What is the minimum function observer order

The design of a minimal order observer which can estimate the state feedback control signal Kx(t) with arbitrarily given observer poles and K<inf>∊</inf>R^pxn, has been tried for years, with the prevailing conclusion that it is an unsolved problem. This paper asserts the following four clear-cut claims. 1) this design problem has been simplified to a set of linear equations K = K<inf>z</inf>diag{c<inf>1</inf>, …, c<inf>r</inf>}D (c<inf>i∊</inf>R^1xm, m = rank(C)) if the observer is strictly proper, where D is already determined and other parameters completely free, and r is the observer order. 2) only this set of linear equations can provide the unified upper bound of r, min{n, v<inf>1</inf>+…+v<inf>p</inf>} and min{n-m, (v<inf>1</inf>-1)+…+(v<inf>p</inf>-1)}, for strictly proper and proper observers, respectively, where v<inf>i</inf> (v<inf>1</inf> ≥ … ≥ v<inf>m</inf>) is the i-th observability index of system (A, B, C, 0). 3) This bound is lower than all other existing ones and is the lowest possible general upper bound. 4) The observer order reduction guaranteed by this bound is very significant even at the computer age.