Guaranteed non-linear estimation using constraint propagation on sets

Abstract

Bounded-error estimation is the estimation of the parameter or state vector of a model from experimental data, under the assumption that some suitably dened errors should belong to some prior feasible sets. When the model outputs are linear in the vector to be estimated, a number of methods are available to enclose all estimates that are consistent with the data into simple sets such as ellipsoids, orthotopes or parallelotopes, thereby providing guaranteed set estimates. In the nonlinear case, the situation is much less developed and there are very few methods that produce such guaranteed estimates. In this paper, the discrete-time problem is cast into the more general framework of constraint satisfaction problems. Algorithms rather classical in the area of interval constraint propagation are extended by replacing intervals by more general subsets of real vector spaces. This makes it possible to propose a new algorithm that contracts the feasible domains for each uncertain variable optimally (i.e., no smaller domain could be obtained) and e¢ciently. The resulting methodology is illustrated on discrete-time nonlinear state estimation. The state at time k is estimated either from past measurement only or from all measurements assumed to be available from the start. Even in the causal case, prior information on the future value of the state and output vectors, due for instance to physical constraints, is readily taken into account.

Interval unions

“…Interval Newton methods to solve this problem are based on the interval mean value theorem applied to (13). Formally, if y ∈ x such that f (y) = 0 then…”

“…Applications of interval analysis comprise packing problems [30], robotics [7,20], localization and map building [12,13], and the protein folding problem [19].…”

Fault tolerance evaluation of nonlinear systems using viability theory

“…A constraint satisfaction problem (CSP) on sets can be formulated as a 3-tuple H = V, D,C [11], where…”

“…It is well known that the solution of this kind of problems has a high complexity [11]. In practice, the sets that define the variable domains in Algorithm 1 are approximated by intervals.…”

Constraint propagation on quadratic constraints

“…Many real word problems are continuous constraint satisfaction problems, often high dimensional ones. Typical applications include robotics C. Grandon & Papegay [10], Merlet [35], localization and map building Jaulin [27], Jaulin et al [28], biomedicine Cruz & Barahona [13], or the protein folding problem Krippahl & Barahona [32].…”