Convergence analysis of central and minimax algorithms in scalar regressor models

In this paper, the estimation of a scalar parameter is considered with given lower and upper bounds of the scalar regressor. We derive non-asymptotic, lower and upper bounds on the convergence rates of the parameter estimate variances of the central and the minimax algorithms for noise probability density functions characterized by a thin tail distribution. This presents an extension of the previous work for constant scalar regressors to arbitrary scalar regressors with magnitude constraints. We expect our results to stimulate further research interests in the statistical analysis of these set-based estimators when the unknown parameter is multi-dimensional and the probability distribution function of the noise is more general than the present setup.     

Guardian map approach to robust stability of linear systems withconstant real parameter uncertainty

New sufficient conditions are derived for stability robustness of linear time-invariant state-space systems with constant real parameter uncertainty. These bounds are obtained by applying a guardian map to the uncertain system matrices. Since this approach is only valid for constant real parameter uncertainty, these bounds do not imply quadratic stability, which guarantees robust stability with respect to time-varying uncertainty but is often conservative with respect to constant real parameter uncertainty. Numerical results are given to compare the new bounds with bounds obtained previously by means of Lyapunov methods     

Relative Loss Bounds for On-Line Density Estimation with the Exponential Family of Distributions

We consider on-line density estimation with a parameterized density from the exponential family. The on-line algorithm receives one example at a time and maintains a parameter that is essentially an average of the past examples. After receiving an example the algorithm incurs a loss, which is the negative log-likelihood of the example with respect to the current parameter of the algorithm. An off-line algorithm can choose the best parameter based on all the examples. We prove bounds on the additional total loss of the on-line algorithm over the total loss of the best off-line parameter. These relative loss bounds hold for an arbitrary sequence of examples. The goal is to design algorithms with the best possible relative loss bounds. We use a Bregman divergence to derive and analyze each algorithm. These divergences are relative entropies between two exponential distributions. We also use our methods to prove relative loss bounds for linear regression.     

On dynamic output feedback robust MPC for constrained quasi-LPV systems

This paper considers the dynamic output feedback robust model predictive control (MPC) of a quasi-linear parameter varying (quasi-LPV) system with bounded noise. In our previous works, for the unknown true state, either its ellipsoidal bounds or its polyhedral bounds were solely applied in the main optimisation problem. The recursive feasibility of the main optimisation problem was guaranteed by a simple refreshment of the ellipsoidal bound, but might be lost by applying the polyhedral bounds. This paper shows how and to what extent the recursive feasibility can be restored when the polyhedral bounds are still utilised. First, we propose a new approach which, at each sampling time, utilises either the ellipsoidal bound or the polyhedral bound in the main optimisation problem, the latter being used if and only if it is contained in the former. Then, we show the sufficient conditions under which the approaches based on polyhedral bounds preserve the property of recursive feasibility. A numerical example is given to illustrate the effectiveness of the controller.     

New robust stability bounds of linear discrete-time systems with time-varying uncertainties

This paper presents new uncertain parameter variation bounds for linear discrete-time systems to preserve asymptotic stability. The Lyapunov method is utilized to treat both structured and unstructured uncertainties, and the results are optimized with respect to a parameter in the inequality used. When applied to examples considered by previous authors, our results give less conservative bounds.     

Parameter identification for models with bounded errors in all variables

In this paper, the problem of parameter identification for models with bounded measurement errors both on the input and on the output is addressed and some corrections to previously published results are presented. In particular, it is shown that only parameter overbounds can in general be computed for systems of the form y = (φ + δφ)θ + δy when the bounded measurement errors δφ and δy are correlated. Since ARMAX and bilinear systems can be represented in this form, it turns out that tight parameter bounds are in general not available for these systems. Finally, we show that it is possible to check a posteriori whether the obtained bounds are tight or not.     

On achievable accuracy in edge localization

Edge localization occurs when an edge detector determines the location of an edge in an image. The authors use statistical parameter estimation techniques to derive bounds on achievable accuracy in edge localization. These bounds, known as the Cramer-Rao bounds, reveal the effect on localization of factors such as signal-to-noise ratio (SNR), extent of edge observed, scale of smoothing filter, and a priori uncertainty about edge intensity. By using continuous values for both image coordinates and intensity, the authors focus on the effect of these factors prior to sampling and quantization. They also analyze the Canny algorithm and show that for high SNR, its mean squared error is only a factor of two higher than the lower limit established by the Cramer-Rao bound. Although this is very good, the authors show that for high SNR, the maximum-likelihood estimator, which is also derived, virtually achieves the lower bound     

结合免疫优化和LS-SVRM观测器的非线性系统自适应鲁棒控制

针对一类单输入单输出不确定非线性控制系统提出了一种自适应鲁棒控制算法. 由于最小均方支持向量回归机(LS-SVRM)的最终解可以化为一个具有线性约束的二次规划问题, 不存在局部极小, 所以该算法在不要求假设系统的状态向量是可测的条件下通过设计基于LS-SVRM的观测器来估计系统的状态向量; 同时在算法中假设LS-SVRM的最优逼近参数向量和标称参数向量之差的范数和逼近误差的界限是未知的, 因此可通过对未知界限估计的调节来提高系统的鲁棒性. 考虑到LS-SVRM本身参数对LS-SVRM性能的影响, 本文应用一种新的免疫优化算法对LS-SVRM的参数进行优化, 从而提高LS-SVRM的逼近能力. 理论研究和仿真例子证实了所提方法的可行性和有效性.     

Allowable parameter variations and robustness recovery in LQG regulators

In this paper, the bounds of allowable parameter variations in the LQG regulator for fixed weighting matrices are obtained and the asymptotic properties of these bounds with respect to weighting matrices are analysed. It is shown that the guaranteed bounds of allowable parameter variations, which are independent of weighting matrices, are often small but not necessarily small under some conditions. It is also shown that under some conditions, the allowable bounds of the LQG regulator can become as large as those of the LQ regulator. In addition, a loop transfer recovery method for perturbed systems is derived under which the LQG regulator may possess the same robustness as the LQ regulator. Examples are given to validate these results.     

Robust stability analysis of networked systems with varying delays

This article is concerned with networked controlled systems (NCS) with uncertain, varying, bounded transmission delays and asynchronous discrete-time static control laws. It is first shown that the delay variation gives rise to a discrete-time uncertain NCS model; robust stability analysis is carried out via a linear matrix inequality approach which, when combined with a directed parameter search, yields an estimate of robust stability bounds against any variations of the maximum allowable delay (constrained within one sampling period) that the closed-loop system can tolerate. The derived bounds are compared with other techniques relying on the singular values of the perturbed NCS-model. The presented simulation results prove the efficacy of the proposed control scheme.     

LEAP EFFECTS AMONG ROBUST ROOT TRAJECTORIES

The set of closed-loop eigenvalues in the presence of the worst uncertainty is termed robust root trajectory. The open-loop gain or some other system parameter is changed monotonically. The admissible uncertainties are considered along their envelope. The worst uncertainty is denoted as that one where a specified function of the eigenvalues leads to the maximum disadvantageous effect on the closed-loop stability performance. Leaps in the dynamic behavior occur even for infinitesimally small changes in the gain parameter when system uncertainties are considered that are located especially at their bounds.     

Set-membership approach for identification of parameter and prediction uncertainty in power-law relationships: The case of sediment yield

Power laws are used to describe a large variety of natural and industrial phenomena. Consequently, they are used in a wide range of scientific research and management applications. This paper focuses on the identification of bounds on the parameter and prediction uncertainty in a power-law relation from experimental data, assuming known bounds on the error between model output and observations. The prediction uncertainty bounds can subsequently be used as constraints, for example in optimisation and scenario studies. The set-membership approach involves identification and removal of outliers, estimation of the feasible parameter set, evaluation of the feasible model-output set and tuning of the specified bounds on model-output error. As an example the procedure is applied to data of scattered sediment yield versus catchment area (Wasson, 1994). The key result is an un-falsified relationship between sediment yield and catchment area with uncertainty bounds on its parameters. The set-membership results are compared with the results from a conventional least-squares approach with first-order variance propagation, assuming a zero-mean, symmetrical error distribution.     

Frequency domain criteria for strict aperiodicity of linear interval systems

Interval systems arise when systems experience parameter variations. This paper derives frequency domain criteria for interval systems to be strictly aperiodic. The frequency domain criteria only requires one frequency domain plot to check for strict aperiodicity of linear interval systems. Furthermore, the largest allowable perturbation bounds can be graphically estimated from the same frequency domain plot.     

Event-based input and state estimation for linear discrete time-varying systems

In this paper, the joint input and state estimation problem is considered for linear discrete-time stochastic systems. An event-based transmission scheme is proposed with which the current measurement is released to the estimator only when the difference from the previously transmitted one is greater than a prescribed threshold. The purpose of this paper is to design an event-based recursive input and state estimator such that the estimation error covariances have guaranteed upper bounds at all times. The estimator gains are calculated by solving two constrained optimisation problems and the upper bounds of the estimation error covariances are obtained in form of the solution to Riccati-like difference equations. Special efforts are made on the choices of appropriate scalar parameter sequences in order to reduce the upper bounds. In the special case of linear time-invariant system, sufficient conditions are acquired under which the upper bound of the error covariance of the state estimation is asymptomatically bounded. Numerical simulations are conducted to illustrate the effectiveness of the proposed estimation algorithm.     

Linear system stability under parameter uncertainties

A sufficient condition for linear system stability under parameter variations is presented. It yields parameter bounds which are not conservative and the test is easily implemented on a digital computer.     

Robust nonlinear dynamic systems — The old question with a new answer

In this paper several upper bounds for parameter variations of a class of nonlinear (feedback) dynamic systems are presented. These sufficient conditions are used iteratively in a computer-aided algorithm to reach to the largest upper bounds or the stability robustness measure of such systems.     

Lower bounds on the information rate of secret sharing schemes with homogeneous access structure

We present some new lower bounds on the optimal information rate and on the optimal average information rate of secret sharing schemes with homogeneous access structure. These bounds are found by using some covering constructions and a new parameter, the k-degree of a participant, that is introduced in this paper. Our bounds improve the previous ones in almost all cases.     

Improved solutions for ill-conditioned problems involved in set-membership estimation for fault detection and isolation

Set-membership (SM) estimation implies that the computed solution sets are guaranteed to contain all the feasible estimates consistent with the bounds specified in the model. Two issues often involved in the solution of SM estimation problems and their application to engineering case studies are considered in this paper. The first one is the estimation of derivatives from noisy signals, which in a bounded uncertainty framework means obtaining an enclosure by lower and upper bounds. In this paper, we improve existing methods for enclosing derivatives using Higher-Order Sliding Modes (HOSM) differentiators combining filtering. Our approach turns the use of high order derivatives more efficiently especially when the signal to differentiate has slow dynamics. The second issue of interest is solving linear interval equation systems, which is often an ill-conditioned problem. This problem is reformulated as a Constraint Satisfaction Problem and solved by the combination of the constraint propagation Forward Backward algorithm and the SIVIA algorithm. The two proposed methods are tested on illustrative examples. The two methods are then used in a fault detection and isolation algorithm based on SM parameter estimation that is applied to detect abnormal parameter values in a biological case study.