Robust identification/invalidation in an LPV framework

A robust linear parameter varying (LPV) identification/invalidation method is presented. Starting from a given initial model, the proposed method modifies it and produces an LPV model consistent with the assumed uncertainty/noise bounds and the experimental information. This procedure may complement existing nominal LPV identification algorithms, by adding the uncertainty and noise bounds which produces a set of models consistent with the experimental evidence. Unlike standard invalidation results, the proposed method allows the computation of the necessary changes to the initial model in order to place it within the consistency set. Similar to previous LPV identification procedures, the initial parameter dependency is fixed in advance, but here a methodology to modify this dependency is presented. In addition, all calculations are made on state‐space matrices which simplifies further controller design computations. The application of the proposed method to the identification of nonlinear systems is also discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust performance optimization based on stochastic model errors: the stable case

Robust control aims to account for model uncertainty in design. Traditional methods for robust control typically assume knowledge of hard bounds on the system frequency response. However, this does not match well with system identification procedures which typically yield statistical confidence bounds on the estimated model. This paper explores a new procedure for obtaining a better match between robust control and system identification by using stochastic confidence bounds for robust control design. Given a nominal design, we set up an optimization problem which is aimed at reducing the statistical variability, measured in a mean square sense, from the nominal sensitivity. The proposed procedure is straightforward and leads to an easily computable solution for the final robust controller in the case of a stable plant and modest plant uncertainty. An illustrative example is provided which shows the advantages of the method. Copyright © 2002 John Wiley & Sons, Ltd.     

Modeling the system of suboptimal robust tracking under unknown upper bounds on the uncertainties and external disturbances

This paper addresses a problem of suboptimal robust tracking for a discrete-time plant under unknown upper bounds on external disturbances. The transfer function of the nominal plant model is assumed to be known, whereas the upper bounds on the external disturbance, measurement noise, and coprime factor perturbations are assumed to be unknown. The results of numerical modeling of the algorithm of suboptimal robust tracking based on relaxed verification of the estimates of upper bound in the closed loop and the use of control criterion associated with such verification as the identification criterion are presented and analyzed. The results of numerical modeling illustrate efficiency of the proposed method of control design.     

A recursive algorithm for estimating multiple models continuous transfer function with non-uniform sampling

A multiple model recursive least squares algorithm combined with a first-order low-pass filter transformation method, named λ-transform, is proposed for the simultaneous identification of multiple model orders continuous transfer functions from non-uniformly sampled input–output data. The λ-transformation is shown to be equivalent to a canonical transformation between discrete z-domain and δ-domain using the negative value of the λ-transform filter time-constant instead of the sampling interval parameter. The proposed algorithm deals with oversampling, sampling jitter or non-uniform sample intervals without the need for extra digital anti-aliasing pre-filtering, downsampling or interpolation algorithms, producing multiple models with a cost function that facilitates automatic selection of best-fitted models. Besides, measurement noise is noted as beneficial, bringing up an inherent bias toward low-order models. Simulated examples and a drum-boiler level experimental result exhibiting non-minimum phase behaviour illustrate the application of the proposed method.     

Robust H ∞ model predictive control for uncertain systems using relaxation matrices

In this paper, a robust H _∞ model predictive control (MPC) technique is proposed for time-varying uncertain discrete-time systems in the presence of input constraints and disturbances. We formulate a minimization problem of the upper bound of finite horizon cost function subject to the terminal inequality for an induced l ₂-norm bound. In order to improve system performance, we propose an LMI condition for the terminal inequality by using relaxation matrices. The LMI condition guarantees induced l ₂-norm bounds of the system despite system uncertainty and disturbance. A numerical example shows the effectiveness of the proposed method.     

Scheduling with Conflicts on Bipartite and Interval Graphs

In this paper, we consider the on-line scheduling of jobs that may be competing for mutually exclusive resources. We model the conflicts between jobs with a conflict graph, so that the set of all concurrently running jobs must form an independent set in the graph. This model is natural and general enough to have applications in a variety of settings; however, we are motivated by the following two specific applications: traffic intersection control and session scheduling in high speed local area networks with spatial reuse. Our results focus on two special classes of graphs motivated by our applications: bipartite graphs and interval graphs. The cost function we use is maximum response time. In all of the upper bounds, we devise algorithms which maintain a set of invariants which bound the accumulation of jobs on cliques (in the case of bipartite graphs, edges) in the graph. The lower bounds show that the invariants maintained by the algorithms are tight to within a constant factor. For a specific graph which arises in the traffic intersection control problem, we show a simple algorithm which achieves the optimal competitive ratio.     

Robust tracking with unknown upper bounds on the perturbations and measurement noise

We consider the robust tracking problem for a discrete time control object with unknown upper bounds on the perturbations. The transition function in the nominal model of the controllable object is assumed to be known, while upper bounds on the external perturbation, measurement noise, and operator perturbations with respect to both output and control are assumed to be unknown or rough. Current estimates of upper bounds on the perturbations are obtained from measurement data with the quality criterion of the tracking problem as the identification criterion.     

Worst case identification of continuous time systems via interpolation

We consider a worst case robust control oriented identification problem recently studied by several authors. This problem is one of identification in the continuous time setting. We give a more general formulation of this problem. The available a priori information in this paper consists of a lower bound on the relative stability of the plant, a frequency dependent upper bound on a certain gain associated with the plant, and an upper bound on the noise level. The available experimental information consists of a finite number of noisy plant point frequency response samples. The objective is to identify, from the given a priori and experimental information, an uncertain model that includes a stable nominal plant model and a bound on the modeling error measured in norm. Our main contributions include both a new identification algorithm and several new ‘explicit’ lower and upper bounds on the identification error. The proposed algorithm belongs to the class of ‘interpolatory algorithms’ which are known to possess a desirable optimality property under a certain criterion. The error bounds presented improve upon the previously available ones in the aspects of both providing a more accurate estimate of the identification error as well as establishing a faster convergence rate for the proposed algorithm.     

L 2-optimal identification of MIMO errors-in-variables models from the v-gap geometrical interpretation

An L ₂-optimal identification method is extended to cope with MIMO errors-in-variables (EIV) model estimation based on a geometrical interpretation for the v-gap metric. The L ₂-optimal approximate models are composed of system and noise models and characterised by a normalised right graph symbol (NRGS) and its complementary inner factor (CIF), respectively. This metric can be evaluated as the supreme of sine values of the maximal principal angles between NRGS frequency responses of two concerned models. In order to make full use of the angular cosine formula for complex vectors to reduce computational loads, a CIF of the NRGS of the perturbed model is introduced and thus, the system parameter optimisation can be efficiently solved by sequential quadratic programming methods. With the estimated system model, the associated noise model can be built by right multiplication of an inner matrix. Finally, a simulation example demonstrates the effectiveness of the proposed identification method.     

SM identification of approximating models for H∞ robust control

Set Membership (SM) H_∞ identification of mixed parametric and non-parametric models is investigated, aimed to estimate a low-order approximating model and an identification error, giving a measure of the unmodelled dynamics in a form well suited for H_∞ control methodologies. In particular, the problem of estimating the parameters of the parametric part and the H_∞ bound on the modelling error is solved using frequency domain data, supposing l_∞ bounded measurement errors and that the system to be identified is exponentially stable. The effectiveness of the proposed procedure is tested on some numerical examples, showing the advantages of the proposed methods over the existing non-parametric H_∞ identification approaches, in terms of lower model order and of tightness in the modelling error bounds. Copyright © 1999 John Wiley & Sons, Ltd.     

基于自适应低秩去噪的磁共振图像重构

本文提出了一种基于自适应低秩去噪的磁共振图像重构算法.该方法使用去噪近似消息传递算法重构磁共振图像,将自适应加权Schatten-p范数最小化方法（Weighted Schatten p-Norm Minimization,WSNM）作为其降噪模型,研究图像的重构性能.根据算法迭代过程中估计的噪声标准差自适应的设定WSNM的图像块大小及相似块个数.实验表明,与近几年提出的磁共振图像重构算法比较,本文提出的算法可以获得更高的峰值信噪比（Peak Signal to Noise Ratio,PSNR）和更低的相对${L_2}$范数误差（Relative${L_2}$Norm Error,RLNE）,得到更好的重建效果.     

Enforcing stability constraints in set-membership identification of linear dynamic systems

In this paper, we consider the identification of linear systems, a priori known to be stable, from input–output data corrupted by bounded noise. By taking explicitly into account a priori information on system stability, a formal definition of the feasible parameter set for a stable linear system is provided. On the basis of a detailed analysis of the geometrical structure of the feasible set, convex relaxation techniques are presented to solve nonconvex optimization problems arising in the computation of parameter uncertainty intervals. Properties of the computed relaxed bounds are discussed. A simulated example is presented to show the effectiveness of the proposed technique.     

H∞ identification of “soft” uncertainty models

The paper investigates the problem of identifying uncertainty models of causal, SISO, LTI, discrete-time, BIBO stable, unknown systems, using frequency domain measurements corrupted by Gaussian noise of known covariance. Additive uncertainty models are looked for, consisting of a nominal model and an additive dynamic perturbation accounting for the modeling error. The nominal model is chosen within a class of affinely parametrized models with transfer function of given (possibly low) order. An estimate of the parameters minimizing the H_∞ modeling error is obtained by minimizing an upper bound of the worst-case (with respect to the modeling error) second moment of the estimation error. Then, a bound in the frequency domain guaranteeing to include, with probability α, the frequency response error between the estimated nominal model and the unknown system is derived.     

μ-synthesis of an electromagnetic suspension system

This paper deals with μ-synthesis of an electromagnetic suspension system. First, an issue of modeling a real physical electromagnetic suspension system is discussed. We derive a nominal model as well as a set of models in which the real system is assumed to reside. Different model structures and possible model parameter values are fully employed to determine unstructured additive plant perturbations, which directly yield uncertainty frequency weighting function. Second, based on the set of plant models, we setup robust performance control objectives. Third, we make use of the D-K iteration approach for the controller design. Finally, implementing the controller with a digital signal processor, experiments are carried out. With these experimental results, we show robust performance of the designed control system     

Error quantification of the normalised right graph symbol for an errors-in-variables system

This paper proposes a novel method to quantify the error of a nominal normalized right graph symbol (NRGS) for an errors-in-variables (EIV) system corrupted with bounded noise. Following an identification framework for estimation of a perturbation model set, a worst-case v-gap error bound for the estimated nominal NRGS can be first determined from \textit{a priori} and \textit{a posteriori} information on the underlying EIV system. Then, an NRGS perturbation model set can be derived from a close relation between the v-gap metric of two models and ${\rm H}_\infty$-norm of their NRGSs' difference. The obtained NRGS perturbation model set paves the way for robust controller design using an ${\rm H}_\infty$ loop-shaping method because it is a standard form of the well-known NCF (normalized coprime factor) perturbation model set. Finally, a numerical simulation is used to demonstrate the effectiveness of the proposed identification method.     

Robust H ∞ control of stochastic linear switched systems with dwell time

The theory of H _∞ control of switched systems is extended to stochastic systems with state‐multiplicative noise. Sufficient conditions are obtained for the mean square stability of these systems where dwell time constraint is imposed on the switching. Both nominal and uncertain polytopic systems are considered. A Lyapunov function, in a quadratic form, is assigned to each subsystem that is nonincreasing at the switching instants. During the dwell time, this function varies piecewise linearly in time following the last switch, and it becomes time invariant afterwards. Asymptotic stochastic stability of the set of subsystems is thus ensured by requiring the expected value of the infinitesimal generator of this function to be negative between switchings, resulting in conditions for stability in the form of LMIs. These conditions are extended to the case where the subsystems encounter polytopic‐type parameter uncertainties. The method proposed is applied to the problem of finding an upper bound on the stochastic L₂‐gain of the system. A solution to the robust state‐feedback control problem is then derived, which is based on a modification of the L₂‐gain bound result. Two examples are given that demonstrate the applicability of the proposed theory. Copyright © 2013 John Wiley & Sons, Ltd.     

H∞ Robust Control Design for an Arm Manipulator

This paper deals with the application of the optimal H control design technique to derive a controller for a 3 DOF arm manipulator affected by nonmodeled dynamics, nonlinearities, actuator dynamics and sensor noise. The control objectives were to obtain a robust stable with robust performance controlled system. A family of nonparametric models for a finite number of the arm configurations was obtained. Then, a multivariable nominal model was proposed and derived together with additive nonparametric uncertainty bounds. Weighting functions were used to normalize the H norm of the uncertainty and to act on the command, sensor, and error signals. Finally, a controller was computed by solving the corresponding H optimization problem. The controller was tried on the nominal and worst case configuration models. The results showed robust stability and robust performance of the controlled system.     

Robust control oriented identification of errors-in-variables models based on normalised coprime factors

A robust control oriented identification approach is proposed to deal with the identification of errors-in-variables models (EIVMs), which are corrupted with input and output noises. Based on normalised coprime factor model (NCFM) representations, a frequency-domain perturbed NCFM for an EIVM is derived according to a geometrical explanation for the v-gap metric. As a result, identification of the EIVM is converted into that of the NCFM. Besides an identified nominal NCFM, its worst case error has to be quantified. Unlike other traditional control-oriented identification methods, the v-gap metric is employed to measure the uncertainties including a priori information on the disturbing noises and the worst case error for the resulting nominal NCFM. Since this metric is also used as an optimisation criterion, the associate parameter estimation problem can be effectively solved by linear matrix inequalities. Finally, a numerical simulation shows the effectiveness of the proposed method.     

Intelligent robust control design of a precise positioning system

This paper addresses an intelligent uncertainty function to improve the robust stability and performance of H _∞ controlled system in terms of reduced conservatism. The system is identified, output performance and control signal requirements are controlled by proper selection of performance and control weighting functions. Adaptive Neuro Fuzzy Inference System (ANFIS) learns the uncertainty bounds of model uncertainty that results from unmodeled dynamics and parameter variations, then the developed uncertainty weighting function will be included in the synthesis of the H _∞ controller. ν-gap measure is utilized to validate the intelligent identified uncertainty bounds and measure the stability of the designed H _∞ controlled system as well. Experimental results on a servo motion system reveal the advantages of combining intelligent uncertainty identification and robust control. Improved performance is achieved. The proposed approach also allows for iterative experiment design.