Kalman Filter for Process Noise Free Systems

Abstract The purpose of this paper is to study the stability properties of the Kalman filter applied to process noise free systems. Due to the absence of the process noise (also known as state noise), such a system cannot be controllable regarding the process noise. The stability of the Kalman filter is typically based on the boundedness of the error covariance matrix governed by a dynamic Riccati equation (DRE), which is usually ensured by the controllability regarding the process noise and by the observability. This classical result cannot be applied to the considered case because of the lack of controllability. It is shown in this paper that, under the observability condition, the solution of the DRE is bounded and converges to a solution of the algebraic Riccati equation (ARE), regardless of the stability of the considered system. In the case of asymptotically unstable systems, the solution of the ARE is not unique. In this case the relationship between the DRE and the ARE is clarified. It is also shown that the DRE based Kalman filter is always valid, whereas the ARE based Kalman filter cannot be applied when the process noise free system has marginally stable modes. A numerical example is presented to verify the correctness of the theoretical analysis.