Nonlinear black-box models in system identification: Mathematical foundations

We discuss several aspects of the mathematical foundations of the nonlinear black-box identification problem. We shall see that the quality of the identification procedure is always a result of a certain trade-off between the expressive power of the model we try to identify (the larger the number of parameters used to describe the model, the more flexible is the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this trade-off is the simple fact that a good approximation technique can be the basis of a good identification algorithm. From this point of view, we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and ‘neuron’ approximations, and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretical developments for the practically implemented versions of the ‘spatially adaptive’ algorithms.     

基于输入输出模型的模糊神经网络滑模控制

1　引言在实际系统中,一般难以取到系统的状态.因此,如何仅利用输入输出模型来控制系统一直是控制理论工作者关注的话题.Narendra等人[1]通过加入两个‘状态’滤波器设计了一种基于输入输出的模型参考自适应控制方案,提出了一个很好的思想.在其后的数十年中,基于输入输出?..     

Nonlinear systems identification using dynamic multi-time scale neural networks

In this paper, two Neural Network (NN) identifiers are proposed for nonlinear systems identification via dynamic neural networks with different time scales including both fast and slow phenomena. The first NN identifier uses the output signals from the actual system for the system identification. The on-line update laws for dynamic neural networks have been developed using the Lyapunov function and singularly perturbed techniques. In the second NN identifier, all the output signals from nonlinear system are replaced with the state variables of the neuron networks. The on-line identification algorithm with dead-zone function is proposed to improve nonlinear system identification performance. Compared with other dynamic neural network identification methods, the proposed identification methods exhibit improved identification performance. Three examples are given to demonstrate the effectiveness of the theoretical results.     

Nonlinear system identification via direct weight optimization

A general framework for estimating nonlinear functions and systems is described and analyzed in this paper. Identification of a system is seen as estimation of a predictor function. The considered predictor function estimate at a particular point is defined to be affine in the observed outputs and the estimate is defined by the weights in this expression. For each given point, the maximal mean-square error (or an upper bound) of the function estimate over a class of possible true functions is minimized with respect to the weights, which is a convex optimization problem. This gives different types of algorithms depending on the chosen function class. It is shown how the classical linear least squares is obtained as a special case and how unknown-but-bounded disturbances can be handled.Most of the paper deals with the method applied to locally smooth predictor functions. It is shown how this leads to local estimators with a finite bandwidth, meaning that only observations in a neighborhood of the target point will be used in the estimate. The size of this neighborhood (the bandwidth) is automatically computed and reflects the noise level in the data and the smoothness priors.The approach is applied to a number of dynamical systems to illustrate its potential.     

Simplifying a neuro-fuzzy model

Neuro-fuzzy modeling allows a fuzzy system to be refined by neural training, thus avoiding lenghty trial-and-error phases in defining both membership functions and inference rules. An approach to obtain simple neuro-fuzzy models is proposed, which reduces the number of rules by means of a systematic procedure that consists in successively removing a rule and updating the remaining rules in such a way that the overall input-output behavior is kept approximately unchanged over the entire training set. A formulation of the proper update is described and a criterion for choosing the rules to be removed is also provided. Initial experimental results show the effectiveness of the proposed method in reducing the complexity of a neuro-fuzzy system by using its input-output data.     

Nonlinear adaptive control system design with asymptotically stable parameter estimation error

The paper presents a new general method for nonlinear adaptive system design with asymptotic stability of the parameter estimation error. The advantages of the approach include asymptotic unknown parameter estimation without persistent excitation and capability to directly control the estimates transient response time. The method proposed modifies the basic parameter estimation dynamics designed via a known nonlinear adaptive control approach. The modification is based on the generalised prediction error, a priori constraints with a hierarchical parameter projection algorithm, and the stable data accumulation concepts. The data accumulation principle is the main tool for achieving asymptotic unknown parameter estimation. It relies on the parametric identifiability system property introduced. Necessary and sufficient conditions for exponential stability of the data accumulation dynamics are derived. The approach is applied in a nonlinear adaptive speed tracking vector control of a three-phase induction motor.     

Efficient nonparametric identification for high-precision motion systems: A practical comparison based on a medical X-ray system

The need for accurate knowledge of complex dynamical behavior for high-performance mechatronic systems led to the development of a vast amount of nonparametric system identification approaches over the recent years. The aim of this paper is to compare several proposed methods based on experiments on a physical complex mechanical system to bridge the gap between identification theory and practical applications in industry where basic identification approaches are often the norm. Typical practical implications such as operation under closed-loop control, multivariable coupled behavior and nonlinear effects are included in the analysis. Finally, a possible approach for fast and reliable identification is illustrated based on measurement results of an interventional medical X-ray system.     

Nonlinear system identification with limited time data

With disturbances modeled by arbitrary solutions to a linear homogeneous differential equation, a least squares-equation error method is developed for parameter identification using data over a limited time interval which has application to certain classes of nonlinear and time varying systems. Examples include the Duffing, Hammerstein, Mathieu and Van der Pol equations together with a class of bilinear systems. The technique seeks to determine the parameters characterizing the disturbance modes in addition to the system parameters, based on the input-output data collected over the finite time interval. The approach circumvents the need to estimate unknown initial conditions through the use of a certain projection operator. Computational considerations are discussed and simulation results are summarized for the Van der Pol equation.     

动态非线性连续时间系统的小波神经网络辨识

将小波神经网络应用于动态非线性连续时间系统的辨识, 同时为了使神经网络的训练达到全局最优和加速小波神经网络训练的收敛速度, 提出了信赖域算法, 并研究了信赖域算法的收敛性. 随后进行了算例仿真, 证明了所提辨识方法的有效性.     

Delay identification for a class of nonlinear systems

The problem of state estimation for nonlinear systems with unknown state delays is still an open problem. In this paper, we propose to add a delay identifier to suitable high-gain observers in order to achieve simultaneous estimation of state and delay. In the case of one constant delay in the state, we provide sufficient conditions to guarantee the exponential convergence to zero of the errors, globally with respect to the initial choice of the system state variables and locally with respect to the initial delay estimation. We validate the method through an example concerning chaotic oscillators.     

空间分布系统的空间模糊建模

本文对空间分布系统提出了一种新的模糊建模方法.首先,在3-D模糊集的基础上给出一种改进的空间模糊集,包含了传统的模糊集和添加的一维空间信息.在空间轴方向上,将传统模糊隶属度函数沿输入变量的随空间的物理变化曲线进行扩展,通过隶属度的连续变化描述输入变量在空间中的变化.其次,基于空间模糊集,采用Mamdani模糊模型形式,设计了对空间分布系统的空间模糊模型的建模方法.最后,通过仿真算例对方法进行了验证.     

Nonlinear system identification of a small-scale unmanned helicopter

This paper presents a comprehensive method for identifying the nonlinear model of a small-scale unmanned helicopter. The model structure is obtained by first principles derivation, and the model parameters are determined by direct measurement and system identification. A new adaptive genetic algorithm is proposed to identify the parameters that cannot be directly measured. To simplify the identification process, the overall system is divided into two subsystems for identification: the heave–yaw dynamics and the lateral–longitudinal dynamics. On the basis of the input–output data collected from actual flight experiments, these two subsystems are identified using the proposed algorithm. The effectiveness of the identified model is verified by comparing the response of the simulation model with the actual response during the flight experiments. Results show that the identified model can accurately predict the response of the small-scale helicopter. Furthermore, the identified model is used for the design of an attitude controller. The experiment results show that the identified model is suitable for controller design.     

On a method for model structure selection in system identification

Bohlin has recently suggested an appealing method for model structure selection. It is shown that the method can be interpreted as a modified F-test. The minimal loss in a larger model structure is substituted by an estimate. The estimate is obtained by minimization of a quadratic approximation of the loss function.     

Numerical uncertainty management in user and student modeling: An overview of systems and issues

A rapidly growing number of user and student modeling systems have employed numerical techniques for uncertainty management. The three major paradigms are those of Bayesian networks, the Dempster-Shafer theory of evidence, and fuzzy logic. In this overview, each of the first three main sections focuses on one of these paradigms. It first introduces the basic concepts by showing how they can be applied to a relatively simple user modeling problem. It then surveys systems that have applied techniques from the paradigm to user or student modeling, characterizing each system within a common framework. The final main section discusses several aspects of the usability of these techniques for user or student modeling, such as their knowledge engineering requirements, their need for computational resources, and the communicability of their results.     

Noise modeling concepts in nonlinear state estimation

It is quite common to assume that uncertainty enters through additive white noise sources when using recursive state estimation algorithms. Also unknown and time-varying parameters are often modeled similarly by augmenting the states with a parameter vector. Further, it is common to reflect initial model uncertainty through the choice of the initial covariance matrices for the states and parameters.In this paper we study noise modeling based on a hypothesis that it is important to model noise correctly. In practice this implies a critical view on the dominating ‘additive noise paradigm’ as a means to model uncertainty. Alternative concepts of modeling the noise are investigated, and it is shown that modeling noise by introducing it in the system auxiliary variables and control inputs may have a positive impact on estimation performance.     

非线性系统辨识

本文综述了非线性系统辨识问题，包括描述非线性系统的模型结构的辨识，模型参数的估计，并对可能的发展方向提出了作者的观点，最后介绍非线性系统辨识的若干应用。     

Multilevel fuzzy relational systems: structure and identification

 Existing fuzzy relational equations (FRE) typically possess an evident single-level structure, where no consequence part of the rule being modeled, is used as a fact to another rule. Corresponding to multistage fuzzy reasoning, a natural extension of traditional fuzzy relational systems (FRS) is to introduce some intermediate levels of processing governed by enhanced FRE's so that the structure resulted becomes multilevel or multistage. Three basic multilevel FRS structures, namely, incremental, aggregated, and cascaded, are considered in this paper and they correspond to different reasoning mechanisms being frequently used by human beings in daily life. While the research works on multilevel FRS are sparse and our ability to solve a system of multilevel FRE's in a purely analytical manner is very limited, we address the identification problem from an optimization approach and introduce three fuzzy neural models. The proposed models consist of single-level FRS modules that are arranged in different hierarchical manners. Each module can be realized by Lin and Lee's fuzzy neural model for implementing the Mamdani fuzzy inference. We have particularly addressed the problem of how to distribute the input variables to different (levels of) relational modules for the incremental and aggregated models. In addition, the new models can learn a complete multistage fuzzy rule set from stipulated data pairs using structural and parameter learning. The effectiveness of the multilevel models has been demonstrated through various benchmarking problems. It can be generally concluded that the new models are distinctive in learning, generalization, and robustness.     

Nonlinear system identification with continuous piecewise linear neural network

This paper considers system identification using domain partition based continuous piecewise linear neural network (DP-CPLNN), which is newly proposed. DP-CPLNN has the capability of representing any continuous piecewise linear (CPWL) function, hence its identification performance can be expected. Another attractive feature of DP-CPLNN is the geometrical property of its parameters. Applying this property, this paper proposes an identification method including domain partition and parameter training. In numerical experiments, DP-CPLNN with this method outperforms hinging hyperplanes and high-level canonical piecewise linear representation, which are two widely used CPWL models, showing the flexibility of DP-CPLNN and the effectiveness of the proposed algorithm in nonlinear identification.     

Neural-networks-based nonlinear dynamic modeling for automotive engines

This paper presents a procedure for using neural networks to identify the nonlinear dynamic model of the intake manifold and the throttle body processes in an automotive engine. A dynamic neural network called external recurrent neural network, is used for dynamic mapping and model construction. Dynamic Levenberg–Marquardt algorithm is then applied to the weight-estimation problem. Modeling results indicate that the neural-network-based models have a rather simple structure. Early results also confirm that the neural-network-based modeling of the manifold dynamics can result in a model that is comparable if not better than the first-principle-based models. In addition, it was verified that the neural model has good generalization capabilities.