Direct Parameter Estimation Method Research Articles

Nonlinear subspace system identification based on output-only measurements

In this paper a new deterministic subspace approach is introduced for multi-dimensional nonlinear system identification based on output-only measurements. The strategy is to excite an arbitrary degree of freedom (DOF) and estimate linear and nonlinear parameters of other DOFs within continuous-time state space models. The nonlinearity is characterized in terms of the measured response and its derivatives in locations away from the excitation. Then, basis functions are assigned to predict the nonlinear restoring force and exploited instead of the input data in the identification process. It is demonstrated that projecting the output row space onto these functions, cancels the subspace corresponding to the nonlinear dynamics and the remaining space contains the dynamics of the underlying linear system as well as the contributions of unmeasured input and noise data to the system response. The recent concept and required assumptions are presented through a few mathematical statements and proofs. To ensure the validity of the proposed approach, several discrete and continuous systems with different structural nonlinearities are studied. The estimated parameters are compared with those obtained from traditional subspace identification algorithms and the direct parameter estimation method. The results highlight the robustness of the proposed approach in dealing with noise contaminated measurements.

Nonlinear System Identification of Folding Fins with Freeplay Using Direct Parameter Estimation

Folding fins are widely adopted in missiles for the efficient use of space during storage and transportation, while nonlinear behavior of freeplay is inevitable due to the factors such as mismachining tolerance, assembly error, and abrasion. The problem of nonlinear system identification of folding fins with freeplay is considered in this paper. A direct parameter estimation method which can identify the nonlinear system with freeplay under base excitation is proposed and subsequently applied to establish the nonlinear dynamic model of a folding fin. The best set of coefficients is selected by using the significance test, allowing the proposed method to detect and locate the most relevant nonlinearities of the practical structure. Experimental results demonstrate that the proposed method is able to decouple the linear and nonlinear dynamics of a nonlinear structure and estimate natural frequencies of the derived linear system along with nonlinear internal forces in one computational step, even if no a priori knowledge of the type of nonlinearities is given.

Design of delayed fractional state variable filter for parameter estimation of fractional nonlinear models

This paper presents a novel direct parameter estimation method for continuous-time fractional nonlinear models. This is achieved by adapting a filter-based approach that uses the delayed fractional state variable filter for estimating the nonlinear model parameters directly from the measured sampled input–output data. A class of fractional nonlinear ordinary differential equation models is considered, where the nonlinear terms are linear with respect to the parameters. The nonlinear model equations are reformulated such that it allows a linear estimator to be used for estimating the model parameters. The required fractional time derivatives of measured input–output data are computed by a proposed delayed fractional state variable filter. The filter comprises of a cascade of all-pass filters and a fractional Butterworth filter, which forms the core part of the proposed parameter estimation method. The presented approaches for designing the fractional Butterworth filter are the so-called, square root base and compartmental fractional Butterworth design. According to the results, the parameters of the fractional-order nonlinear ordinary differential model converge to the true values and the estimator performs efficiently for the output error noise structure.

Impact of seed loading ratio on the growth kinetics of mono-ammonium phosphate under isothermal batch crystallization

The effect of seed load ratio on the growth kinetics of Mono-ammonium phosphate (MAP) under isothermal batch crystallization was investigated quantitatively. A direct parameter estimation method was proposed and applied to extract the growth kinetic parameters from a simple crystallization model using our experimental solution concentration decline data. The method assured the globally best parameters to be obtained and was found less sensitive to experimental errors. The linear growth constants kg and the growth order g were found to be in the range of 1,000-2,600 μm·min−1 and 0.93-1.12, respectively, for MAP crystallized at 40 °C. Both parameters decreased significantly with increase of seed load ratio and kg even showed a strong linear decline trend. The effective crystallization time also decreased with the seed mass. The proposed methodology could be extended to study the effect of other operation variables such as temperature and initial supersaturation on the crystal growth rate.

직접적인 매개변수 추정방법을 이용한 새로운 수정된 Neyman-Scott 구형펄스모형 개발 연구

Direct parameter estimation method is verified with various models based on Neyman-Scott rectangular pulse model (NSRPM). Also, newly modified NSRPM (NMSRPM) that uses normal distribution is developed. Precipitation data observed by Korea Meteorological Administration (KMA) for 47 years is applied for parameter estimation. For model performance verification, we used statistics, wet ratio and precipitation accumulate distribution of precipitation generated. The comparison of statistics indicates that absolute relative error (ARE)s of the results from NSRPM and modified NSRPM (MNSRPM) are increasing on July, August, and September and ARE of NMNSRPM shows 10.11% that is the smallest ARE among the three models. NMNSRPM simulates the characteristics of precipitation statistics well. By comparing the wet ratio, MNSRPM shows the smallest ARE that is 16.35% and by using the graphical analysis, we found that these three models underestimate the wet ratio. The three models show about 2% of ARE of precipitation accumulate probability. Those results show that the three models simulate precipitation accumulate probability well. As the results, it is found that the parameters of NSRPM, MNSRPM and NMNSRPM are able to be estimated by the direct parameter estimation method. From the results listed above, we concluded that the direct parameter estimation is able to be applied to various models based on NSRPM. NMNSRPM shows good performance compared with developed model-NSRPM and MNSRPM and the models based on NSRPM can be developed by the direct parameter estimation method.

Neyman-Scott 구형 펄스모형의 직접적인 매개변수 추정연구

SRPM (Neyman-Scott Rectangular Pulse Model)은 수문학분야에서 널리 쓰이고 있는 강수생성모형이다. NSRPM을 구축하기 위해서는 총 5개의 매개변수를 추정하여야 한다. 일반적으로 사용되는 모멘트를 이용하여 매개변수를 추정할 경우, 사용되는 목적함수의 증가에 따라 추정되는 매개변수의 결과가 평탄해지고 목적함수를 추가하거나 조정하기 위해서는 복잡한 수식을 다시 계산해야 하며 추정된 매개변수가 무작위변수 생성 모형에 따라 상이한 결과를 나타내는 단점이 있다. 본 연구에서는 직접적인 매개변수 추정방법을 제시하여 모멘트를 이용한 매개변수 추정의 단점을 극복하고자 하였다. 직접적인 추정방법을 적용하기 위하여 NSRPM의 강수 생성 개수에 따른 통계치 변화를 모의하여 직접적인 추정을 위한 모형을 구축하였다. 기상청 청주 지상관측소의 관측 강수 자료를 사용하여 모멘트를 이용하여 추정된 매개변수와 직접적인 방법을 이용하여 매개변수를 추정하였다. 총 4 개의 무작위변수 알고리즘을 적용하여 강수를 생성하였고, 관측 강수 시계열을 이용하여 정확도를 비교하였다. 비교 결과 직접적인 추정방법이 모멘트를 이용한 매개변수 추정방법보다 안정적이고 높은 정확도를 보이는 매개변수를 추정하는 것으로 나타났다. NSRPM (Neyman-Scott Rectangular Pulse Model) is one of the common model for generating future precipitation time series in stochastical hydrology. There are 5 parameters to compose the NSRPM model for generating precipitation time series. Generally parameter estimation using moment has some problems related with increased objective functions and shows different results in accordance with random variable generating models. In this study, direct parameter estimation method was proposed to cover with disadvantages of parameter estimation using moment. To apply the direct parameter estimation, generating stochastical data variance in accordance with numbers of precipitation events of NSRPM was done. Both kinds of methods were applied at the Cheongju gauge station data. Precipitation time series were generated using 4 different random variable generator, and compared with observed time series to check the accuracies. As a results, direct method showed more stable and better results.

A two-step method to identify parameters of piecewise linear systems

Based on the direct parameter estimation method and the Legendre series approximation, a two-step method is presented to estimate by measured data the physical parameters, i.e., mass, damping, stiffness and knot, of nonlinear systems with symmetrical piecewise linear restoring force. At first, the piecewise linear restoring force of the system is approximated by Legendre series. A least squares direct parameter estimation method is adopted to identify the mass and damping parameters of the system and the corresponding coefficients of the Legendre series. Secondly, the physical parameters of the piecewise linear restoring force, i.e., the primary and the secondary stiffness and the knots, are extracted from the estimated Legendre coefficients. Neither iteration nor search process is needed in this method. The validity of the method is demonstrated by numerical simulation on a single-degree-of-freedom and a three-degree-of-freedom system with both noise-free data and noise-polluted data.

Estimation of sludge sedimentation parameters from single batch settling curves

In this paper a comparison is made between two means of obtaining the parameters for the settling velocity models that are at the core of the solid flux theory: (i) the traditional approach using zone settling velocity ( V ZS) data obtained from a dilution experiment and (ii) a new direct parameter estimation method relying on a single batch settling curve (SBSC). For four distinct sludges, settling curves were recorded at different sludge concentrations ( X) and the Vesilind parameters were calculated in the traditional way. The value of the resulting model was evaluated by cross-validating it on its ability to describe complete SBSC's. Provided the dynamics of the sludge blanket descent were moderate a settler model incorporating these traditional Vesilind parameters could reasonably match the experimental batch settling curves. However, when the dynamics of the sludge blanket descent were fast, the Vesilind model failed. It was tried whether other settling velocity models could result in better fits to the SBSC. Here, the Cho model turned out to be the most effective one. When cross-validating the Cho model on the dilution experiment V ZS-data, however, it was found to be less performing than the Vesilind model in describing the relationship between V ZS and X. The fact that the Vesilind model is superior to the Cho model in describing such relationships, while the Cho model is better in describing complete settling curves, clearly points out that current settling models are still very empirical in nature. An important finding was that with all settling velocity models tested, practical identifiability problems appeared, indicating the need for better experimental designs. Finally, the flux curves associated with the SBSC-estimated Cho parameters were compared with the traditional Vesilind flux curves. Although both types of flux curves generally showed similar trends, the reliability of SBSC-based flux curve predictions is at present insufficient to warrant replacement of the traditional estimation of settling characteristics.