The filter design from data (FD2) problem: Nonlinear Set Membership approach

In the paper, we consider the problem of designing a filter that, operating on noisy measurements of input u and output y of a dynamical system, gives estimates (possibly optimal in some sense) of some other variable of interest z. A large body of literature exists, which investigates this problem assuming that the system equations relating u, y and z are known. However, in most practical situations, the system equations are not (completely) known, but a data set composed of noisy measurements of u, y and z is available. In such situations, a two-step procedure is typically adopted: a model is identified from the set of measured data, and the filter is designed on the basis of the identified model. In this paper, we propose an alternative solution, which uses the available data set of measured u, y and z not for the identification of system dynamics, but for the direct design of the filter. Such a direct design is investigated within the Nonlinear Set Membership framework. In the case of full observability, an almost optimal filter is derived, where optimality refers to minimizing a worst-case estimation error. In the case of partial observability, conditions are given for which the direct design is guaranteed to give bounded estimation error. Three examples are presented, related to the Lorenz chaotic system the first two, and to an automotive application the third one.     

The Wonham filter under uncertainty: A game–theoretic approach

The paper is devoted to a state filtering problem of Markov jump processes given the continuous and/or counting observations. All the transition intensity matrix, observation plan and counting intensity are parameterized by a random vector with uncertain distribution on a known support set.The estimation problem is formulated in minimax settings with a conditional optimality criterion. We reduce the initial minimax problem to a dual problem of constrained quadratic optimization. The corresponding numerical algorithm of minimax filtering is presented as well as its illustrative implementation in the monitoring of a TCP link status under uncertainty.     

Unscented Kalman filter for continuous‐time nonlinear fractional‐order systems with process and measurement noises

This study proposes the design of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system involving the process noise and the measurement noise. The nonlinear fractional‐order system is discretized to get the difference equation. According to the unscented transformation, the design method of unscented Kalman filter for a continuous‐time nonlinear fractional‐order system is provided. Compared with the extended Kalman filter, the proposed method can obtain a more accurate estimation effect. For fractional‐order systems containing non‐differentiable nonlinear functions, the method proposed in this paper is still effective. The unknown parameters are also discussed by the augmented vector method to achieve the state estimation and parameter identification. Finally, two examples are offered to verify the effectiveness of the proposed unscented Kalman filter for nonlinear fractional‐order systems.     

Robust filter design for piecewise discrete‐time systems with time‐varying delays

A novel delay‐dependent filtering design approach is developed for a class of linear piecewise discrete‐time systems with convex‐bounded parametric uncertainties and time‐varying delays. The time‐delays appear in the state as well as the output and measurement channels. The filter has a linear full‐order structure and guarantees the desired estimation accuracy over the entire uncertainty polytope. The desired accuracy is assessed in terms of either ??_∞‐performance or ??₂–??_∞ criteria. A new parametrization procedure based on a combined Finsler's Lemma and piecewise Lyapunov–Krasovskii functional is established to yield sufficient conditions for delay‐dependent filter feasibility. The filter gains are determined by solving a convex optimization problem over linear matrix inequalities. In comparison to the existing design methods, the developed methodology yields the least conservative measures since all previous overdesign limitations are almost eliminated. By means of simulation examples, the advantages of the developed technique are readily demonstrated. Copyright © 2009 John Wiley & Sons, Ltd.     

A nonlinear set‐membership filter for on‐line applications

A guaranteed estimator for a general class of nonlinear systems and on‐line usage is developed and analysed. This filter bounds the linearization error, then applies a linear set‐membership filter such that stability guarantees hold for nonlinear systems. A tight bound on the linearization error is found using interval analysis. This filter recursively estimates an ellipsoidal set in which the true state lies. General assumptions include the use of bounded noises and twice continuously differentiable dynamics. When the system is uniformly observable, it is proven that the nonlinear set‐membership filter is stable. In addition, if no noise is present and the initial error is small, the error between the centre of the estimated set and the true value converges to zero. The result is an estimator which is computationally attractive and can be implemented robustly in real‐time. The proposed method is applied to a two‐state example to demonstrate the theoretical results. Copyright © 2003 John Wiley & Sons, Ltd.     

A novel modeling and controlling approach for high‐order nonlinear systems

This paper proposes a completely data–driven control law for a class of high–order nonlinear systems based on their virtual characteristic models. With sampled–data techniques and estimation methods, a novel lower–order adaptive characteristic model is constructed to reduce the system complexities. Moreover, a corresponding sliding mode control law is designed, which can guarantee tracking errors of the resulting closed‐loop system converging into a predefined bound in finite time. The practical examples are provided to illuminate the effectiveness of the proposed approaches.     

Event‐triggered filter design for Markovian jump delay systems with nonlinear perturbation using quantized measurement

This paper deals with the problem of mixed passivity and H_∞ filter design for a class of Markovian jump delay systems with nonlinear perturbation under event‐triggered scheme and quantization. Firstly, based on an integral inequality, a new sufficient condition for the stochastic stability and performance analysis of the filtering error system is proposed. Secondly, a mode‐dependent condition for the solvability of the filter design problem is given in terms of linear matrix inequalities (LMIs). The filter parameters can be derived using feasible solutions of the presented LMIs. Finally, three numerical examples are given to illustrate the effectiveness and advantages of the proposed filter design method.     

A UD factorization‐based nonlinear adaptive set‐membership filter for ellipsoidal estimation

The extended set‐membership filter (ESMF) for nonlinear ellipsoidal estimation suffers from numerical instability, computation complexity as well as the difficulty in filter parameter selection. In this paper, a UD factorization‐based adaptive set‐membership filter is developed and applied to nonlinear joint estimation of both time‐varying states and parameters. As a result of using the proposed UD factorization, combined with a new sequential and selective measurement update strategy, the numerical stability and real‐time applicability of conventional ESMF are substantially improved. Furthermore, an adaptive selection scheme of the filter parameters is derived to reduce the computation complexity and achieve sub‐optimal estimation. Simulation results have shown the efficiency and robustness of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

H ∞ observer design for uncertain nonlinear discrete‐time systems with time‐delay: LMI optimization approach

We present a robust H _∞ observer for a class of nonlinear discrete‐time systems. The class under study includes an unknown time‐varying delay limited by upper and lower bounds, as well as time‐varying parametric uncertainties. We design a nonlinear H _∞ observer, by using the upper and lower bounds of the delay, that guarantees asymptotic stability of the estimation error dynamics and is also robust against time‐varying parametric uncertainties. The described problem is converted to a standard optimization problem, which can be solved in terms of linear matrix inequalities (LMIs). Then, we expand the problem to a multi‐objective optimization problem in which the maximum admissible Lipschitz constant and the minimum disturbance attenuation level are the problem objectives. Finally, the proposed observer is illustrated with two examples. Copyright © 2014 John Wiley & Sons, Ltd.     

A novel data reduction method for Takagi–Sugeno fuzzy system design based on statistical design of experiment

This paper introduces a simple, systematic and effective method for designing Takagi–Sugeno (T–S) fuzzy systems utilizing a significantly smaller training data set versus existing methods. Creating proper training data is usually not an easy task and requires spending considerable time and resources. The proposed method first uses a three-level factorial design to partition the output space. Next the least square technique is used to estimate each of the partitioned output spaces. The membership functions are introduced with only three variables (min, max and number of membership functions). Fuzzy rules are generated with respect to the partitioned output surfaces and the membership functions. The proposed method is applied to two benchmark problems, controlling an inverted pendulum as well as modeling a nonlinear function. In the case of the inverted pendulum simulation results demonstrate significant improvement. In the case of nonlinear function modeling we demonstrated sufficient accuracy with only 9 training data, which represents 98% reduction in the number of training data compare to other method. Additionally, the proposed method offers extremely low computation time allowing it to be used with adaptive type systems.     

Optimal‐switched extended H∞ filter for nonlinear systems with stochastic uncertainties

The extended H_∞ filter (EH_∞F) is a conservative solution with infinite‐horizon robustness for the state estimation problem regarding nonlinear systems with stochastic uncertainties, which leads to excessive costs in terms of filtering optimality and reduces the estimation precision, particularly when uncertainties related to external disturbances and noise appear intermittently. In order to restore the filtering optimality lost due to the conservativeness of the EH_∞F design, we developed an optimal‐switched (OS) filtering mechanism based on the standard EH_∞F to obtain an optimal‐switched extended H_∞ filter (OS‐EH_∞F). The OS mechanism has an error‐tolerant switched (ETS) structure, which switches the filtering mode between optimal and H_∞ robust by setting a switching threshold with redundancy to uncertainties, and a robustness‐optimality cost function (ROCF) is introduced to determine the threshold and optimize the ETS structure online. The ROCF is the weighted sum of the quantified filtering robustness and optimality. When a weight is given, the proposed OS‐EH_∞F can obtain the optimal state estimates while maintaining the filtering robustness at an invariant ratio. A simulation example of space target tracking has demonstrated the superior estimation performance of the OS‐EH_∞F compared with some other typical filters, thereby verifying the effectiveness of using the weight to evaluate the estimation result of the filters.     

Time‐varying filter design for semi‐Markov jump linear systems with intermittent transmission

This paper focuses on the filter design problem for semi‐Markov jump linear systems. The system outputs are transmitted to the filter via networks, and it is assumed that the transmission is imperfect with data packet dropouts subject to the Bernoulli random binary distribution. A σ‐error mean square stability criterion is first derived for the underlying systems. On the basis of the criterion, the H_∞ performance analysis is conducted. By constructing a time‐varying Lyapunov function, a time‐varying H_∞ filter scheme is investigated. Because the presented approach can cover the mode‐dependent and mode‐independent time‐invariant H_∞ filter schemes as special cases, the conservatism of the derived results is less than those of the time‐invariant filter schemes. An active suspension system with activator uncertainties is lastly presented to illustrate the effectiveness and feasibility of the derived theoretical results. Copyright © 2017 John Wiley & Sons, Ltd.     

Robust multiple‐fault detection filter

A new robust multiple‐fault detection and identification algorithm is determined. Different from other algorithms which explicitly force the geometric structure by using eigenstructure assignment or geometric theory, this algorithm is derived from solving an optimization problem. The output error is divided into several subspaces. For each subspace, the transmission from one fault, denoted the associated target fault, is maximized while the transmission from other faults, denoted the associated nuisance fault, is minimized. Therefore, each projected residual of the robust multiple‐fault detection filter is affected primarily by one fault and minimally by other faults. The transmission from process and sensor noises is also minimized so that the filter is robust with respect to these disturbances. It is shown that, in the limit where the weighting on each associated nuisance fault transmission goes to infinity, the filter recovers the geometric structure of the restricted diagonal detection filter of which the Beard–Jones detection filter and unknown input observer are special cases. Filter designs can be obtained for both time‐invariant and time‐varying systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Reliable H ∞ filter design for T–S fuzzy model‐based networked control systems with random sensor failure

This paper investigates robust and reliable H _∞ filter design for a class of nonlinear networked control systems: (i) a T‐S fuzzy model with its own uncertainties is used to approximate the nonlinear dynamics of the plant, (ii) a new sensor failure model with uncertainties is proposed, and (iii) the signal transfer of the closed‐loop system is under a networked communication scheme and therefore is subject to time delay, packet loss, and/or packet out of order. Four new theorems are proved to cover the conditions for the robust mean square stability of the systems under study in terms of LMIs, and a decoupling method for the filter design is developed. Two examples, one of them is based on a model of an inverted pendulum, are provided to demonstrate the design method. Copyright © 2011 John Wiley & Sons, Ltd.     

Tracking rigid bodies using only position data: A shadowing filter approach based on newtonian dynamics

Tracking a moving object like a ship, vehicle, aircraft or even an animal or human is challenging especially when given only noisy observations of the path. To successfully track such an object the method used needs to infer the unknown information, for example the acceleration, from noisy position data. Here we present such a method based on a shadowing filter algorithm. In comparison with other filters such as Kalman and Particle filters the shadowing filter solves the tracking problem based on deterministic dynamics instead of statistics. The algorithm presented shows how to track rigid bodies having an unknown moment of inertia and we validate the performance of the filter and explore how the two important parameters of the filter impact on its performance.     

Matrix inequality‐based observer design for a class of distributed transport‐reaction systems

The problem of designing a globally exponentially convergent observer for a class of (linear in transport and nonlinear in generation) semi‐linear parabolic distributed systems is addressed within a matrix inequality framework, yielding (i) sufficient convergence conditions with physical meaning and (ii) the weight of a Lyapunov functional as design degree of freedom. The proposed approach is illustrated and tested with a representative case example in chemical reaction engineering. Copyright © 2013 John Wiley & Sons, Ltd.     

The format of problem representation for in‐game learning supports

A promising method to support game‐based learning is to facilitate learners' externalization of cognitive and metacognitive processes. Externalizing Problem Representation (EPR) refers to a cognitive behaviour in which a learner constructs her own representations overtly. The purpose of this study is to investigate whether learning supports promoting EPR enhance qualitative understanding and quantitative proficiency in ratios and proportional relationships in a learning game (i.e., E‐Rebuild) context. Specifically, this study investigated the effects of representation format in problem representation on qualitative understanding and quantitative proficiency in a learning game context. The results of this study indicate that (a) symbolic learning supports better facilitate comprehension of math concepts and their relations than iconic learning supports in video game contexts, (b) symbolic learning supports better facilitate players' reflection for implicit understanding and promote their math problem‐solving skills, (c) participants in the symbolic learning support group increased significantly in qualitative understanding but not in quantitative proficiency after gameplay, and (d) participants in the iconic learning support group experienced significant growth in quantitative proficiency but not in qualitative understanding after gameplay.     

Exponential H∞ filter design for stochastic time‐varying delay systems with Markovian jumping parameters

In this paper, the exponential H_∞ filter design problem is investigated for a general class of stochastic time‐varying delay system with Markovian jumping parameters. The stochastic uncertainties appear in both the dynamic and the measurement equations and the state delay is assumed to be time‐varying. Attention is focused on the design of mean‐square exponentially stable and Markovian jump filter such that the filtering error systems are mean‐square exponentially stable and the estimation error satisfies a given H_∞ performance. By introducing some slack matrix variables, delay‐dependent sufficient conditions for the solvability of the above problem are presented in terms of linear matrix inequalities (LMIs). In addition, the decay rate can be a given positive value without any other constraints. When the proposed LMIs are feasible, an explicit expression of the desired H_∞ filter can be given. A numerical example is provided to illustrate the effectiveness of the proposed design approach. Copyright © 2009 John Wiley & Sons, Ltd.