Set-Membership Filtering with State Constraints

Abstract

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 ...

Regression with Set-Valued Categorical Predictors

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Set membership filtering for discrete saturated time‐varying networked systems with incomplete measurements under round‐robin protocol

“…The set membership filtering has been concerned by many scholars that the system noise is unknown but bounded (UBB) [7][8][9]. The set membership filtering does not require knowledge of the statistical characteristics of noise, only the bounds of the noise, and can ensure that all states are included in the estimated set.…”

Kalman-Filter-Based Unconstrained and Constrained Extremum-Seeking Guidance on SO(3)

“…Using an LMI approach to Kalman filtering, state constraints, or set membership can easily be introduced. An LMI-based approach is used for set membership filtering for equality, inequality, and linearized nonlinear constraints in [21,22]. Unlike the gain-projected Kalman filter, the LMI-based filter modifies the Kalman gain so that the state estimate is constrained during the correction step, and thus no extra constraining step is needed to ensure that the state estimates satisfy the constraints.…”

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“…Using [21,22] as inspiration, a novel LMI-based Kalman filter is introduced to estimate the constrained gradient for the extremum-seeking guidance problem. The filter has a different structure, different assumptions on the system's characteristics, and a different derivation from that in [21], and it can handle both the linear inequality and norm constraints placed on the gradient estimates. These attributes make the LMI-based filter well suited for estimating the constrained gradient in extremum-seeking applications.…”