Regularized Structured Total Least Norm for the Identification of Bilinear Systems in the Errors-in-Variables Framework

Coordinate transformation is the process of converting spatial data from a source coordinate to a target coordinate system. A set of control points, measured in the two coordinate systems, is used to estimate the transformation parameters. In general, more control points are measured, and the over-determined system is adjusted using the least squares method. However, the standard least squares method assumes that errors exist only in the measurements made at one coordinate system, or at the observation vector (y). This is not the case in many physical systems where errors exist in all the measurements made in both the source coordinate and the target coordinate systems. The Structured Total Least Norm (STLN) method is a relatively new mathematical concept developed to solve estimation problems of so-called Error-In-Variables (EIV) models. The method is specifically suitable for dealing with transformation problems, since it can handle the special structure of the data matrix (A). The STLN method is uniquely used to compute the parameters of common linear coordinate transformations (affine and similarity). A numerical example is presented to demonstrate the superiority of this technique in terms of accuracy and to compare the standard LS method, the generalized LS algorithm, and the STLN approach.

<div class="htmlview paragraph">As an important class of nonlinear system, polytopic bilinear system is investigated. Combined with the properties of convex polytope, the nonlinear control for polytopic bilinear system is formulated by synthesizing nonlinear <i>H</i><sub>∞</sub> controller which is designed for polytopic bilinear system at vertices. For a semi-active suspension system with controllable damping and variant stiffness elements, it is easily modeled as a polytopic bilinear system model. In this case, the desired nonlinear control properties are pursued in making effective use of the changeable damping property while the variant stiffness is taken as the affine parameter of polytopic model. Therefore, polytopic bilinear system model could be reduced to a feasible problem by polytopic convex decomposition. Then the control problem of bilinear system model is to find a solution of nonlinear <i>H</i><sub>∞</sub> control. Here, the merits in using nonlinear <i>H</i><sub>∞</sub> state feedback control are that it can confine the damping force output to 1<sup>st</sup> and 3<sup>rd</sup> phase planes where changeable damping force is produced and its nonlinear control output makes a fast state origin convergence in keeping small vibration and motion. In this paper, a practical control design demonstration for a polytopic bilinear suspension system is described after the theory description. The simulation study and experimental results demonstrate the effective control performance in improving ride comfort.</div>

Identification of Input-Output Bilinear Systems

Objectives To solve the problem that the coefficient matrix and the observation vector have structured characteristics for many practical applications in the field of surveying and mapping, that is, the coefficient matrix and the observation vector contain fixed quantities (or even fixed columns) and random quantities, and the random quantities at different positions are linearly related. Methods Starting from the errors-in-variables (EIV) function model, the augmented matrix composed of the coefficient matrix and the observation vector is expressed as an affine function form, and the function model is reconstructed by the variable projection method. Then, a structured total least squares (STLS) estimation algorithm is derived by the Lagrange method. Results The example results show that the proposed algorithm is consistent with the existing weighted or structured total least squares estimation algorithms that can solve structured problem in the coefficient matrix and the observation vector. Compared with the weighted total least squares (WTLS) estimation algorithms, the proposed algorithm only needs to establish a positive definite weight matrix of independent random variables and reduces the number of estimates. Compared with other STLS estimation algorithms, the proposed algorithm takes into account the overall structure of the coefficient matrix and the observation vector. Conclusions This shows the effectiveness of the proposed algorithm. The STLS estimation and the WTLS estimation ensure the statistical optimality from the perspective of the function model and the stochastic model, respectively.

A procedure based on neural networks for the classification of linear and nonlinear systems is presented, using excitation and response data under swept sine excitation. Special attention is paid to the classification and identification of linear and bilinear systems, the latter being considered since they exhibit typical characteristics of cracked systems. The computer simulations show that: (1) using the procedure presented in this paper the trained classification network can reliably classify a linear system and different nonlinear systems; (2) the output of the trained identification neural network for a linear system and a bilinear system can be used as a quantitative indicator of characteristics of bilinear systems having different stiffness ratios (k(x>0)/k(x<0)) with respect to the bilinear system used in the training stage; (3) for two-degree-of-freedom systems, the trained network can not only determine the existence of a bilinear stiffness and the magnitude of its stiffness ratio, but also specify which stiffness is bilinear, i.e. indicate its position. These results provide a possibility of using the trained neural networks to detect and locate structural cracks which have the characteristics of bilinear systems.

Identification of bilinear systems

Total least squares (TLS) is a technique that solves the traditional least squares (LS) problem for an errors-in-variables (EIV) model, in which both the observation vector and the design matrix are contaminated by random errors. Four- and seven-parameter models of coordinate transformation are typical EIV model. To determine which one of TLS and LS is more effective, taking the four- and seven-parameter models of Global Navigation Satellite System (GNSS) coordinate transformation with different coincidence pointsas examples, the relative effectiveness of the two methods was compared through simulation experiments. The results showed that in the EIV model, the errors-in-variables-only (EIVO) model and the errors-in-observations-only (EIOO) model, TLS is slightly inferior to LS in the four-parameter model coordinate transformation, and TLS is equivalent to LS in the seven-parameter model coordinate transformation. Consequently, in the four- and seven-parameter model coordinate transformation, TLS has no obvious advantage over LS.

This study proposes a new multi-input multi-output optimal bilinear signal control method in which a bilinear dynamic model approximation is used to capture the nonlinear dynamics of the urban traffic networks. With signal green time splits as the control input and traffic delay changes as the output for each intersections in the network, a bilinear system model was developed, which, on the basis of linear system modeling, takes interactions among traffic delays and signal timing splits into consideration. Based on the bilinear system modeling framework, we conducted two steps in each time interval to derive traffic control strategies: (1) we used the normalized least-squared algorithm to estimate system parameters; and (2) we solved an online optimization problem to obtain the updated traffic control inputs for the signal timing that minimizes future traffic delays. We evaluated the proposed method in a microscopic traffic simulation environment (VISSIM) with a 35-intersection network of Bellevue city in Washington. Two different traffic demand patterns: (1) normal traffic demands; and (2) time-varying traffic demands were simulated to compare the performance of different control strategies. Experimental results show that (1) the proposed bilinear system model can better describe traffic system dynamics than linear-model based methods, such as our previously developed linear-quadratic regulator control; and (2) the proposed method outperforms the state-of-the-art signal control strategies, namely the max-pressure and the self-organizing traffic light control methods. We have also shown that the proposed method is applicable to all other possible network layouts and signal controller phasing structures. IEEE

Recursive Identification of Continuous Bilinear Systems

This paper considers the iterative parameter estimation for a dual‐rate sampled‐data bilinear system with autoregressive moving average noise. Through combining the auxiliary model identification idea with the data filtering technique, this paper derives two filtering auxiliary model gradient‐based iterative algorithms by using two different filters. The key is to construct an auxiliary model for predicting the unavailable outputs, and to transform the dual‐rate bilinear system identification model into two sub‐identification models. Finally, an auxiliary model gradient‐based iterative (AM‐GI) algorithm is presented for comparison. The simulation results indicate that the proposed algorithms are effective for identifying the dual‐rate sampled‐data bilinear systems, and can generate more accurate parameter estimates and have a higher computational efficiency than the AM‐GI algorithm.

Existing Gauss-Newton methods for solving the problem of 3D similarity transformation are all based on the method of numerical partial derivatives, which can linearize the 3D similarity transformation model by using Taylor expansion for approximated variables to the first order derivative. And by using the linearized 3D transformation model, all of the unknown variables can be obtained by an iterative process. In existing models, the model considering random errors of design matrix is called errors-in-variables (EIV) model. In contrast to the Gauss-Newton methods, a Gauss-quasi-Newton method for solving the problem of 3D similarity transformation is proposed in this paper. In the Gauss-quasi-Newton method, 3D similarity transformation model is linearized by a differential method, and based on the differential linearization model, an iterative robust estimation can be carried out by considering both random errors and gross errors in 3D coordinates of common points. The new method proposed in this paper can be applied to 3D similarity transformation of arbitrary rotational angles. Finally, the new method is validated by three cases, and results showed that the new method is effective and feasible.

Most practioners of inverse problems use least squares or maximum likelihood (MLE) to estimate parameters with the assumption that the errors are normally distributed. When there are errors both in the measured responses and in the independent variables, or in the model itself, more information is needed and these approaches may not lead to the best estimates. A review of the error-in-variables (EIV) models shows that other approaches are necessary and in some cases Bayesian inference is to be preferred.

Model recognition and validation for an off-road vehicle electrohydraulic steering controller

A survey of nonlinear system identification algorithms and related topics is presented by extracting significant results from the literature and presenting these in an organised and systematic way. Algorithms based on the functional expansions of Wiener and Volterra, the identification of block-oriented and bilinear systems, the selection of input signals, structure detection, parameter estimation and recent results from catastrophe theory and included. The limitations, relationships and applicability of the methods are discussed throughout.

Buildings, consuming over 70% of the electricity in the U.S., play significant roles in smart grid infrastructure. The automatic operation of buildings and their subsystems in responding to signals from a smart grid is essential to reduce energy consumption and demand, as well as improve the resilience to power disruptions. In order to achieve such automatic operation, high fidelity and computationally efficiency building energy forecasting models under different weather and operation conditions are needed. Currently, data-driven (black box) models and hybrid (grey box) models are commonly used in model based building control operation. However, typical black box models often require long training period and are bounded to weather and operation conditions during the training period. On the other hand, creating a grey box model often requires long calculation time due to parameter optimization process and expert knowledge during the model structure determining and simplification process. An earlier study by the authors proposed a system identification approach to develop computationally efficient and accurate building energy forecasting models. This paper attempts to extend this early study and to quantitatively evaluate how the most important characteristics of a building energy system: its nonlinearity and response time, affect the system identification process and model accuracy. Two commercial building: a small-size and a medium-size commercial building, with varying chiller nonlinearity, are simulated using EnergyPlus in lieu of real buildings for model development and validation. The system identification method proposed in the early study is applied to these two buildings that have varying nonlinearity and response time. Adaption of the proposed system identification method based on systems’ nonlinearity and response time is proposed in this study. The energy forecasting results demonstrate that the adaption is capable of significantly improve the performance of the system identification model.