Continuous-time Nonlinear Dynamic Systems Research Articles

Cleaning the Periodicity in Discrete- and Continuous-Time Nonlinear Dynamical Systems

We investigate periodicity suppression in two-dimensional parameter-spaces of discrete- and continuous-time nonlinear dynamical systems, modeled respectively by a two-dimensional map and a set of three first-order ordinary differential equations. We show for both cases that, by varying the amplitude of an external periodic forcing with a fixed angular frequency, windows of periodicity embedded in a chaotic region may be totally suppressed.

Dimensionality Reduction of Dynamical Systems with Parameters: A Geometric Approach

A method for obtaining a low-dimensional description of a family of attractors produced by continuous-time nonlinear dynamical systems with static parameters is developed. A geometric approach is taken, which allows for the reduction of general state space manifolds, not restricted to $\mathbb{R}^n$. An existing secant-based projection method, utilizing optimization over Grassmann manifolds, is extended for use with multiple parameter values and data sets. A family of reduced vector fields is obtained, with parameterization by the original parameter space, that reproduces the projected attractors and their dynamics in a low-dimensional space. We illustrate the method with several examples. The Rössler system demonstrates the accurate reproduction of period-doubling bifurcations; a forced damped double pendulum demonstrates a nonautonomous system with angular variables; and the Brusselator demonstrates application to a high-dimensional system.

Convex Computation of the Maximum Controlled Invariant Set For Polynomial Control Systems

We characterize the maximum controlled invariant (MCI) set for discrete- as well as continuous-time nonlinear dynamical systems as the solution of an infinite-dimensional linear programming problem. For systems with polynomial dynamics and compact semialgebraic state and control constraints, we describe a hierarchy of finite-dimensional linear matrix inequality (LMI) relaxations whose optimal values converge to the volume of the MCI set; dual to these LMI relaxations are sum-of-squares (SOS) problems providing a converging sequence of outer approximations to the MCI set. The approach is simple and readily applicable in the sense that the approximations are the outcome of a single semidefinite program with no additional input apart from the problem description. A number of numerical examples illustrate the approach.

A $Q$-Modification Neuroadaptive Control Architecture for Discrete-Time Systems

This brief extends the new neuroadaptive control framework for continuous-time nonlinear uncertain dynamical systems based on a Q -modification architecture to discrete-time systems. As in the continuous-time case, the discrete-time update laws involve auxiliary terms, or Q-modification terms, predicated on an estimate of the unknown neural network weights which in turn involve a set of auxiliary equations characterizing a set of affine hyperplanes. In addition, we show that the Q -modification terms in the discrete-time update law are designed to minimize an error criterion involving a sum of squares of the distances between the update weights and the family of affine hyperplanes.

Computation of the real logarithm for a discrete-time nonlinear system

For a given nonlinear discrete-time dynamical system, the problem is considered of finding, if any, a continuous-time nonlinear dynamical system such that the given system is its exact sampled-data representation. Constructive solutions to the problem are given by geometric tools such as symmetries.

On Lyapunov stability of interconnected nonlinear systems: recursive integration methodology

This paper evolved from an endeavor to construct a Lyapunov function of interconnected nonlinear systems described by ordinary differential equations. This construction is an intractable problem and crucial part of the design when applied to problems of high dimension with intricate structure. The design obtained is applicable if the system under investigation is a generalization of the strict feedback form and whose dependency graphs satisfies the decomposition into lower triangular form. This design is mainly developed for moderated nonlinear continuous time dynamical systems. However, as we show in the example, it is emphasized that the present results can be extended to discontinuous systems provided they can be approximated by smooth modifications. A key step in our approach which based on back integrating procedure is to decompose the interconnected system into subsystems using graph theoretic decomposition. In the proposed methodology, the postulated Lyapunov function is obtained by backward integration the composite system trajectory. This novel approach leads straightforward and by one shot methods to both Lyapunov function of the composite system and its time derivative. A practical example is presented to illustrate the effectiveness of the results.

A new fuzzy [formula omitted] filter design for nonlinear continuous-time dynamic systems with time-varying delays

This paper focuses on the problem of H ∞ filter design for nonlinear systems with time-varying delays. A new approach of filter design is proposed. The proposed results may improve the existing ones due to estimate the upper bound of the derivative of Lyapunov functional without ignoring some useful terms. A sufficient condition for the existence of such a filter is established by using the linear matrix inequality (LMI) approach. Finally, two examples are given to demonstrate the effectiveness and the benefits of the proposed method.

Continuous-time and continuous–discrete-time unscented Rauch–Tung–Striebel smoothers

This article considers the application of the unscented transformation to approximate fixed-interval optimal smoothing of continuous-time non-linear stochastic dynamic systems. The proposed methodology can be applied to systems, where the dynamics can be modeled with non-linear stochastic differential equations and the noise corrupted measurements are obtained continuously or at discrete times. The smoothing algorithm is based on computing the continuous-time limit of the recently proposed unscented Rauch–Tung–Striebel smoother, which is an approximate optimal smoothing algorithm for discrete-time stochastic dynamic systems.

Piecewise linear implementation of nonlinear dynamical systems: from theory to practice

It is shown how to implement by a mixed-signal circuit a continuous-time dynamical system. The chosen case study is the Hindmarsh-Rose model of a biological neuron, but the design strategy can be applied to a large class of continuous-time nonlinear dynamical systems. The system nonlinearities are first approximated by using piecewise-linear functions and then digitally implemented on a field programmable gate array. The linear part of the system is completely analogue and is implemented by using operational amplifiers. Measurement results show that the circuit can reproduce the main dynamics of a biologically plausible neuron.

Observer-based digital tracker for the manoeuvring target described by the continuous-time non-linear dynamic system

The current paper develops an effective and robust digital guidance law to intercept the highly manoeuvring target described by the continuous-time non-linear dynamic system. First, an optimal linearization model for the non-linear system is constructed at the operating point of interest on the trajectory. Then, high-gain optimal linear quadratic analogue tracker and observer are designed by utilizing the optimal linearization model and the optimal control theory, so that the effect of the unpredictable acceleration of the target can be substantially attenuated and even can be disregarded. For reducing the control effort without raising the prescribed interception time, and for practical implementation of the designed tracker, the prediction-based digital redesign method is utilized to obtain relatively low-gain digital tracker and observer from the well-designed high-gain optimal analogue tracker and observer. With the aid of the global positioning system, the proposed observer-based digital tracker is able to successfully intercept a highly manoeuvring target without the constraint on initial trajectory, and can effectively deal with various barriers during the interception. Illustrative examples are given to demonstrate the effectiveness and robustness of the proposed digital guidance law.

Black-box identification and simulation of continuous-time nonlinear systems with random processes

This paper proposes a methodology for black-box identification and simulation of continuous-time nonlinear dynamical systems based on random process modeling and kriging. It is assumed that the (finite-dimensional) state vector is observed with noise at regularly or irregularly spaced instants. The proposed identification method consists of two steps. The first step is the estimation of the time derivatives of the state vector. The second step consists in the approximation of the controlled vector field. For the simulation of the system, a new integration scheme is proposed. This integration scheme makes it possible to deal consistently with the error of approximation of the vector field.

Design of Asymptotic Estimators: An Approach Based on Neural Networks and Nonlinear Programming

A methodology to design state estimators for a class of nonlinear continuous-time dynamic systems that is based on neural networks and nonlinear programming is proposed. The estimator has the structure of a Luenberger observer with a linear gain and a parameterized (in general, nonlinear) function, whose argument is an innovation term representing the difference between the current measurement and its prediction. The problem of the estimator design consists in finding the values of the gain and of the parameters that guarantee the asymptotic stability of the estimation error. Toward this end, if a neural network is used to take on this function, the parameters (i.e., the neural weights) are chosen, together with the gain, by constraining the derivative of a quadratic Lyapunov function for the estimation error to be negative definite on a given compact set. It is proved that it is sufficient to impose the negative definiteness of such a derivative only on a suitably dense grid of sampling points. The gain is determined by solving a Lyapunov equation. The neural weights are searched for via nonlinear programming by minimizing a cost penalizing grid-point constraints that are not satisfied. Techniques based on low-discrepancy sequences are applied to deal with a small number of sampling points, and, hence, to reduce the computational burden required to optimize the parameters. Numerical results are reported and comparisons with those obtained by the extended Kalman filter are made.

Variable structure neural networks for adaptive control of nonlinear systems using the stochastic approximation

This paper is concerned with the adaptive control of continuous-time nonlinear dynamical systems using neural networks. Referred to as a variable structure neural network, a novel neural network architecture, is proposed and shown to be useful in approximating the unknown nonlinearities of dynamical systems. In the variable structure neural network, the number of basis functions can be either increased or decreased with time according specified design strategies so that the network will not overfit or underfit the data set. Based on the Gaussian radial basis function (GBRF) variable neural network, an adaptive control scheme is presented. The location of the centers and the determination of the widths of the GBRFs are analysed using a new method inspired from the adaptive diffuse element method combined with a pruning algorithm. In the standard problem of a feedback based control, the cost to be minimized is a function of the output derivative. When the cost function depends on the output error, the gradient method cannot be applied to adjust the neural network parameters. In this case, the stochastic approximation approach allows the computation of the cost function derivatives. The developed weight adaptive laws use a stochastic approximation algorithm. This algorithm consists of the use of the Kiefer–Wolfowitz method.

Adaptive sliding inverse control of a class of non-linear systems preceded by unknown non-symmetrical dead-zone

This paper deals with the adaptive control of a class of continuous-time non-linear dynamic systems preceded by unknown non-symmetrical, non-equal slope dead-zones. By exploring the properties of the dead-zone model intuitively and mathematically, a dead-zone inverse is constructed. Based on this inverse construction, an adaptive sliding controller is designed. Lyapunov stability analysis shows that the proposed adaptive control law ensures global stability of the adaptive system and achieves desired tracking precision. Simulation results attained for an uncertain non-linear system are presented to illustrate and further validate the effectiveness of the proposed approach.

ROBUST FUZZY FAULT DETECTION FOR CONTINUOUS-TIME NONLINEAR DYNAMIC SYSTEMS

: A new fault detection scheme for continuous-time nonlinear dynamic system is studied. The nonlinear system is modelled by Takagi-Sugeno fuzzy model. A fuzzy observer-based approach is presented to detect the fault occurred in the dynamic system. The generated residual signal is robust with respect to undesirable effects of unknown inputs and modelling errors but sensitive to faults. By applying H∞ optimization techniques, a sufficient condition to solvability of the formulated problem is established in terms of Linear Matrix Inequalities. The algorithm for residual evaluation function and threshold computation are also given. Finally, an example is given to illustrate the effectiveness of the proposed design techniques.

불확실성을 갖는 단일 링크 유연로봇의 지능형 디지털 제어

In this paper, we propose a systematic method of a fuzzy-model-based controller for continuous-time nonlinear dynamical systems which may contain uncertainties. The continuous-time uncertain TS fuzzy model is first constructed to represent the uncertain nonlinear system. A parallel distributed compensation (PDC) technique is then used to design a fuzzy model based controller for both stabilization and tracking. Finally, the designed continuous-time controller is converted to an equivalent discrete-time controller by using an intelligent digital redesign method. This new design technique provides a systematic and effective framework for integration of the fuzzy model based control theory and the advanced digital redesign technique for nonlinear dynamical systems with uncertainties. Finally, the single link flexible-joint robot arm is used as an illustrative example to show the effectiveness and the feasibility of the developed design method.

Commande adaptative floue d'une classe de systèmes non linéaires multivariables

In this paper, a direct fuzzy adaptive control scheme for a class of uncertain continuous-time nonlinear multi-input multi-output dynamic systems is proposed. Within this scheme, the fuzzy systems are implemented to generate directly the control input signals without dynamic system estimation. In this approach, the fuzzy system parameters are updated by using a proportional-plus-integral adaptive law. It is proved that the proposed adaptive scheme guarantees the boundedness of all signals in the closed-loop system. Simulations performed on a two-link robot manipulator illustrate the approach and exhibit its performances.

Robust adaptive control of a class of nonlinear systems with unknown dead-zone

This paper deals with the adaptive control of a class of continuous-time nonlinear dynamic systems preceded by an unknown dead-zone. By using a new description of a dead-zone and by exploring the properties of this dead-zone model intuitively and mathematically, a robust adaptive control scheme is developed without constructing the dead-zone inverse. The new control scheme ensures global stability of the adaptive system and achieves desired tracking precision. Simulations performed on a typical nonlinear system illustrate and clarify the validity of this approach.

불확실성을 갖는 동적 시스템에 대한 지능형 디지털 재설계

본 논문에서는 불확실성을 포함한 연속시간 비선형 동적 시스템에 대한 퍼지 모델 기반 제어기의 지능형 디지털 재설계를 위한 조직적인 방법을 제안한다. 불확실 비선형 시스템을 표현하기 위해 연속시간 불확실 TS 퍼지 모델이 구성되었다. 안정화와 추적을 위한 퍼지 모델 기반 제어기 설계를 위해 EPDC 기법이 이용되었다. 설계된 연속시간 제어기는 지능형 디지털 재설계 기법을 이용함으로써 이산시간 제어기로 전환되었다. 이 새로운 설계 기법은 퍼지 모델 기반 제어 이론과 불확실성을 가진 비선형 동적 시스템에 대한 진보된 디지털 재설계 기법의 통합을 위한 조직적이고, 효과적인 틀을 제공한다. 마지막으로, 단일 링크 유연 로봇 시스템의 모의 실험을 이용해 개발된 설계 방법의 효용성과 실행 가능성을 입증한다. In this paper, we propose a systematic method for intelligent digital redesign of a fuzzy-model-based controller for continuous-time nonlinear dynamical systems which may also contain uncertainties. The continuous-time uncertain TS fuzzy model is first constructed to represent the uncertain nonlinear systems. An extended parallel distributed compensation(EPDC) technique is then used to design a fuzzy-model-based controller for both stabilization and tracking. The designed continuous-time controller is then converted to an equivalent discrete-time controller by using an integrated intelligent digital redesign method. This new design technique provides a systematic and effective framework for integration of the fuzzy-model-based control theory and the advanced digital redesign technique for nonlinear dynamical systems with uncertainties. Finally, The single link flexible-joint robot arm is used as an illustrative example to show the effectiveness and the feasibility of the developed design method.

FAULT DIAGNOSIS OF DYNAMIC SYSTEMS BY USING A QUICK IDENTIFICATION METHOD AND ITS APPLICATION TO SHIP PROPULSION SYSTEM

The paper is concerned with a method of identification-based fault diagnosis for nonlinear continuous-time dynamic systems. A quick identification method is introduced in order to monitor online physical parameters specifying fault modes in the system. A decentralized fault diagnosis system which can accommodate for complex fault modes due to multi-simultaneous changes in system parameters is proposed based on the quick identification method. The effectiveness of the method has been confirmed through simulation studies on a ship propulsion system constructed as a standard model for benchmark test of fault diagnosis.