced-order models obtained via the equation error method

Some observations are made on the asymptotic stability and controllability, of reduced-order models which are obtained via the minimization of the equation error method.

Advanced methods of recursive time-series analysis

Two of the most advanced procedures for recursively estimating the parameters in linear, observation space models of stochastic dynamic systems are the prediction error (PE) and optimal generalized equation error (OGEE) methods. This paper discusses the relationship between these methods in the case of the transfer function (TF) or Box-Jenkins model ; and compares their performance in terms of optimality, computational complexity and practical robustness. While the methods are quite similar for the TF model, it is shown that the instrumental-variable inspired OGEE approach yields algorithms that are computationally simpler and more robust when applied in practical situations.

Recursive output error identification algorithms theory and evaluation

Several recurisve algorithms for parametric identification of discrete time systems derived from Model Reference Adaptive System (M.R.A.S.) techniques are analysed. All these algorithms belong to the class of output error methods which have been very little discussed previously in the identification literature. These algorithms are analysed in the deterministic and stochastic environment using the Equivalent Feedback Representation (E.F.R.) and Ordinary Differential Equation (O.D.E.) methods respectively. A comparative evaluation of these algorithms is presented. This comparison is made also with respect to various widely used recursive algorithms belonging to the ‘equation error method’ (extended least squares, approximate maximum likelihood).

Convergence Analysis of M.R.A.S. Schemes Used for Recursive Identification

Several recursive algorithms for parametric identification of discrete time systems derived from M.R.A.S. techniques in a deterministic environment areanalysed. All these algorithms belong to the class of output error method which have been very little discussed previously in the identification litterature. These algorithms are analysed in the stochastic environment using the O.D.E. method. A summary of the convergence properties of these algorithms is also given for the deterministic case. A comparative evaluation of these algorithms is presented with respect to recursive algorithms belonging to "equation error method" (least squares, extended least squares, approximate maxlirnum likelihood.

Modified Equation Error Methods for Parameter Estimation

Two parameter estimation algori tt.ms are described for detern1ining the stability and control derivatives of an aircraft and an automobile from the measured input and output time histories, The algorithms are based on the equation error method.The first method called MEEM - acronym for Modified Equation Error Method - is a simple, fast and reasonably accurate one step method. In the second method SPECO - acronym for System Parameter Estimation through Constrained Optimization - the cumulative equation error is minimized by a constrained optimization technique. The inforrnation regerding the polarity of certain parameters on the basis of physical consideration is used as constraints in the technique, Good results are obtained when applied to the extractior. of stability and control derivatives of an aircraft and an automobile even from arbitrary zero starting values for the iteration. Eut the program is slow like all other direct search rr,ethods and hence perhaps only useful for off-line parameter estimation.

Modified equation error method for aircraft parameter estimation

The problem of estimation of aircraft parameters is of considerable significance in the field of flight data reduction. Modified equation error method (MEEM) has been developed for the extraction of stability and control derivatives from given input and output ‘ flight ’ test data. The estimates as given by MEEM compare extremely well with a very sophisticated technique like the method of maximum likelihood estimation. It is found that the algorithm MEEM is simple and takes less time compared to the involved methods like the output error method. Further it is sufficient to measure only the derivatives of the states (linear and/or angular accelerations).

A novel technique for obtaining aerodynamic data using simple, free flight trajectory measurements

Free flight testing of missile designs is generally necessary to confirm that accurate estimates have been made of effects due to the change in size from wind tunnel model to full scale missile and of effects arising from the flight dynamic behaviour of the missile. The amount of such testing is generally limited by the high costs involved and so every effort is made to obtain the maximum information from each flight trial. During recent years parameter estimation techniques have been developed to provide superior methods for analysis of free flight tests. They enable the analyst to extract the required data with greater accuracy from fewer tests than has previously been possible. The old equation error methods relied on measuring all the variables characterising the projectile motion and using the equations of motion to obtain a relationship between aerodynamic forces and moments and translational and angular velocities of the vehicle.

Convergence analysis of parametric identification methods

A certain class of methods to select suitable models of dynamical stochastic systems from measured input-output data is considered. The methods are based on a comparison between the measured outputs and the outputs of a candidate model. Depending on the set of models that is used, such methods are known under a variety of names, like output-error methods, equation-error methods, maximum-likelihood methods, etc. General results are proved concerning the models that are selected asymptotically as the number of observed data tends to infinity. For these results it is not assumed that the true system necessarily can be exactly represented within the chosen set of models. In the particular case when the model set contains the system, general consistency results are obtained and commented upon. Rather than to seek an exact description of the system, it is usually more realistic to be content with a suitable approximation of the true system with reasonable complexity properties. Here, the consequences of such a viewpoint are discussed.

Maximum Likelihood Identification of Aircraft Stability and Control Derivatives

Application of a generalized identification method to flight test data analysis. The method is based on the maximum likelihood (ML) criterion and includes output error and equation error methods as special cases. Both the linear and nonlinear models with and without process noise are considered. The flight test data from lateral maneuvers of HL-10 and M2/F3 lifting bodies are processed to determine the lateral stability and control derivatives, instrumentation accuracies, and biases. A comparison is made between the results of the output error method and the ML method for M2/F3 data containing gusts. It is shown that better fits to time histories are obtained by using the ML method. The nonlinear model considered corresponds to the longitudinal equations of the X-22 VTOL aircraft. The data are obtained from a computer simulation and contain both process and measurement noise. The applicability of the ML method to nonlinear models with both process and measurement noise is demonstrated.

Real-Time Identification Using Adaptive Discrete Model

This paper presents a new method for real time identification using a discrete adaptive model. Synthesis of adaptive loops for model parameters is achieved by using the hyperstability concepts. Attention is focused on a special treatment of error between model and process outputs which take into account the feedback realized by adaptive loops. Algorithms are written for identification with parallel (output error method) and series parallel (equation error method) models. Performance of these methods are compared and simulation results of parameters variance and biasing in presence of noise are given. Moreover effects on convergence speed are described. Experimental results have been obtained on a heat-exchanger. In conclusion, an extension of this method to adaptive filtering is proposed and an application to very high noise-signal ratio measurements is given.

A generalized method for the identification of aircraft stability and control derivatives from flight test data.

This paper discusses the application of a generalized identification method for flight test data analysis. The method is based on the maximum likelihood (ML) criterion and includes output error and equation error methods as special cases. Both the linear and nonlinear models with and without process noise are considered. The flight test data from lateral maneuvers of HL-10 and M2/F3 lifting bodies are processed to determine the lateral stability and control derivatives, instrumentation accuracies and biases. A comparison is made between the results of the output error method and the generalized ML method for M2/F3 data containing gusts. It is shown that better fits to time histories are obtained by using the generalized ML method.