The goal of this technical note is to design interval observers for a class of nonlinear continuous-time systems. The first part of this work shows that it is usually possible to design an interval observer for linear systems by means of linear time-invariant changes of coordinates even if the system is not cooperative. This result is extended to a class of nonlinear systems using partial exact linearisations. The proposed observers guarantee to enclose the set of system states that is consistent with the model, the disturbances and the measurement noise. Moreover, it is only assumed that the measurement noise and the disturbances are bounded without any additional information such as stationarity, uncorrelation or type of distribution. The proposed observer is illustrated through numerical simulations.

Interval State Estimation for a Class of Nonlinear Systems

We propose a new design method of asymptotic observers for a class of nonlinear mechanical systems: Lagrangian systems with configuration (position) measurements. Our main contribution is to introduce a state (position and velocity) observer that is invariant under any changes of the configuration coordinates. The observer dynamics equations, as the Euler-Lagrange equations, are intrinsic. The design method uses the Riemannian structure defined by the kinetic energy on the configuration manifold. The local convergence is proved by showing that the Jacobian of the observer dynamics is negative definite (contraction) for a particular metric defined on the state-space, a metric derived from the kinetic energy and the observer gains. From a practical point of view, such intrinsic observers can be approximated, when the estimated configuration is close to the true one, by an explicit set of differential equations involving the Riemannian curvature tensor. These equations can be automatically generated via symbolic differentiations of the metric and potential up to order two. Numerical simulations for the ball and beam system, an example where the scalar curvature is always negative, show the effectiveness of such approximation when the measured positions are noisy or include high frequency neglected dynamics.

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An intrinsic observer for a class of Lagrangian systems

In this paper, an observer-based sliding mode control (SMC) problem is investigated for a class of uncertain delta operator systems with nonlinear exogenous disturbance. A novel robust stability condition is obtained for a sliding mode dynamics by using Lyapunov theory in delta domain. Based on a designed sliding mode observer, a sliding mode controller is synthesized by employing SMC theory combined with reaching law technique. The robust asymptotical stability problem is also discussed for the closed-loop system composed of the observer dynamics and the state estimation error dynamics. Furthermore, the reachability of sliding surfaces is also investigated in state-estimate space and estimation error space, respectively. Finally, a numerical example is given to illustrate the feasibility and effectiveness of the developed method.

Observer-based sliding mode control for a class of discrete systems via delta operator approach

Full-order nonlinear observer-based excitation controller design for interconnected power systems via exact linearization approach

In this note, we address the problem of transforming a nonlinear system into nonlinear observer canonical form in the extended state-space with the aid of dynamic system extension and introduction of virtual outputs. As an intermediate step for the general problem, we consider a restricted structure of dynamic extension, which is obtained, roughly speaking, by adding the chains of integrators to the outputs of original system. We propose sufficient conditions which can be verified using the system dynamics expressed in their original coordinates. An illustrative example is included that demonstrates the advantage of the proposed method over the conventional method.

Nonlinear observer design by dynamic observer error linearization

Presents a state observer for a class of nonlinear systems based on the input output linearization. While the previous result presented state observers for nonlinear systems of full relative degree, we proposed a procedure fur the design of nonlinear state observers which do not require the hypothesis of full relative degree. Assuming that there exists a global state observer for internal dynamics and that some functions are globally Lipschitz, we can design a globally convergent state observer. It is also shown that if the zero dynamics are locally exponentially stable, then there exists a local state observer. An example is given to illustrate the proposed design of nonlinear state observers.

Input output linearization approach to state observer design for nonlinear system

In this paper, we present a state observer for single-input/single-output nonlinear systems which fail to have well defined relative degree. It is shown that there exists a local state observer for a nonlinear system if it has robust relative degree n. The proposed observer utilizes the coordinate change which transforms a system into an approximate normal form. The proposed method is applied to a ball and beam system, and simulation results show that substantial improvement in the performance was achieved compared with other local observers.

N.H. Jo

Papers

Input output linearization approach to state observer design for nonlinear system

A state observer for nonlinear systems and its application to ball and beam system