Wide Class Of Nonlinear Systems Research Articles

Multistability in dynamical systems induced by weak periodic perturbations.

It is shown that weak resonant perturbations at subharmonic frequencies can induce multistability in a wide class of nonlinear systems, which display the period-doubling route into chaos or possess isolated subharmonic branches. The number of attractors induced depends on the subharmonic frequency, amplitude, and phase of periodic perturbations, as well as an initial dynamical state of nonlinear systems. Experimental and numerical evidences are given on the basis of a loss-modulated CO2 laser.

Robust Output Feedback Tracking for a Class of Nonlinear Systems with Nonlinear Parameterization

The problem of global robust output tracking is solved for a class of uncertain nonlinear systems with nonlinear parameterization using output feedback. The technique proposed extends the existing output tracking results to a wider class of nonlinear systems without the use of an internal model. By increasing a single parameter in the controller, the tracking error can be arbitrarily reduced. The controller also guarantees the boundedness of the internal states. Due to the triangular structure of the system, the controller design is based on the backstepping technique.

Passivation and control of partially known SISO nonlinear systems via dynamic neural networks

In this paper, an adaptive technique is suggested to provide the passivity property for a class of partially known SISO nonlinear systems. A simple Dynamic Neural Network (DNN), containing only two neurons and without any hidden-layers, is used to identify the unknown nonlinear system. By means of a Lyapunov-like analysis the new learning law for this DNN, guarantying both successful identification and passivation effects, is derived. Based on this adaptive DNN model, an adaptive feedback controller, serving for wide class of nonlinear systems with an <em>a priori</em> incomplete model description, is designed. Two typical examples illustrate the effectiveness of the suggested approach.

Manipulator Joint Friction Identification using Genetic Algorithm and its Experimental Verification

Like many other mechanical dynamic systems, flexible manipulator systems experience stiction or sticking friction, which may cause input-dependent instabilities. Manipulator performance can be enha nced by identifying friction but it is hard and expensive to measure friction by direct and precise sensing of contact displacements and forces. This study addresses the problem of identifying flexible manipulator joint friction. A dynamic model of a two-link flexible manipulator based upon finite element and Lagrange's method is constructed. The dynamic model includes the effects of joint compliances and actuator dynamics. Friction is also incorporated in the dynamic model to account for stick-slip at the joints. Next, the friction parameters are to be determined. The identification problem is posed as an optimization problem to be solved using nonlinear programming methods. A genetic algorithm is used to increase the convergence rate and the chances of finding the global optimum. The identified friction parameters are experimentally verified and it is expected that the identification technique is applicable to a system parameter identification problem associated with a wide class of nonlinear systems.

Stable Neural Controllers for Nonlinear Dynamic Systems

In this paper, a stability based approach is introduced to design neural controllers for nonlinear systems. The requisite control input is generated as the output of a neural network, which is trained off-line such that the time derivative of a positive definite function of the state variables becomes negative at all points. By using the successfully trained networks as controllers, the closed-loop system can be made stable. The stability framework introduced permits the generation of more efficient algorithms that lead to a larger region of stability for a wide class of nonlinear systems.

Towards Real-Time, Non-Linear Quadratic Optimal Regulation

A novel non-linear design method based on linear quadratic optimal control theory is presented that applies to a wide class of non-linear systems. The method is easy to apply and results in a near optimal solution that has the potential to be implemented in real-time in the sense that the solution is causal, in contrast to the conventional, quadratic regulator solution, which is anti-causal (backwards in time) and must be computed off-line. The key feature of the design method is the introduction of state-dependence in the weight matrices of the usual linear quadratic cost function, leading to a non-linear control policy, even for linear dynamics. To demonstrate the method, a simple vehicle suspension model with a cubic damping force is used, in conjunction with non-linear state penalties that better reflect the engineering objectives of active vehicle vibration suppression. A number of simulations is conducted and compared with a passively mounted vehicle.

A Nonlinear Observer Approach: Experience with a Hydraulic System

A fault detection method is proposed for a fairly wide class of nonlinear systems based on a nonlinear observer. Other methods proposed in the field are related to this method or are special cases of it. The observer is designed such that the error dynamics are independent ofstate, input, output and unknown disturbances. Conditions are given for the observer to exist and to be robust with respect to an unknown input. Applications are made to a hydraulic system.

A bound for the magnitude characteristics of nonlinear output frequency response functions Part 1. Analysis and computation

A bound for the magnitude frequency domain characteristics associated with the outputs of a wide class of nonlinear systems is derived as a relatively simple function of the generalized frequency response functions and properties of the system inputs. It is shown how the practical computation of the new bound can be easily performed for nonlinear systems with finite but arbitrary order nonlinearities and worked examples are included. The paper is divided into two parts. In Part 1, an expression for the output magnitude bound is derived, properties of the result are discussed and general procedures for the practical computation of the bound are developed. In Part 2 the practical computations associated with applying the bound to the polynomial nonlinear autoregressive model with exogenous input are discussed.

Gain bounds of higher-order nonlinear transfer functions

Bounds on the gain for high-order nonlinear transfer functions of a wide class of nonlinear systems are derived. It is shown that the gain bounds can be expressed explicitly in terms of the coefficients of discrete-time nonlinear models. The bounds can be used to check the convergence of the transfer function series and hence to determine the truncation order in practical applications.

Intermediate-term earthquake prediction.

An earthquake of magnitude M and linear source dimension L(M) is preceded within a few years by certain patterns of seismicity in the magnitude range down to about (M - 3) in an area of linear dimension about 5L-10L. Prediction algorithms based on such patterns may allow one to predict approximately 80% of strong earthquakes with alarms occupying altogether 20-30% of the time-space considered. An area of alarm can be narrowed down to 2L-3L when observations include lower magnitudes, down to about (M - 4). In spite of their limited accuracy, such predictions open a possibility to prevent considerable damage. The following findings may provide for further development of prediction methods: (i) long-range correlations in fault system dynamics and accordingly large size of the areas over which different observed fields could be averaged and analyzed jointly, (ii) specific symptoms of an approaching strong earthquake, (iii) the partial similarity of these symptoms worldwide, (iv) the fact that some of them are not Earth specific: we probably encountered in seismicity the symptoms of instability common for a wide class of nonlinear systems.

Computational complexity of Volterra based nonlinear compensators

The Volterra series can be used to represent a wide class of nonlinear systems with memory. A Volterra inverse can be used to apply post (or pre)-distortion for the purpose of nonlinear compensation. The authors determine and compare the computational complexity of two such compensators: one based on the analytical pth order Volterra inverse, the other on the adaptive Volterra inverse.

Chaos Identification of Nonlinear Vibrating System Using Neural Networks and Its Application.

This research is concerned with an application of chaos to the identification method of a nonlinear dynamical system using neural networks. In conventional control theory, it is difficult to identify a mathematical model of practical system because of system complexity and the existence of nonlinearity. Instead of theoretical methods, neural networks with different architectures have been applied to the identification and control of a wide class of nonlinear systems. In this paper, we propose a learning control scheme to realize the identification and control of a chaos system, using multilayered neural networks, in which the control performance is satisfied for an unknown controlled object by repeated trials. First, the effectiveness of neural networks is shown in the identification problem of Duffing's chaotic time series data and the possibility of short-term predictability in chaotic system is discussed. Second, a new control technique is proposed which identifies nonlinear systems as nonlinear mapping in the forward direction and employs its inverse mapping as a controller. As a result, it is shown that the network dientified by chaotic time series data is applicable to the prediction and control problem.

Partial-state global stabilization for general triangular systems

We deal with the partial-state global stabilization problem for a wide class of nonlinear systems which contains those having triangular structure. The sufficient conditions we propose guarantee global stabilization by means of an output smooth feedback integrator.

Noninteraction and Stability via Invertible Feedback Laws and Some Existence Conditions

For a wide class of nonlinear systems with more inputs than outputs, the authors show that the asymptotic stability of suitable dynamics and the asymptotic stabilizability via dynamic state feedback of suitable systems are necessary to achieve noninteraction and stability via invertible feedback laws. These conditions generalize some recent results obtained for the same class of systems. Some interesting existence conditions are also given, and relationships with the above necessary conditions are identified.

Homoclinic manifolds and numerical chaos in the nonlinear Schrödinger equation

Crossings of homoclinic manifolds is a well known mechanism underlying observed chaos in low dimensional systems. We discuss an analogous situation as it pertains to the numerical simulation of a well known integrable partial differential equation, the nonlinear Schrödinger equation. In various parameter regimes, depending on the initial data, numerical chaos is observed due to either truncation effects or errors on the order of roundoff. The explanation of the underlying cause of the chaos being due to crossing of homoclinic manifolds induced by the numerical errors is elucidated. The nonlinear Schrödinger equation is prototypical of a much wider class of nonlinear systems in which computational chaos can be a significant factor.

Monitoring the Combustion Set of a Gas Turbine

In this paper, we investigate the problem of monitoring the combustion chambers of the gas turbine of an electrical power generator. We describe a general approach for detecting small changes, which can be applied to a wide class of nonlinear systems, can deal with a reduced order and biased signature of the monitored system, and is robust with respect to changes in the functioning mode. We also describe a statistical method for diagnosing the detected changes in the combustion chambers. We report about the processing of a significant amount of real data from two different types of turbines.

Disturbance rejection in nonlinear systems using neural networks

Neural networks with different architectures have been successfully used for the identification and control of a wide class of nonlinear systems. The problem of rejection of input disturbances, when such networks are used in practical problems is considered. A large class of disturbances, which can be modeled as the outputs of unforced linear or nonlinear dynamic systems, is treated. The objective is to determine the identification model and the control law to minimize the effect of the disturbance at the output. In all cases, the method used involves expansion of the state space of the disturbance-free plant in an attempt to eliminate the effect of the disturbance. Several stages of increasing complexity of the problem are discussed in detail. Theoretical justification is provided for the existence of solutions to the problem of complete rejection of the disturbance in special cases. This provides the rationale for using similar techniques in situations where such theoretical analysis is not available.

Robust receding horizon control of constrained nonlinear systems

We present a method for the construction of a robust dual-mode, receding horizon controller which can be employed for a wide class of nonlinear systems with state and control constraints and model error. The controller is dual-mode. In a neighborhood of the origin, the control action is generated by a linear feedback controller designed for the linearized system. Outside this neighborhood, receding horizon control is employed. Existing receding horizon controllers for nonlinear, continuous time systems, which are guaranteed to stabilize the nonlinear system to which they are applied, require the exact solution, at every instant, of an optimal control problem with terminal equality constraints. These requirements are considerably relaxed in the dual-mode receding horizon controller presented in this paper. Stability is achieved by imposing a terminal inequality, rather than an equality, constraint. Only approximate minimization is required. A variable time horizon is permitted. Robustness is achieved by employing conservative state and stability constraint sets, thereby permitting a margin of error. The resultant dual-mode controller requires considerably less online computation than existing receding horizon controllers for nonlinear, constrained systems.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Necessary conditions for nonlinear block noninteracting control with stability via dynamic state-feedback

In this paper we give some necessary conditions to achieve noninteraction and stability via dynamic state-feedback for a wide class of nonlinear systems with block-partitioned outputs. Moreover, a class of regular dynamic state-feedback is defined and it is shown that the asymptotic stability of certain dynamics, together with the asymptotic stabilizability of certain system, is necessary to achieve noninteraction and stability via regular dynamic state-feedback. These two conditions, plus the asymptotic stabilizability of a suitable system, are shown to be also sufficient to solve the problem.

Coprime factorization over a class of nonlinear systems

In this paper, we consider the problem of generalizing elements of linear coprime factorization theory to a nonlinear context. The idea is to work with a suitably wide class of nonlinear systems to cover many practical situations, yet not cope with so broad a class as to disallow useful generalizations to the linear results. In particular, we work with nonlinear systems characterized in terms of (possibly time-varying) state-dependent matrices A(x), B(x), C(x), D(x) and an initial state x0. (This class clearly does contain the class of finite-dimensional linear (time-varying) systems.) We achieve first right coprime factorizations for idealized situations. To achieve stable left factorizations we specialize to the case where the matrices are output-dependent. Alternatively, we work with systems, perhaps augmented by a direct feedthrough term, where the input is reconstructible from the output. For nonlinear feedback control systems, with plant and controller having stable left factorizations, then under appropriate regularity-conditions earlier results have allowed the generation of the class of stabilizing controllers for a system in terms of an arbitrary stable system (parameter). Plant uncertainties, including unknown initial conditions are modelled by means of a Yula–Kucera-type parametrization approach developed for nonlinear systems. Certain robust stabilization results are also shown, and simulations demonstrate the regulation of nonlinear plants using the techniques developed. All the results are presented in such a way that specialization for the case of linear systems is immediate.