Set-valued Estimation Research Articles

Multiple-model adaptive control using set-valued observers

SUMMARY A multiple-model adaptive control methodology is proposed that is able to provide stability and performance guarantees, for uncertain linear parameter-varying plants. The identification problem is addressed by taking advantage of recent advances in model falsification using set-valued observers (SVOs). These SVOs provide set-valued estimates of the state of the system, according to its dynamic model. If such estimate is the empty set, the underlying dynamic model is invalidated, and a different controller is connected to the loop. The behavior of the proposed control algorithm is demonstrated in simulation, by resorting to a mass–spring–dashpot system. As a caveat, the computational burden associated with the SVOs can be prohibitive under some circumstances. Copyright © 2013 John Wiley & Sons, Ltd.

Estimation of linear positive systems with unknown time-varying delays

This paper considers the estimation problem for linear positive systems with time-varying unknown delays. Similar to set-valued estimation approaches, we provide a confident region within which the trajectory of the observed positive system always evolves. Guaranteed upper and lower estimates for the instantaneous states are characterized by means of a special kind of extended Luenberger-type interval observer. We provide constructive conditions for its existence and establish the asymptotic convergence of its associated interval error. In addition, we give an LP-based method which allows one to construct the proposed interval observer solely from the data of the system.

Distributed set-valued estimation in sensor networks with limited communication data rate

This paper is concerned with a distributed set-valued estimation problem in noisy sensor networks with limited communication data rate. The sensors iterate a sample-quantize-exchange process: firstly, the sensors synchronously take noisy measurements of the plant; secondly, owing to the communication channel constraint, each sensor makes the manipulation of quantization on its measurement, estimate and performance bound by using a static logarithmic quantizer; thirdly, the quantized information is exchanged among the neighbors. By assuming hard bounds on the process noises and measurement noises, a distributed set-valued estimation strategy is proposed for each sensor to generate a set of estimates in state space which contains the true state of the plant. A sufficient condition for the existence of the distributed set-valued estimators is derived, and the estimator parameters are determined by solving a convex optimization problem. Two illustrative examples are provided to demonstrate the effectiveness of the proposed theoretical results.

Stochastic Model Validation and Estimation for Linear Discrete-Time Systems with Partial Prior Information

The problem of recursive estimation and model validation for linear discrete-time systems with partial prior information is examined. More specifically, an underlying linear discrete-time system is considered where the statistics of the driving noise is assumed to be known only partially; i.e. a class of noise inputs is given from which the underlying actual noise is assumed to be chosen. A set-valued estimator is then derived and the conditional expectation is shown to belong to an ellipsoidal set consistent with the measurements and the underlying noise description. When the underlying noise is consistent with the underlying partial model and a sequence of realized measurements is given then the ellipsoidal, set-valued, estimate is computable using a Kalman filter-type algorithm. The estimator inherently solves a stochastic model validation problem whereby it is possible to estimate the consistency between the assumed model, knowledge on the partial prior noise statistics and the measured data.

Minimizing Euclidian State Estimation Error for Linear Uncertain Dynamic Systems Based on Multisensor and Multi-Algorithm Fusion

In this paper, a multisensor linear dynamic system with model uncertainty and bounded noises is considered. Based on previously developed set-valued estimation methods in terms of convex optimization, we propose several efficient algorithms of centralized sensor fusion, distributed sensor fusion, and multi-algorithm fusion to minimize the Euclidian estimation error of the state vector. Obviously, an ellipsoid/box with a larger “size” cannot be in general guaranteed to contain another ellipsoid/box with a smaller “size” since centers and shapes of the two ellipsoids/boxes may be different from each other. This fact and the complementary advantages of multiple sensors and multiple algorithms motivate us to construct multiple estimation ellipsoids/boxes squashed along each entry of the state vector as much as possible respectively by using the technique of multiple differently weighted objectives. Then intersection fusion of these estimation ellipsoids/boxes yields a final Euclidian-error-minimized state estimate. Numerical examples show that the new method with multi-algorithm at both the sensors and the fusion center can significantly reduce the Euclidian estimation error of the state.

Online Model Identification for Set-valued State Estimators With Discrete-Time Measurements

AbstractAlthough Kalman Filters provide an optimal solution to the state estimation problem for linear systems, they require knowledge of an accurate description of the model of the system. More robust approaches to Kalman Filtering for systems with uncertain models have been developed, such as set-valued state estimators, which identify the set of possible states that the system may be in, given a description of its uncertainties. The theory behind this class of filters contemplates the possibility of determining if a given system model is feasible with respect to its measurements, but does not explore the possibility of explicitly identifying the system online. This work provides insight into the advantages of performing simultaneous model identification for the class of set-valued estimators, by introducing an Adaptive Set-Valued Estimator, and presenting a practical example of its usage. The results of this robust estimator are then compared to those obtained when using a classical set-valued estimator.

Construction of set-valued estimates of reachable sets for some nonlinear dynamical systems with impulsive control

A method of constructing ellipsoidal estimates of reachable sets is proposed for a nonlinear dynamical system with a scalar impulsive control and uncertainty in initial data. A special discontinuous change of time is used to transform the impulsive system under consideration into an ordinary differential inclusion without impulsive components. To estimate reachable sets of the obtained nonlinear differential inclusion, results from the theory of ellipsoidal estimation and theory of evolution equations of set-valued states of dynamical systems under uncertainty are used.

Vision-Based Target Tracking and Surveillance With Robust Set-Valued State Estimation

Tracking a target from a video stream (or a sequence of image frames) involves nonlinear measurements in Cartesian coordinates. However, the target dynamics, modeled in Cartesian coordinates, result in a linear system. We present a robust linear filter based on an analytical nonlinear to linear measurement conversion algorithm. Using ideas from robust control theory, a rigorous theoretical analysis is given which guarantees that the state estimation error for the filter is bounded, i.e., a measure against filter divergence is obtained. In fact, an ellipsoidal set-valued estimate is obtained which is guaranteed to contain the true target location with an arbitrarily high probability. The algorithm is particularly suited to visual surveillance and tracking applications involving targets moving on a plane.

Distributed State Estimation for Hybrid and Discrete Event Systems Using l-Complete Approximations

The topic of this paper is distributed state estimation for time-invariant systems with finite input and output spaces. We assume that the system under investigation can be realised by a hybrid I/S/O-machine, where some of the discrete states may also represent failure modes. Our approach is based on previous work, e.g., Moor and Raisch (1999); Moor et al. (2002), where l-complete approximations were proposed as discrete event abstractions for hybrid dynamical systems. In particular, it has been shown that l-complete approximations can be used to provide set-valued estimates for the unknown system state. Estimates are conservative in the sense that the true state can be guaranteed to be contained in the set-valued estimate. In this contribution, we show that for a class of hybrid systems the same estimate can be obtained via a distributed, or decentralised, approach involving several less complex approximations, which are run in parallel. For a larger class of systems, it can be shown that this approach provides an outer approximation of the estimate provided by a monolithic l-complete estimator. The proposed procedure implies significant computational savings during estimator synthesis, with an only modest increase in on-line effort. The latter is a result of “assembling” the global estimate from the available local estimates. The resulting computational trade-off is explicitly discussed.

Fault Detection and Isolation of an Aircraft Using Set-Valued Observers

AbstractThis paper describes an application of a new fault detection and isolation (FDI) technique based on set-valued observers (SVOs) to a linear parameter varying (LPV) longitudinal aircraft dynamic model. The FDI strategy adopted herein computes and uses the set-valued estimates of the SVOs to eliminate models of the plant that are not compatible with the set of observations provided by the aircraft sensor suite and actuation data. The design of the SVOs takes into account model uncertainty and disturbances, thus avoiding false alarms due to such perturbations. The behavior of the proposed solution is assessed in simulation, by deliberately generating hard and soft sensor/actuator faults. The results show that the faults take, in general, only a few iterations to be detected and isolated, therefore paving the way for the use of the proposed methodology in practical applications.

Statistical aspects of fuzzy monotone set-valued stochastic processes. Application to birth-and-growth processes

The paper considers a particular family of fuzzy monotone set-valued stochastic processes. The proposed setting allows us to investigate suitable α -level sets of such processes, modeling birth-and-growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set-valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.

Set-Valued Estimation of Freeway Traffic Density

Abstract The information about traffic density on a freeway is required by real time traffic responsive ramp flow controllers. Currently, density cannot be measured directly and has to be estimated from traffic flow and/or velocity measurements. This paper describes the algorithm for the set-valued density estimation using the macroscopic first order Cell Transmission Model (CTM) together with combined flow and velocity measurements. It is assumed that freeway cell capacities, on-ramp demands and measurement noise values are not known exactly, but lie within certain intervals. Monotonicity of CTM allows to reduce density estimation to a series of interval arithmetic operations.

ON THE HAMILTONIAN TECHNIQUES FOR DESIGNING NONLINEAR OBSERVERS UNDER SET-MEMBERSHIP UNCERTAINTY

This paper deals with the problem of constructing observers for nonlinear systems subjected to unknown but bounded disturbances. The problem is therefore treated within the framework of the so-called guaranteed (set membership) approach. The solutions are thus given in set-theoretic terms as “information sets” consistent with the model, the measurements and the given constraints on the uncertain items. The novelty in the approach is the application of Hamiltonian techniques which allow to treat nonlinear systems. The observation process is described by an HJB (Hamilton - Jacobi - Bellman) equation for a certain value function (the “information state”) whose level sets are the set-valued estimates. Presented here are two types of HJB equation with some comparison theorems which allow to replace the rather complicated solutions to these equations by their upper and lower estimates, which are simpler to calculate. The application of information states in input design for guaranteed parameter identification is finally considered.

Set-valued solutions to impulsive differential inclusions

This paper deals with the state estimation problem for impulsive control systems described by differential inclusions with measures. The problem is studied under uncertainty conditions with set-membership description of uncertain variables which are taken to be unknown but bounded with given bounds. Such problems arise from mathematical models of dynamical and physical systems for which we have an incomplete description of their generalized coordinates (e.g. the model may contain unpredictable errors without their statistical description). In this setting instead of an isolated trajectory of the dynamical control system we have a tube of such trajectories and the phase state vector should be replaced by the set of its possible values. The techniques of constructing the trajectory tubes and their cross-sections that may be considered as set-valued state estimates to differential inclusions with impulses are studied.

Accuracy of Construction of Approximating Models under Bounded Measurement Noises

If the "output" of a system is measured under an additive noise with predefined set-valued estimate, then there is an optimal size for the vector of parameters of the approximating model under which the guaranteed approximation error is minimal. This result is extended to the case in which the a priori noise estimate depends on the measured parameter.

Set-valued estimation approach to recursive robust H∞ filtering

A recursive robust H∞ filtering algorithm is proposed for a discrete-time uncertain linear system subject to the sum quadratic energy constraint. This type of uncertainty description can accommodate a large class of uncertainties. A set-valued estimation approach is used to tackle the problem. To this end, an augmented energy constraint is produced by combining an energy constraint on the H∞-norm condition of the error dynamics and an inequality relationship between the uncertainty input and output. The robust H∞ filtering problem is formulated as finding the set of estimates that satisfy the augmented constraint. The solutions are given in terms of ellipsoids whose centres are the minimums of the indefinite quadratic function defined by the augmented constraint. Krein space estimation theory is utilised to efficiently deal with the minimisation problem of the indefinite quadratic function it is shown that the robust H∞ filter is simply a special form of, the Krein space Kalman filter. The proposed robust filter has basically the same structure as the information form of a Kalman filter and therefore needs only a small computational effort is required in its implementation. In addition, it can be reduced into versions of robust and nominal filters by tuning the relevant parameters. Numerical examples are presented that verify that; (i) the proposed filter guarantees robustness in the presence of parametric uncertainties; and (ii) its bounding ellipsoidal sets of filtered estimates always contain true states.

THE JOINT MODEL AND STATE ESTIMATION PROBLEM UNDER SET-MEMBERSHIP UNCERTAINTY

The present paper deals with combined estimation of the state and of the system model for a continuous-time system under uncertain perturbations which may arise not only in the inputs but also in the system parameters. This produces model uncertainty of the multiplicative type. The uncertain items are assumed either unknown but bounded by hard or soft bounds or bounded with unspecified bound.They are obtained in recurrent form, through appropriate versions of the Principle of Optimality which produces Dynamic Programming schemes. Also discussed is the case when the perturbations are mixed – partly unknown but bounded and partly with unspecified bound. The result is given by a pointwise or set-valued estimator for the state space variable as well as for the transition function of the investigated system.

A Measurement Feedback Switching Control of Systems with State and Control Constraints

This paper considers measurement feedback switching control of systems with pointwise-in-time state and control constraints. Prior works have proved that switching control strategies have ability to resolve a common conflict in the design of state regulators : meeting performance requirements such as fast response and good disturbance rejection while simultaneously avoiding violation of specified state and control constraints. However, all of such switching control strategies are derived under consideration of full state feedback. In this paper, measurement feedback switching control law is proposed. The main idea is to utilize a set-valued estimation technique which produces a set of possible state vectors consistent with measured output data. Admissibility conditions on the initial state are satisfied, proposed output feedback switching control law is proved to fulfill the constraints and stability requirements. The disadvantage of this approach is that the online computational burden may severely limit its applicability, especially for systems with fast dynamics.

Observer Based Partial Switching Control of Systems with State and Control Constraints

This paper considers output-feedback switching control of systems with pointwise-in-time state and control constraints. In the case of full state feedback, the ability of switching control laws for resolving a common conflict in the design of state regulators has been proved. Under measurement feedback, previous work of the authors derived the switching control law utilizing a set-valued state estimation technique. However, the online computational burden in constructing the set-valued estimates of the plant state may severely limit its applicability. Proposed methods in this paper alleviate this burden. The main idea is to construct an upper bound of certain quadratic form of unmeasured error signals. The resulting measurement feedback switching control law is then constructed in conjunction with this upper bound as an estimation error, rather than set-valued estimate of the state.

Modulation of estimators and confidence sets

An unknown signal plus white noise is observed at $n$ discrete time points. Within a large convex class of linear estimators of $\xi$, we choose the estimator $\hat{\xi}$ that minimizes estimated quadratic risk. By construction, $\hat{\xi}$ is nonlinear. This estimation is done after orthogonal transformation of the data to a reasonable coordinate system. The procedure adaptively tapers the coefficients of the transformed data. If the class of candidate estimators satisfies a uniform entropy condition, then $\hat{\xi}$ is asymptotically minimax in Pinsker’s sense over certain ellipsoids in the parameter space and shares one such asymptotic minimax property with the James–Stein estimator. We describe computational algorithms for $\hat{\xi}$ and construct confidence sets for the unknown signal. These confidence sets are centered at $\hat{\xi}$, have correct asymptotic coverage probability and have relatively small risk as set-valued estimators of $\xi$.