Ellipsoidal Set Research Articles

Set-Valued Feedback Control and Its Application to Event-Triggered Sampled-Data Systems

Motivated by challenges in feedback control with event-triggered control input updates, this article studies an ellipsoidal set-valued feedback control problem for linear time-invariant (LTI) discrete-time (DT) systems, and its application to event-triggered control systems. First, a state feedback controller based on the set-valued information is designed for LTI DT systems with disturbances. Then, the stability of the closed-loop system is analyzed, and the state of the closed-loop system is shown to be uniformly ultimately bounded in an ellipsoidal set. Based on the proposed set-valued feedback controller, we consider the case when the event-triggering condition (ETC) is described by an ellipsoidal set for an event-triggered control system. To ensure the requirements on control performance and communication rates simultaneously, an optimal ETC design problem is formulated and its optimal solution is provided. Finally, simulation examples are given to show the effectiveness of the proposed method.

Managing supply risk: Robust procurement strategy for capacity improvement

We consider a buyer’s procurement strategy from a supplier with random capacity. The supplier can exert an unobservable effort to improve its capacity. The buyer is ambiguous about the parameters of the supplier’s capacity distribution. We introduce an ellipsoidal uncertainty set to represent the uncertain parameters and analyze how the buyer determines the optimal order quantity and the wholesale price. Using the worst-case criterion, we completely characterize the structure of the optimal robust procurement strategy. We show that the optimal robust wholesale price becomes larger as the degree of parameter uncertainty increases; and the buyer’s worst-case expected profit is decreasing in the degree of parameter uncertainty. We then establish an upper bound for the impact that parameter uncertainty may have on the buyer’s profitability. This upper bound can serve as a measure for the buyer to evaluate the benefit of reducing parameter uncertainty. Finally, we extend the results by considering general Lp-normuncertainty sets, general capacity distributions, and random demand. Our findings provide managerial guidelines on the design of the robust procurement strategy in practice.

An ellipsoidal Newton’s iteration method of nonlinear structural systems with uncertain-but-bounded parameters

This paper presents an ellipsoidal Newton’s iteration method for predicting the response of nonlinear structural systems with uncertain-but-bounded parameters. In the study, the uncertainty in parameters is expressed in terms of an ellipsoid set in an appropriate vector space and the bounds for the solution set of the nonlinear equations are aiming to be calculated effectively. In the framework of the convex set theory and Taylor series expansion, the ellipsoidal Newton’s iteration scheme is established. Two various models of the scheme depending on the different models of quantifying the region of the iterative solution in iterative calculation are discussed. The bounds of the solution are updated iteratively by using the maximum and minimum values of the solution increment, which can be obtained by solving the optimization problem. The convergence of the scheme is proved and the general procedure for its implementation is also presented. Three numerical examples are employed to illustrate the feasibility and accuracy of the proposed method in evaluating the bounds of nonlinear structural systems with uncertain-but-bounded parameters in comparison with the Monte-Carlo Simulation and the point-based iteration method.

Guaranteed set-membership state estimation of an octorotor's position for radar applications

In the context of state estimation of dynamical systems subject to bounded perturbations and measurement noises, this paper proposes an application of a guaranteed ellipsoidal-based set-membership state estimation technique to estimate the linear position of an octorotor used for radar applications. The size of the ellipsoidal set containing the real state is minimized at each sample time taking into account the measurements performed by the drone's sensors. Three case studies highlight the efficiency of the estimation technique in finding guaranteed bounds for the octorotor's linear position. The computed guaranteed bounds in the linear trajectory are exploited to find the maximum operating frequency of the radar, a necessary information in radar applications.

Directional expansion‐based fault diagnosis algorithm using orthotopic and ellipsoidal filtering

For solving the problem of fault diagnosis for systems with unknown but bounded noises, a directional expansion-based fault diagnosis algorithm using orthotopic and ellipsoidal filtering is proposed in this study. In this fault diagnosis process, whether the system is faulty or not is checked by detecting whether the ellipsoidal feasible set is empty. When the ellipsoidal feasible set is empty, the specific faulty components are isolated based on the empty state of the test ellipsoidal sets. The orthotopic set is expanded in the faulty directions alone to minimise the set containing the fault parameter vector. Finally, comparative analyses of the proposed fault diagnosis algorithm and global expansion-based fault diagnosis algorithm using the orthotopic and ellipsoidal filtering in both numerical and case simulations prove the effectiveness and practicability of the proposed algorithm and show the advantages of this algorithm in view of convergence rate and fault identification speed.

Generalized Farkas Lemma with Adjustable Variables and Two-Stage Robust Linear Programs

In this paper, we establish strong duality between affinely adjustable two-stage robust linear programs and their dual semidefinite programs under a general uncertainty set, that covers most of the commonly used uncertainty sets of robust optimization. This is achieved by first deriving a new version of Farkas’ lemma for a parametric linear inequality system with affinely adjustable variables. Our strong duality theorem not only shows that the primal and dual program values are equal, but also allows one to find the value of a two-stage robust linear program by solving a semidefinite linear program. In the case of an ellipsoidal uncertainty set, it yields a corresponding strong duality result with a second-order cone program as its dual. To illustrate the efficacy of our results, we show how optimal storage cost of an adjustable two-stage lot-sizing problem under a ball uncertainty set can be found by solving its dual semidefinite program, using a commonly available software.

Robust extended Kalman filtering for non‐linear systems with unknown input: a UBB model approach

This study presents an unknown-but-bounded (UBB) model approach to robust extended Kalman filtering (REKF) problem for the simultaneous input and state estimation of non-linear systems with parametric uncertainties. An augmented non-linear state-space model is suggested to estimate the unknown input concurrently with the state variables without any delay. Due to more satisfactory physical assumptions and interpretations, the initial state vector and the disturbances are considered as UBB white processes, rather than the conventional stochastic white processes. Hence, they are modelled as ellipsoidal sets, and a recursive UBB-REKF is developed. The suggested algorithm aims to guarantee an optimal upper bound for the estimation error covariance considering the parametric uncertainties and the linearisation errors. Finally, the effectiveness and remarkable estimation accuracy of the proposed UBB-REKF is illustrated in a manoeuvring target tracking problem.

Trust Your Data or Not—StQP Remains StQP: Community Detection via Robust Standard Quadratic Optimization

We consider the robust standard quadratic optimization problem (RStQP), in which an uncertain (possibly indefinite) quadratic form is optimized over the standard simplex. Following most approaches, we model the uncertainty sets by balls, polyhedra, or spectrahedra, more generally, by ellipsoids or order intervals intersected with subcones of the copositive matrix cone. We show that the copositive relaxation gap of the RStQP equals the minimax gap under some mild assumptions on the curvature of the aforementioned uncertainty sets and present conditions under which the RStQP reduces to the standard quadratic optimization problem. These conditions also ensure that the copositive relaxation of an RStQP is exact. The theoretical findings are accompanied by the results of computational experiments for a specific application from the domain of graph clustering, more precisely, community detection in (social) networks. The results indicate that the cardinality of communities tend to increase for ellipsoidal uncertainty sets and to decrease for spectrahedral uncertainty sets.

Consensus Modeling with Asymmetric Cost Based on Data-Driven Robust Optimization

The robust optimization method has progressively become a research hot spot as a valuable means for dealing with parameter uncertainty in optimization problems. Based on the asymmetric cost consensus model, this paper considers the uncertainties of the experts’ unit adjustment costs under the background of group decision making. At the same time, four uncertain level parameters are introduced. For three types of minimum cost consensus models with direction restrictions, including MCCM-DC, $$\varepsilon $$ -MCCM-DC and threshold-based (TB)-MCCM-DC, the robust cost consensus models corresponding to four types of uncertainty sets (Box set, Ellipsoid set, Polyhedron set and Interval-Polyhedron set) are established. Sensitivity analysis is carried out under different parameter conditions to determine the robustness of the solutions obtained from robust optimization models. The robust optimization models are then compared to the minimum cost models for consensus. The example results show that the Interval-Polyhedron set’s robust models have the smallest total costs and strongest robustness. Decision makers can choose the combination of uncertainty sets and uncertain levels according to their risk preferences to minimize the total cost. Finally, in order to reduce the conservatism of the classical robust optimization method, the pricing information of the new product MACUBE 550 is used to build a data-driven robust optimization model. Ellipsoid uncertainty set is proved to better trade-off the average performance and robust performance through different measurement indicators. Therefore, the uncertainty set can be selected according to the needs of the group.

Robust transmission expansion planning with uncertain generations and loads using full probabilistic information

Recent advances in Transmission Network Expansion Planning (TNEP) have demonstrated that two-stage adaptive robust optimization (ARO) renders the expansion planning problem tractable for real systems and, at the same time, constitutes a relevant approach to deal with uncertain demand and generation capacity in the TNEP problem. However, the use of cardinality and polyhedral uncertainty sets does not consider the correlation of uncertain parameters and makes the interpretation of final results difficult from a probabilistic view-point. The recent use of level II structural-reliability and decomposition techniques allows the incorporation of ellipsoidal uncertainty sets, which permits the use of expected values and the variance-covariance matrix of uncertain parameters (First-Order Second-Moment, FOSM). However, it also constitutes an approximation. This paper presents an iterative method which, using FOSM and appropriately modifying the reliability index associated with the desired objective function quantile, transforms the model into a level III structural-reliability method that allows the consideration of the complete probabilistic structure of the uncertain parameters. Numerical results from a realistic case study are presented and discussed.

Constrained Uncertain System Stabilization with Enlargement of Invariant Sets

An enhanced method able to perform accurate stability of constrained uncertain systems is presented. The main objective of this method is to compute a sequence of feedback control laws which stabilizes the closed-loop system. The proposed approach is based on robust model predictive control (RMPC) and enhanced maximized sets algorithm (EMSA), which are applied to improve the performance of the closed-loop system and achieve less conservative results. In fact, the proposed approach is split into two parts. The first is a method of enhanced maximized ellipsoidal invariant sets (EMES) based on a semidefinite programming problem. The second is an enhanced maximized polyhedral set (EMPS) which consists of appending new vertices to their convex hull to minimize the distance between each new vertex and the polyhedral set vertices to ensure state constraints. Simulation results on two examples, an uncertain nonisothermal CSTR and an angular positioning system, demonstrate the effectiveness of the proposed methodology when compared to other works related to a similar subject. According to the performance evaluation, we recorded higher feedback gain provided by smallest maximized invariant sets compared to recently studied methods, which shows the best region of stability. Therefore, the proposed algorithm can achieve less conservative results.

Robust data envelopment analysis via ellipsoidal uncertainty sets with application to the Italian banking industry

This paper extends the conventional DEA models to a robust DEA (RDEA) framework by proposing new models for evaluating the efficiency of a set of homogeneous decision-making units (DMUs) under ellipsoidal uncertainty sets. Four main contributions are made: (1) we propose new RDEA models based on two uncertainty sets: an ellipsoidal set that models unbounded and correlated uncertainties and an interval-based ellipsoidal uncertainty set that models bounded and correlated uncertainties, and study the relationship between the RDEA models of these two sets, (2) we provide a robust classification scheme where DMUs can be classified into fully robust efficient, partially robust efficient and robust inefficient, (3) the proposed models are extended to the additive DEA model and its efficacy is analyzed with two imprecise additive DEA models in the literature, and finally, (4) we apply the proposed models to study the performance of banks in the Italian banking industry. We show that few banks which were resilient in their performance can be robustly classified as partially efficient or fully efficient in an uncertain environment.

A minimax regret approach for robust multi-objective portfolio selection problems with ellipsoidal uncertainty sets

Since security return cannot be accurately estimated using past data, in this paper it is assumed to take values in a given ellipsoidal uncertainty set. This paper aims to discuss a robust multi-objective portfolio selection problem based on the minimax regret criterion under an ellipsoidal uncertainty sets, in which the two objective functions are the portfolio return to be maximized and the mean absolute deviation as a risk measure to be minimized. The robust counterpart formulation for the proposed model is firstly presented, then an algorithm based on the relaxation procedure is designed to solve the robust counterpart formulation with second-order cone constraints and infinite constraints. Finally, a practical example based on real market data is presented to illustrate the effectiveness of the proposed model and the algorithm. Compared with the traditional robust portfolio model based on minimax robustness, the robust minimax regret optimal solutions proposed in this paper have better performance on several evaluation criteria.

Self-triggered output-feedback control of LTI systems subject to disturbances and noise

Self-triggered control (STC) and periodic event-triggered control (PETC) are aperiodic sampling techniques aiming at reducing control data communication when compared to periodic sampling. In both techniques, the effects of measurement noise in continuous-time systems with output feedback are unaddressed. In this work we prove that additive noise does not hinder stability of output-feedback PETC of linear time-invariant (LTI) systems. Then we build an STC strategy that estimates PETC’s worst-case triggering times. To accomplish this, we use set-based methods, more specifically ellipsoidal sets, which describe uncertainties on state, disturbances and noise. Ellipsoidal reachability is then used to predict worst-case triggering condition violations, ultimately determining the next communication time. The ellipsoidal state estimate is recursively updated using guaranteed state estimation (GSE) methods. The proposed STC is designed to be computationally tractable at the expense of some added conservatism. It is expected to be a practical STC implementation for a broad range of applications.

Smallest Ellipsoid Containing $p$-Sum of Ellipsoids With Application to Reachability Analysis

In this article the problem of ellipsoidal bounding of convex set-valued data, where the convex set is obtained by the p-sum of finitely many ellipsoids, for any real p ≥ 1 is studied. The notion of p-sum appears in the Brunn-Minkowski-Firey theory in convex analysis, and generalizes several well-known set-valued operations, such as the Minkowski sum of the summand convex sets (here, ellipsoids). We derive an outer ellipsoidal parameterization for the p-sum of a given set of ellipsoids, and compute the tightest such parameterization for two optimality criteria: minimum trace and minimum volume. For such optimal parameterizations, several known results in the system-control literature are recovered as special cases of our general formula. For the minimum volume criterion, our analysis leads to a fixed point recursion over a scalar that parameterizes the shape matrix of the outer ellipsoid. This recursion is proved to be contractive, and found to converge fast in practice. We apply these results to compute the forward reach sets for a linear control system subject to different convex set-valued uncertainty models for the initial condition and control, generated by varying p ∈ [1,∞]. Our numerical results show that the proposed fixed point algorithm offers more than two orders of magnitude speed-up in computational time for p=1, compared to the existing semidefinite programming approach without significant effect on the numerical accuracy. For p > 1, the reach set computation results reported here are novel. Our results are expected to be useful in real-time safety critical applications, such as decision making, for collision avoidance of autonomous vehicles, where the computational time scale for reach set calculation needs to be much smaller than the vehicular dynamics time scale.

How considering multiple criteria, uncertainty scenarios and biological interactions may influence the optimal silvicultural strategy for a mixed forest

Assessing pre-defined strategies remains the status quo for studies supporting silvicultural decision-making for future forest management, yet, such strategies may not fully address decision-makers' preferences and uncertainty attitudes. We develop a continuous stand-level optimisation approach that integrates multiple decision criteria, uncertain input data based on ellipsoidal uncertainty sets and biological interactions. The optimisation aims to derive silvicultural strategies that closely align with the objectives and uncertainty attitudes of decision-makers. The novel approach optimises tree species composition and harvesting regimes simultaneously. In our example, the decision criteria are the soil expectation value (SEV), the volume of timber harvested, the sum of cash flows and the average amount of carbon stored in the forest. We use input data for Norway spruce (Picea abies), Silver fir (Abies alba) and European beech (Fagus sylvatica), and integrate biological stand-level interactions represented by a) enhanced survival of tree species in mixed forests and b) the growth response of trees remaining after partial harvesting. The resulting optimal silvicultural strategies ranged from a clear-cutting system (when maximising SEV and ignoring uncertainty) to continuous cover forestry (maximising SEV and considering uncertainty). Our analyses did not support single species forestry – even the clear-cutting system suggested a mixed rather than pure forest. Silvicultural strategies that consider multiple criteria stored up to 47% more carbon than the clear-cutting system, but their SEVs were up to 39% lower. Biological interactions influence the optimal stand composition and harvesting regime, while establishment costs and the discount rate affect the level achieved for each management criterion. Lower survival due to changes in climate hardly influenced the model results. Our optimisation approach is flexible and may integrate many more and different criteria. It is useful to derive silvicultural strategies to guide science-based recommendations for various forest decision-makers.

Observer-based output feedback robust MPC via zonotopic set-membership state estimation for LPV systems with bounded disturbances and noises

For linear parameter varying (LPV) systems with bounded disturbances and noises, this paper presents a synthesis approach to observer-based output feedback robust model predictive control (RMPC) via the zonotopic set-membership state estimation. In the off-line optimization problem, the state observer sub-gains with a minimal ellipsoidal robust positively invariant (RPI) set for the zonotopic estimation error set are designed. In the on-line control optimization problem, bounds of the current and future zontopic estimation error sets are outer approximated by scaling the minimal ellipsoidal RPI set. Then, an on-line constrained optimization problem that considers bounds of the predicted zonotopic estimation error sets is solved to guarantee robust stability of the closed-loop observer system. In the on-line receding horizon optimization, the zonotopic estimation error set is refreshed based on zonotopic properties, and its outer approximated ellipsoidal set is optimized via the technique of S-procedure. By properly considering the relationship between the zonotopic estimation error set and its outer approximated ellipsoidal set, recursive feasibility of the optimization problem can be ensured. The on-line optimization problem guarantees that the nominal estimation error system and nominal closed-loop observer system converge to the origin. When bounded disturbances and noises are considered, the estimation error system and closed-loop observer system respectively bounded within time-varying RPI sets and robust control invariant (RCI) sets are steered to a neighborhood of the origin such that robust stability of the controlled system is guaranteed. A simulation example is given to illustrate the proposed method.

Distributed guaranteed two-target tracking over heterogeneous sensor networks under bounded noises and adversarial attacks

This paper is concerned with the distributed guaranteed estimation for tracking two interacting mobile targets over a multi-sensor network in the presence of unknown-but-bounded noises and various adversarial attacks. First, a heterogeneous sensor network framework in terms of two distinct groups of sensors is employed to monitor the two targets. Each sensor in either group possesses different sensing, processing and communicating capabilities, thereby leading to distinct communication topologies among intra- and inter-group sensors. Second, a unified attack model is established to skillfully accommodate multiple adversarial attacks including node manipulation attacks and deception attacks. Third, two different groups of distributed consensus-based estimators are delicately constructed to deal with the network heterogeneity. Criteria for designing the desired estimators are then derived such that the true states of the two maneuvering targets are guaranteed to be enclosed by the calculated estimate ellipsoids at each time step regardless of the noises and attacks. Furthermore, two tractable optimization algorithms, in both single- and two-target tracking cases, are applied to recursively calculate the smallest possible ellipsoidal estimate sets. Finally, numerical verification of distributed two-vehicle tracking is carried out to demonstrate the effectiveness and applicability of the obtained results.

Can Robust Optimization Offer Improved Portfolio Performance? An Empirical Study of Indian market

The emergence of robust optimization has been driven primarily by the necessity to address the demerits of the Markowitz model. There has been a noteworthy debate regarding consideration of robust approaches as superior or at par with the Markowitz model, in terms of portfolio performance. In order to address this skepticism, we perform empirical analysis of three robust optimization models, namely the ones based on box, ellipsoidal and separable uncertainty sets. We conclude that robust approaches can be considered as a viable alternative to the Markowitz model, not only in simulated data but also in a real market setup, involving the Indian indices of S&P BSE 30 and S&P BSE 100. Finally, we offer qualitative and quantitative justification regarding the practical usefulness of robust optimization approaches from the point of view of number of stocks, sample size and types of data.

Stabilization of constrained switched systems via multiple Lyapunov R-functions

This paper investigates the stabilization problem of switched linear systems subject to state constraints, input saturation and external disturbance. Within the set-theoretic framework of R-function, the multiple Lyapunov R-functions (MLRFs) are presented, and merge both the multiple polyhedral and the multiple quadratic Lyapunov functions. The advantages of the MLRFs are that the external level sets are designed in accordance to the state constraints and inner level sets arbitrary can be made as close as possible to the smooth one. Based on the MLRFs method, stabilization conditions for the constrained switched system under a gradient-based switching law are derived. The union of the invariant truncated ellipsoidal sets is provided as an alternative approximation to the domain of attraction of the constrained switched system. Each truncated ellipsoidal set is the intersection of a polyhedral set and an ellipsoidal set, which come from the polyhedral Lyapunov function and the quadratic Lyapunov function, respectively. The sub-optimality bounds are investigated by a policy iteration. Two examples are simulated to show the benefits of the proposed strategy.