Konrad Reif

Modification of the extended Kalman filter with an additive term of instability

In this paper we propose an observer for nonlinear systems similar to the extended Kalman filter. The observer gain is computed by a Riccati differential equation assuming a more instable system. The purpose of this modification is two-fold: first the degree of stability can be assigned in advance and secondly this modification allows an effective treatment of the nonlinearities.

Application of homotopy methods to nonlinear control problems

The purpose of this article is to give a concise introduction to the use of homotopy methods for treating nonlinear control problems. The theoretical foundations are briefly sketched, especially some important topics concerning linearisation along trajectories. The main contribution is that for a certain control the deviation of the actual state from the target state is decaying exponentially with time. The proof of this statement contains an explicit design instruction for the required control. This design instruction is discussed in some detail and explicitly carried out for an example system. We conclude with a short comparison of two usual nonlinear control methods with the proposed method by means of the example system.

Linearisation Along Trajectories and the Extended Kalman Filter

Abstract In this paper it is proved that under certain conditions the well known extended Kalman filter is an exponential observer for nonlinear systems in the sense of Kou, Elliott and Tarn (1975). A deterministic framework is presented, which does not require the large amount of preliminary material on stochastic processes. Linearisation along trajectories enables us to use concepts of linear time-varying systems for nonlinear systems. The close relations between control along trajectories, output injection and nonlinear observers are explored. For the synthesis of an appropriate output injection the Riccati differential equation is used. The exponential stability of the estimation error is examined with a Lyapunov function. Via this Lyapunov function the solution of the Riccati differential equation gets a clear meaning in the context of the extended Kalman filter, where a stochastic interpretation of this quantity is difficult.

Ein stochastischer Algorithmus zur Bildgenerierung durch einen zweidimensionalen Markoff-Erneuerungsprozess

Zusammenfassung In diesem Beitrag wird ein stochastischer Algorithmus zur Bildgenerierung durch einen zweidimensionalen Markov Renewal Process (MRP) betrachtet. Die Zustandsraumauswahl und Gittereigenschaften, die die Gitterform (eine Form der Konturstruktur) voraussetzt, werden manuell in Voraus vom Operator bestimmt. Der Algorithmus lässt verschiedene Typen der Gitterformstrukturen zu. Horizontale, senkrechte und diagonale Elemente bei einer 8-fach-Nachbarschaft sind erlaubt. Er bietet eine hervorragende Grundlage für Entwurf, Analyse, Validierung und Qualitätssicherung von Bildverarbeitungsalgorithmen.

Nonlinear feedback stabilization by tangential linearization

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.

The exponential stability of the invariant-norm PCA algorithm

This brief investigates a recently proposed principal component analysis (PCA) algorithm. We prove that that the solutions of the corresponding differential equations converge to the principal eigenvectors of the autocorrelation matrix and calculate an exponential decaying bound for the error.

Stabilität des zeitkontinuierlichen erweiterten Kaiman-Filters

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.

Analog approach for the eigen-decomposition of positive definite matrices☆

Abstract This paper proposes an analog approach for performing the eigen-decomposition of positive definite matrices. We show analytically and by simulations that the proposed circuit is guaranteed to converge to the desired eigenvectors and eigenvalues of positive definite matrices.

Feedback control of nonlinear systems based on tangent linearisation using the multiple shooting method

In this paper we propose a design method for a feedback stabilisation of a nonlinear system which is based on the tangential linearisation of the resulting solution curve in state space. The solution is found by treating a linear control problem for a linear time-varying system together with the original nonlinear system equation. For the linear control design any standard technique can be used, e.g. LQ optimal control. The closed-loop stability of the resulting system is mathematically proved. Also the numerical algorithm for solving the resulting boundary value problem which is based on the multiple shooting method is sketched. A highly unstable nonlinear example system is used to show the practical application of the proposed method and the results are compared to the standard optimal controller.

Nonlinear control using homotopy methods

Homotopy and continuation methods are commonly used tools in numerical analysis to solve nonlinear algebraic equations and optimisation problems. In this paper they are used to generalise the feedback controller design based on the well-known linearisation around an operating point. The result is a nonlinear feedback control law with guaranteed closed loop stability properties. The advantages are very unrestrictive conditions on the plant to be controlled and the possibility to upgrade a reliable linear controller.