The authors analyze the error behavior for the discrete-time extended Kalman filter for general nonlinear systems in a stochastic framework. In particular, it is shown that the estimation error remains bounded if the system satisfies the nonlinear observability rank condition and the initial estimation error as well as the disturbing noise terms are small enough. This result is verified by numerical simulations for an example system.

Stochastic stability of the discrete-time extended Kalman filter

Applying the unscented Kalman filter for nonlinear state estimation

We analyze the convergence aspects of the invariant extended Kalman filter (IEKF), when the latter is used as a deterministic nonlinear observer on Lie groups, for continuous-time systems with discrete observations. One of the main features of invariant observers for left-invariant systems on Lie groups is that the estimation error is autonomous. In this paper we first generalize this result by characterizing the (much broader) class of systems for which this property holds. For those systems, the Lie logarithm of the error turns out to obey a linear differential equation. Then, we leverage this “log-linear” property of the error evolution, to prove for those systems the local stability of the IEKF around any trajectory, under the standard conditions of the linear case. One mobile robotics example and one inertial navigation example illustrate the interest of the approach. Simulations evidence the fact that the EKF is capable of diverging in some challenging situations, where the IEKF with identical tuning keeps converging.

https://hal-mines-paristech.archives-ouvertes.fr/hal-01692380/document

The Invariant Extended Kalman Filter as a Stable Observer

We analyze the behavior of the extended Kalman filter as a state estimator for nonlinear deterministic systems. Using the direct method of Lyapunov, we prove that under certain conditions, the extended Kalman filter is an exponential observer, i.e., the dynamics of the estimation error is exponentially stable. Furthermore, we discuss a generalization of the Kalman filter with exponential data weighting to nonlinear systems.

The extended Kalman filter as an exponential observer for nonlinear systems

We analyze the convergence aspects of the invariant extended Kalman filter (IEKF), when the latter is used as a deterministic non-linear observer on Lie groups, for continuous-time systems with discrete observations. One of the main features of invariant observers for left-invariant systems on Lie groups is that the estimation error is autonomous. In this paper we first generalize this result by characterizing the (much broader) class of systems for which this property holds. Then, we leverage the result to prove for those systems the local stability of the IEKF around any trajectory, under the standard conditions of the linear case. One mobile robotics example and one inertial navigation example illustrate the interest of the approach. Simulations evidence the fact that the EKF is capable of diverging in some challenging situations, where the IEKF with identical tuning keeps converging.

The invariant extended Kalman filter as a stable observer

Brief Paper: An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability

Linearisation Along Trajectories and the Extended Kalman Filter

In this article an observer is introduced for state estimation of nonlinear deterministic systems. The observer gain is computed by a Riccati differential equation similar to the extended Kalman filter. The modification of the Riccati differential equation for the proposed observer is based on the application of H,-filtering and robust control methods, which allows an effective treatment of the nonlinearities. Thus, the high performance of the extended Kalman filter can be further improved.

An Observer for Nonlinear Systems Based on H,-Filtering Techniques

In this article we propose a design method for feedback stabilization of a nonlinear system based on tangential linearization along the solution curve in the state space. The tangent vector along this solution satisfies a linear time-varying system, which can be controlled by standard design techniques, e.g. LQ optimal control. We present a method to obtain stabilizing feedback for the nonlinear system from the (LQ optimal) feedback of the tangential linearization. The closed-loop stability of the resulting system is mathematically proved. The proposed design is then applied to a highly nonlinear and unstable electro-magnetic example system and compared with other nonlinear control design methods by means of this example.

Nonlinear feedback stabilization by tangential linearization

In diesem Aufsatz wird das erweiterte Kaiman-Filter in seiner Funktion als ein allgemeiner Zustandsschätzer für nichtlineare stochastische Systeme untersucht. Es wird gezeigt, daß der dabei auftretende Schätzfehler unter gewissen Bedingungen im Quadratmittel beschränkt ist. Diese Bedingungen beinhalten insbesonderey daß der Anfangsschätzfehler und die störenden Rauschterme klein genug sind. Die angegebenen theoretischen Ergebnisse sind durch numerische Simulationen veranschaulicht, wobei an einigen ausgewählten Fällen beschränkte sowie divergente Schätzfehler präsentiert werden.

Konrad Reif

Papers

Brief Paper: An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability

Linearisation Along Trajectories and the Extended Kalman Filter

An Observer for Nonlinear Systems Based on H,-Filtering Techniques

Nonlinear feedback stabilization by tangential linearization

Stabilität des zeitkontinuierlichen erweiterten Kaiman-Filters