Set-valued State Estimation Research Articles

Set-valued state estimators for a class of uncertain discrete-time nonlinear systems

Set-membership state estimation problem for a certain class of discrete-time nonlinear systems is tackled in this work. First, this problem is formulated as a reachability analysis problem. Then, to avoid the wrapping effect, explicit expression of the general solution of this class of systems is used to design new set-valued state estimation schemes. Moreover, for practical considerations, two periodically re-initialized algorithms based on both interval analysis and zonotope computation are introduced. The merit of the proposed approach is illustrated on a numerical example and its performance is compared to that of other methods borrowed from the literature.

Set-Valued State Estimator With Sparse and Delayed Measurements for Uncertain Discrete-Time Linear Systems

Set-Valued State Estimator With Sparse and Delayed Measurements for Uncertain Discrete-Time Linear Systems

Set-valued state estimation of nonlinear discrete-time systems and its application to attack detection

Set-valued state estimation of nonlinear discrete-time systems and its application to attack detection

Tight set-valued state estimation by combining reachability analysis and set-filtering approaches

In this work, a new approach to design set-valued state estimator for linear discrete-time systems subject to additive and bounded process and measurement uncertainties is proposed. First, the system state equation is rewritten to obtain a stable numerical scheme on which an explicit reachability method is developed, based on zonotopic set computation and a re-initialization procedure. Then, to enhance the accuracy of the computed reachable set, a set-filtering technique is designed based on the system output equation and its intrinsic invariant relationships. The implementation of this filtering method is based on interval analysis coupled to contractor algorithms. The convergence property of the proposed set-valued state estimator is shown under the classical detectability assumption of linear systems. Some simulation results are presented to show the merit of the proposed new set-membership state estimation approach.

Exact Set-valued Estimation using Constrained Convex Generators for uncertain Linear Systems

Set-valued state estimation when in the presence of uncertainties in the model have been addressed in the literature essentially following three main approaches: i) interval arithmetic of the uncertain dynamics with the estimates; ii) factorizing the uncertainty into matrices with unity rank; and, iii) performing the convex hull for the vertices of the uncertainty space. Approach i) and ii) introduce a lot of conservatism because both disregard the relationship of the parameters with the entries of the dynamics matrix. On the other hand, approach iii) has a large growth on the number of variables required to represent the set or is approximated losing its main advantage in comparison with i) and ii). In this paper, with the application of autonomous vehicles in GPS-denied areas that resort to beacon signals for localization, we develop an exact (meaning no added conservatism) and optimal (smallest growth in the number of variables) closed-form definition for the convex hull of Convex Constrained Generators (CCGs). This results in a more efficient method to represent the minimum volume convex set corresponding to the state estimation. Given that reductions methods are still lacking in the literature for CCGs, we employ an approximation using ray-shooting that is comparable in terms of accuracy with methods for Constrained Zonotopes as the ones implemented in CORA. Simulations illustrate the greater accuracy of CCGs with the proposed convex hull operation in comparison to Constrained Zonotopes.

Experimental Validation of an Ellipsoidal State Estimation Procedure for a Magnetic Levitation System

Set-based state estimation procedures have the advantage of enclosing all possible system states under the assumption of bounded measurement uncertainty, the structural correctness of dynamic systems models, and the representation of external disturbances and imperfectly known parameters by finitely large sets. In contrast to stochastic counterparts, often employing one of the available variants of Kalman filters, set-based approaches are less widely used. The reason for this observation is the fact that naive implementations often suffer from a non-negligible degree of overestimation and that (unless certain monotonicity properties are satisfied) set-based computations come with a notable increase of the computational complexity, resulting among others from required interval splitting procedures. This paper tries to resolve both issues by means of an ellipsoidal implementation of a discrete-time set-valued state estimation procedure that is validated experimentally and compared with an Unscented Kalman Filter (UKF) for a laboratory-scale magnetic levitation system.

GPS Spoofing Mitigation and Timing Risk Analysis in Networked Phasor Measurement Units via Stochastic Reachability

<h3>Abstract</h3> To address phasor measurement unit (PMU) vulnerability to spoofing, we propose the use of a set-valued state estimation technique known as stochastic reachability (SR)-based distributed Kalman filter (DKF) that computes secure global positioning system (GPS) timing across a network of receivers. Utilizing SR, we estimate not only GPS time but also its stochastic reachable set, which is parameterized by probabilistic zonotope (p-Zonotope). While requiring known measurement error bounds in only non-spoofed conditions, we designed a two-tiered approach. We first performed measurement-level spoofing mitigation via deviation of a measurement innovation from its expected p-Zonotope. We then performed state-level timing risk analysis via a determination of the intersection probability of the estimated p-Zonotope with an unsafe set that violates IEEE C37.118.1a-2014 standards. Finally, we validated our SR-DKF algorithm by subjecting it to a simulated receiver network to coordinate signal-level spoofing. We demonstrate improved timing accuracy and successful spoofing mitigation via the use of our SR-DKF algorithm. We also validated the robustness of the estimated timing risk as the number of receivers were varied.

Set-valued state estimation of nonlinear discrete-time systems with nonlinear invariants based on constrained zonotopes

This paper presents new methods for set-valued state estimation of discrete-time nonlinear systems whose trajectories are known to satisfy nonlinear equality constraints, called invariants (e.g., conservation laws). Set-valued estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. Most existing methods employ a standard prediction-update framework with set-based prediction and update steps based on various set representations and techniques. However, achieving accurate enclosures for nonlinear systems remains a significant challenge. This paper presents new methods based on constrained zonotopes that improve the standard prediction-update framework for systems with invariants by adding a consistency step. This new step uses invariants to reduce conservatism and is enabled by new algorithms for refining constrained zonotopes based on nonlinear constraints. This paper also presents significant improvements to existing prediction and update steps for constrained zonotopes. Specifically, new update algorithms are developed that allow nonlinear measurement equations for the first time, and existing prediction methods based on conservative approximation techniques are modified to allow a more flexible choice of the approximation point, which can lead to tighter enclosures. Numerical results demonstrate that the resulting methods can provide significantly tighter enclosures than existing zonotope-based methods while maintaining comparable efficiency.

DYNAMIC SYSTEMS STATE, DISTURBANCES AND NOISES SET-VALUED ESTIMATION UNDER CONDITIONS OF INCOMPLETE INFORMATION

The paper considers the problem of set-valued dynamic systems state estimation under conditions of uncertainty, when the sets of disturbances and noises possible values are known and statistical information about them is absent or cannot be obtained. An algorithm for feasible set polyhedral approximation is described, when the sets of possible values of disturbances and noises are polyhedra. The algorithm is based on the implicit description of the information set with linear equations and inequalities systems and solving a number of linear programming problems. Methods for increasing the estimation accuracy by taking into account additional information about disturbances and noises models are considered. Set-valued estimation of the dynamical system state vector is described when the disturbances are given as a system of functions with unknown coefficients. In this case, due to the use of information that the coefficients are constant, the dynamic system state estimates are more accurate than in the case when the disturbances are known up to a set of possible values. A numerical example is presented to demonstrate the algorithm performance. Aim. The aim of the research is to develop dynamic system state, disturbance and noises set-valued estimation algorithms. Research methods. Methods of optimization theory, filtering, linear algebra, MATLAB software package were used in the work. Results. Dynamic system state estimation algorithm was described. The algorithm takes into account additional information about disturbances and noises models. A method of feasible set polyhedral approximation is described, which makes it possible to obtain a set-valued estimate of a state vector, a vector of disturbances and noises, and an evolution of reachable sets. It can be used in the adaptive estimation and control algorithms development. The algorithm for set-valued estimation of the system state vector and coefficients in the disturbance decomposition as a system of given functions is developed. Conclusion. An algorithm for feasible set polyhedral approximation was described.The numerical example was performed and the analysis of the estimateswas presented.

Robust set‐membership state estimator against outliers in data

Based on interval computation, a set-membership state estimator capable to manage a certain type of outliers in measurements is proposed for uncertain discrete-time linear systems. To achieve this purpose, two set-valued filtering techniques are implemented in the presented state estimation algorithm. The setting up of these techniques offers two main advantages. On one hand, the convergence of the estimated state enclosures width is guaranteed and, on the other hand, the algorithm robustness against outliers in data is ensured. That is, unlike former methods, the proposed set-valued state estimator preserves the framing property despite the presence of some false values in available sensors data. To show the efficiency and the performance of the introduced set-valued state estimator, it is compared, through two numerical examples, with an optimal interval observer selected from the literature for its high performance.

Set-based state estimation and fault diagnosis of linear discrete-time descriptor systems using constrained zonotopes

This paper presents new methods for set-valued state estimation and active fault diagnosis of linear descriptor systems. The algorithms are based on constrained zonotopes, a generalization of zonotopes capable of describing strongly asymmetric convex sets, while retaining the computational advantages of zonotopes. Additionally, unlike other set representations like intervals, zonotopes, ellipsoids, paralletopes, among others, linear static constraints on the state variables, typical of descriptor systems, can be directly incorporated in the mathematical description of constrained zonotopes. Therefore, the proposed methods lead to more accurate results in state estimation in comparison to existing methods based on the previous sets without requiring rank assumptions on the structure of the descriptor system and with a fair trade-off between accuracy and efficiency. These advantages are highlighted in two numerical examples.

Guaranteed methods based on constrained zonotopes for set-valued state estimation of nonlinear discrete-time systems

This paper presents new methods for set-valued state estimation of nonlinear discrete-time systems with unknown-but-bounded uncertainties. A single time step involves propagating an enclosure of the system states through the nonlinear dynamics (prediction), and then enclosing the intersection of this set with a bounded-error measurement (update). When these enclosures are represented by simple sets such as intervals, ellipsoids, parallelotopes, and zonotopes, certain set operations can be very conservative. Yet, using general convex polytopes is much more computationally demanding. To address this, we present in this paper two new methods, a mean value extension and a first-order Taylor extension, for efficiently propagating constrained zonotopes through nonlinear mappings. These extend existing methods for zonotopes in a consistent way. Examples show that these extensions yield tighter prediction enclosures than zonotopic estimation methods, while largely retaining the computational benefits of zonotopes. Moreover, they enable tighter update enclosures because constrained zonotopes can represent intersections much more accurately than zonotopes.

Self-Triggered and Event-Triggered Set-Valued Observers

This paper addresses the problem of reducing the required network load and computational power for the implementation of Set-Valued Observers (SVOs) in Networked Control System (NCS). Event- and self-triggered strategies for NCS, modeled as discrete-time Linear Parameter-Varying (LPV) systems, are studied by showing how the triggering condition can be selected. The methodology provided can be applied to determine when it is required to perform a full (“classical”) computation of the SVOs, while providing low-complexity state overbounds for the remaining time, at the expenses of temporarily reducing the estimation accuracy. As part of the procedure, an algorithm is provided to compute a suitable centrally symmetric polytope that allows to find hyper-parallelepiped and ellipsoidal overbounds to the exact set-valued state estimates calculated by the SVOs. By construction, the proposed triggering techniques do not influence the convergence of the SVOs, as at some subsequent time instants, set-valued estimates are computed using the conventional SVOs. Results are provided for the triggering frequency of the self-triggered strategy and two interesting cases: distributed systems when the dynamics of all nodes are equal up to a reordering of the matrix; and when the probability distribution of the parameters influencing the dynamics is known. The performance of the proposed algorithm is demonstrated in simulation by using a time-sensitive example.

Stochastic and deterministic fault detection for randomized gossip algorithms

This paper addresses the problem of detecting faults in linear randomized gossip algorithms, where the selection of the dynamics matrix is stochastic. A fault is a disturbance signal injected by an attacker to corrupt the states of the nodes. We propose the use of Set-Valued Observers (SVOs) to detect if the state observations are compatible with the system dynamics for the worst case in a deterministic setting. The concept of Stochastic Set-Valued Observers (SSVOs) is also introduced to construct a set that is guaranteed to contain all possible states with, at least, a pre-specified desired probability. The proposed algorithm is stable in the sense that it requires a finite number of vertices to represent polytopic sets and it allows for the computation of the largest magnitude of the disturbance that an attacker can inject in the network without being detected. Results are presented to reduce the computational cost of this approach and, in particular, by considering only local information and representing the remainder of the network as a disturbance. The case of a consensus algorithm is discussed leading to the conclusion that, by using the proposed SVOs, finite-time consensus is achieved in non-faulty environments. A novel algorithm is proposed that produces less conservative set-valued state estimates by having nodes exchanging local estimates. The algorithm inherits all the previous properties and also enables finite-time consensus computation regardless of the value of the horizon.

Set-Valued State Estimation and Attack Detection for Uncertain Descriptor Systems

A set-valued descriptor state estimation algorithm is given for determining all those descriptor state values consistent with a given uncertain descriptor system model, sensor model and a general class of uncertainty model. An approach for detecting faults or attacks signals that are corrupting the uncertain descriptor system and/or the set of sensor readings is also provided.

Robust Filtering for Uncertain Nonlinear Systems Satisfying a Sum Quadratic Constraint

This paper develops a robust extended Kalman filter for nonlinear uncertain systems with a deterministic description of noise and uncertainty. The system state dynamics are formulated in reverse time and the uncertainties are modeled in terms of sum quadratic constraints. The robust filtering problem is formulated as a set-valued state estimation problem which is recast into an optimal control problem. The solution of the resulting Hamilton-Jacobi-Bellman equation is obtained by using a quadratic approximation. This leads to an approximate information state for the filtering problem which is computed recursively through a difference Riccati equation derived by linearizing the observation equation and using a quadratic approximation to the system dynamics. The reduction of the robust estimator to the standard Kalman filter in the uncertainty-free linear case is also discussed.

Decentralized robust set-valued state estimation in networked multiple sensor systems

This paper addresses a decentralized robust set-valued state estimation problem for a class of uncertain systems via a data-rate constrained sensor network. The uncertainties of the systems satisfy an energy-type constraint known as an integral quadratic constraint. The sensor network consists of spatially distributed sensors and a fusion center where set-valued state estimation is carried out. The communications from the sensors to the fusion center are through data-rate constrained communication channels. We propose a state estimation scheme which involves coders that are implemented in the sensors, and a decoder–estimator that is located at the fusion center. Their construction is based on the robust Kalman filtering techniques. The robust set-valued state estimation results of this paper involve the solution of a jump Riccati differential equation and the solution of a set of jump state equations.

A Discrete-Time Robust Extended Kalman Filter for Uncertain Systems With Sum Quadratic Constraints

This technical note outlines the formulation of a novel discrete-time robust extended Kalman filter for uncertain systems with uncertainties described in terms of sum quadratic constraints. The robust filter is an approximate set-valued state estimator which is robust in the sense that it can handle modeling uncertainties in addition to exogenous noise. Riccati and filter difference equations are obtained as an approximate solution to a reverse-time optimal control problem defining the set-valued state estimator. In order to obtain a solution to the set-valued state estimation problem, the discrete-time system dynamics are modeled backwards in time.

Model validation for IQC uncertain systems with fixed initial conditions

The paper considers a model validation problem for a class of uncertain systems in which the uncertainty is described by an integral quadratic constraint and the uncertain system has zero initial condition. The problem is solved by applying methods from linear quadratic optimal control with a fixed endpoint. This leads to a method for model validation which is based around a robust Kalman filter structure. The paper also yields a result on robust set-valued state estimation for uncertain systems in which the initial condition is known to be zero.

Set-valued state estimation for uncertain continuous-time systems via limited capacity communication channels

This paper addresses a problem of set-valued state estimation for uncertain continuous-time systems via limited capacity communication channels. The uncertainty of the systems satisfies an integral quadratic constraint. Using results from the robust Kalman filtering, we design a coder/decoder-estimator pair that allows us to construct set-valued state estimate of the systems via communication channels.