Guaranteed State Estimation Research Articles

Reducing the wrapping effect of set computation via Delaunay triangulation for guaranteed state estimation of nonlinear discrete-time systems

Set computation methods have been widely used to compute reachable sets, design invariant sets and estimate system state for dynamic systems. The wrapping effect of such set computation methods plays an essential role in the accuracy of their solutions. This paper studies the wrapping effect of existing interval, zonotopic and polytopic set computation methods and proposes novel approaches to reduce the wrapping effect for these set computation methods based on the task of computing the dynamic evolution of a nonlinear uncertain discrete-time system with a set as the initial state. The proposed novel approaches include the partition of a polytopic set via Delaunay triangulation and also the representation of a polytopic set by the union of small zonotopes for the following set propagation. The proposed novel approaches with the reduced wrapping effect has been further applied to state estimation of a nonlinear uncertain discrete-time system with improved accuracy. Similar to bisection for interval and zonotopic sets, Delaunay triangulation has been introduced as a set partition tool for polytopic sets, which has opened new research directions in terms of novel set partition, set representation and set propagation for reducing the wrapping effect of set computation.

Tight Remainder-Form Decomposition Functions With Applications to Constrained Reachability and Guaranteed State Estimation

This paper proposes a tractable family of remainder-form mixed-monotone decomposition functions that are useful for over-approximating the image set of nonlinear mappings in reachability and estimation problems. Our approach applies to a new class of nonsmooth, discontinuous nonlinear systems that we call either-sided locally Lipschitz semicontinuous (ELLS) systems, which we show to be a strict superset of locally Lipschitz continuous (LLC) systems, thus expanding the set of systems that are formally known to be mixed-monotone. In addition, we derive lower and upper bounds for the over-approximation error and show that the lower bound is achieved with our proposed approach, i.e., our approach constructs the tightest, tractable remainder-form mixed-monotone decomposition function. Moreover, we introduce a set inversion algorithm that along with the proposed decomposition functions, can be used for constrained reachability analysis and guaranteed state estimation for continuous- and discrete-time systems with bounded noise.

Privacy-preserving set-based estimation using partially homomorphic encryption

The set-based estimation has gained a lot of attention due to its ability to guarantee state enclosures for safety-critical systems. However, collecting measurements from distributed sensors often requires outsourcing the set-based operations to an aggregator node, raising many privacy concerns. To address this problem, we present set-based estimation protocols using partially homomorphic encryption that preserve the privacy of the measurements and sets bounding the estimates. We consider a linear discrete-time dynamical system with bounded modeling and measurement uncertainties. Sets are represented by zonotopes and constrained zonotopes as they can compactly represent high-dimensional sets and are closed under linear maps and Minkowski addition. By selectively encrypting parameters of the set representations, we establish the notion of encrypted sets and intersect sets in the encrypted domain, which enables guaranteed state estimation while ensuring privacy. In particular, we show that our protocols achieve computational privacy using the cryptographic notion of computational indistinguishability. We demonstrate the efficiency of our approach by localizing a real mobile quadcopter using ultra-wideband wireless devices.

Event-Triggered State Estimation for Fractional-Order Neural Networks

This paper is concerned with the problem of event-triggered state estimation for a class of fractional-order neural networks. An event-triggering strategy is proposed to reduce the transmission frequency of the output measurement signals with guaranteed state estimation performance requirements. Based on the Lyapunov method and properties of fractional-order calculus, a sufficient criterion is established for deriving the Mittag–Leffler stability of the estimation error system. By making full use of the properties of Caputo operator and Mittag–Leffler function, the evolution dynamics of measured error is analyzed so as to exclude the unexpected Zeno phenomenon in the event-triggering strategy. Finally, two numerical examples and simulations are provided to show the effectiveness of the theoretical results.

Comparison of guaranteed state estimators for linear time-invariant systems

Guaranteed state estimation computes the set of possible states of dynamical systems given the bounds of model uncertainties, disturbances, and noises. For the first time, we evaluate and compare a broad class of guaranteed state estimators for linear time-invariant systems regarding scalability, accuracy, and real-time applicability. In particular, we consider strip-based observers, set-propagation observers, and interval observers. The performance of most guaranteed state estimators is significantly affected by the chosen set representation. Zonotopes have emerged as a popular set representation since important operations such as linear maps and Minkowski sum can be computed exactly and efficiently. Furthermore, we provide comparisons with ellipsoids and constrained zonotopes as set representations. The comparison is conducted on various models for state estimation of autonomous vehicles.

Guaranteed State Estimation Using a Bundle of Interval Observers with Adaptive Gains Applied to the Induction Machine.

The scope of this paper is the design of an interval observer bundle for the guaranteed state estimation of an uncertain induction machine with linear, time-varying dynamics. These guarantees are of particular interest in the case of safety-critical systems. In many cases, interval observers provide large intervals for which the usability becomes impractical. Hence, based on a reduced-order hybrid interval observer structure, the guaranteed enclosure within intervals of the magnetizing current’s estimates is improved using a bundle of interval observers. One advantage of such an interval observer bundle is the possibility to reinitialize the interval observers at specified timesteps during runtime with smaller initial intervals, based on previously observed system states, resulting in decreasing interval widths. Thus, unstable observer dynamics are considered so as to take advantage of their transient behavior, whereby the overall stability of the interval estimation is maintained. An algorithm is presented to determine the parametrization of reduced-order interval observers. To this, an adaptive observer gain is introduced with which the system states are observed optimally by considering a minimal interval width at variable operating points. Furthermore, real-time capability and validation of the proposed methods are shown. The results are discussed with simulations as well as experimental data obtained with a test bench.

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.

About control of guaranteed estimation

The control problem by parameters in the course of the guaranteed state estimation of linear non-stationary systems is considered. It is supposed that unknown disturbances in the system and the observation channel are limited by norm in the space of square integrable functions and the initial state of the system is also unknown. The process of guaranteed state estimation includes the solution of a matrix Riccati equation that contains some parameters, which may be chosen at any instant of time by the first player (an observer) and the second player (an opponent of the observer). The purposes of players are diametrically opposite: the observer aims to minimize diameter of information set at the end of observation process, and the second player on the contrary aims to maximize it. This problem is interpreted as a differential game with two players for the Riccati equation. All the choosing parameters are limited to compact sets in appropriate spaces of matrices. The payoff of the game is interpreted through the Euclidean norm of the inverse Riccati matrix at the end of the process. A specific case of the problem with constant matrices is considered. Methods of minimax optimization, the theory of optimal control, and the theory of differential games are used. Examples are also given.

Performance guaranteed state estimation for renewable penetration with improved meters

State estimation (SE) aims at monitoring the transmission and distribution networks to achieve stable and reliable grid operations. While the SE problem is non-convex, a local search method is used to achieve global optimum based on the belief that power system states are with limited variation in a short time scale. However, the recent and rapid deployment of renewables leads to strong power and state fluctuations in power grids, making local search method prone to local optimums with large estimation errors. Here, the authors propose to analyse SE with small measurement noises because highly accurate smart sensors are deployed in the past few years. By exploring a special structure of the performance metrics, the authors reformulate the SE problem in an extended state space. The new formulation is convex under the no-measurement-noise assumption. In order to use such a method when there are noises, the authors prove that a perturbation of globally optimal solution is asymptotically bounded by the measurement noise level. This prevents local optimums, which can create a large estimation error. Simulation results demonstrate that the method has a better performance compared to both weighted least square and the recent semidefinite programming-based approaches, especially when the noise is small.

Guaranteed State Estimation for Nonlinear Discrete-Time Systems via Indirectly Implemented Polytopic Set Computation

This paper proposes a new set-membership technique to implement polytopic set computation for nonlinear discrete-time systems indirectly. The proposed set-membership technique is applied to solve the guaranteed state estimation problem for nonlinear discrete-time systems with a bounded description of noises and parameters. A common practice for this problem in the literature is to search an optimal zonotope to bound the intersection of the evolved uncertain state trajectory and the region of the state space consistent with the observed output at each observation update. The new approach keeps the polytopic set resulting from the intersection intact and computes the evolution of this intact polytopic set for the next time step through representing the polytopic set exactly by the intersection of zonotopes. Such an approach avoids the overapproximation bounding process at each observation update, and thus, a more accurate state estimation can be obtained. An illustrative example is provided to demonstrate the effectiveness of the proposed guaranteed state estimation via indirectly implemented polytopic set computation.

Interval observer for LPV systems with unknown inputs

This study addresses the problem of guaranteed state and unknown input (UI) estimation for a class of linear parameter varying systems affected by UIs and other bounded perturbations. The context of interval observer in the cooperative framework is adopted. Thus, lower and upper bounds of the state and UI are provided. No assumption is made on the UI. The approach is based on a rank condition allowing to decouple the UI from the state estimation errors (so that the state estimation is guaranteed, regardless of the variation of the UI). The convergence of the proposed interval observer is studied by Lyapunov theory and stability conditions are expressed in terms of linear matrix inequalities which ease the design of the gain matrices of the observer. Finally, illustrative examples are given for discussion and comparison with respect to the existing results.

Effective recursive parallelotopic bounding for robust output-feedback control

In this paper an approach is studied to guaranteed (set-membership) state estimation for robust output-feedback model predictive control (MPC) with hard input and state constraints. Uncertainties are assumed to arise in a dynamic system from unknown initial conditions of state variables and due to unknown-but-bounded measurement noise. The uncertainty in the state variables is represented as a parallelotopic set. The employed state-estimation algorithm recursively outbounds the feasible set that is given by an intersection of model predictions with obtained measurement information. Along with the well-known minimum-volume criterion for parallelotopic outbounding, three alternative criteria are proposed and studied. The aim is to identify the best outbounding approach for improving performance of the robust MPC.

On the Accuracy and Convergence of the Minimax Filtering Algorithm for Chaotic Signals

The article is focused on the filtering problem for chaotic signals. The original discrete signal is generated by a one-dimensional chaotic system, and the measured signal is corrupted by additive errors. The goal is to estimate the unknown system states from measurements. The minimax filtering algorithm is developed in the context of guaranteed state estimation that is based on a set-membership description of uncertainty. It assumes that the unknown variables (system states and measurement errors) are bounded by intervals (sets of possible values). The proposed algorithm is a recursive procedure based on interval analysis. It computes interval estimates that are guaranteed to contain the true states of the system (true values of the original signal). The computation of the interval estimate consist of three steps (prediction, measurement, and correction) that are similar to the computation of the information set for linear dynamical systems. The point estimates are obtained by an algorithm that is similar to the Kalman filter. This paper studies the accuracy and convergence properties of the minimax filter. The aims of this study are the following: to confirm the effectiveness of the proposed algorithm for computation of the point estimates, to compare the results of the minimax filter and the unscented Kalman filter, and to derive the sufficient conditions for obtaining the exact value of the state. The computational scheme of the minimax filter and numerical simulations are given for the logistic map.

Zonotopic state estimation and fault detection for systems with time-invariant uncertainties

This paper proposes a robust guaranteed state estimation method with application to fault detection by combining H∞ observer design with zonotopic analysis for discrete-time systems with both time-variant and time-invariant uncertainties. In order to improve the estimation accuracy, based on the H∞ technique, the observer design is achieved by solving a linear matrix inequality. The main contribution of this method lies in that the time invariance of some uncertainties is considered to reduce the conservatism of interval estimation. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.

Interval observers for LPV systems and application to the guaranteed state estimation of an induction machine

The scope of this paper is the design of interval observers for linear parameter-varying systems subject to input and output uncertainties. To this, two coupled dynamical systems are designed which compute an upper bound and a lower bound of the state which enclose the real value with a bounded interval width. Besides of a full-order interval observer, a reduced-order interval observer is proposed to reduce the computational effort. In addition to sufficient conditions and the mathematical description of the interval observers, an optimization-based design procedure for their practical application is presented. The performance of the interval observers is shown by the guaranteed state estimation of an induction machine.

Improved set-membership guaranteed state estimation for uncertain non-linear systems

This paper presents a new improved set-membership algorithm for guaranteed state estimation for non-linear discrete-time systems with an unknown but bounded description of disturbance and noise. In this paper, the non-linear model is linearized first and the difference of convex (DC) programming method is used to outer-bound the linearization error, which is regarded as a disturbance of the system, and the improved set-membership algorithm guaranteed state estimation for the linearization system. An example is given to illustrate the proposed algorithm.

One Improvement on Zonotope Guaranteed Parameter Estimation

This paper studies the guaranteed state estimation in terms of zonotope, and does some improvements for nonlinear discrete time system with a bounded description of noise and parameters. Firstly we extend the Taylor series with respect to two variables so that the mean value extension which is used to compute an interval enclosure can be improved and extended. Secondly based on the improved mean value extension, a generalization of classical method is proposed as it considers uncertainty in the model of system. Thirdly we give one iterative process in one algorithm to obtain a bound of the exact uncertain state set. Finally the simulation example results confirm the identification theoretical results.

A new set-membership estimation method based on zonotopes and ellipsoids

This paper presents a new approach for guaranteed state estimation based on zonotopes and ellipsoids for linear discrete-time systems with unknown but bounded perturbations and noises. At each sample time, a predicted state set is calculated by zonotopes and an outer bounding ellipsoid of the predicted state set is calculated. State estimation is calculated directly by intersects with an outer bounding ellipsoid and strip. The precision of the state estimation will increase as output order improves by this method. Two examples have been provided to clarify the algorithm.

Guaranteed State Estimation Algorithms for Linear Systems in the Presence of Bounded Noise

Guaranteed State Estimation Algorithms for Linear Systems in the Presence of Bounded Noise

A New Approach for Guaranteed Ellipsoidal State Estimation

This paper proposes a new ellipsoid-based guaranteed state estimation approach for linear discrete-time systems with bounded perturbations and bounded measurement noise. This approach is based on the minimization of the radius of the ellipsoidal state estimation set. Firstly, the ellipsoidal state estimation is computed by off-line solving a Linear Matrix Inequality optimization problem. Secondly, a new online method is developed in order to improve the accuracy of the estimation but it leads to an increase of the online computation load. A new scaling technique is proposed to reduce the computation time, while keeping a good accuracy of the state estimation. An illustrative example is analyzed in order to show the advantages of the proposed approach.