Set-membership Filtering Research Articles

Set‐Membership Filtering Based Model Predictive Control for Trajectory Tracking of Automated Guided Vehicles

ABSTRACTThis article considers the trajectory tracking problem of automated guided vehicles with unknown‐but‐bounded noises and proposes a set‐membership filtering (SMF) based model predictive control strategy. A 2‐DOF vehicle model is first established with considering the unknown‐but‐bounded process and measurement noises. Then, a trajectory tracking control scheme is proposed which consists of two parts, that is, a set‐membership observer and a model predictive controller. The proposed set‐membership observer is able to deal with the unknown‐but‐bounded noises and obtain the state estimation ellipsoid for the considered automated guided vehicle. Further, the designed model predictive controller utilizes the estimated ellipsoid to compute the control sequence. Finally, simulation results demonstrate the effectiveness and superiority of the proposed method.

Underwater multizonotope terrain‐aided navigation method with coarse map based on set‐membership filter

AbstractTerrain‐aided navigation (TAN) is a viable method to achieve long‐term underwater navigation for long‐range autonomous underwater vehicles (AUVs). However, the high‐accuracy positioning results of most TAN systems rely on precise a priori seabed terrain maps, which restricts their applicability to a few areas with accurate bathymetric measurements of the seabed terrain. This article introduces a TAN system based on the General Bathymetric Chart of the Oceans (GEBCO) data set for global marine applications. Specifically, to address the low accuracy and poor robustness of the TAN system with imprecise bathymetric measurement and low‐resolution data from the GEBCO data set, this article proposes a multizonotope TAN method based on set‐membership filter (SMF) theory. The SMF theory is employed to handle the unknown distribution of the measurement noise from the GEBCO data set, introducing a multizonotope measurement update model to achieve more precise navigational results while addressing the perceptual ambiguity caused by self‐similar terrain. The smoothness of the terrain is incorporated as a parameter in the generation ranges of multizonotope, enabling adaptive adjustment based on terrain smoothness to reduce costs and enhance navigational performance. The accuracy and robustness of the proposed method are verified through all shipboard experiments, publicly available data sets, and AUV experiments. Compared with state‐of‐the‐art TAN methods, the average and maximum positioning errors have decreased by 64.83% and 48.84%, respectively. Finally, based on the experimental results, a preliminary distribution of suitable areas in the oceans is provided.

Ellipsoidal estimation of motion parameters of a non-cooperative space vehicle from visual information

The use of near-Earth space is currently complicated by presence of space debris objects in Earth’s orbit, which include spent stages of launch vehicles, inoperative satellites, and other large and small artificial objects. One approach to solving the problem of space debris involves docking and capturing a non-cooperative space object or spacecraft (target) with a so-called service spacecraft (chaser) for further actions to repair, refuel or change its orbit. Rendezvous and docking are complicated by the rotation of uncontrolled space objects caused by various factors. To perform this task, it is necessary to know the parameters of the orbital, rotational and relative motion of the target. The parameters of the orbital motion of such objects are usually known quite accurately from measurements from the Earth. This paper examines the case of a tumbling non-cooperative target located in an elliptical orbit. It is assumed that the target relative position and orientation are measured by the computer vision system (CVS) of the chaser. In this case, the position and orientation of the graphical reference frame (GRF) associated with the known 3-D graphical model of the target are determined relative to the reference frame associated with the chaser. The specific type of CVS is not considered. It is assumed that the chaser can carry out some maneuvers near the target and all parameters of the chaser angular motion are known. Thus, the attitude of the GRF relative to inertial reference frame (IRF) can be determined. The measured parameters are not enough to ensure safe rendezvous and docking with the target. To complete this task, it is necessary to determine all kinematic and dynamic parameters of the relative motion between the spacecraft. The rest of the required parameters are estimated. Orientation and rotation parameterization is done using quaternions. The angular motion equation of the target is considered in the GRF. This makes the angular velocity estimation faster and the inertia tensor estimation more stable. Stochastic characteristics of measurement errors are considered to be unknown and are not used. The only information about the errors is the bounds of their values. To determine the relative motion parameters, we use a new dynamic set-membership filter with ellipsoidal estimates. The filter can be successfully implemented on low-power on-board processors. The properties of the proposed algorithm are demonstrated using numerical simulation. The results obtained are expected to be used in the development of a navigation system for the rendezvous and docking, developed by a group of Ukrainian space industry enterprises under the leadership of the LLC “Kurs-orbital” (https://kursorbital.com/).

Neural-Network-Based Set-Membership Filtering Under WTOD Protocols via a Novel Event-Triggered Compensation Mechanism

Neural-Network-Based Set-Membership Filtering Under WTOD Protocols via a Novel Event-Triggered Compensation Mechanism

Online Monitoring for Human Sit-to-Stand Movement Based on Karush-Kuhn-Tucker Optimized Zonotope Set-Membership Filter.

As the global aging population continues to grow, there has been a significant increase in the number of fall-related injuries among the elderly, primarily due to reduced muscle strength and balance control, especially during sit-to-stand (STS) movements. Intelligent wearable robots have the potential to provide fall prevention assistance to individuals at risk, but an accurate and timely assessment of human movement stability is essential. This article presents a fall prediction algorithm for STS movements based on the Karush-Kuhn-Tucker (KKT) optimized zonotope set-membership filter (KKT-ZSMF), enabling real-time assessment of human stability. To quantify the feasible stability region of human STS movement, a mathematical model is proposed based on dynamic stability theory. Additionally, an online fall-prediction approach is developed, utilizing the zonotope set-membership filter to iteratively update the set that represents the instantaneous stability region. The approach incorporates a KKT optimization algorithm to compute the optimal convex hull, thereby enhancing the accuracy and efficiency of the set-membership filter. Experimental validation is conducted with the participation of 13 subjects including 5 elderly subjects, comparing the performance of the proposed KKT-ZSMF algorithm with other relevant methods. The results confirm the accuracy and real-time performance of the KKT-ZSMF algorithm for predicting human STS movement stability, achieving an overall prediction accuracy of 93.49% and a runtime of no more than 7.91 ms. These findings demonstrate the suitability of the algorithm for fall prevention assistance in daily activities.

Ellipsoidal estimation of parameters of rotational and translational motion of a non-cooperative space vehicle from visual information


 The use of near-Earth space is currently complicated by the presence of space debris objects in Earthʼs orbit, which include spent stages of launch vehicles, inoperative spacecraft, and other large and small objects associated with human activity in space. One of the elements of solving the problem of space debris is the docking and capture of an uncontrolled non-cooperative space object or spacecraft by a so-called on-orbit servicing spacecraft to carry out further actions to repair it, refuel or change its orbit. The situation is complicated by the fact that, under the influence of various factors, uncontrolled space objects are in a state of rotation. The parameters of the orbital motion of such objects are known quite accurately from measurements from the Earth. To carry out safe approach and docking, knowledge of the parameters of rotational motion, as well as the parameters of relative motion, is also required. The most general case of motion of a non-cooperative tumbling spacecraft located in an elliptical orbit is considered. It is assumed that the three-dimensional graphic model of such spacecraft is known. The servicing spacecraft (SSC) is equipped with a mono camera that takes pictures of the non-cooperative spacecraft (NSC). Based on a comparison of the characteristic features of photographs and images obtained using the graphical model, the computer vision system (CVS) determines the distance vector to the so-called graphical coordinate system, rigidly fixed on the NSC and the quaternion of its relative attitude. The specific type of CVS is not considered. It is assumed that the SSC carries out some maneuvers near the satellite. All parameters of the SSC angular motion are assumed to be known. This work considers the most general case of the relative motion of SSC and NSC. Using quaternion calculus, all basic kinematic and dynamic equations are obtained. The measured parameters are not enough to ensure safe rendezvous and docking with the NSC. The stochastic characteristics of errors of the CVS measurement are not assumed to be known and, accordingly, are not used. Only their maximum values are specified for them. We consider the use of new dynamic set-membership filter using ellipsoids to solve the problem of determining the parameters of the relative motion of the NSC which is in free uncontrolled motion. The filter can be implemented under conditions of the limited computational capability available on onboard processors. The relative motion parameters include the distance vector between the centers of mass (c.m.) of the NSC and the SSC, the relative velocity vector, the quaternion of the orientation of the main axes of inertia of the satellite relative to the inertial coordinate system, the ratio of the moments of inertia of the satellite, the vector of the position of the NSC c.m. in the graphical coordinate system. The properties of the proposed algorithm are demonstrated using numerical simulation. The results obtained are expected to be used in the development, creation and testing of a navigation system for the rendezvous and docking of the SSC, developed by a group of Ukrainian space industry enterprises under the leadership of the LLC «Kurs-orbital» (https://kursorbital.com/).

Set-based state estimation for discrete-time constrained nonlinear systems: An approach based on constrained zonotopes and DC programming

This paper introduces a novel state estimator designed for discrete-time nonlinear dynamical systems that encompass unknown-but-bounded uncertainties, along with state linear inequality and nonlinear equality constraints. Our algorithm is based on constrained zonotopes (CZs) and employs a DC programming approach (where DC stands for difference of convex functions). Recently, techniques such as mean value extension and first-order Taylor extension have been adapted from zonotopes to facilitate the propagation of CZs across nonlinear mappings. While the resulting algorithms, known as CZMV and CZFO, achieve higher precision compared to the original zonotopic versions, they still exhibit sensitivity to the wrapping and dependency effects inherent in interval arithmetic. These interval-related challenges can be mitigated through the use of DC programming, as it allows for the determination of approximation error bounds by solving optimization problems. One direct advantage of this technique is the elimination of the dependency effect. Our set-membership filter, referred to as CZDC, provides an alternative solution to CZMV and CZFO. To showcase the effectiveness of our proposed approach, we conducted experiments using CZDC on two numerical examples.

Confidence set-membership FIR filter for discrete time-variant systems

In this paper, a confidence set-membership filter based on the finite impulse response (FIR) structure is proposed for time-variant systems that are simultaneously subject to both the unknown but bounded noises and the Gaussian noises. At each step, the filter provides a confidence state set that includes the true state with an adjustable confidence level. The FIR structure improves the robustness and brings more degrees of freedom to the filter design. Multiple recent measurements within a moving horizon are fully utilized to construct and optimize each confidence state set. Under two different size criteria, an analytical solution and an LMI-based numerical solution for the filter gain are derived by minimizing the size of the confidence state set, respectively. Finally, an example is given to illustrate the effectiveness and superiority of the algorithms.

Outlier-Robust Gaze Signal Filtering Framework Based on Eye-Movement Modality Recognition and Set-Membership Approach.

High-quality gaze signals are crucial in many biomedical fields that utilize them. However, the limited studies on gaze signal filtering can hardly address the outliers and non-Gaussian noise in gaze data simultaneously. Our objective is to design a generic filtering framework capable of reducing the noise and eliminating outliers of the gaze signal. In this study, we design an eye-movement modality-based zonotope set-membership filtering framework (EM-ZSMF) to suppress the noise and outliers of the gaze signal. This framework consists of an eye-movement modality recognition model (EG-NET), an eye-movement modality-based gaze movement model (EMGM), and a zonotope set-membership filter (ZSMF). The eye-movement modality determines the EMGM, and the ZSMF combined with the EMGM completes the filtering of the gaze signal. Moreover, this study establishes an eye-movement modality and gaze filtering dataset (ERGF) that can be utilized for the evaluation of future work integrating eye-movement modality with gaze signal filtering. The eye-movement modality recognition experiments demonstrated that our proposed EG-NET achieved the best Cohen's kappa compared with previous studies. The gaze data filtering experiments showed that the proposed EM-ZSMF reduced the gaze signal noise and eliminated outliers effectively, and achieved the best performance (RMSEs and RMS) compared with previous methods. The proposed EM-ZSMF effectively recognizes eye-movement modalities, reduces gaze signal noise and, eliminates outliers. To the best of the authors' knowledge, this is the first attempt to simultaneously solve the problem of non-Gaussian noise and outliers in gaze signals. The proposed framework has the potential for application in any eye image-based eye trackers and contributes to the development of eye-tracking technology.

Set-membership filtering for two-dimensional shift-varying systems with stochastic communication protocol and uniform quantization

This paper is concerned with the set-membership filtering problem for the two-dimensional shift-varying systems subject to the stochastic communication protocol and uniform quantization effects. To prevent the data transmission from collisions, only one sensor can transmit the measured information to the filter at each sampling shift instant, and the selected sensor is determined by the scheduling strategy of the stochastic communication protocol. On the contrary, the uniform quantization mechanism has been employed to mitigate the influence of quantization error on filtering performance. Incorporating the stochastic communication protocol and uniform quantization mechanism, this paper proposes a set-membership filter design framework for the two-dimensional shift-varying systems with unknown-but-bounded noises. Sufficient conditions are derived for the existence of desired set-membership filter by utilizing double mathematical induction, such that the estimation error resides within the ellipsoidal set. Moreover, the optimal filtering algorithm is given by minimizing the ellipsoidal constraints. Finally, several examples are provided to illustrate the effectiveness of the proposed filter design algorithm.

Set membership filtering for discrete saturated time‐varying networked systems with incomplete measurements under round‐robin protocol

AbstractIn this paper, the set membership filtering problem is investigated for a class of discrete networked systems with time delay and state saturation under the round‐robin (RR) protocol, where the system noise is unknown but bounded. In order to reduce the communication burden, the RR protocol is used to schedule the measurement information of the system. In addition, the incompleteness of the measurement information is considered. A set membership filter is established based on incomplete measurements. The objective is to give a criterion that the filtering error is confined to the ellipsoidal set by using the recursive linear matrix inequality (RLMI) method. Moreover, a convex optimization method is proposed to minimize the obtained ellipsoids. Finally, a numerical simulation illustrates that the designed set membership filtering scheme is effective.

Finite-horizon distributed set-membership filtering with dynamical bias and DoS attacks under binary encoding schemes

In this work, we cope with the design issue of finite-horizon distributed set-membership filters concerning a set of time-varying systems (TVSs) over sensor networks (SNs). The system to be discussed is suffering from dynamical bias and unknown but bounded noise. The binary encoding schemes (BESs) are utilized in the data transmission through SNs from neighbors to the intelligent sensor nodes, which exhibit robustness to some degree due to usage of the binary data. Both random bit errors caused by channel noises and denial of service (DoS) attacks reflecting network vulnerability are considered that may arise and lead to signal distortion in the transmission process of the binary bit string. In the framework of adopting BESs, our aim is to realize the distributed set-membership filtering for TVSs over SNs limited by dynamical bias and DoS attacks. Sufficient conditions, relying on the solvability of a class of recursive linear matrix inequalities, are obtained for the presence of the expected time-varying distributed set-membership filters. Furthermore, an additional optimization issue is constructed to find the minimal ellipsoids (on the basis of matrix trace) to obtain the best filtering performance. Eventually, a numerical simulation instance is provided to verify the availability of the presented design strategy of finite-horizon distributed set-membership filters.

A component-based coding–decoding approach to set-membership filtering for time-varying systems under constrained bit rate

This paper is concerned with the set-membership filtering (SMF) problem for a class of discrete time-varying systems with unknown-but-bounded noises under bit rate constraints. The communication between sensor nodes and filters is implemented through a wireless digital communication network with limited bandwidth. A bit rate constraint is first established to quantify the extent to which the network is constrained. A component-based coding–decoding procedure is proposed that enables individual decoder to decode messages from different components scattering in different physical locations. Based on this procedure, a decoded-measurement-based recursive SMF scheme with a prediction–correction structure is put forward. The desired parameters of the set-membership filter can be calculated recursively by the proposed recursive SMF scheme. Furthermore, the co-design issue of the bit rate allocation protocol and the filter gain is converted into the mixed-integer nonlinear programming problem that is solved by means of the particle swarm optimization and the recursive filtering algorithms. Finally, numerical simulations on two scenarios are conducted to validate the effectiveness of the proposed SMF approach.

A quantized set‐membership estimation‐based heading control method of unmanned surface vessels under unknown‐but‐bounded wave‐induced disturbances

AbstractThis paper is concerned with the heading tracking control problem for a network‐based unmanned surface vessel (USV) under wave‐induced disturbances. Considering the characteristics of the wave‐induced disturbances, the unknown‐but‐bounded (UBB) noise model is adopted. Meanwhile, signal quantization is taken into account due to the limited capacity of the communication channel. Based on these considerations, a novel set‐membership based control scheme is proposed, which consists of a set‐membership filter (SMF) and a set‐membership controller (SMC). In the proposed control scheme, the SMF is designed for providing an ellipsoidal estimation set for the USV at each sampling step, and then the SMC uses the estimation set for producing control input for the USV in order to track the reference. Sufficient conditions are derived for the considered USV system to be asymptotically stable and satisfy the modified set‐membership performance constraint. Simulation results verifies the effectiveness of the proposed control scheme.

Nonlinear set membership filter with state estimation constraints via consensus-ADMM

This paper considers the state estimation problem for nonlinear dynamic systems with unknown but bounded noises. Set membership filter (SMF) is a popular algorithm to solve this problem. In the set membership setting, we investigate the filter problem where the state estimation requires to be constrained by a linear or nonlinear equality. We propose a consensus alternating direction method of multipliers (ADMM) based SMF algorithm for nonlinear dynamic systems. To deal with the difficulty of nonlinearity, instead of linearizing the nonlinear system, a semi-infinite programming (SIP) approach is used to transform the nonlinear system into a linear one, which allows us to obtain a more accurate estimation ellipsoid. For the solution of the SIP, an ADMM algorithm is proposed to handle the state estimation constraints, and each iteration of the algorithm can be solved efficiently. Finally, the proposed filter is applied to typical numerical examples to demonstrate its effectiveness.

Distributed Set-Membership Fusion Filtering for Nonlinear 2-D Systems Over Sensor Networks: An Encoding-Decoding Scheme.

In this article, the distributed set-membership fusion filtering problem is investigated for a class of nonlinear 2-D shift-varying systems subject to unknown-but-bounded noises over sensor networks. The sensors are communicated with their neighbors according to a given topology through wireless networks of limited bandwidth. With the purpose of relieving the communication burden as well as enhancing the transmission security, a logarithmic-type encoding-decoding mechanism is introduced for each sensor node so as to encode the transmitted data with a finite number of bits. A distributed set-membership filter is designed to determine the local ellipsoidal set that contains the system state by only utilizing the data from the local sensor node and its neighbors, where the proposed filter scheme is truly distributed with desirable scalability. Then, a new ellipsoid-based fusion rule is developed for the designed set-membership filters in order to form the fused ellipsoidal set that has a globally smaller volume than all local ellipsoidal sets. With the aid of the mathematical induction technique, the set theory, and the convex optimization approach, sufficient conditions are derived for the existence of the desired distributed set-membership filters and the fusion weights. Then, the filter parameters and the fusion weights are acquired by solving a set of constrained optimization problems. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed fusion filtering algorithm.

A non-uniform event-triggered distributed filtering scheme for discrete-time nonlinear systems over wireless sensor networks

This paper investigates the distributed set-membership filtering problem for discrete-time nonlinear systems subject to one-sided Lipschitz (OSL) and quadratic inner-boundedness (QIB) constraints over wireless sensor networks (WSNs). The investigation takes into account both system and sensor nonlinearities under unknown but bounded (UBB) perturbations. Moreover, a novel non-uniform event-triggering condition is introduced to achieve resource conservation while maintaining filter accuracy. The proposed triggering condition is more general and can be customized as per design requirements. Furthermore, a recursive condition for set-membership filter design is derived that guarantees the existence of optimal bounding ellipsoids for true system states. In contrast to the existing methods, the proposed approach considers set-membership filtering for discrete-time systems subject to OSL system and sensor nonlinearities over a WSN and deals with non-uniform amplitude-based triggering. Finally, a numerical example is presented to validate filter’s efficacy.

Set-membership filtering for two-dimensional systems with dynamic event-triggered mechanism

In this paper, the set-membership filtering problem is investigated for a class of two-dimensional (2-D) shift-varying systems described by the general Fornasini-Marchesini second model with a dynamic event-triggered mechanism. The sensors are communicated with the remote filter through a communication network of limited bandwidth. In order to improve the efficiency of utilizing the network resources, a dynamic event-triggered mechanism is exploited for the 2-D shift-varying system, and a set-membership filter is then designed for 2-D systems subject to unknown-but-bounded noises so that the actual system state is guaranteed to reside within an optimized ellipsoidal set. By means of the mathematical induction technique and the set theory, sufficient conditions are derived to ensure the existence of the desired set-membership filter. Furthermore, the filter gains are obtained by solving a set of optimization problems subject to ellipsoidal constraints. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed filter design method.

Fuzzy Set-Membership Filtering for Discrete-Time Nonlinear Systems

In this article, the problem of state estimation is addressed for discrete-time nonlinear systems subject to additive unknown-but-bounded noises by using fuzzy set-membership filtering. First, an improved T-S fuzzy model is introduced to achieve highly accurate approximation via an affine model under each fuzzy rule. Then, compared to traditional prediction-based ones, two types of fuzzy set-membership filters are proposed to effectively improve filtering performance, where the structure of both filters consists of two parts: prediction and filtering. Under the locally Lipschitz continuous condition of membership functions, unknown membership values in the estimation error system can be treated as multiplicative noises with respect to the estimation error. Real-time recursive algorithms are given to find the minimal ellipsoid containing the true state. Finally, the proposed optimization approaches are validated via numerical simulations of a one-dimensional and a three-dimensional discrete-time nonlinear systems.

Dual Set-Membership State Estimation for Power Distribution Networks Under Event-Triggered Mechanism

This article is concerned with the set-membership state estimation problem for power distribution networks (PDNs) over a resource-constrained communication network under the influence of unknown but bounded (UBB) noises. Firstly, in order to alleviate the pressure of information communication network (ICN) while meeting the state estimation requirements, the event-triggered mechanism is adopted to send data containing more valid information to estimation center, reasonably reducing the signal transmission frequency. Secondly, an event-triggered dual set-membership filter (ET-DSMF) is designed to improve the performance of state estimation. The proposed filter performs a discrete approximation for a semi-infinite programming problem by the random sampling technique, and a compact linearization error bounding ellipsoid is obtained by solving the dual problem of the nonlinear programming. Subsequently, a sufficient condition for the existence of the estimated ellipsoid is derived depending on the mathematical induction method. The key time-varying filter gain matrix and optimal estimated ellipsoid are determined recursively by solving a convex optimization problem, according to the minimum trace criterion. Finally, the effectiveness of the proposed filtering algorithm is demonstrated by performing simulation experiments on the IEEE 13 distribution test system.