Mobile Robotics, 2nd Edition

6Identification

The aim of identification is to estimate unmeasured quantities from other measured values, with high precision. In the particular case in which the quantity to estimate is the state vector of an invariant linear system, state observers using pole placement (or Luenberger observers) can be considered efficient tools for identification. In this chapter, we will present several basic concepts of estimation, with the aim of introducing Kalman filtering in the next chapter. In summary, this filtering can be seen as state observation for dynamic linear systems with time-variable coefficients. However, in contrast to more standard observers using a pole placement method, Kalman filtering uses the probabilistic properties of signals. Here we will consider the static (as opposed to the dynamic) case. The unknowns to estimate are all stored in a vector of parameters p, while the measurements are stored in a vector of measurements y. In order to perform this estimation, we will mainly look at the so-called least squares approach which seeks to find the vector p that minimizes the sum of the squares of the errors.

6.1. Quadratic functions

In the case in which the dependency between the vectors p and y is linear, the least squares method is used to minimize a quadratic function. This paragraph recalls several concepts attached to these functions, which are of a particular nature.

6.1.1. Definition

A quadratic function f : ℝⁿ → ℝ is a function of the form:

where Q is ...