What is the minimum function observer order?

Abstract

The design of a minimal order observer which can estimate the state feedback control signal Kx( t) with arbitrarily given observer poles and K, has been worked on for many years, with the prevailing conclusion that it is an unsolved problem. This paper asserts for the first time that this design problem has been simplified to a set of linear equations K = K z diag{c 1,…,c r } D, where D is fully determined and other parameters are completely free, and where r is the observer order. This paper also asserts that only this set of linear equations can provide the unified upper bound of r, min{ n, v 1 + … + v p } and min { n − m,( v 1 − 1) + … + ( v p − 1)}, for strictly proper and proper observers, respectively, where n, m, p and v i ( i = 1,…, p) are the plant order, number of outputs, number of inputs, and the descending order observability indexes, respectively. This general upper bound is lower than all other existing ones and is the lowest possible general upper bound.