Set-valued Filtering And Smoothing

Morrell, Darryl; Stirling, Wynn C.

doi:10.1109/acssc.1988.753990

Cited by 19 publications

(32 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, if P k+1 satisfies LMI (13), and x k+1 is determined by (14), then one-step-ahead state x k+1 resides in its state estimation ellipsoid (x k+1 ¡x k+1 ) T P…”

Section: Theorem 1 For the System (1)-(2) Subject To The Constraint (mentioning

confidence: 99%

“…tr(P k+1 ) subject to (13) (32) andx k+1 is determined by (14), where ¦(x k , u k ) and ¦ 1 (x k ) are defined in (15) and (16), respectively. REMARK 3 We can see from Theorem 1 that the inequalities (13) are linear to the variables P k+1 , G k , L k , and N k , ¿ 1 , ¿ 2 , ¿ 3 .…”

Section: ¡1mentioning

confidence: 99%

“…The topic of set-membership filtering has attracted a growing research interest, since it is based only on the knowledge of the hard bounds of the process and measurement noises [3,6,8,10,13,14,[16][17][18][19][20][21]29]. The idea of set-membership filtering is to provide all possible state estimates that are characterised by the set of state estimates consistent with both the observations received and the unknown but bounded process and measurement noises [3,10,20].…”

Section: Introductionmentioning

confidence: 99%

“…Hence the set-membership filtering problem aims to find the smallest characterisation of the feasible set of the states, rather than providing the most possible states under some optimality criteria, for example, Kalman filtering [2,27,33,34,36,38] and H 1 filtering [28,30,32]. Set-membership filtering is also called set-value filtering as the actual estimate is a set in state space rather than a single vector [14], [17], [21]. 0018-9251/09/$26.00 c°2009 IEEE Set-membership filtering problem was first considered by Witsenhausen [29].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Set-Membership Filtering with State Constraints

Yang

2009

IEEE Trans. Aerosp. Electron. Syst.

View full text Add to dashboard Cite

In this paper, the problem of set-membership filtering is considered for discrete-time systems with equality and inequality constraints between their state variables. We formulate the problem of set-membership filtering as finding the set of estimates that belong to an ellipsoid. A centre and a shape matrix of the ellipsoid are used to describe the set of estimates and the solution to the set of estimates is obtained in terms of matrix inequality.Unknown but bounded process and measurement noises are handled under the inequality constraints by using S-procedure. I. INTRODUCTIONThe topic of set-membership filtering has attracted a growing research interest, since it is based only on the knowledge of the hard bounds of the process and measurement noises [3,6,8,10,13,14,[16][17][18][19][20][21]29]. The idea of set-membership filtering is to provide all possible state estimates that are characterised by the set of state estimates consistent with both the observations received and the unknown but bounded process and measurement noises [3,10,20]. The set-membership filtering can find a region in the state-space that guarantees to contain the unknown true state vector [13]. Hence the set-membership filtering problem aims to find the smallest characterisation of the feasible set of the states, rather than providing the most possible states under some optimality criteria, for example, Kalman filtering [2,27,33,34,36,38] Schweppe [20] for the set-membership filtering problem under the energy-type constraint. The solution to a set-membership filtering problem with the individual instantaneous constraints was determined by describing a bounding ellipsoid to the set of possible states [3]. The resulting filter is similar to that proposed by Schweppe [20], but it has an important advantage that the gain matrix does not depend on the particular output observations and is therefore precomputable. Recently, attempts have been made to deal with the set-membership filtering problems for uncertain systems. For example, a combinational ellipsoidal bounded uncertain system was considered in [16]. The sum quadratic uncertain systems have been studied in [17]-[19]. For systems with both bounded noise and parametric uncertainty, a technique-based semi-definite optimization method has been proposed in [8] to handle several inequalities. It has led to a simple and neat algorithm. We adopt this technique in this paper.However, in practical applications such as vehicle tracking, there are some hard constraints on the vehicle position when the vehicle is travelling on a known road (straight line or curve). Such tracking problem can be regarded as a filtering problem incorporating a state constraint with the road network information from digital maps [11,22,37]. This paper intends to study the set-membership filtering problem incorporating state constraints. The filtering problems with state constraints have been studied within the Kalman filter framework [9,24,31]. There have been several approaches to address this problem, which can be ...

show abstract

“…Thus, if P k+1 satisfies LMI (13), and x k+1 is determined by (14), then one-step-ahead state x k+1 resides in its state estimation ellipsoid (x k+1 ¡x k+1 ) T P…”

Section: Theorem 1 For the System (1)-(2) Subject To The Constraint (mentioning

confidence: 99%

Section: ¡1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Set-Membership Filtering with State Constraints

Yang

2009

IEEE Trans. Aerosp. Electron. Syst.

View full text Add to dashboard Cite

show abstract

“…While focussing on convex sets of functions, this way of generalizing classical probability theory has been studied in [13]. Also, approaches to Bayesian state estimation such as [14] and [15] are based on sets of densities. All of these concepts are aimed at providing wider possibilities of modeling uncertainty in order to overcome the limitations of a purely stochastic point of view.…”

Section: A Generic State Prediction With Sets Of Densitiesmentioning

confidence: 99%

Nonlinear Model Predictive Control considering stochastic and systematic uncertainties with sets of densities

Hekler

Lyons

Noack

et al. 2010

2010 IEEE International Conference on Control Applications

View full text Add to dashboard Cite

Abstract-In Model Predictive Control, the quality of control is highly dependent upon the model of the system under control. Therefore, a precise deterministic model is desirable. However, in real-world applications, modeling accuracy is typically limited and systems are generally affected by disturbances. Hence, it is important to systematically consider these uncertainties and to model them correctly. In this paper, we present a novel Nonlinear Model Predictive Control method for systems affected by two different types of perturbations that are modeled as being either stochastic or unknown but bounded quantities. We derive a formal generalization of the Nonlinear Model Predictive Control principle for considering both types of uncertainties simultaneously, which is achieved by using sets of probability densities. In doing so, a more robust and reliable control is obtained. The capabilities and benefits of our approach are demonstrated in real-world experiments with miniature walking robots.

show abstract

Density-ratio robustness in dynamic state estimation

Benavoli

Zaffalon

2013

Mechanical Systems and Signal Processing

View full text Add to dashboard Cite

The filtering problem is addressed by taking into account imprecision in the knowledge about the probabilistic relationships involved. Imprecision is modelled in this paper by a particular closed convex set of probabilities that is known with the name of density ratio class or constant odds-ratio (COR) model. The contributions of this paper are the following. First, we shall define an optimality criterion based on the squared-loss function for the estimates derived from a general closed convex set of distributions. Second, after revising the properties of the density ratio class in the context of parametric estimation, we shall extend these properties to state estimation accounting for system dynamics. Furthermore, for the case in which the nominal density of the COR model is a multivariate Gaussian, we shall derive closed-form solutions for the set of optimal estimates and for the credible region. Third, we discuss how to perform Monte Carlo integrations to compute lower and upper expectations from a COR set of densities. Then we shall derive a procedure that, employing Monte Carlo sampling techniques, allows us to propagate in time both the lower and upper state expectation functionals and, thus, to derive an efficient solution of the filtering problem. Finally, we empirically compare the proposed estimator with the Kalman filter. This shows that our solution is more robust to the presence of modelling errors in the system and that, hence, appears to be a more realistic approach than the Kalman filter in such a case.

show abstract

Set-valued Filtering And Smoothing

Cited by 19 publications

References 9 publications

Set-Membership Filtering with State Constraints

Set-Membership Filtering with State Constraints

Nonlinear Model Predictive Control considering stochastic and systematic uncertainties with sets of densities

Density-ratio robustness in dynamic state estimation

Contact Info

Product

Resources

About