Control theoretic approach to inertial navigation systems

Abstract

In this work, the analysis of inertial navigation systems (INS) is approached from a control theory point of view. Linear error models are presented and discussed and their eigenvalues are computed in several special cases. It is shown that the exact expressions derived for the eigenvalues differ slightly from the commonly used expressions. The observability of INS during initial alignment and calibration at rest is analyzed. A transformation that is based on physical insight is introduced that enables us to determine the unobservable subspace and states rather easily by inspection of the new dynamics matrix. Finally, the relationship between system observability and quality of estima- tion is presented. ever, as will be shown in the sequel, the inclusion of the vertical channel does alter the eigenvalues slightly. In the examination of system observability, we use a straightforward transforma- tion into observable and unobservable subsystems that, in turn, expose the states that hamper the estimation of INS errors during the initial alignment and calibration phase of operation. This approach was adopted successfully in the past by Kor- tum3 who considered the problem of INS platform alignment in which the measurements were the horizontal accelerometer outputs, whereas in the present case the measurements are the INS horizontal velocity components. In addition, the compari- son of this approach to the classical one that is presented in the present paper, as well as the discussion of uniqueness and the relationship between observability and quality of estimation, provide additional insight into the observability issue. It is hoped that the examination of INS as a unified system from a control theory point of view will shed more light on the system and contribute additional insight into the analysis of INS. In the next section, we describe the INS linear error model that will be the investigated plant. In Sec. Ill, we investigate the eigenvalues of INS in various phases of operation, and in Sec. IV, the issues of controllability and observability of the system are discussed. The relation between system observability and the ability to estimate its states during initial alignment is discussed in Sec. V. Finally, in Sec. VI, the conclusions are presented.