Euler Approximate Model Research Articles

Robust sensitive fault detection and estimation for single-rate and multirate nonlinear sampled-data systems

This paper investigates the problem of fault detection and estimation for nonlinear sampled-data systems in the presence of unknown exogenous inputs. Both cases of single-rate and multirate sampling are treated and general observer-based frameworks are provided using discrete-time approximation to estimate fault signals of any sources. We show that these frameworks are input-to-error stable with respect to the estimation error and can simultaneously enhance robustness against unknown inputs and sensitivity to faults in a mixed H−∕H∞ sense for the unknown exact discrete-time model of the plant. Our results are then applied to a class of Lipschitz nonlinear systems with a refined Euler approximate model to derive sampled-data fault estimation techniques where stability and H−∕H∞ optimization are ensured using linear matrix inequalities (LMIs). Simulation results of a flexible joint robot illustrate the effectiveness of the proposed methodologies.

Sufficient LMI stability conditions for Lur’e type systems governed by a control law designed on their Euler approximate model

In this paper, the stability issue of Lur’e systems governed by a control law stabilising their forward Euler approximate model is investigated. More specifically, the considered control law is obtained by exploiting the advantages of a new Lur’e type Lyapunov function with disconnected level sets. This Lyapunov function is adapted to discrete-time Lur’e systems and to the structure of the forward Euler approximate model. The main result consists of linear matrix inequality conditions allowing to guarantee that the continuous-time Lur’e system associated with the proposed digital control law is globally asymptotically stable. The relevance of this approach is illustrated using a numerical example.

Robust Fault Reconstruction in Discrete-Time Lipschitz Nonlinear Systems via Euler-Approximate Proportional Integral Observers

The problem of observer-based robust fault reconstruction for a class of nonlinear sampled-data systems is investigated. A discrete-time Lipschitz nonlinear system is first established, and its Euler-approximate model is described; then, an Euler-approximate proportional integral observer (EPIO) is constructed such that simultaneous reconstruction of system states and actuator faults are guaranteed. The presented EPIO possesses the disturbance-decoupling ability because its architecture is similar to that of a nonlinear unknown input observer. The robust stability of the EPIO and convergence of fault-reconstructing errors are proved using Lyapunov stability theory together withH∞techniques. The design of the EPIO is reformulated into convex optimization problem involving linear matrix inequalities (LMIs) such that its gain matrices can be conveniently calculated using standard LMI tools. In addition, to guarantee the implementation of the EPIO on the exact model, sufficient conditions of its semiglobal practical convergence are provided explicitly. Finally, a single-link flexible robot is employed to verify the effectiveness of the proposed fault-reconstructing method.

Straight-Line Trajectory Tracking Control for Sampled-Data Underactuated Ships

Straight-line trajectory tracking control of sampled-data underactuated ships with both state and output feedback controllers is considered and analyzed by resorting to a nonlinear sampled-data control theory. State feedback controllers and reduced-order observers are designed on the basis of the Euler approximate models. Then they are combined to obtain output feedback controllers that semiglobally practically uniformly asymptotically stabilize the tracking error dynamics. Both simulation and experimental results are given to show the efficiency of the designed controllers.

Sampled-Data Backstepping Control of a Quadrotor Unmanned Aerial Vehicle

Sampled-data backstepping control of a quadrotor UAV is presented in this paper. The discrete time controllers have been designed on the basis of a continuous time system model and an approximate discrete time equivalent system model (Euler Approximate model) obtained using the Euler method. The performance of the controllers obtained using the two sampled-data approaches has been compared in the presence of wing gusts and modeling uncertainties. Simulations have revealed that the backstepping controller designed on the basis of an approximate discrete time model has better performance in the presence of wind gusts and modeling uncertainties. In addition, the closed loop system has a larger region of attraction.

Design of novel observers for robust fault detection in discrete-time Lipschitz non-linear systems with Euler-approximate models

This paper addresses the problem of observer-based robust fault detection in a class of Lipschitz non-linear systems. A Euler-approximate model for continuous-time Lipschitz non-linear systems is first established; then, a discrete time non-linear observer is designed such that the dynamic output-estimation error, which is assigned as a residual signal, asymptotically converges to zero if no actuator faults and external disturbances exist in the system. The new observer has similarity to a non-linear unknown input observer (UIO). Compared with the existing UIOs, the design of the presented observer requires fewer gain matrices and equation constraints; less computation load is therefore needed. On the other hand, the new observer is designed based on the Euler-approximate model. To ensure its implementation on the exact model, sufficient conditions for semiglobal practical convergence of the proposed observer are explicitly provided. With external disturbances, a nonlinear H∞ observer is constructed to achieve robust actuator fault detection. Observer design problem can be systematically solved using linear matrix inequality (LMI)-based optimisation technique. Lastly, a single-link flexible robot is employed to illustrate the effectiveness of the proposed observer-based FD scheme.

A Simultaneous Stabilization Problem of Linear Systems and their Discretized Models

We present a method to design a feedback controller which stabilizes a continuous-time linear system and its discretized models simultaneously. These discretized models include the Euler approximate discrete-time model and the exact discrete-time model. We present some conditions on solvability of the simultaneous stabilization problem, and a procedure for designing such a controller. We also give three examples to illustrate our research results.

Backstepping designs for the stabilisation of nonlinear sampled-data systems via approximate discrete-time model

The problems of backstepping, adaptive backstepping and reduced order observer based output feedback control of sampled-data nonlinear systems in strict feedback form are considered. Controller design methods based on the Euler approximate model are presented for these problems. The controllers are designed to compensate the effects of the discrepancy between the Euler approximate model and exact discrete time model, parameter estimation error in adaptive control and observer error in output feedback control, which behave as disturbance. It is shown that the obtained controllers semi-globally practically asymptotically stabilise the plant model under standard assumptions. Then numerical examples are given to illustrate the design methods.

Design and performance analysis of deterministic learning of sampled-data nonlinear systems

Abstract In this paper, we extend the deterministic learning theory to sampled-data nonlinear systems. Based on the Euler approximate model, the adaptive neural network identifier with a normalized learning algorithm is proposed. It is proven that by properly setting the sampling period, the overall system can be guaranteed to be stable and partial neural network weights can exponentially converge to their optimal values under the satisfaction of the partial persistent excitation (PE) condition. Consequently, locally accurate learning of the nonlinear dynamics can be achieved, and the knowledge can be represented by using constant-weight neural networks. Furthermore, we present a performance analysis for the learning algorithm by developing explicit bounds on the learning rate and accuracy. Several factors that influence learning, including the PE level, the learning gain, and the sampling period, are investigated. Simulation studies are included to demonstrate the effectiveness of the approach.

Fault-tolerant control for a class of nonlinear sampled-data systems via a Euler approximate observer

In this paper, a fault-tolerant control (FTC) scheme is developed for a class of nonlinear sampled-data systems. First, a Euler approximate discrete model is used to describe the plant under the sampling. Under this model, an observer-based fault estimation method is proposed. To guarantee the accuracy of both the state and fault estimation values, the conditions to make the approximate model consistent with the exact model are presented. Then, an active fault-tolerant controller, which has a constraint condition for the sampling time, is designed to make the faulty system stable. Finally, an inverted pendulum is used to show the efficiency of the proposed method.

Nonlinear Sampled-Data Stabilization of Dynamically Positioned Ships

In this brief, we consider the nonlinear sampled-data stabilization of dynamically positioned ships. Using the nonlinear sampled-data control theory developed by Nesic and the integrator backstepping technique for the Euler approximate model, we design semiglobally practically asymptotically (SPA) stabilizing controllers. We design both state feedback and output feedback controllers. We give a numerical example to illustrate the design methods.

Robust backstepping for the Euler approximate model of sampled-data strict-feedback systems

Stabilization of uncertain sampled-data strict-feedback systems is addressed. The stability study is carried out on the Euler approximation of the exact discretized model of the plant. Firstly, a class of state-feedback controllers is developed that guarantees an input-to-state stability property for the closed-loop system. Additionally, assuming some hypotheses on the uncertain terms hold, a practical asymptotic stability property is ensured by designing an appropriate class of controllers.

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.

Design of static and dynamic output feedback controllers through Euler approximate models: uncertain systems with norm-bounded uncertainties

Design of static and dynamic output feedback controllers through Euler approximate models: uncertain systems with norm-bounded uncertainties

Stabilization of sampled-data nonlinear systems via backstepping on their Euler approximate model

Two integrator backstepping designs are presented for digitally controlled continuous-time plants in special form. The controller designs are based on the Euler approximate discrete-time model of the plant and the obtained control algorithms are novel. The two control laws yield, respectively, semiglobal-practical stabilization and global asymptotic stabilization of the Euler model. Both designs achieve semiglobal-practical stabilization (in the sampling period that is regarded as a design parameter) of the closed-loop sampled-data system. A simulation example illustrates that the obtained controllers may sometimes be superior to backstepping controllers based on the continuous-time plant model that are implemented digitally.

DISCRETE-TIME IDA-PBC DESIGN FOR SEPARABLE HAMILTONIAN SYSTEMS

We develop a discrete-time counterpart of IDA-PBC design for separable Hamiltonian systems. Euler approximate models are used to obtain the discrete-time control laws, by replacing “differential” in the continuous-time design with “difference” in the discrete-time design. This approach results in a type of discrete-time controller that can be seen as a simple modification of an emulation controller obtained by sample and hold of the continuous-time IDA-PBC controller. However, due to the crucial issue of Hamiltonian conservation in IDA-PBC design, this simple modification, which results in an “almost” Hamiltonian conservation in discrete-time, yields a significant improvement to the performance of the sampled-data control system.