Least-squares identification for ARMAX models without the positivereal condition

Recursive identification problems of linear stochastic feedback control systems described by ARMAX models are studied without imposing the strictly positive real condition on the noise model. An increasing lag least squares is used in the first step to estimate the noise process, while the parameter estimate is formed in the second step by an extended least squares. In the algorithm, there is an increase in computational cost in comparison to traditional algorithms. However the results of this note do not need any a priori information or conditions on the noise model except that of stability, and they are applicable to the identification of general feedback control systems     

On the role of prefiltering in nonlinear system identification

Data prefiltering is often used in linear system identification to increase model accuracy in a specified frequency band, as prefiltering is equivalent to a frequency weighting on the prediction error function. However, this interpretation applies only to a strictly linear setting of the identification problem. In this note, the role of data and error prefiltering in nonlinear system identification is analyzed and a frequency domain interpretation is provided, based on the Volterra series representation of nonlinear systems. Simulation results illustrate the conclusions of the analysis.     

On worst-case approximation of feasible system sets via orthonormal basis functions

This note deals with the approximation of sets of linear time-invariant systems via orthonormal basis functions. This problem is relevant to conditional set membership identification, where a set of feasible systems is available from observed data, and a reduced-complexity model must be estimated. The basis of the model class is made of impulse responses of linear filters. The objective of the note is to select the basis function poles according to a worst-case optimality criterion. Suboptimal conditional identification algorithms are introduced and tight bounds are provided on the associated identification errors.     

Optimal asymptotic identification under bounded disturbances

The intrinsic limitation of worst-case identification of linear time-invariant systems using data corrupted by bounded disturbances, when the unknown plant is known to belong to a given model set, is studied. This is done by analyzing the optimal worst-case asymptotic error achievable by performing experiments using any bounded input and estimating the plant using any identification algorithm. It is shown that under some topological conditions on the model set, there is an identification algorithm which is asymptotically optimal for any input, and the optimal asymptotic error is characterized as a function of the inputs. These results, which hold for any error metric and disturbance norm, are applied to three specific identification problems: identification of stable systems in the l₁ norm, identification of stable rational systems in the H_∞ norm and identification of unstable rational systems in the gap metric. For each of these problems, the general characterization of optimal asymptotic error is used to find near-optimal inputs to minimize the error     

Optimal estimation for continuous-time systems with delayed measurements

This note focuses on the traditional problem of the Kalman filtering for linear continuous-time systems. Although the problem has been studied widely in the past decades, little work has been done for the time-delayed systems and some fundamental problems remain to be solved. This note proposes a new tool, namely, reorganization innovation analysis approach , to investigate the filtering problem for systems with delayed measurements. The Kalman filter is given in terms of the solution of standard Riccati equations. The performance is clearly demonstrated through analytical results and simulation. The solved problem in this note is related with some more complicated problems such as H/sub /spl infin// fixed-lag smoothing, H/sub /spl infin// control with preview and control with input delays.     

Optimality, approximation, and complexity in set membership H/sub /spl infin// identification

Investigates the set membership identification of time-invariant, discrete-time, exponentially stable, possibly infinite-dimensional, linear systems from time or frequency-domain data, corrupted by deterministic noise. The aim is to deliver not a single model, but a set of models whose size in H/sub /spl infin// norm measures the uncertainty in the identification. The main focus of the note is on the optimality properties for finite data and on the tradeoff between optimality and complexity of approximated low order model sets. A method is given for evaluating convergent and computationally efficient inner and outer approximations of the value set for a given frequency. Such approximations allow one to compute, within any desired accuracy, the identification error of any identified model, and to evaluate an optimal model at any given number of frequencies. By suitably approximating these values, model sets with nominal models in RH/sub /spl infin// are then derived, whose order is selected by trading off between model set complexity and identification accuracy degradation. This degradation is evaluated by computing the optimality level, defined as the ratio between the reduced model identification error and the optimal one. A numerical example demonstrates the effectiveness of the presented results.     

Strong consistency of recursive identification for Hammerstein systems with discontinuous piecewise-linear memoryless block

This note deals with identification of Hammerstein systems with discontinuous piecewise-linear memoryless block followed by a linear subsystem. Recursive algorithms are proposed for estimating coefficients of the linear subsystem and six unknown parameters contained in the nonlinear static block. By taking a sequence of iid random variables with uniform distribution to serve as the system input, strong consistency is proved for all estimates given in the note. The theoretical results are verified by computer simulation.     

Direct adaptive pole placement with application to nonminimum phase systems

This note presents the first direct adaptive control structure for nonminimum phase systems where controller parameter identification can be posed as a standard linear parameter estimation problem. In particular, a linear equation error model is formulated for estimating both the controller parameters and some additional auxiliary parameters. The key idea involves insertion of the Bezout polynomial identity into the polynomial design equation which arises in arbitrary pole placement.     

基于Volterra级数的一类非线性系统性能评估

化工过程多具有非线性特征,针对用线性系统性能评估方法处理非线性系统会存在过估计的情况,研究了一类叠加线性干扰的非线性系统的性能评估问题。通过使用Volterra级数近似非线性环节,把最小方差性能评估问题转化成一类模型辨识问题,并从辨识误差中得到非线性系统的最小方差估计值。通过数值仿真,将新方法所得结果和现有线性性能评估方法进行了比较,验证了设计算法的优越性。     

Robust H/sub 2/ and H/sub /spl infin// filters for uncertain LFT systems

In this note, robust H/sub 2/ and H/sub /spl infin// filter design problems are considered. The uncertainties, unstructured or structured, are norm bounded and represented by linear fractional transformation (LFT). The main result is that after upper-bounding the objectives, the problems of minimizing the upper bounds are converted to finite dimensional convex optimization problems involving linear matrix inequalities (LMIs). These are extensions of well-known results for systems with polytopic uncertainty. It is shown that for the unstructured, norm bounded uncertainty case, the upper bounds are directly minimized (without further overbounding), yielding less conservative results than previously available. An elementary numerical example is given to illustrate the results.     

Derivative computations for the log likelihood function

This note improves on a method for computing derivatives of the log likelihood function used in the identification of discrete, linear time-invariant Gaussian systems. A formula for the derivative is obtained in terms of the derivatives of the matrices used to define the system under identification.     

Closed‐Loop Identification for Control of Linear Parameter Varying Systems

This paper deals with system identification for control of linear parameter varying systems. In practical applications, it is often important to be able to identify small plant changes in an incremental manner without shutting down the system and/or disconnecting the controller; unfortunately, closed‐loop system identification is more difficult than open‐loop identification. In this paper we prove that the so‐called Hansen scheme, a technique known from linear time‐invariant systems theory for transforming closed‐loop system identification problems into open‐loop‐like problems, can be extended to accommodate linear parameter varying systems as well. We investigate the identified subsystem's parameter dependency and observe that, under mild assumptions, the identified subsystem is affine in the parameter vector. Various identification methods are compared in direct and Hansen Scheme setups in simulation studies, and the application of the Hansen Scheme is seen to improve the identification performance.     

Identification of IIR Wiener systems with spline nonlinearities that have variable knots

An algorithm is developed for the identification of Wiener systems, linear dynamic elements followed by static nonlinearities. In this case, the linear element is modeled using a recursive digital filter, while the static nonlinearity is represented by a spline of arbitrary but fixed degree. The primary contribution in this note is the use of variable knot splines, which allow for the use of splines with relatively few knot points, in the context of Wiener system identification. The model output is shown to be nonlinear in the filter parameters and in the knot points, but linear in the remaining spline parameters. Thus, a separable least squares algorithm is used to estimate the model parameters. Monte-Carlo simulations are used to compare the performance of the algorithm identifying models with linear and cubic spline nonlinearities, with a similar technique using polynomial nonlinearities.     

Input design via LMIs admitting frequency-wise model specifications in confidence regions

A framework for reformulating input design problems in prediction error identification as convex optimization problems is presented. For linear time-invariant single input/single output systems, this framework unifies and extends existing results on open-loop input design that are based on the finite dimensional asymptotic covariance matrix of the parameter estimates. Basic methods for parametrizing the input spectrum are provided and conditions on these parametrizations that guarantee that all possible covariance matrices for the asymptotic distribution of the parameter estimates can be generated are provided. A wide range of model quality constraints can be handled. In particular, different frequency-by-frequency constraints can be used. This opens up new applications of input design in areas such as robust control. Furthermore, quality specifications can be imposed on all models in a confidence region. Thus, allowing for statements such as "with at least 99% probability the model quality specifications will be satisfied".     

RBF-ARX模型在倒立摆系统建模中的应用

针对直线一级倒立摆控制系统的非线性特性,采用RBF-ARX模型对倒立摆系统的全局非线性动态特性进行建模.讨论了RBF-ARX模型结构的选取,模型参数辨识,RBF参数优化等问题.并且分别比较了该倒立摆系统的RBF-ARX模型与全局线性ARX模型,以及将RBF-ARX在某一工作点局部线性化后的模型与局部线性ARX模型的预测输出和模型误差,验证了RBF-ARX模型在倒立摆系统建模和辨识中的有效性.     

Nonlinear system identification and control of chemical processes using fast orthogonal search

The use of the fast orthogonal search (FOS) method is presented for model estimation and control of nonlinear chemical processes. FOS provides a nonlinear approximation used in an inner-loop that allows for simpler linear control methods to be used as an outer-loop controller. It is a straightforward, simple-to-use method for linearization of systems based on orthogonal system identification. The control concept is derived from the method of inverse dynamic control (IDC). The novel combination of this with the FOS method of system identification results in a very efficient and effective method of control. The method is demonstrated and tested on two nonlinear chemical process control simulations and the results are shown to compare very favourably to published results on the same problems.     

Quadratic stability and stabilization of dynamic interval systems

This note presents necessary and sufficient conditions for the quadratic stability and stabilization of dynamic interval systems. The results are obtained in terms of linear matrix inequalities (LMIs) and extended to the quadratic stability and stabilization of linear systems with uncertain parameters. With the powerful LMI toolbox, it is very convenient to solve these problems. The illustrative examples show that this method is effective and less conservative to check the robust stability and to design the stabilizing controller for dynamic interval systems.     

Nonparametric kernel algorithm for recovery of functions from noisy measurements with applications

This note presents a kernel algorithm for recovery of a regression function from noisy data. Conditions are provided that assure pointwise convergence in the mean square and almost sure senses. An application to a class of linear system identification problems is discussed.     

On minimum spanning blocks in discrete linear systems

The question of spanning blocks in linear discrete systems arises in many adaptive identification and control problems and is related to the convergence of these algorithms. Specifically, in block-invariant adaptive control, it is of importance to know the minimum length of these spanning blocks. Establishing this minimum length is the topic of the present note. It is found to be equal to the sum of the dimensions of the system's state space and its controllable subspace.     

A parametric extension of mixed time/frequency robustidentification

A parametric extension to the time/frequency robust identification framework is presented. The results can be applied to stable linear time-invariant systems on which time and/or frequency experiments have been performed. The parametric portion of the model should be affine in the unknown parameters, which includes practical applications such as flexible structures. The consistency problem is cast as a constrained finite-dimensional convex optimization problem that can be formulated as a linear matrix inequality. The proposed procedure provides an interpolating identification algorithm, convergent and optimal up to a factor of two (with respect to central algorithms)