Use of prior bounds on time constants and steady‐state gain in recursive parameter bounding

Computation of parameter bounds of a linear dynamical system, given input–output observations and bounds on model‐output error, has been developed as an alternative to classical parameter estimation using least squares, maximum likelihood or the prediction error method. When bounds on time‐domain plant behaviour are known in advance, they can be used to develop prior parameter bounds for discrete‐time rational transfer‐function parameters. These bounds can be used to initialize standard parameter‐bounding algorithms which process input–output observations to update the exact polytope feasible set or one of its outer bounding approximations such as an ellipsoid, orthotope or parallelotope. This paper presents a method to compute such prior bounds from bounds on time constants and steady‐state (dc) gain, often available from the physics of the system or from previous experience. The method finds subsets making up the prior feasible parameter set, recursively in model order, for any configuration of the pole ranges. An analysis leading to measures of the value of prior bounds, in terms of their chances of remaining active when new bounds derived from observations are imposed, is presented. A simulation study compares polytope updating with and without such initial bounds. The simulations investigate the influence of the tightness of time‐constant and steady‐state‐gain bounds in reducing the volume of the feasible sets obtained as observations are processed. The effects of initial bound tightness and signal‐to‐noise ratio on survival time of the prior bounds are also examined. Copyright © 2002 John Wiley & Sons, Ltd.     

On-line parameter interval estimation using recursive least squares

A bank of recursive least-squares (RLS) estimators is proposed for the estimation of the uncertainty intervals of the parameters of an equation error model (or RLS model) where the equation error is assumed to lie between known upper and lower bounds. It is shown that the off-line least-squares method gives the maximum and minimum parameter values that could have produced the recorded input-output sequence. By modifying the RLS estimator in two ways, it is possible to recursively compute inner and outer bounds of the uncertainty intergals. It is shown that the inner bound is asymptotically tight.     

Adaptive state feedback control by orthogonal approximation functions

For a class of feedback linearizable systems a state feedback adaptive control based on orthogonal approximation functions is designed, under the assumption of matching conditions for the uncertainties and of known bounds on a given compact set for the unknown non‐linear function. By virtue of Bessel inequality, the bound on the unknown non‐linear function gives a bound on the norm of the optimal weight vector for any choice of the number of approximating functions, which allows us to design a robust state feedback adaptive scheme with parameter projections. The resulting control algorithm has several advantages over available schemes: it does not require a priori bounds on the approximation error and on the optimal weight vector; it is repeatable, since the set of initial conditions for the state and the parameter estimates from which a class of reference signals is tracked is explicitly given; it characterizes the L_∞ and L₂ performance of the tracking error in terms of both the approximation and the parameter estimation error; it gives full flexibility in the choice of the number of approximating orthogonal functions. Copyright © 2002 John Wiley & Sons, Ltd.     

基于粒子群优化的鲁棒Minimax估计

针对未知但有界噪声干扰下的动态系统提出一种基于粒子群优化的鲁棒minimax估计方法.方法的基本思想是将鲁棒minimax估计问题转化为参数空间上的优化问题,然后采用一种改进粒子群优化算法获得模型参数的最优估计.仿真结果显示该方法可以在估计模型参数的同时准确估计误差界的大小.     

A consistently convergent OBE algorithm with automatic estimation of error bounds

Conventional optimal bounding ellipsoid (OBE) algorithms require a priori knowledge of error bounds which is unknown in most applications. Conservative (overestimated) error bounds used in practice may lead to inconsistent parameter estimation. This paper presents an enhanced OBE algorithm that is proven to be consistently convergent without a priori knowledge of error bounds. Only a lower bound of the ‘tail probability’ of the disturbance process is required. Simulations comparing conventional and the enhanced OBE algorithm is provided. © 1998 John Wiley & Sons, Ltd.     

Optimum flat delay filter characteristics

The problem of finding optimum filter characteristics that produce flat delay in the passband is formulated as a convex programming problem for minimum-phase lowpass filters. The flatness of the group delay is indicated by an error norm, which is a measure of the deviation of delay characteristic from a constant reference level. When the attenuation specifications are relaxed, it is possible to achieve a constant delay characteristic and the problem is solved analytically. When the attenuation constraints are too tight to achieve the constant delay characteristic, the problem is solved numerically. If the least squares error criterion is used, the problem is a quadratic programming problem. It reduces to a linear programming problem for the minimax criterion. Some resulting optimum attenuation characteristics and their associated error norms are given. The tabulated error norms represent the lowest theoretical limits that lowpass, minimum-phase filters can ever attain. This is confirmed by comparing the limits against the performance of some well-known filters.     

Simultaneous input and parameter estimation with input observers and set‐membership parameter bounding: theory and an automotive application

The paper addresses an on‐line, simultaneous input and parameter estimation problem for a first‐order system affected by measurement noise. This problem is motivated by practical applications in the area of engine control. Our approach combines an input observer for the unknown input with a set‐membership algorithm to estimate the parameter. The set‐membership algorithm takes advantage of a priori available information such as (i) known bounds on the unknown input, measurement noise and time rate of change of the unknown input; (ii) the form of the input observer in which the unknown parameter affects only the observer output; and (iii) the input observer error bounds for the case when the parameter is known exactly. The asymptotic properties of the algorithm as the observer gain increases are delineated. It is shown that for accurate estimation the unknown input needs to approach the known bounds a sufficient number of times (these time instants need not be known). Powertrain control applications are discussed and a simulation example based on application to engine control is reported. A generalization of the basic ideas to higher order systems is also elaborated. Copyright © 2006 John Wiley & Sons, Ltd.     

Finite‐dimensional approximate modeling with error bounds of flexible vibrating systems based on partial eigenstructures

An approach to control‐oriented uncertainty modeling is presented for a class of elastic vibrating systems such as flexible structures, beams and strings, described by partial differential equations. Uncertainty bounding techniques are developed using the upper and lower bounds of the unknown eigenparameters. The result forms a basis for a finite‐dimensional controller design in which closed loop stability and performance are guaranteed. A feasible set of systems is defined of all systems governed by a class of differential equations with certain norm bounds of the unknown input and output operators and with partially known bounds of the eigenparameters. Then the perturbation magnitude covering the feasible set is evaluated in the frequency domain where a standard truncated modal model is chosen as the nominal one. An upper bound to the truncated error magnitude which is calculated by linear programming is proposed. It is demonstrated that all the parameters formulating a feasible set are derived by finite element analysis for a flexible beam example, and feasibility of the proposed scheme is also illustrated by numerical bounding results. © 2006 Wiley Periodicals, Inc. Electr Eng Jpn, 155(2): 36–44, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20272     

Robust time‐varying Kalman estimators for systems with packet dropouts and uncertain‐variance multiplicative and linearly correlated additive white noises

This paper is concerned with robust estimation problem for a class of time‐varying networked systems with uncertain‐variance multiplicative and linearly correlated additive white noises, and packet dropouts. By augmented state method and fictitious noise technique, the original system is converted into one with only uncertain noise variances. According to the minimax robust estimation principle, based on the worst‐case system with conservative upper bounds of uncertain noise variance, the robust time‐varying Kalman estimators (filter, predictor, and smoother) are presented. A unified approach of designing the robust Kalman estimators is presented based on the robust Kalman predictor. Their robustness is proved by the Lyapunov equation approach in the sense that their actual estimation error variances are guaranteed to have the corresponding minimal upper bounds for all admissible uncertainties. Their accuracy relations are proved. The corresponding robust steady‐state Kalman estimators are also presented, and the convergence in a realization between the time‐varying and steady‐state robust Kalman estimators is proved. Finally, a simulation example applied to uninterruptible power system shows the correctness and effectiveness of the proposed results.     

Robust H∞ filtering for switched linear discrete‐time systems with polytopic uncertainties

In this paper, the problem of robust H_∞ filtering for switched linear discrete‐time systems with polytopic uncertainties is investigated. Based on the mode‐switching idea and parameter‐dependent stability result, a robust switched linear filter is designed such that the corresponding filtering error system achieves robust asymptotic stability and guarantees a prescribed H_∞ performance index for all admissible uncertainties. The existence condition of such filter is derived and formulated in terms of a set of linear matrix inequalities (LMIs) by the introduction of slack variables to eliminate the cross coupling of system matrices and Lyapunov matrices among different subsystems. The desired filter can be constructed by solving the corresponding convex optimization problem, which also provides an optimal H_∞ noise‐attenuation level bound for the resultant filtering error system. A numerical example is given to show the effectiveness and the potential of the proposed techniques. Copyright © 2006 John Wiley & Sons, Ltd.     

Directly computable L2 and L∞ performance bounds for morse's dynamic certainty equivalence adaptive controller

In this paper we consider a model reference adaptive control scheme where the classical error augmentation and standard tuning error normalization are avoided through the use of Morse's high-order tuner. We consider the particular scheme of Morse where the concept of dynamic certainty equivalence is used to reduce the error equation to one that involves only first-order dynamics. With such an error equation, it is first shown that one can directly obtain computable L∞ and L∞ bounds on the tracking error. This is an improvement over some earlier results where either only local L∞ bounds were obtained or the calculation of the global bounds required additional computation. Second, inserting an adaptive gain into Morse's high-order tuner, we show that fast adaptation improves both the L₂ and L∞ bounds on the tracking error, in the sense that the effect of the parametric uncertainty on these bounds is attenuated. Finally, using a simple example, we demonstrate how an earlier attempt to use the adaptive gain to simultaneously attenuate the effect of the parametric uncertainty as well as the initial conditions on the L₂ bound for the tracking error has led to an incorrect result.     

A clarifying analysis of feedback error learning in an LTI framework

Feedback error learning (FEL) is a proposed technique for reference‐feedforward adaptive control. FEL in a linear and time‐invariant (LTI) framework has been studied recently; the studies can be seen as proposed solutions to a ‘feedforward MRAC’ problem. This paper reanalyzes two suggested schemes with new interpretations and conclusions. It motivates the suggestion of an alternative scheme for reference‐feedforward adaptive control, based on a certainty‐equivalence approach. The suggested scheme differs from the analyzed ones by a slight change in both the estimator and the control law. Boundedness and error convergence are then guaranteed when the estimator uses normalization combined with parameter projection onto a convex set where stability of the estimated closed‐loop system holds. Copyright © 2008 John Wiley & Sons, Ltd.     

Weighted fusion robust steady-state estimators for multisensor networked systems with one-step random delay and inconsecutive packet dropouts

In this paper, the weighted fusion robust steady-state Kalman filtering problem is studied for a class of multisensor networked systems with mixed uncertainties. The uncertainties include same multiplicative noises in system parameter matrices, uncertain noise variances, as well as the one-step random delay and inconsecutive packet dropouts, which modeled by sequences of Bernoulli variables with different probabilities. By defining a new observation vector and applying the augmented method, the system under study is converted into one with only uncertain noise variances. The sufficient conditions for the existence of steady-state estimators are given. According to the minimax robust estimation principle, based on the worst-case subsystems with conservative upper bounds of uncertain noise variances, the robust local steady-state Kalman estimators (predictor, filter, and smoother) are proposed. Applying the optimal fusion algorithm weighted by matrices and the covariance intersection fusion algorithm, the two kinds of robust fusion steady-state Kalman estimators are derived in a unified framework. The robustness of the proposed fusion estimators is proved by applying the permutation matrices and the global Lyapunov equations method, such that, for all admissible uncertainties, the actual steady-state estimation error variances of the estimators are guaranteed to have the corresponding minimal upper bounds. The accuracy relations among the robust local and fusion steady-state Kalman estimators are proved. An example with application to autoregressive moving average signal processing is proposed, which shows that the robust local and fusion signal estimation problems can be solved by the state estimation problems. Simulation example verifies the effectiveness and correctness of the proposed results.     

Robust ISI equalization for uncertain channels via minimax optimal filtering

We consider a problem of equalization for intersymbol interference (ISI) occurring during data transmission through an imperfectly known channel. It is shown that this problem can be effectively addressed using the minimax filtering approach developed in the paper. This approach leads to a numerically tractable methodology for robust equalizer design which guarantees an optimal upper bound on the equalization error. Copyright © 2006 John Wiley & Sons, Ltd.     

Hierarchical fusion robust Kalman filter for clustering sensor network time‐varying systems with uncertain noise variances

For the clustering time‐varying sensor network systems with uncertain noise variances, according to the minimax robust estimation principle, based on the worst‐case conservative system with conservative upper bounds of noise variances, applying the optimal Kalman filtering, the two‐level hierarchical fusion time‐varying robust Kalman filter is presented, where the first‐level fusers consist of the local decentralized robust fusers for the clusters, and the second‐level fuser is a global decentralized robust fuser for the cluster heads. It can reduce the communication load and save energy resources of sensors. Its robustness is proved by the proposed Lyapunov equation method. The concept of robust accuracy is presented, and the robust accuracy relations of the local, decentralized, and centralized fused robust Kalman filters are proved. Specially, the corresponding steady‐state robust local and fused Kalman filters are also presented, and the convergence in a realization between the time‐varying and steady‐state robust Kalman filters is proved by the dynamic error system analysis method. A simulation example shows correctness and effectiveness. Copyright © 2013 John Wiley & Sons, Ltd.     

具有有界扰动的奇异摄动系统的稳健性分析

本文对高频寄生、外界扰动及高频输入信号共存的奇异摄动系统进行了稳健性研究,采用新的自适应律,给出了状态偏差,寄生变量和失配参数向量的有限上界.     

Construction of a robust adaptive regulator for time-varying discrete time systems

A robust adaptive regulator is constructed for single-input/single-output discrete time systems modelled by a linear time-varying difference equation that includes an error term to incorporate model errors and/or disturbances. It is assumed that the parameters of the nominal model belong to a known bounded convex set and that the ‘frozen time’ nominal model is stabilizable for all possible parameter values. The estimation of the parameters of the nominal model is carried out using a standard gradient-type algorithm with a projection operation. An adaptive regulator is then constructed from the solution to a finite time Riccati equation. It is shown that the resulting closed-loop system is globally stable if the mean of the parameter time variations is sufficiently small and if the model error is sufficiently small, but where the disturbances applied to the plant may be arbitrarily large.     

同步发电机辨识参数对数据误差的敏感性分析

为了从理论上描述参数辨识结果受数据误差的影响程度,针对同步发电机六阶模型,给出了一种参数对数据误差的敏感性分析方法。基于稳态参数与暂态参数的辨识模型,分别构造了同步发电机稳态参数、暂态参数与数据误差的关系式,给出了参数对数据误差的上界估计。基于实测PMU数据,利用该分析方法,以电流误差为例,进行了计算参数对数据误差估计式的实例分析。结果表明,同步发电机参数受到电流数据误差的影响程度依次为稳态、暂态、次暂态参数。     

Self-adaptive finite-element techniques for the computation ofinhomogeneous Poissonian fields

It is shown how complementary energy methods can be used both with the finite-element method to provide error bounds on solutions and as a basis for a self-adaptive mesh-generation strategy applicable to inhomogeneous Poissonian field problems. The self-adaptive approach allows very simple initial mesh structures to be defined, and yet the automatically generated mesh is such that optimal computer accuracy is guaranteed. This technique, suitable for solving electrostatic-field problems, is also highly computer efficient, particularly if the parameter of interest is system energy. Under such circumstances a close estimate to the exact solution can be obtained with the aid of two simple solutions using only a low number of degrees of freedom. This is a direct consequence of the error-bounded nature of the complementary forms. Results are presented which illustrate the operation of these techniques and their effectiveness     

Robust adaptive stabilization for time-varying systems with unmodelled dynamics and disturbances

In this paper, robust adaptive stabilization is discussed for time-varying discrete time systems with disturbances and unmodelled dynamics. Both bounded and unbounded stochastic disturbances are considered. It is assumed that the parameters of the nominal model belong to a bounded convex set and that the ‘frozen time’ nominal model is stabilizable for all possible parameter values. Requiring neither external excitation nor stable invertibility of the nominal model, an adaptive regulator is constructed on the basis of the solution to a finite time Riccati equation and a projected gradient estimator. It is shown that the closed-loop system is stable if both the time average of the parameter variations and the model error are sufficiently small.