Set-membership Framework Research Articles

Validated computation of the local truncation error of Runge–Kutta methods with automatic differentiation

A novel approach to bound the local truncation error of explicit and implicit Runge–Kutta methods is presented. This approach takes its roots in the modern theory of Runge–Kutta methods, namely the order condition theorem, defined by John Butcher in the 1960s. More precisely, our work is an instance, for Runge–Kutta methods, of the generic algorithm defined by Ferenc Bartha and Hans Munthe-Kaas in 2014 which computes B-series with automatic differentiation techniques. In particular, this specialized algorithm is combined with set-membership framework to define validated numerical integration methods based on Runge–Kutta methods.

Set‐membership normalised least M‐estimate spline adaptive filtering algorithm in impulsive noise

This letter proposes a set-membership normalised least M-estimate algorithm based on Wiener spline adaptive filter (SAF). The proposed algorithm combines the set-membership framework and least-M estimate scheme, thus achieving faster convergence rate and effective suppression of impulsive noise on the filter weight and control point adaptation. Simulation results demonstrate that the proposed one exhibits more robust performance compared to the conventional SAF algorithms in an impulsive noise environment.

Ellipsoidal bounding set-membership identification approach for robust fault diagnosis with application to mobile robots

A robust fault diagnosis approach is developed by incorporating a set-membership identification (SMI) method. A class of systems with linear models in the form of fault related parameters is investigated, with model uncertainties and parameter variations taken into account explicitly and treated as bounded errors. An ellipsoid bounding set-membership identification algorithm is proposed to propagate bounded uncertainties rigorously and the guaranteed feasible set of faults parameters enveloping true parameter values is given. Faults arised from abrupt parameter variations can be detected and isolated on-line by consistency check between predicted and observed parameter sets obtained in the identification procedure. The proposed approach provides the improved robustness with its ability to distinguish real faults from model uncertainties, which comes with the inherent guaranteed robustness of the set-membership framework. Efforts are also made in this work to balance between conservativeness and computation complexity of the overall algorithm. Simulation results for the mobile robot with several slipping faults scenarios demonstrate the correctness of the proposed approach for faults detection and isolation (FDI).

Actuator Fault Compensation in a Set-membership Framework for Linear Parameter-Varying Systems

This paper presents an actuator fault compensation approach for a class of Linear Parameter-Varying (LPV) systems with noisy measurements. The proposed method is based on interval estimation assuming that the fault vector and the external disturbances are unknown but bounded. The main idea consists in designing a control law, based on a linear state feedback, to guarantee closed-loop stability. An additive control, based on fault bounds, is used to compensate the impact of actuator faults on system performances. The closed-loop stability of the robust fault compensation scheme is established in the Lyapunov sense. Finally, the theoretical results are illustrated using a numerical example.

Network-Based Data-Driven Filtering With Bounded Noises and Packet Dropouts

This paper is concerned with the problem of a network-based data-driven filter design for discrete-time linear systems with bounded noises and packet dropouts. One favorable feature is that the designed filter can be directly employed without identifying the unknown system model. To compensate the negative effects of packet dropouts, an output predictor is first designed to reconstruct the missing data based on the received outputs and the inputs of the system. The asymptotic convergence of the output prediction error is established, of which the rate can be adjusted by the parameter. Then utilizing the predicted outputs and the received measurements, an almost-optimal data-driven filter with tractability is proposed within the set membership (SM) framework and the bound on the worst case estimation error is derived. Finally, two illustrative examples, including a comparison example and an application example, are presented to show the advantages of the proposed design and the effectiveness of the theoretical results.

Set-membership identifiability of nonlinear models and related parameter estimation properties

Abstract Identifiability guarantees that the mathematical model of a dynamic system is well defined in the sense that it maps unambiguously its parameters to the output trajectories. This paper casts identifiability in a set-membership (SM) framework and relates recently introduced properties, namely, SM-identifiability, μ-SM-identifiability, and ε-SM-identifiability, to the properties of parameter estimation problems. Soundness and ε-consistency are proposed to characterize these problems and the solution returned by the algorithm used to solve them. This paper also contributes by carefully motivating and comparing SM-identifiability, μ-SM-identifiability and ε-SM-identifiability with related properties found in the literature, and by providing a method based on differential algebra to check these properties.

Data-driven Filter Design for Linear Systems with Quantized Measurements

This paper is concerned with the problem of data-driven filter design for linear systems with bounded noise by using quantized measurements. Since the mathematical model of the plant studied is unavailable, most of the existing model-based filter design approaches cannot be used to solve this problem. Another challenge lies in the fact that all the measurement data accessible is quantized. To solve this issue, a quasi-feasible filter set within the set membership framework is proposed, and a data-driven optimal worst-case filter is designed. Furthermore, an l2-l∞ almost-optimal worst-case filter design algorithm is presented by means of linear programming technique. A numerical example is given to illustrate the effectiveness of the proposed algorithms.

Set membership identification of switched linear systems with known number of subsystems

This paper addresses the problem of robust identification of a class of discrete-time linear hybrid systems, switched linear models, in a set membership framework. Given a finite collection of input/output data from a noisy process the objective is twofold: (i) establish whether this data was generated by a system that switches amongst an a priori known number of subsystems, and (ii) in that case identify a suitable set of linear models along with a switching sequence that can explain the available experimental information. Our main result shows that these problems are equivalent to minimizing the rank of a matrix whose entries are affine in the optimization variables, subject to a convex constraint imposing that these variables are the moments of an (unknown) Borel measure with finite support. The use of well known (tight) convex relaxations of rank allows for further reducing the problem to a semidefinite optimization that can be efficiently solved. In the second part of the paper we extend these results to handle sensor failures that result in corrupted input/output measurements. Assuming that these failures are infrequent, we show that the problem can be recast into an optimization form where the objective is to simultaneously minimize the rank of a matrix and the number of nonzero rows of a second one. In both cases, appealing to well known convex relaxations of rank and sparsity leads to overall semidefinite optimization problems that can be efficiently solved. These results are illustrated with multiple examples showing substantially improved identification performance in the presence of noise and sensor faults.

Set-Membership Adaptive Localization Algorithm with Time-Varying Error Bounds for Underwater Wireless Sensor Networks

This paper presents a set-membership adaptive localization algorithm with time-varying error bounds for underwater wireless sensor networks (UWSNs). In large-scale UWSNs, the nonstationary underwater environments, the insufficient prior information of hybrid noise, the small sample size of available distance measurements, and the node mobility all pose severe challenges for localization, and most current schemes are not applicable. Unlike most of the existing approaches, we tackle the multihop localization problem in a set-membership framework based on the consideration that the distance measurement uncertainty can be cast into an unknown but bounded (UBB) context. The principle of our scheme is firstly to use the bootstrap method to build confidence intervals and error bounds from a small sample set of distance measurements and then to determine the positions by a low-complexity interval analysis method as well as an adaptive localization update specification with time-varying error bounds. Simulation results show that our proposed scheme is an effective and efficient localization approach in large-scale UWSNs.

Set‐membership localisation algorithm based on adaptive error bounds for large‐scale underwater wireless sensor networks

Proposed is a new set-membership approach based on adaptive error bounds to solve multi-hop localisation problems in the cases where the prior information is insufficient and only a very few distance measurements are available, which are commonly encountered in large-scale underwater wireless sensor networks. Unlike most of the existing schemes, the localisation problem is solved in a set-membership framework which is based on the consideration that the distance measurement uncertainty can be cast into an unknown but bounded context. For the first time, an adaptive bounds specification is introduced into multi-hop localisation for eliminating the risk of overbounding or underbounding. To reduce the computational burden, boxes are utilised to approximate the irregular intersection of all feasible solutions. Then the feasible sets and coordinates of ordinary nodes are obtained by using the interval analysis method. Simulation results confirm the effectiveness of the algorithm.

A Sparsification Approach to Set Membership Identification of Switched Affine Systems

This paper addresses the problem of robust identification of a class of discrete-time affine hybrid systems, switched affine models, in a set membership framework. Given a finite collection of noisy input/output data and some minimal a priori information about the set of admissible plants, the objective is to identify a suitable set of affine models along with a switching sequence that can explain the available experimental information, while minimizing either the number of switches or subsystems. For the case where it is desired to minimize the number of switches, the key idea of the paper is to reduce this problem to a sparsification form, where the goal is to maximize sparsity of a suitably constructed vector sequence. Our main result shows that in the case of ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> bounded noise, this sparsification problem can be exactly solved via convex optimization. In the general case where the noise is only known to belong to a convex set N, the problem is generically NP-hard. However, as we show in the paper, efficient convex relaxations can be obtained by exploiting recent results on sparse signal recovery. Similarly, we present both a sparsification formulation and a convex relaxation for the (known to be NP hard) case where it is desired to minimize the number of subsystems. These results are illustrated using two non-trivial problems arising in computer vision applications: video-shot and dynamic texture segmentation.

Unified Set Membership theory for identification, prediction and filtering of nonlinear systems

The problem of making inferences from data measured on nonlinear systems is investigated within a Set Membership (SM) framework and it is shown that identification, prediction and filtering can be treated as specific instances of the general presented theory. The SM framework presents an alternative view to the Parametric Statistical (PS) framework, more widely used for studying the above specific problems. In particular, in the SM framework, a bound only on the gradient of the model regression function is assumed, at difference from PS methods which assume the choice of a parametric functional form of the regression function. Moreover, the SM theory assumes only that the noise is bounded, in contrast with PS approaches, which rely on noise assumptions such as stationarity, uncorrelation, type of distribution, etc. The basic notions and results of the general inference making theory are presented. Moreover, some of the main results that can be obtained for the specific inferences of identification, prediction and filtering are reviewed. Concluding comments on the presented results are also reported, focused on the discussion of two basic questions: what may be gained in identification, prediction and filtering of nonlinear systems by using the presented SM framework instead of the widely diffused PS framework? why SM methods could provide stronger results than the PS methods, requiring weaker assumptions on system and on noise?

Analysis of consensus protocols with bounded measurement errors

This paper analyzes two classes of consensus algorithms in the presence of bounded measurement errors. The considered protocols adopt an updating rule based either on constant or vanishing weights. Under the assumption of bounded error, the consensus problem is cast in a set-membership framework, and the agreement of the team is studied by analyzing the evolution of the feasible state set. Bounds on the asymptotic difference between the states of the agents are explicitly derived, in terms of the bounds on the measurement noise and the values of the weight matrix.

Direct design from data of optimal filters for LPV systems

In real-world applications, the traditional approach to filter design is based on a two-step procedure: (1) model identification from data; (2) filter design from the identified model. However, the two-step procedure is in general not optimal and evaluating the effects of the modeling error (occurring in the identification step) on the estimation error of the designed filter is a largely open problem. In this paper, a new approach to filter design for LPV systems is proposed, based on the direct design of the filter from data, avoiding the intermediate step of model identification. This approach, developed within a Set Membership framework, allows the design of optimal filters and the evaluation of non-conservative bounds on the estimation error. An applicative example is presented, related to the estimation of vehicle yaw rate, a variable used by safety control systems to improve the vehicle stability.

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.

Set-Membership Fuzzy Filtering for Nonlinear Discrete-Time Systems

This paper is concerned with the set-membership filtering (SMF) problem for discrete-time nonlinear systems. We employ the Takagi-Sugeno (T-S) fuzzy model to approximate the nonlinear systems over the true value of state and to overcome the difficulty with the linearization over a state estimate set rather than a state estimate point in the set-membership framework. Based on the T-S fuzzy model, we develop a new nonlinear SMF estimation method by using the fuzzy modeling approach and the S-procedure technique to determine a state estimation ellipsoid that is a set of states compatible with the measurements, the unknown-but-bounded process and measurement noises, and the modeling approximation errors. A recursive algorithm is derived for computing the ellipsoid that guarantees to contain the true state. A smallest possible estimate set is recursively computed by solving the semidefinite programming problem. An illustrative example shows the effectiveness of the proposed method for a class of discrete-time nonlinear systems via fuzzy switch.

Reliable Discrimination of Models Based on EIS Data: I. Calibration on Equivalent Electrical Circuits

We present a test to discriminate equivalent electrical models via electrochemical impedance spectroscopy (EIS) data collected on a frequency response analyzer (FRA). Derived in the set-membership framework, this test is an interesting alternative to classical estimators, as it does not rely on statistical thresholding and bypasses many numerical limitations (initialization, convergence) of nonlinear least-squares methods. Models are systematically discarded when their associated impedance is proved not to satisfy both the experimental data and the uncertainty range of the FRA. In this paper, we detail the methodology adapted to FRA measurements and apply the test on known electrical circuits to evaluate the required uncertainty ranges of different commercial impedance meters.

Optimal and suboptimal algorithms in set membership identification

We discuss in this paper optimality properties of identification algorithms in a set membership framework. We deal with restricted-complexity (conditional) identification, where approximations (models) to a possibly complex system are selected from a low dimensional space. We discuss the worst- and average-case settings. In the worst-case setting, we present results on optimality, or suboptimality, of algorithms based on computing the unconditional or conditional Chebyshev centres of an uncertainty set. In the average-case setting, we show that the optimal algorithm is given by the projection of the unconditional Chebyshev centre. We show explicit formulas for its average errors, allowing us to see the contribution of all problem parameters to the minimal error. We discuss the case of weighted average errors corresponding to non-uniform distributions over uncertainty sets, and show how the weights influence the minimal identification error.

Localization of Autonomous Underwater Vehicles by Floating Acoustic Buoys: A Set-Membership Approach

This paper addresses localization of autonomous underwater vehicles (AUVs) from acoustic time-of-flight measurements received by a field of surface floating buoys. It is assumed that measurements are corrupted by unknown-but-bounded errors, with known bounds. The localization problem is tackled in a set-membership framework and an algorithm is presented, which produces as output the set of admissible AUV positions in a three-dimensional (3-D) space. The algorithm is tailored for a shallow water situation (water depth less than 500 m), and accounts for realistic variations of the sound speed profile in sea water. The approach is validated by simulations in which uncertainty models have been obtained from field data at sea. Localization performance of the algorithm are shown comparable with those previously reported in the literature by other approaches who assume knowledge of the statistics of measurement uncertainties. Moreover, guaranteed uncertainty regions associated to nominal position estimates are provided. The proposed algorithms can be used as a viable alternative to more traditional approaches in realistic at-sea conditions.

Set Membership identification of nonlinear systems

In the paper the problem of identifying nonlinear dynamic systems, described in nonlinear regression form, is considered, using finite and noise-corrupted measurements. Most methods in the literature are based on the estimation of a model within a finitely parametrized model class describing the functional form of involved nonlinearities. A key problem in these methods is the proper choice of the model class, typically realized by a search, from the simplest to more complex ones (linear, bilinear, polynomial, neural networks, etc.). In this paper an alternative approach, based on a Set Membership framework is presented, not requiring assumptions on the functional form of the regression function describing the relations between measured input and output, but assuming only some information on its regularity, given by bounds on its gradient. In this way, the problem of considering approximate functional forms is circumvented. Moreover, noise is assumed to be bounded, in contrast with statistical methods, which rely on assumptions such as stationarity, ergodicity, uncorrelation, type of distribution, etc., whose validity may be difficult to test reliably and is lost in presence of approximate modeling. In this paper, necessary and sufficient conditions are given for the validation of the considered assumptions. An optimal interval estimate of the regression function is obtained, providing its uncertainty range for any assigned regressor values. The set estimate allows to derive an optimal identification algorithm, giving estimates with minimal guaranteed L p error on the assigned domain of the regressors. The properties of the optimal estimate are investigated and its worst-case L p identification error is evaluated. The presented approach is tested and compared with other nonlinear methods on the identification of a water heater, a mechanical system with input saturation and a vehicle with controlled suspensions.