Discrete-time Nonlinear Observer Research Articles

Stability of a class of discrete-time nonlinear recursive observers

This work provides a framework for nominal and robust stability analysis for a class of discrete-time nonlinear recursive observers (DNRO). Given that the system has linear output mapping, local observability and Jacobian matrices satisfying certain conditions, the nominal and robust stability of the DNRO is defined by the property of estimation error dynamics and is analyzed using Lyapunov theory. Moreover, a simultaneous state and parameter estimation scheme is shown to be Input-to-State Stable (ISS), and adaptively reduce plant-model mismatch on-line. Three design strategies of the DNRO that satisfy the stability results are given as examples, including the widely used extended Kalman filter, extended Luenberger observer, and the fixed gain observer.

Modular Design of Discrete-time Nonlinear Observers for State and Disturbance Estimation

This work studies the problem of design of discrete-time nonlinear observers in the presence of exogenous disturbances. In particular, the present work proposes a systematic design method for discrete-time nonlinear state and disturbance observers in modular form, consisting of an observer for the disturbance-free part of the system, along with a disturbance observer and a state-estimate corrector. The modular observer is first defined and characterized in a general setting and then, a systematic design method is developed on the basis of exact linearization with eigenvalue assignment. Sufficient conditions for feasibility of the proposed nonlinear modular observer design are derived.

Map Invariance and the State Reconstruction Problem for Nonlinear Discrete-time Systems

The role of map invariance is examined within the context of the dynamic state reconstruction problem for nonlinear discrete-time systems. In particular, the key notion of invariant manifold for maps in nonlinear discrete-time dynamics is shown to be conceptually insightful and technically quite effective to address important issues related to the deterministic observerbased nonlinear state estimation problem in the discrete-time domain. As a necessary first methodological step, the problem of quantitatively characterizing the asymptotic long-term behavior of nonlinear discrete-time systems with a skew-product structure using the notion of map invariance is revisited. The formulation of this problem can be naturally realized through a system of invariance functional equations (FEs), for which a set of existence and uniqueness conditions of a solution is provided. Under a certain set of conditions, it is shown that the invariant manifold computed attracts all system trajectories/orbits, and therefore, the asymptotic long-term dynamic behavior of the system is determined through the restriction of the discrete-time system dynamics on the invariant manifold. Within the above analytical framework, the nonlinear full-order observer design problem in the discrete-time domain is considered, appropriately formulated and an interpretation of previous work on the problem is attempted through the notion of invariant manifolds for maps. Furthermore, this framework allows the development of a new approach to the nonlinear reduced-order observer design problem for multiple-output systems in the discrete-time domain, which is also presented in the present work. Finally, the performance of the proposed nonlinear reduced-order discrete-time observer is assessed in an illustrative bioreactor example through simulations.

The Construction of Nonlinear Discrete-time Observers with Local Lipschitz Conditions

In this paper, we present a new methodology for the construction of discrete-time observers for nonlinear discrete-time systems which satisfy Lipschitz condition on bounded sets. The problem under consideration is mathematically addressed through the existence of a continuous mapping in the state space that maps the linear system with an output injection to the observing system provided that the observing orbit is bounded. The existence of such a continuous mapping is proved by Banach's fixed point theorem by making use of a cut-off function, which is used to construct the observer. The proposed method can be directly carried over to the continuous-time cases. An example for discrete-time Henon system with non-smooth output is provided to illustrate the theoretical outcomes and to show the effectiveness of the proposed observer.

Necessary and Sufficient Conditions for State Equivalence to a Nonlinear Discrete-Time Observer Canonical Form

In this technical note, we obtain necessary and sufficient conditions for a multi-input, multi-output, discrete-time nonlinear system to be state equivalent to a nonlinear observer form, and for an uncontrolled multi-output system to be state equivalent to a linear observer form. We adopt a geometric approach, and the proofs are constructive with respect to the required coordinate change.

On the non-linear discrete-time observer design problem

A set of conditions is presented, under which a non-linear discrete-time observer inducing linear estimation error dynamics exists for non-linear continuous (C0) discrete-time systems. In particular, the existence of a homeomorphism in state space is established that maps the orbits of a linear system with an output injection term onto the observing system, implying the existence of an invariant attracting manifold for the extended system. Within the aforementioned framework, it is shown that the discrete-time version of the Hartman-Grobman Theorem can be naturally reproduced as a special case. Finally, the performance of the proposed non-linear discrete-time observer is evaluated in two representative case studies, where two different dynamical systems of the Lozi and H?non-type are considered that exhibit non-linear and chaotic behaviour.

Circle-criterion approach to discrete-time nonlinear observer design

This paper addresses the design of discrete-time nonlinear observers through the circle criterion. The new design method is mainly devoted to either globally Lipschitz systems or bounded-state systems whose nonlinearities can be decomposed into a linear combination of positive-slope nonlinearities. The observer design is not restricted to systems with positive-slope nonlinearities, but it encompasses systems with non-positive-slope nonlinearities too. Stability conditions of the observation error are given in terms of numerically tractable linear matrix inequalities. Illustrative examples are presented in order to highlight the main features and advantages of the new proposed technique over some classical designs.

Discrete-time nonlinear observer design with general criteria

A class of discrete-time nonlinear system and measurement equations having incrementally conic nonlinearities and finite energy disturbances is considered. A linear matrix inequality-based design approach is presented, which guarantees the satisfaction of a variety of performance criteria ranging from simple estimation error boundedness to dissipativity. Some simulation examples are included to illustrate and provide support to the proposed design methodology.

A Direct Method for the Construction of Nonlinear Discrete-Time Observer With Linearizable Error Dynamics

We provide a direct method for the design of nonlinear discrete-time observers by a construction of a change of variables. An explicit expression of the change of variables is given. Some simulations for chaotic systems, such as Lozi system and He/spl acute/non system, are provided to illustrate the proposed method.

Nonlinear discrete-time observer design with linearizable error dynamics

A necessary and sufficient condition for the existence of a discrete-time nonlinear observer with linearizable error dynamics is provided. The result can be applied to any real analytic nonlinear system whose linear part is observable. The necessary and sufficient condition is the solvability of a nonlinear functional equation. Furthermore, the well-known Siegel's theorem on the linearizability of a mapping is naturally reproduced in a corollary. The proposed observer design method is constructive and can be applied approximately to any sufficiently smooth, linearly observable system yielding a local observer with approximately linear error dynamics.

Reduced-Order Nonlinear Discrete-Time Observer Design Using Functional Equations

The present research work proposes a new approach to the reduced-order nonlinear discrete-time observer design problem. Based on the early ideas that influenced the development of the linear Luenberger observer, the proposed approach develops a nonlinear analogue. The formulation of the reduced-order nonlinear discrete-time observer design problem is realized via a system of first-order linear nonhomogeneous functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived using results from functional equations theory. The solution to the above system of functional equations can be proven to be locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package.

Discrete-time nonlinear observer design using functional equations

The present research work proposes a new approach to the discrete-time nonlinear observer design problem. Based on the early ideas that influenced the development of the linear Luenberger observer, the proposed approach develops a nonlinear analogue. The formulation of the discrete-time nonlinear observer design problem is realized via a system of first-order linear nonhomogeneous functional equations, and a rather general set of necessary and sufficient conditions for solvability is derived using results from functional equations theory. The solution to the above system of functional equations can be proven to be locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package.

Anti-saturation discrete-time nonlinear controllers and observers

In this paper, a nonlinear controller design method based on a nonlinear state space model is proposed. By adding differentiable saturation elements, this controller design scheme can deal with the saturation of the manipulated variables. Linear approximation is successively executed at each sampling time. The velocity-type controller and observer are designed based on the linear model. For the calculation of linear approximation, the information of the states at the previous sampling time must be known. A nonlinear observer using integration of nonlinear state equations is adopted for the state estimation. The observer gain of the linear observer is applied to the nonlinear observer. This control system can stabilize open-loop unstable processes and non-minimum phase systems. The performance is demonstrated by its successful application to control problems of a CSTR, which has two stable points and one unstable steady-state point.

Remarks on linearization of discrete-time autonomous systems and nonlinear observer design

This paper presents necessary and sufficient conditions under which a discrete-time autonomous system with outputs is locally state equivalent to an observable linear system or a system in the nonlinear observer form (Krener and Isidori, 1983). In particular, an open problem raised in Lee and Nam (1991), namely the observer linearization problem, is solved for a nonlinear system which may not be invertible (i.e., the mapping f may not be a local diffeomorphism ). As a consequence, the nonlinear observer design problem is solved by means of exact linearization techniques.