Nonlinear Observer Design Problem Research Articles

Invariant observer design for scan matching-aided localization: A tutorial

Abstract Aided localization, a nonlinear observer design problem, is critical to a range of mobile robotics applications. This paper provides a tutorial presentation of the required background, design steps and calculations of a scan matching-aided localization observer using the novel invariant (symmetry-preserving) observer design methodology. The Invariant Extended Kalman Filter approach is used to systematically obtain the gains of the resulting design, and compared against a conventional, non-invariant Extended Kalman Filter design.

Reduced‐order observer design for one‐sided Lipschitz time‐delay systems subject to unknown inputs

This study addresses the observer design problem for a class of one-sided Lipschitz time-delay systems in the presence of unknown inputs. The non-linearities are assumed to satisfy the one-sided Lipschitz and quadratically inner-bounded conditions; hence, a wider class of non-linear systems is investigated in this work. A novel approach for the non-linear observer design problem subject to time delays and disturbances is proposed. Both H ∞ observer design and asymptotic observer design with reduced-order are introduced. To deal with the time-delay issue, Wirtinger-based integral inequality, which encompasses the Jensen inequality, is employed to derive less conservative synthesis conditions in linear matrix inequalities form. Two numerical examples are given to illustrate the effectiveness and the edge of the authors' results over other relevant works in the literature.

An LPV based robust peak-to-peak state estimation for genetic regulatory networks with time varying delay

This paper addresses the nonlinear observer design problem for gene regulatory networks with time-varying delay, focusing on the case of unstable GRNs with oscillatory behavior. Currently available approaches are conservative due to presence of nonlinear terms, which should be dealt with. In addition, nonlinear terms are only known approximately in practice and therefore previous works may lead to undesirable performance. To address conservativeness issue, we have provided an LPV approach. Besides, to diminish effects of uncertain nonlinear terms on observer performance, a peak-to-peak state estimation problem is considered. By defining appropriate Lyapunov–Krasovskii functional, sufficient conditions are derived that guarantee the desirable performance. The superiority of the proposed method to existing approaches is illustrated by means of a numerical example. Moreover, good performance of state estimation in presence of uncertainty is demonstrated by simulations.

A discrete-time observer design for spacecraft attitude determination using an orthogonality-preserving algorithm

This paper studies a nonlinear discrete-time partial state observer design problem for rigid spacecraft systems, particularly for spacecraft attitude estimation. The construction is inspired by an orthogonality-preserving numerical algorithm, and it yields a semiglobal practical asymptotic observer. We show that the separation principle holds, and hence stabilization using output feedback is possible. An example is provided to illustrate the usefulness of our results in a sampled-data implementation of an attitude control for a rigid spacecraft using output feedback and to illustrate the advantages of the proposed observer design, especially compared to a sample and hold version of a similar type observer designed in continuous-time.

Robust compound observers design for Lipschitz non-linear systems with unknown input disturbance

In this paper, we present a new concept of the robust compound observer for Lipschitz non-linear system with unknown input disturbance, which can simultaneously estimate the system states and unknown input disturbances. The robust compound observer is designed by using the structure principle of the general unknown input observer and the conditions for the existence of the robust compound observer are established. Combined with LMI methods, the designing freedoms are released and LMI existence condition and some verification are given for the compound observers which can asymptotically estimate the system state and unknown input disturbance to any desired accuracy. The presented robust compound observer matrices are calculated using MATLAB LMI toolbox to make the difficult non-linear observer design problem an easy task. Numerical simulation results are presented to validate the approach and the robustness to the system non-linearity and uncertainty.

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.

Observer Design for a Class of Discrete-time Nonlinear Systems

Necessary and sufficient conditions are presented under which a discrete-time autonomous system with outputs is locally diffeomorphic to an output-scaled linear observable system or an output-scaled nonlinear system in the observer form. As a consequence of such characterizations, the nonlinear observer design problem is studied for a broader class of discrete-time nonlinear systems by using the exact linearization technique that is based on the differential geometric approach.

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.

Nonlinear observer design for state and disturbance estimation

A new systematic framework for nonlinear observer design that allows the concurrent estimation of the process state variables together with key unknown process or sensor disturbances is proposed. The nonlinear observer design problem is addressed within a similar methodological framework as the one introduced in [N. Kazantzis, C. Kravaris, Nonlinear observer design using Lyapunov's auxiliary theorem, Systems Control Lett. 34 (1998) 241; A.J. Krener, M. Xiao, Nonlinear observer design in the Siegel domain, SIAM J. Control Optim. 41 (2002) 932.] for state estimation purposes only. From a mathematical standpoint, the problem under consideration is addressed through a system of first-order singular PDEs for which a rather general set of solvability conditions is derived. A nonlinear observer is then designed with a state-dependent gain that is computed from the solution of the system of singular PDEs. Under the aforementioned conditions, both state and disturbance estimation errors converge to zero with assignable rates. The convergence properties of the proposed nonlinear observer are tested through simulation studies in an illustrative example involving a biological reactor.

C(sI-A)-1B가 최소위상이 될 LMI 조건을 이용한 해석과 설계

We derive a linear matrix inequality(LMI) condition guaranteeing that any invariant zeros of a triple (A, B, C) lie in the open left half plane of the complex plane, i.e. <TEX>$C(sI-A)^{-1}B$</TEX> is minimum phase. The LMI condition is equivalent to a certain constrained Lyapunov matrix equation which can be found in many results relating to stability analysis or control design. We show that the LMI condition can be used to simplify various control engineering problems such as a dynamic output feedback control problem, a variable structure static output feedback control problem, and a nonlinear system observer design problem. Finally, we give some numerical examples.

Nonlinear observer design in the presence of delayed output measurements

The present work proposes a new approach to the nonlinear observer design problem in the presence of delayed output measurements. The proposed nonlinear observer possesses a state-dependent gain which is computed from the solution of a system of first-order singular partial differential equations, and in particular, consists of a chain of state observation algorithms that reconstruct the unmeasurable state vector at different delayed time-instants within the time-delay window introduced by the available output measurements. Therefore, the proposed nonlinear observer exhibits a chain-like structure that explicitly reflects and takes into account the magnitude of the output delay. Furthermore, a set of conditions is derived under which convergence of the estimation error to zero is established. Finally, the performance of the proposed observer and its convergence properties are evaluated in an illustrative biological reactor example.

Nonlinear observer design for the slow states of a singularly perturbed system

The present work proposes a new approach to the nonlinear observer design problem in the presence of two-time-scale multiplicity. In particular, nonlinear processes are considered that exhibit fast and unmeasurable slow dynamic modes, and the latter need to be accurately reconstructed through the use of a state observer. The proposed observer is designed on the basis of the reduced-order process dynamics that evolve on the system’s slow manifold, and the dynamic behavior of the estimation error is analyzed and mathematically characterized in the presence of the unmodeled fast process dynamics. It is shown, that within the proposed nonlinear observer design framework, the observation error generated by neglecting the fast process dynamics is of order O ( ɛ ) , where ɛ is the perturbation parameter and a measure of the relative speed/time-constant of the fast and the slow component of the process dynamics. Furthermore, the analysis conducted establishes robustness of the proposed observer design method with respect to fast unmodeled process dynamics. Finally, the performance of the proposed method and the convergence properties of the reduced-order nonlinear observer designed are evaluated in an illustrative biological reactor example.

On designing observers for time-delay systems with non-linear disturbances

In this paper, the observer design problem is studied for a class of time-delay non-linear systems. The system under consideration is subject to delayed state and non-linear disturbances. The time-delay is allowed to be time-varying, and the non-linearities are assumed to satisfy global Lipschitz conditions. The problem addressed is the design of state observers such that, for the admissible time-delay as well as non-linear disturbances, the dynamics of the observation error is globally exponentially stable. An effective algebraic matrix inequality approach is developed to solve the non-linear observer design problem. Specifically, some conditions for the existence of the desired observers are derived, and an explicit expression of desired observers is given in terms of some free parameters. A simulation example is included to illustrate the practical applicability of the proposed theory.

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.

Nonlinear observer design using Lyapunov’s auxiliary theorem

The present work proposes a new approach to the nonlinear observer design problem. Based on the early ideas that influenced the development of the linear Luenberger observer theory, the proposed approach develops a nonlinear analogue. The formulation of the observer design problem is realized via a system of singular first-order linear PDEs, and a rather general set of necessary and sufficient conditions for solvability is derived by using Lyapunov’s auxiliary theorem. The solution to the above system of PDEs is locally analytic and this enables the development of a series solution method, that is easily programmable with the aid of a symbolic software package.

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.

Non-linear observer design by observer canonical forms

This paper studies the non-linear observer design problem by observer canonical forms. We present necessary and sufficient conditions in terms of the ranks of matrices for the existence of a non-linear state coordinate change under which a time-variable system is transformed into an observer canonical form. These conditions give the multiple versions of Li and Tao (1986) for the single-output case. Then a comparison is made with the Lie algebraic conditions obtained by Krener and Respondek (1985), and modified by the authors in a recent paper (Xia and Gao 1987).