Least-squares-based iterative and gradient-based iterative estimation algorithms for bilinear systems

For bilinear systems with colored noise, this paper gives the input–output representation of the bilinear systems through eliminating the state variables in the model and derives a three-stage gradient-based iterative algorithm and a three-stage least-squares-based iterative algorithm for identifying the parameters of the input–output representation by means of the hierarchical identification principle. A gradient-based iterative (GI) algorithm is given for comparison. Compared with the GI algorithm, the proposed algorithms have lower computational burden and faster convergence speed. The simulation results indicate that the proposed algorithms are more effective for identifying bilinear systems.     

Identification of nonlinear hysteretic parameters by enhanced response sensitivity approach

Hysteresis is a ubiquitous phenomenon describing the special nonlinear memory-based relation between the input and the output in many physical systems. Identifying the hysteretic parameters is the first step towards practical application of hysteretic models. In this paper, a general framework for parameter identification of nonlinear hysteretic models is developed based on the enhanced response sensitivity approach. To do so, three typical hysteretic models—Bouc–Wen model, bilinear model with kinematic hardening and bilinear model with equal yielding force are analyzed at first and the general way to model a structure with such hysteretic components is established thereafter. Then, the enhanced response sensitivity approach is presented for inverse parameter identification where the key lies in the sensitivity analysis and the trust-region constraint. Particularly, smoothing procedure is introduced to overcome the non-differentiability of bilinear hysteretic functions for sensitivity analysis of bilinear models. Numerical examples are studied to testify the feasibility and performance of the proposed approach.     

Identification of discrete-time bilinear systems through equivalent linear models

This paper presents a novel method for identification of discrete-time, time-invariant state-space models of bilinear dynamical systems using the steady-state portion of a single input/multiple output time-history measurements. These measurements are recorded by exciting the system with a linear combination of sine and cosine functions of user-selected frequencies enriched by a subtle amount of random component. The proposed method relies on conversion of the bilinear system into an equivalent linear model (ELM) by an accurate approximation of the state in the bilinear term using a set of sine and cosine basis functions whose frequencies are obtained as combinations of the input frequencies. Observer/Kalman Filter Identification (OKID), a?linear time invariant (LTI) system identification algorithm, is used to identify the aforementioned ELM from which the original bilinear model is recovered. A?numerical example is also provided.     

Multistage least squares based iterative estimation for feedback nonlinear systems with moving average noises using the hierarchical identification principle

This paper develops a multistage least squares based iterative algorithm to estimate the parameters of feedback nonlinear systems with moving average noise from input–output data. Since that the identification model is bilinear on the unknown parameter space, the solution is to decompose a system into several subsystems with each of which is linear about its parameter vector, then to replace the unknown noise terms in the information vectors with their corresponding estimates at the previous iteration of each subsystem, and estimate each subsystem, respectively. The simulation results show that the proposed algorithm can work well.     

On the use of neural networks for identification of linear and nonlinear systems

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.Visiting scholar, from People's Republic of China.     

Continuous-Time Bilinear System Identification

The objective of this paper is to describe a new method for identification of a continuous-time multi-input and multi-output bilinear system. The approach is to make judicious use of the linear-model properties of the bilinear system when subjected to a constant input. Two steps are required in the identification process. The first step is to use a set of pulse responses resulting from a constant input of one sample period to identify the state matrix, the output matrix, and the direct transmission matrix. The second step is to use another set of pulse responses with the same constant input over multiple sample periods to identify the input matrix and the coefficient matrices associated with the coupling terms between the state and the inputs. Numerical examples are given to illustrate the concept and the computational algorithm for the identification method.     

A Volterra Series Approach to the Approximation of Stochastic Nonlinear Dynamics

A response approximation method for stochastically excited, nonlinear, dynamic systems is presented. Herein, the output of the nonlinear system isapproximated by a finite-order Volterra series. The original nonlinear system is replaced by a bilinear system in order to determine the kernels of this series. The parameters of the bilinear system are determined by minimizing, in a statistical sense,the difference between the original system and the bilinear system. Application to a piecewise linear modelof a beam with a nonlinear one-sided supportillustrates the effectiveness of this approach in approximatingtruly nonlinear, stochastic response phenomena in both the statistical momentsand the power spectral density of the response of this system in case ofa white noise excitation.     

Newton iterative identification for a class of output nonlinear systems with moving average noises

This paper discusses iterative identification problems for a class of output nonlinear systems (i.e., Wiener nonlinear systems) with moving average noises from input–output measurement data, based on the Newton iterative method. The basic idea is to decompose a nonlinear system into two subsystems, to replace the unknown variables in the information vectors with their corresponding estimates at the previous iteration, and to present a Newton iterative identification method using the hierarchical identification principle. The numerical simulation results indicate that the proposed algorithms are effective.     

Nonlinear model predictive control based on piecewise linear Hammerstein models

This paper develops a nonlinear model predictive control (MPC) algorithm for dynamic systems represented by piecewise linear (PWL) Hammerstein models. At each sampling instant, the predicted output trajectory is linearized online at an assumed input trajectory such that the control actions can be easily calculated by solving a quadratic programming optimization problem, and such linearization and optimization may be repeated a few times for good linear approximation accuracy. A three-step procedure is developed to linearize a PWL function, where the derivatives of a PWL function are obtained by a computationally efficient look-up table approach. Unlike many existing MPC algorithms for Hammerstein systems, it does not require the inversion of static nonlinearity and can directly cope with input constraints even in multivariable systems. Two benchmark chemical reactors are studied to illustrate the effectiveness of the proposed algorithm.     

Least squares algorithm for an input nonlinear system with a dynamic subspace state space model

For a Hammerstein input nonlinear system with a subspace state space linear element, this paper transforms the system into a bilinear identification model by using the property of the shift operator to the state space model and presents a recursive and an iterative least squares algorithms to generate parameter estimates and state estimates by using the hierarchical identification principle and by replacing the unknown state variables with their estimates. The proposed approaches are computationally more efficient than the over-parameterization model based least squares method.     

Newton iterative identification method for an input nonlinear finite impulse response system with moving average noise using the key variables separation technique

This paper studies parameter identification problems for input nonlinear finite impulse response systems with moving average noise (i.e., input nonlinear finite impulse response moving average systems). Since the identification model of the system contains the product of the parameters of the nonlinear part and the linear part, we use the key variables separation technique and express the output of the system as the linear combination of all parameters, and then derive a Newton iterative identification method. The simulation results show that the proposed algorithm is effective.     

An interaction measure for control configuration selection for multivariable bilinear systems

Control configuration selection is the procedure of choosing the appropriate input and output pairs for the design of decoupled (SISO or block) controllers for multivariable systems. This step is an important prerequisite for a successful industrial control strategy. In industrial practice it is often the case that systems which need to be controlled are non-linear, and linear models are insufficient to describe the behavior of the processes. The focus of this paper is on the problem of control configuration selection for a class of non-linear systems which is known as bilinear systems. A gramian-based interaction measure for control configuration selection of MIMO bilinear processes is described. In general, most of the results on the control configuration selection, which have been proposed so far, can only support linear systems. The proposed gramian-based interaction measure not only supports bilinear processes but also can be used to propose a richer sparse or block diagonal controller structure. The method is illustrated further with the help of some illustrative examples.     

Continuous-time bilinear system identification using single experiment with multiple pulses

A novel method is presented for the identification of a continuous-time bilinear system from the input?Coutput data generated by a single experiment with multiple pulses. In contrast to the conventional approach utilizing multiple experiments, the current work documents the advantage of using a single experiment and sets up a procedure to obtain bilinear system models. The special pulse inputs employed by earlier research can be avoided and accurate identification of the continuous-time system model is possible by performing a single experiment incorporating a class of control input sequences combining pulses with free-decay response. The algorithm presented herein is more attractive in practice for the identification of bilinear systems. Numerical examples presented demonstrate the methodology developed in the paper.     

Gradient-based iterative identification for Wiener nonlinear systems with non-uniform sampling

This paper focuses on the identification problem of Wiener nonlinear systems with non-uniform sampling. The mathematical model for the Wiener nonlinear system is established from the non-uniformly sampled input–output data. In order to solve the identification problem of the Wiener nonlinear system with the unmeasurable variables in the information vector, the gradient-based iterative algorithm is presented by replacing the unmeasurable variables with their corresponding iterative estimates. Finally, the simulation results indicate that the proposed algorithm is effective.     

一类加权全局迭代参数卡尔曼滤波算法

结合参数卡尔曼滤波算法和全局迭代推广卡尔曼滤波算法本文提出了加权全局迭代参数卡尔曼滤波算法。参数卡尔曼滤波算法可避免系统参数和状态变量之间的非线性耦合 ,同时通过带有目标函数的全局迭代算法保证能够获取到稳定、收敛的识别结果。分别针对线性结构模型和随动强化双线性结构模型进行了仿真参数识别。结果显示 ,不加权的全局迭代参数卡尔曼滤波算法对线性系统是有效的 ,而对非线性系统必须使用加权的全局迭代参数卡尔曼滤波算法。当信噪比较大 ,迭代无法得到收敛的结果时 ,目标函数保证了较好识别结果的获得     

Recursive parameter identification of the dynamical models for bilinear state space systems

This paper investigates the recursive parameter and state estimation algorithms for a special class of nonlinear systems (i.e., bilinear state space systems). A state observer-based stochastic gradient (O-SG) algorithm is presented for the bilinear state space systems by using the gradient search. In order to improve the parameter estimation accuracy and the convergence rate of the O-SG algorithm, a state observer-based multi-innovation stochastic gradient algorithm and a state observer-based recursive least squares identification algorithm are derived by means of the multi-innovation theory. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed algorithms.     

Wavelet basis expansion-based Volterra kernel function identification through multilevel excitations

Volterra series is a powerful mathematical tool for nonlinear system analysis, which extends the convolution integral for linear system to nonlinear system. There is a wide range of nonlinear engineering systems and structures which can be modeled as Volterra series. One question involved in modeling a functional relationship between the input and output of a system using Volterra series is to identify the Volterra kernel functions. In this article, a wavelet balance method-based approach is proposed to identify the Volterra kernel functions from observations of the in- and outgoing signals. The basic routine of the approach is that, from the system outputs under multilevel excitations, the Volterra series outputs of different orders are first estimated with the wavelet balance method, and then the Volterra kernel functions of different orders are separately estimated through their corresponding Volterra series outputs by expanding them with four-order B-spline wavelet on the interval. The simulation studies verify the effectiveness of the proposed Volterra kernel identification method.     

Discrete bright–dark soliton solutions and parameters controlling for the coupled Ablowitz–Ladik equation

Analysis of piecewise-linear nonlinear dynamical systems is critical for a variety of civil, mechanical, and aerospace structures that contain gaps or prestress that are caused by cracks, delamination, joints or interfaces among components. Recently, a technique referred to as bilinear amplitude approximation (BAA) was developed to estimate the response of bilinear systems that have no gap or prestress. The method is based on an idea that the dynamics of a bilinear system can be treated as a combination of linear responses in two time intervals both of which the system behaves as a distinct linear system: (1) the open state and (2) the closed or sliding state. Both geometric and momentum constraints are then applied as compatibility conditions between the states to couple the linear vibrational response for each time interval. In order to estimate the response for more general cases where there are either gaps or prestress in the system, a generalized BAA method is proposed in this paper. The new method requires inclusion of contact stiffness and damping to model contact behavior in the sliding state, and new equilibrium positions for each state to establish proper coordinates. The new method also finds the bilinear frequency of the system, which cannot be computed using the bilinear frequency approximation method previously developed since that method is only accurate for the zero gap and no prestress case. The generalized BAA method is demonstrated on a single degree of freedom system, a three degree of freedom system, and a cracked cantilever beam model for various gap sizes and prestress levels.     

Gradient-based parameter estimation for input nonlinear systems with ARMA noises based on the auxiliary model

This paper presents a gradient-based iterative identification algorithm and an auxiliary-model-based multi-innovation generalized extended stochastic gradient algorithm for input nonlinear systems with autoregressive moving average (ARMA) noises, i.e., the input nonlinear Box–Jenkins (IN–BJ) systems. The estimation errors given by the gradient-based iterative algorithm are smaller than the generalized extended stochastic gradient algorithm under same data lengths. A simulation example is provided.     

非线性系统参数辩识的一种频域模型

本文基于对非线性系统的可分离性假设，将非线性弹性力和阻尼力分别分解为物理坐标下各点间相对位移和相对速度的幂级数函数，导出了一般多自由度非线性系统在恒幅激励下的广义频率响应函数与输入输出之间的迭代关系式，提出了非线性系统中基本线性部分的概念，进而了一种在实验条件下的系统物理参数辩识方法。