the behavior of function J(x) for the QL estimator is the same as WLS if the measurement error yi(x) is within given bounds.
Figure 2.5 Objective function for the QL estimator as a function of the error (one measurement).
The objective function value of QL estimator for a given measurement has a quadratic shape (like WLS estimator) if the measurement error value is within a tolerance T; otherwise, this objective function component has an absolute value shape like LAV estimator (see Figure 2.5). Note that a tolerance T must be selected in advance.
2.5.4.2 QL Mathematical Programming Formulation
The QL mathematical programming formulation is
(2.37a)
subject to
(2.37c)
(2.37d)
If the general formulation (2.36) is to be recast as a mathematical programming problem, it is necessary to include (i) a binary variable vector b, (ii) a positive variable vector s, and (iii) two sets of constraints (2.37b). The resulting formulation (2.37) is a mixed integer nonlinear problem.
Again, the set of constraints (2.37e) of problem (2.37) can be relaxed, leading to a nonlinear problem, whose optimal solution generally meets constraints (2.37e).
2.5.5 Least Median of Squares
Least median of squares (LMS) is a robust estimator [2]. The objective function to be minimized is the squared measurement error whose value is the median of all squared measurement errors. The key idea underlying this technique is that the median of a set of values is a more robust estimate than the mean.
2.5.5.1 LMS General Formulation
The general formulation of the LMS estimator is
(2.38a)
subject to
(2.38b)
(2.38c)
(2.38d)
where the function median(x1, x2, …, xn) computes the median value of set {x1, x2, …, xn}.
Reference [25] proposes a mathematical programming formulation for LMS estimator to be applied to a linear estimator. In this chapter, this formulation is applied to the state estimation problem, and the mathematical programming formulation of LMS estimator is presented below.
2.5.5.2 LMS Mathematical Programming Formulation
The LMS mathematical programming formulation is
subject to
(2.39c)
(2.39d)
(2.39f)
where parameter M is a sufficiently large constant and parameter ν identifies the median and can be computed as [26]
where n is the number of state variables and function int(x) denotes the integer part of x.
The rationale for formulation (2.39) can be graphically explained. Figure 2.6 depicts the set of measurement errors yi(x), which are sorted by value. The colored zone is delimited by the bounds [−TLMS, TLMS] and includes the smaller measurement errors up to the position ν.
Figure 2.6 Graphical representation of the LMS estimator.
The objective function (2.39a) minimizes the variable TLMS, i.e. the width of the colored zone. The vector constraints (2.39b) and (2.39e)
