Observer Design —— A Survey

This paper surveys the results of observer design for linear time-invariant(L-T-I) deterministic irreducible open-loop systems(OLS), the most basic type of OLS. An observer estimates Kxx(t) signal where K is a constant and x(t) is the state vector of the OLS. Thus, an observer can be used as a feedback controller that implements state feedback control(SFC) or Kxx(t)-control,and observer design is therefore utterly important in all feedback control designs of state space theory. In this survey, the observer design results are divided into three categories and for three respective main purposes. The first category of observers estimate signal Kxx(t) only with a given K, and this survey has four conclusions: 1) Function observer that estimates Kxx(t) directly is more general than state observer that estimates x(t), and may be designed with order lower than that of state observer, and the additional design objective is to minimize observer order; 2) The function observer design problem has already been simplified to the solving of a single set of linear equations only while seeking the lowest possible number of rows of the solution matrix, and an apparently most effective and general algorithm of solving such a problem can guarantee unified upper and lower bounds of the observer order; 3) Because such a single set of linear equations is the simplest possible theoretical formulation of the design problem and such theoretical observer order bounds are the lowest possible, and because the general, simple, and explicit theoretical formula for the function observer order itself do not exist, the theoretical part of this design problem is solved; 4) Because the function observer order is generically near its upper bound, further improvement on the computational design algorithm so that the corresponding observer order can be further reduced,is generically not worthwhile. The second category of observers further realize the loop transfer function and robustness properties of the direct SFC, and the conclusion of this survey is also fourfold: 1) To fully realize the loop transfer function of a practically designed Kxx(t)-control, the observer must be an output feedback controller(OFC) which has zero gain to OLS input; 2) If parameter K is separately designed before the observer design, as in the separation principle which has been followed by almost all people for over half of a century, then OFC that estimates Kxx(t) does not exist for almost all OLS s; 3) As a result, a synthesized design principle that designs an OFC first and is valid for almost all OLS s, is proposed and fully developed, the corresponding K will be designed afterwards and will be constrained by the OFC order as well as the OFC parameters; 4) Although the Kxx(t)-control is constrained in this new design principle and is therefore called the "generalized SFC"(as compared to the existing SFC in which K is unconstrained), it is still strong enough for most OLS s and this new design principle overcomes many fundamental drawbacks of the existing separation principle. The third category of observers estimate Kxx(t) signal at special applications such as fault detection and identification and systems with time delay effects. Using directly the result of OFC that estimates Kxx(t) of the second category, these observers can be generally and satisfactorily designed.