Ellipsoidal Set Research Articles

An adjustable Predictive&Prescriptive method for the RO-based optimal power flow problem

Traditionally before solving the optimal power flow considering uncertainty (OPF–U) problem, the predicted value of uncertainty parameters, such as wind power, e.g., is derived from data using a statistics approach or machine learning. Based on the predicted uncertainty parameters, the solution to the OPF-U problem can be obtained by the prescriptive analytics technique, such as robust optimization (RO). However, it is unclarified how the prediction error in predictive analytics affects solving the OPF-U problem in prescriptive analytics. We propose an adjustable framework method combining machine learning and RO for the OPF-U problem. The k-nearest neighbor is applied to obtain k samples around the predicted value from sufficient historical data. And the optimization results from a minimum volume ellipsoid set containing the k samples are applied to construct KMV set. Then a robust fluctuation region with an adjustable budget level is gained from the KMV set by a two-term exponential formula, which can be embedded into a two-stage RO model. Computational experiments under test cases of different uncertainty scales show the robustness and adjustability of the proposed fluctuation region are better than the state-of-the-art box and ellipsoidal sets. The solution of the proposed two-stage RO model is more economical than the state-of-the-art RO model. The out-of-sample simulation also demonstrates the proposed adjustable Predictive&Prescriptive method can reduce the computational burden as the scale of the system increases when predictive and prescriptive analytics are separated.

Ellipsoidal Set-Based Extended Command Governor for Constrained Linear Systems With Bounded Disturbances

The purposes of this paper are twofold. The first is to derive a convex formulation for optimizing the fictitious dynamics of the extended command governor strategy recently proposed in [1]. The second purpose is to provide an efficient procedure to solve online the optimization problem of the ellipsoidal set based command governor or the extended command governor. Two numerical examples with comparison to earlier solutions from the literature illustrate the effectiveness of the proposed algorithm.

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.

The detection mechanism for false data injection attack via the ellipsoidal set‐membership approach

AbstractThe false data injection (FDI) attack detection problem in cyber‐physical systems (CPSs) is investigated in this paper. A novel attack detection algorithm is proposed based on the ellipsoidal set‐membership approach. In comparison to the existing FDI attack detection methods, the developed attack detection approach in this paper neither requires predefined thresholds nor specific statistical characteristics of the attacks. In order to guarantee that the estimation ellipsoid contains normal states despite the unknown but bounded (UBB) process and measurement noises, the one‐step ellipsoidal set‐membership estimation method is put forward. In addition, a convex optimization algorithm is introduced to calculate the gain matrix of the observer recursively. Moreover, with the help of the state estimation ellipsoid, the residual ellipsoid can be obtained for attack detection. Whether a detector can detect the FDI attack depends on the relationship between the residual value and residual ellipsoidal set. Finally, the effectiveness of the proposed method is demonstrated by a numerical simulation example.

A Novel Set-Valued Sensor Fault Diagnosis Method for Lithium-Ion Battery Packs in Electric Vehicles

Sensor fault diagnosis is of great significance to ensure safe battery operation. This paper proposes a novel sensor fault diagnosis method that achieves the simultaneous fault detection, fault source and type identification, and fault estimation in a comprehensive way. Specifically, a set-valued observer, featuring a state predictor and a state estimator, is first constructed and designed to guarantee the inclusion of the unavailable actual battery state due to unknown modeling errors and noises at every instant of time. Compared with the traditional observers, a distinct feature of the proposed one lies in that the calculated state predictions and estimations of the battery system at each time step are ellipsoidal sets in state space rather than single vectors. The boundedness of state prediction and estimation errors is formally proved, and the tractable design criteria for determining the real-time optimal prediction and estimation ellipsoids are also derived. As for diagnosis algorithm, fault detection is implemented based on the intersection between the prediction and estimation ellipsoids. Then, a two-layer Pearson correlation coefficient analysis mechanism is developed to identify the source and type of sensor faults. Another set-valued observer based on an augmented battery model is further designed to estimate the fault level. Finally, experimental studies of a battery cell under different sensor fault sources, types and values are elaborated to verify the effectiveness of the proposed method.

The conjugate locus in convex 3-manifolds

In this paper we study the conjugate locus in convex manifolds. Our main tool is Jacobi fields, which we use to define a special coordinate system on the unit sphere of the tangent space; this provides a natural coordinate system to study and classify the singularities of the conjugate locus. We pay particular attention to 3-dimensional manifolds, and describe a novel method for determining conjugate points. We then make a study of a special case: the 3-dimensional (quadraxial) ellipsoid. We emphasise the similarities with the focal sets of 2-dimensional ellipsoids.

Dual-Mode Robust Fuzzy Model Predictive Control of Time-Varying Delayed Uncertain Nonlinear Systems With Perturbations

For time-varying delayed nonlinear systems with parameter uncertainties and persistent disturbances, an online and an offline robust fuzzy model predictive control (RFMPC) algorithms are proposed in this paper. Both methods guarantee the input-to-state stability (ISS) of the system. Furthermore, a novel alternative optimization (AOP) approach and a dual-mode OP/AOP strategy are proposed to prevent the performance deterioration of the online optimal control (OP) approach due to the challenges in addressing the bilinear matrix inequality (BMI) constraints. With the established AOP and dual-mode OP/AOP techniques, the optimization problem constrained by BMIs is transformed into convex, and a significantly more precise approximation of the ellipsoidal minimal robust positively invariant (EmRPI) set can be calculated. Besides, the system can eventually converge into a more compact ellipsoidal set. These two alternative optimization methods can be easily extended to various nonlinear systems. A numerical example and a continuous-time stirring tank (CSTR) example are provided to validate the effectiveness and advantages of the established methodologies.

The use of weather nowcasting convolutional neural network extrapolators in cardiac PET imaging.

Algorithms to predict short-term changes in local weather modalities have been used in meteorology for many years. These algorithms predict the temporospatial change in the movement of weather patterns such as cloud cover or precipitation. This paper extends convolutional neural network models for weather prediction/nowcasting to predict evolution in the extrapolation of sequentially acquired count data seen with cardiac positron tomography (PET) data to expected value temporally rather than spatially. Six different algorithms used for nowcasting were modified and applied to confirm the approach. These algorithms were trained on an image data set of both simulated ellipsoids and simulated cardiac PET data. The peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) were calculated for each of these trained models. They were compared to the BM3D denoising algorithm as a baseline comparison to a standard method of image denoising. Most of the implemented algorithms showed a significant improvement in both PSNR and SSIM when compared with the baseline standard, especially when the algorithms were implemented in combination. The best results were obtained with a combination of the ConvLSTM and TrajGRU algorithms with a PSNR improvement over the standard of 5 and more than double the SSIM metric. This approach of using serially acquired count data to extrapolate a future expected representation through convolutional neural networks has been shown to produce accurate representations of the expected value when compared with a baseline analytic methodology. This paper confirms that algorithms such as these can be used to substantially improve image estimation and shows significant improvement over a baseline standard.

Ellipsoidal tube‐based output feedback robust MPC for linear systems with bounded disturbances and noises

AbstractThis paper designs an ellipsoidal tube‐based output feedback robust model predictive control approach for linear systems with bounded disturbances and noises subject to physical constraints. The ellipsoidal tube‐based output feedback robust model predictive control approach combines the off‐line optimization to compute controller parameters and deal with system uncertainties, and the on‐line optimization to stabilize the nominal system with time‐varying tightened constraints. In the off‐line optimization, the state observer gain, ancillary feedback controller gain, and tightened constraint sets on the nominal system are synthesized in one optimization. Then, an ellipsoidal terminal constraint set with terminal controller gain and terminal cost matrix, and tightened constraint sets related to different sizes of estimation error bounds are computed. In the on‐line model predictive control optimization with time‐varying tightened constraints, a sequence of nominal control inputs is receding horizon optimized to steer the nominal system state to the ellipsoidal terminal constraint set. When the current nominal system state is bounded within the ellipsoidal terminal constraint set, the nominal control inputs are switched to the feedbacks on the closed‐loop nominal system states. Recursive feasibility of the proposed algorithm and robust stability of the controlled system are guaranteed.

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.

Quadratic covariance‐constrained filtering for linear and non‐linear systems with non‐Gaussian noises

AbstractThis study considers a robust quadratic covariance‐constrained filtering problem for discrete time‐varying linear and non‐linear dynamic systems with non‐Gaussian noises. Non‐Gaussian noises are presumed to be unknown, bounded, and limited in a specified ellipsoidal set. In this approach, first, a general standard linear form of the filter is introduced for state estimation in linear dynamic systems. The filter gain is obtained by minimizing the upper bound of the estimation error's covariance matrix. The Lyapunov theory demonstrates the stability of this filter. Second, we extend the proposed filtering approach to non‐linear dynamic systems that are considered as a combination of linear and non‐linear terms. The Lipschitz‐like condition is assumed for the non‐linear part. A new filter structure is proposed in this case and the filter gain is obtained by the same idea to minimize the upper bound of the error's covariance matrix. Finally, four numerical examples are presented to signify the effectiveness and performance of the proposed filters for linear and non‐linear systems.

A Recursive Robust Set-Membership Estimator for WSN-Assisted Moving Targets Tracking With UBB Anchor Location Uncertainty

In this paper, a recursive robust set-membership fusion estimator (RSMFE) is presented to track the moving targets treated as the mobile nodes in a mobile wireless sensor network (MWSN) system with unknown but bounded (UBB) anchor location uncertainty. Firstly, the accurate ellipsoidal set containing the Lagrange remainder of nonlinear distance measurement, which contains the effect of anchor location uncertainty, is estimated and adopted to improve the stability and accuracy of the moving target tracking. Secondly, the recursive RSMFE-based tracking formulae of reduced computation complexity, consisting of the prediction formulae and the fusion update formulae, are derived to estimate the robust minimized or optimal ellipsoidal set containing the actual moving target state by appropriately fusing the measured distances from the nearby anchors subject to the UBB anchor location uncertainty, where the computation complexity of the fusion update formulae is reduced substantially based on the decoupled techniques and the equivalent transformation. Finally, the numerical simulation and experiment of the WSN-assisted moving target tracking system with various anchor location uncertainty settings are performed to verify the effectiveness and superiority of the proposed recursive RSMFE-based moving target tracking or localization method.

Ellipsoidal approximations of the minimal robust positively invariant set

An optimization algorithm is presented in this paper for the minimal robust positively invariant (mRPI) set approximations via sums-of-squares (SOS) optimization. The mRPI set is an effective tool for robust analysis of uncertain systems under bounded disturbances. The approximation of the mRPI set is always characterized by a polyhedron computed after finite time iterations. In this paper, an mRPI set is characterized by an ellipsoidal set while bounded parametric uncertainties act on states. The proposed algorithm optimizes the shape matrix of the ellipsoidal set approximation by minimizing the volume of the ellipsoidal set. The algorithm is designed for discrete-time and continuous-time nonlinear systems respectively. The algorithm has the ability to further minimize the mRPI set by optimizing the state-feedback control law. Examples are employed to validate the effectiveness of the proposed algorithms.

A 3D Mathematical Breast Texture Model With Parameters Automatically Inferred From Clinical Breast CT Images.

A numerical realistic 3D anthropomorphic breast model is useful for evaluating breast imaging applications. A method is proposed to model small and medium-scale fibroglandular and intra-glandular adipose tissues observed in the center part of clinical breast CT images. The method builds upon a previously proposed model formulated as stochastic geometric processes with mathematically tractable parameters. In this work, the medium-scale parameters were automatically and objectively inferred from breast CT images. We hypothesized that a set of random ellipsoids exhibiting cluster interaction is representative to model the medium-scale intra-glandular adipose compartments. The ellipsoids were reconstructed using a multiple birth, death and shift algorithm. Then, a Matérn cluster process was used to fit the reconstructed ellipsoid centers. Finally, distributions of the ellipsoid shapes and orientations were estimated using maximum likelihood estimators. Feasibility was demonstrated on 16 volumes of interests (VOI). To assess the realism of the 3D breast texture model, β and LFE metrics computed in simulated projection images of simulated texture realizations and clinical images were compared. Visual realism was illustrated. For 12 out of 16 VOIs, our hypothesis on clustering interaction process is confirmed. The average β values from simulated texture images (3.7 to 4.2) of the 12 different VOIs are higher than the average β value from 2D clinical images (2.87). LFE of simulated texture images and clinical mammograms are similar. Compared to our previous model, whereby simulation parameters were based upon empirical observations, our inference method substantially augments the ability to generate textures with higher visual realism and larger morphological variety.

Systematic Literature Review on Robust Optimization in Solving Sustainable Development Goals (SDGs) Problems during the COVID-19 Pandemic

Handling uncertainty is important in decision making, especially for SDGs problems. Robust Optimization (RO) is an applied optimization method that can be employed to handle optimization under uncertain data. With SDGs problems, many uncertain data have been considered in decision making. With RO, the data uncertainties are assumed to lay within a compact, convex continuous set. There are three special sets that can be used to represent the data, i.e., box, ellipsoidal, or polyhedral uncertainty sets. These special sets lead the SDGs problems to a computationally tractable optimization model, such that the global optimal solution is attained. However, literature reviews on the application of RO in SDGs decision-making is sparse, especially during the COVID-19 pandemic period. This paper examines the following topics: (1) the purposes of studies of RO and SDGs during the COVID-19 pandemic, (2) the state-of-the-art in RO-SDGs to determine the research objectives, and (3) the SDGs type of problems that have been modeled using RO. A systematic literature review is conducted in this paper, wherein discussion is based on a PRISMA (Preferred Reporting Items for Systematic Review and Meta-Analyses) flowchart. To this end, the database reference searching conducted on the Scopus, Science Direct, and SAGE databases, is completed using the help RStudio software. The analysis was carried out on two datasets, assisted by the output visualization using RStudio software with the “bibliometrix” package, and using the ‘biblioshiny()’ command to create a link to the “shiny web interface”. In this paper, the research gap on application of RO to SDGs problems is analyzed in order to identify the research objectives, methods, and specific RO-SDGs problems. As a result, the application of RO to SDGs problems is rare; this finding provides a motivation to conduct a further study of RO and SDGs during the COVID-19 pandemic. An expansion is presented using the key phrase “Operations Research and Optimization Modeling”, or “OROM”. SDGs in Indonesia may be referenced as an example of the capacity building available through RO/OROM.

Program Arrives Home Smoothly: Uncertainty-Based Routing Scheduling of Home-Based Elderly Care Programs

In China, the home-based elderly care program system plays an important role in meeting the needs of elderly individuals. The routing scheduling of home-based elderly care (RSHEC) programs is closely related to the quality of the home-based elderly care programs. The structural supply problem faced by home-based elderly care programs is a prominent problem, and the RSHEC programs are an important aspect that has rarely been studied. This paper explores RSHEC programs under uncertainty by comprehensively considering the costs of home-based elderly care, such as the fixed costs, time, and transportation costs. First, a deterministic mixed integer programming (MIP) model was constructed to solve the general routing scheduling problem. In addition, to effectively cope with the uncertainty and risk of the modern market, the robust optimization theory and algorithm model are introduced, namely, the mixed integer box set robust optimization (MIBRO) model and the mixed integer ellipsoid set robust optimization (MIERO) model. Finally, MATLAB and the Gurobi package are applied to obtain the solutions of the models. The case verification shows that the MIP model has the lowest total cost under deterministic conditions. However, the MIERO and MIBRO models can achieve more robust RSHEC programs under uncertain conditions. The results prove the effectiveness and feasibility of the optimization model and algorithm, which provides reference value for management decisions regarding home-based elderly care programs.

Robust particle filter for state estimation in presence of bounded but uncertain parameters based on ellipsoidal set membership approach

Accurate estimation of state variables plays an important role in control, monitoring and optimization. Considering that the uncertainty of parameter may back-propagate to the particle and then influence the state estimation, this paper proposes a robust particle filtering method in the presence of bounded uncertain parameters based on the ellipsoidal set membership filtering (ESMF) approach. This proposed method employs ellipsoidal sets to enclose the parameter uncertainty, and each prior particle and posterior particle are determined according to the ellipsoidal calculation based on the ESMF algorithm: The prior particle is characterized by an ellipsoid derived by the ellipsoidal summation of linear transformation ellipsoid, linearization error ellipsoid and system noise ellipsoid; Each posterior particle is obtained by updating the prior particle through the ellipsoidal intersection of the prior particle ellipsoid and the measurement ellipsoid. As a result, the uncertainty of parameter may be incorporated into the state estimation, and the advantages of both ESMF and particle filter are taken by the proposed method. The efficacy of the proposed method is shown by three simulation examples.

Capability Exploration of Extended-State Observer-Based Control Under the Uncertain Case of Disturbance and Actuator Saturation

Considering a class of nonlinear systems subject to actuator saturation and uncertain control input, this article explored the control capability of an extended-state observer-based active disturbance rejection controller in the presence of disturbances and uncertainties. First, the actuator saturation is handled with the convex hull in the extended-state observer-based control scheme, and an ellipsoid invariant set inside the domain of attraction for control systems is obtained using the Lyapunov method. Second, the enlargement problem of the invariant set is also studied, which illustrates that the convex hull of the ellipsoid invariant sets is also an enlargement invariant set. Finally, comprehensive comparisons with existing methods on a brushless dc motor demonstrate the stronger capability of the designed controller on disturbance rejection and actuator saturation. The results show that the disturbance offset and performance index (integrated time and absolute error and integral absolute error) are about half of the classical proportional–integral–derivative(PID) and double-integral sliding-mode control, or the disturbance rejection capability is about twice in this case.

Neural-Network-Based Set-Membership Fault Estimation for 2-D Systems Under Encoding-Decoding Mechanism.

In this article, the simultaneous state and fault estimation problem is investigated for a class of nonlinear 2-D shift-varying systems, where the sensors and the estimator are connected via a communication network of limited bandwidth. With the purpose of relieving the communication burden and enhancing the transmission security, a new encoding-decoding mechanism is put forward so as to encode the transmitted data with a finite number of bits. The aim of the addressed problem is to develop a neural-network (NN)-based set-membership estimator for jointly estimating the system states and the faults, where the estimation errors are guaranteed to reside within an optimized ellipsoidal set. With the aid of the mathematical induction technique and certain convex optimization approaches, sufficient conditions are derived for the existence of the desired set-membership estimator, and the estimator gains and the NN tuning scalars are then presented in terms of the solutions to a set of optimization problems subject to ellipsoidal constraints. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed estimator design method.

A robust optimization approach for platooning of automated ground vehicles in port hinterland corridors

Modern ports face significant challenges as strategic nodes of global supply chains, being responsible for the coordination of inbound and outbound flows at deep-sea and in hinterland transport corridors. Digitization and the adoption of disruptive technologies can help ports to tide over operational challenges. Automated Ground Vehicles (AGVs) are an integral part of operations at many modern ports, especially inside container terminals. With the shift to automated transport outside of the terminal areas, these AGVs may form platoons to establish an efficient port hinterland transport corridor. In this work, we propose a new robust optimization approach to assess the time and cost-efficiency of applying such AGV platoons in a container pickup and delivery problem. We develop a bi-objective mixed-integer programming model, which simultaneously minimizes time and cost elements, and also considers emissions. Each transportation task can be carried out by AGVs or conventional trucks, while the number of available vehicles for each mode is uncertain (as they are used to connect different modalities of container transport). The robust optimization model is based on an ellipsoidal uncertainty set to handle this uncertainty and an augmented epsilon constraint method to obtain Pareto-optimal solutions for this multi-objective problem. The developed framework is evaluated in two case studies: the Port of Rotterdam in The Netherlands and the Port of Valparaíso in Chile, with different traveling distances in corridors to a dry port (200 km) and a pre-terminal (11 km), respectively. The results indicate that the new direct delivery scheme by AGV platoons is significantly more cost- and time-efficient than the benchmark and provides a low-carbon emission transportation mode. While the benefits of decreased dwell times (56% on average) and carbon emissions (on average by 10%) are similar for short and long traveling distances, the savings in cost increase (from 4.9% to 8%) with the increased distance in the Rotterdam case.