Class Of Lipschitz Nonlinear Systems Research Articles

Sampled-Data Stabilization of a Class of Nonlinear Systems With Application in Robotics

The output feedback stabilization problem for the class of nonlinear Lipschitz systems is considered. A discrete-time feedback controller is designed for the sampled-data case, where the output of the plant is only available at discrete points of time and where the objective is to stabilize the system continuously using a discrete-time controller. We show that exact stabilization in this case can be achieved using a direct sampled-data design approach, based on H∞ optimization theory, in which neither the plant model nor the controller need to be discretized a priori. The proposed design is solvable using commercially available software and is shown to have important advantages over the classical emulation approach that has been used to solve similar problems. The applicability of the proposed techniques in the robotics field is thoroughly discussed from both the modeling and design perspectives.

Fault Diagnosis of a Class of Uncertain Nonlinear Systems with Lipschitz Nonlinearities

This paper presents a fault detection and isolation scheme for a class of Lipschitz nonlinear systems with nonlinear and unstructured modeling uncertainty. This significantly extends previous results by considering a more general class of system nonlinearities which are modeled as functions of the system input and partially measurable state variables. A new fault diagnosis method is developed using adaptive estimation techniques. The fault isolability conditions are rigorously established via the use of the so-called fault mismatch functions, which are defined to characterize the mutual difference between pairs of faults.

Robust Reduced Order Observer for Lipschitz Nonlinear Systems

This paper presents a robust reduced order observer for a class of Lipschitz nonlinear systems with external disturbance. Sufficient conditions on the existence of the proposed observer are characterized by linear matrix inequalities. It is also shown that the proposed observer design can reduce the effect on the estimation error of external disturbance up to the prescribed level. Finally, a numerical example is provided to verify the proposed design method.

시간 지연을 갖는 Lipschitz 비선형 시스템의 강인 상태 관측기

This paper presents a robust state observer design for a class of Lipschitz nonlinear systems with time delay and external disturbance. Sufficient conditions on the existence of the proposed observer are characterized by linear matrix inequalities. It is also shown that the proposed observer design can reduce the effect on the estimation error of external disturbance up to the prescribed level in spite of the existence of time delay. Finally, a numerical example is provided to verify the proposed design method.

Lipschitz 비선형 시스템의 강인 저차 상태 관측기

This paper presents a robust reduced order state observer for a class of Lipschitz nonlinear systems with external disturbance. Sufficient conditions on the existence of the proposed observer are characterized by linear matrix inequalities. It is also shown that the proposed observer design can reduce the effect on the estimation error of external disturbance up to the prescribed level. Finally, a numerical example is provided to verify the proposed design method.

LMI optimization approach to robust H∞ observer design and static output feedback stabilization for discrete‐time nonlinear uncertain systems

AbstractA new approach for the design of robust H∞ observers for a class of Lipschitz nonlinear systems with time‐varying uncertainties is proposed based on linear matrix inequalities (LMIs). The admissible Lipschitz constant of the system and the disturbance attenuation level are maximized simultaneously through convex multiobjective optimization. The resulting H∞ observer guarantees asymptotic stability of the estimation error dynamics and is robust against nonlinear additive uncertainty and time‐varying parametric uncertainties. Explicit norm‐wise and element‐wise bounds on the tolerable nonlinear uncertainty are derived. Also, a new method for the robust output feedback stabilization with H∞ performance for a class of uncertain nonlinear systems is proposed. Our solution is based on a noniterative LMI optimization and is less restrictive than the existing solutions. The bounds on the nonlinear uncertainty and multiobjective optimization obtained for the observer are also applicable to the proposed static output feedback stabilizing controller. Copyright © 2008 John Wiley & Sons, Ltd.

Robust [formula omitted] observer design for sampled-data Lipschitz nonlinear systems with exact and Euler approximate models

An LMI approach is proposed for the design of robust H ∞ observers for a class of Lipschitz nonlinear systems. Two type of systems are considered, Lipschitz nonlinear discrete-time systems and Lipschitz nonlinear sampled-data systems with Euler approximate discrete-time models. Observer convergence when the exact discrete-time model of the system is available is shown. Then, practical convergence of the proposed observer is proved using the Euler approximate discrete-time model. As an additional feature, maximizing the admissible Lipschitz constant, the solution of the proposed LMI optimization problem guaranties robustness against some nonlinear uncertainties. The robust H ∞ observer synthesis problem is solved for both cases. The maximum disturbance attenuation level is achieved through LMI optimization.

A sliding mode observer-based strategy for fault detection, isolation, and estimation in a class of Lipschitz nonlinear systems

This article investigates the design and application of a sliding mode observer (SMO) strategy for actuator as well as sensor fault detection, isolation, and estimation (FDIE) problem for a class of uncertain Lipschitz nonlinear systems. Actuator FDIE is addressed by regrouping the system's inputs into a structure suitable for SMO design. Similarly, by filtering the regrouped outputs, a similar system structure can be developed for sensor FDIE problem. Once in the suitable form and under certain assumptions, nonlinear SMOs are proposed for actuator and sensor FDIE. A systematic LMI-based design approach for the proposed SMO is presented. Additionally, the article addresses four problems, namely: (P1) What are the conditions for isolating single and/or multiple faults? (P2) What is the maximum number of faults that can be isolated simultaneously? (P3) How should one design SMO-based FDI approach in order to achieve multiple fault isolation using as few observers as possible? (P4) How can one estimate the shape of the faults? To solve the above problems, a new concept called fault isolation index (FIX) is proposed for actuator and sensor FDIE. It is proved that fault isolation can only be achieved if FIX ≠ 0, and also that the maximum number of faults that can be isolated is equal to FIX. Using the proposed fault isolation strategy and by treating some healthy inputs or outputs as unknown inputs, a systematic FDIE design scheme using a bank of nonlinear SMOs, which provides a solution for the four problems is provided. An example is used to illustrate the proposed ideas. The simulation results show that the proposed FDIE scheme can successfully detect and isolate both slowly and fast-changing actuator faults. It is also shown that accurate estimation of actuator faults can be achieved.

LMI-based sensor fault diagnosis for nonlinear Lipschitz systems

The problem of sensor fault diagnosis in the class of nonlinear Lipschitz systems is considered. A dynamic observer structure is used with the objective to make the residual converge to the faults vector achieving detection and estimation at the same time. It is shown that, unlike the classical constant gain structure, this objective is achievable by minimizing the faults effect on the estimation error of the dynamic observer. The use of appropriate weightings to solve the design problem in a standard convex optimization framework is also demonstrated. An LMI design procedure solvable using commercially available software is presented.

Output Feedback Stabilization for a Class of Lipschitz Nonlinear Systems

In this letter, we provide a solution to the stabilization problem of a class of Lipschitz nonlinear systems by output feedback. Via the newly proposed nonlinearity characterization function (NCF) concept, we propose an effective method in designing an output feedback controller. Under the suggested sufficient condition which is derived by using the NCF, the proposed control scheme achieves the global exponential stabilization.

Nonlinear High-Gain Observer Design via Semidefinite Programming

In this paper, a nonlinear high-gain observer for the class of Lipschitz nonlinear systems is proposed. The proposed observer can be applied to multi-input-multi-output systems, without assuming any normal form representation. The applicability of the proposed observer design to a given Lipschitz nonlinear system can be checked with the help of semidefinite programming (SDP). If the SDP has a solution, a nonlinear observer gain parameterized by the speed of the observer is obtained.

Design of Adaptive Observation Schemes for Lipschitz Nonlinear Systems.

The problem of adaptive observer synthesis for a class of Lipschitz nonlinear systems is investigated. A new algorithm to the design of adaptive observer scheme is proposed and the respective observer gains are obtained through the solution of a Riccati equation with linear constraints. Solvability conditions for such a Constrained Riccati Problem (CRP) are derived and a systematic algorithm for obtaining adaptive observers which guarantee the error convergence of the state estimates is provided.

Robust global L2-gain control of structural non-linear systems

This paper examines the problem of robust L2 gain control of a class of Lipschitz non-linear systems which is in the combined feedback and feedforward (CFF) form. Based on the Lyapunov theory, we develop a simple procedure for constructing a feedback controller which guarantees that the so-called L2-gain from the exogenous input noise to the state is minimized or no larger than a perscribed value. Finally, we stress that the design procedure does not rely on a solution to the Hamilton-Jacobi equation or Riccati equation.

Observers for Lipschitz nonlinear systems

This paper presents some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The stability of the nonlinear observer for such systems is not determined purely by the eigenvalues of the linear stability matrix. The correct necessary and sufficient conditions on the stability matrix that ensure asymptotic stability of the observer are presented. These conditions are then reformulated to obtain a sufficient condition for stability in terms of the eigenvalues and the eigenvectors of the linear stability matrix. The eigenvalues have to be located sufficiently far out into the left half-plane, and the eigenvectors also have to be sufficiently well-conditioned for ensuring asymptotic stability. Based on these results, a systematic computational algorithm is then presented for obtaining the observer gain matrix so as to achieve the objective of asymptotic stability.

Existence and design of observers for nonlinear systems: Relation to distance to unobservability

This paper presents a systematic design methodology and some fundamental insights into observer design for the class of Lipschitz nonlinear systems. The existence of an asymptotically stable observer is shown to be guaranteed when the distance to unobservability is larger than the Lipschitz constant of the nonlinear system. An analytical solution for the observer is provided in this case. A methodology for the use of a coordinate transformation is then developed so as to reduce the Lipschitz constant and increase the distance to unobservability in the new coordinates. The methodology is directly applicable to the important class of feedback linearizable systems. The developed theory is used successfully in the design of an observer for a flexible joint robotic system.