estimation law; and thus it proves that manifold is measurable. In the second layer for is introduced as estimates of coefficients. The mapping error is specified as where . If coefficients are equal to , then mapping error Now it is possible to calculate from above equation and hence the peaking phenomenon is reduced thanks to properties of manifold. The created mapping gives a lot of flexibility in the estimation process. Even it is possible to estimate the value of in the second layer and to calculate the state . Bounding observers Bounding or interval observers constitute a class of observers that provide two estimations of the state simultaneously: one of the estimations provides an upper bound on the real value of the state, whereas the second one provides a lower bound. The real value of the state is then known to be always within these two estimations. These bounds are very important in practical applications, as they make possible to know at each time the precision of the estimation. Mathematically, two Luenberger observers can be used, if is properly selected, using, for example, positive systems properties: one for the upper bound (that ensures that converges to zero from above when , in the absence of noise and uncertainty), and a lower bound (that ensures that converges to zero from below). That is, always See also Moving horizon estimation Kalman filter Extended Kalman filter Positive systems References In-line references General references External links Kalman Filter Explained Simply, Step-by-Step Tutorial of the Kalman Filter with Equations Classical control theory Time series