Wide Class Of Nonlinear Systems Research Articles

Robust Output Feedback Guaranteed Cost Control of Uncertain Fuzzy Systems with Immeasurable Premise Variables

This paper is concerned with robust output feedback stabilization with guaranteed cost of uncertain Takagi-Sugeno fuzzy systems with immeasurable premise variables. When we consider Takagi-Sugeno fuzzy systems, the selection of premise variables plays an important role. If the premise variable is the state of the system, then a fuzzy system describes a wide class of nonlinear systems. However, the state is not measurable in the output feedback control problem. In this case, a control design based on parallel distributed compensation is impossible because a controller depends on immeasurable premise variables. In this paper, we consider the robust output feedback stabilization problem with guaranteed cost where the premise variable is not measurable. We formulate this problem as a robust stabilization of uncertain linear systems. Numerical examples are given to illustrate our theory.

H ∞ output feedback control for fuzzy systems with immeasurable premise variables: Discrete-time case

This paper discusses H ∞ output feedback control of discrete-time Takagi–Sugeno fuzzy systems with immeasurable premise variables. When we consider the output feedback control of Takagi–Sugeno fuzzy systems, the selection of premise variables plays an important role. If the premise variable is the state of the system, then a fuzzy system describes a wide class of nonlinear systems. However, the state is not measurable in the output feedback control problem. In this case, a control design of the underlying nonlinear system based on parallel distributed compensation is infeasible because a controller depends on the immeasurable state variable. In this paper, we introduce a new method to treat fuzzy systems with immeasurable premise variables and consider a design method of H ∞ output feedback control problem. We formulate this fuzzy control problem as a robust H ∞ control of an uncertain system. Numerical examples are given to illustrate our methods.

Analysis of Output Frequencies of Nonlinear Systems

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, an algorithm is derived for the determination of the output frequency ranges of nonlinear systems, which extends previous results on the output frequencies of nonlinear systems to a more general situation. The new results are significant for the analysis of the output frequency response of a wide class of nonlinear systems. </para>

Output frequency response function of nonlinear Volterra systems

An expression for the output frequency response function (OFRF), which defines the explicit analytical relationship between the output spectrum and the system parameters, is derived for nonlinear systems which can be described by a polynomial form differential equation model. An effective algorithm is developed to determine the OFRF directly from system simulation or experimental data. Simulation studies demonstrate the significance of the OFRF concept, and verify the effectiveness of the algorithm which evaluates the OFRF numerically. These new results provide an important basis for the analytical study and design of a wide class of nonlinear systems in the frequency domain.

Robust Output Feedback Stabilization of Uncertain Discrete-Time Fuzzy Systems with Immeasurable Premise Variables

This paper is concerned with robust output feedback stabilization of uncertain discrete-time Takagi-Sugeno fuzzy systems with immeasurable premise variables. When we consider Takagi-Sugeno fuzzy systems, the selection of premise variables plays an important role. The premise variables are usually given, and so the output is selected as the premise variable. In this case, however, a fuzzy system description is limited. If the premise variable is the state of the system, then a fuzzy system describes a wide class of nonlinear systems. In this case, a controller design based on parallel distributed compensation is impossible because the controller depends on immeasurable premise variables. In this paper, we consider the robust output feedback stabilization problem for uncertain fuzzy systems where the premise variable is the state, which is not measurable. We formulate such a problem as a robust stabilization problem of uncertain linear systems. Some numerical examples illustrate our theory.

An Immersion-Based Observer Design for Rank-Observable Nonlinear Systems

In this note, an observer design for a wide class of nonlinear systems is proposed. It is based on some high-gain design for the particular case of state affine systems up to triangular nonlinearity on the one hand, and immersion of nonlinear systems into such a form on the other hand. Conditions for the observer convergence are given in terms of appropriate excitation, and a constructive procedure for the immersion is presented. The methodology is illustrated by examples of application

Impulsive control and synchronization of a new chaotic system

In this paper we discuss the new chaotic system proposed by Huang Wei. We give the sufficient conditions for the stabilization and synchronization of this new chaotic system via impulsive control. For these sufficient conditions that B is not necessarily symmetric. Thus, our results can be used for a wider class of nonlinear systems. Simulation results are provided to show the effectiveness of the proposed method.

Development of bilinear power system representations for small signal stability analysis

In this paper, a new analysis technique for predicting and characterizing nonlinear behavior of stressed power networks is proposed. Making use of an analytical approximation of the system nonlinear model via the use of a truncated Carleman linearization technique, a bilinear state-space model of the power system is developed in which the second and higher order nonlinear terms are explicitly incorporated in the series expansion representation of the system model. The proposed framework enables the study of the dynamic behavior of nonlinear systems, both analytically and numerically, and can be used to represent a wide class of non-linear systems and oscillatory processes. Analytical criteria are developed based on the structural properties of the bilinear state-space model matrices, which predict the existence and stability character of modal interaction in terms of the eigenstructure of the linear system representation. The properties and behavior of the bilinear model are then investigated, and a number of useful results are derived. The present method is quite general and extends readily to higher-dimensional systems. A simplified 2-area, 4-machine system is used to illustrate the proposed procedure. Detailed nonlinear time-domain simulations are conducted to identify the strength of nonlinear behavior arising from interaction of the fundamental modes of oscillation, as well as to check the validity of the analysis.

A single network adaptive critic (SNAC) architecture for optimal control synthesis for a class of nonlinear systems

Even though dynamic programming offers an optimal control solution in a state feedback form, the method is overwhelmed by computational and storage requirements. Approximate dynamic programming implemented with an Adaptive Critic (AC) neural network structure has evolved as a powerful alternative technique that obviates the need for excessive computations and storage requirements in solving optimal control problems. In this paper, an improvement to the AC architecture, called the "Single Network Adaptive Critic (SNAC)" is presented. This approach is applicable to a wide class of nonlinear systems where the optimal control (stationary) equation can be explicitly expressed in terms of the state and costate variables. The selection of this terminology is guided by the fact that it eliminates the use of one neural network (namely the action network) that is part of a typical dual network AC setup. As a consequence, the SNAC architecture offers three potential advantages: a simpler architecture, lesser computational load and elimination of the approximation error associated with the eliminated network. In order to demonstrate these benefits and the control synthesis technique using SNAC, two problems have been solved with the AC and SNAC approaches and their computational performances are compared. One of these problems is a real-life Micro-Electro-Mechanical-system (MEMS) problem, which demonstrates that the SNAC technique is applicable to complex engineering systems.

Analytical Characterization of Hysteresis Loops Described by the Bouc-Wen Model

The Bouc-Wen model is able to capture, in an analytical form, a range of shapes of hysteretic cycles which match the behavior of a wide class of nonlinear systems. The obtained models have been used either to predict the behavior of the physical hysteretic elements or for control purposes. The Bouc-Wen model belongs to a class of smooth analytical models described by differential equations. This gives an appropriate mathematical framework to develop analytical knowledge on relevant properties, which can be exploited for a better practical use, in particular for identification and control purposes. In this context, recent papers have studied input-output and dissipative properties and obtained analytical expressions for hysteretic limit cycles and parameter identification formulae. In this direction, this paper presents new results in the form of analytical expressions which allow to understand the direct influence of the model parameters in the shape of the hysteretic loops, thus giving a theoretically based insight for a precise use of the model in practical applications.

Recursive Observer Design, Homogeneous Approximation, and Nonsmooth Output Feedback Stabilization of Nonlinear Systems

We present a nonsmooth output feedback framework for local and/or global stabilization of a class of nonlinear systems that are not smoothly stabilizable nor uniformly observable. A systematic design method is presented for the construction of stabilizing, dynamic output compensators that are nonsmooth but Houmllder continuous. A new ingredient of the proposed output feedback control scheme is the introduction of a recursive observer design algorithm, making it possible to construct a reduced-order observer step-by-step, in a naturally augmented manner. Such a nonsmooth design leads to a number of new results on output feedback stabilization of nonlinear systems. One of them is the global stabilizability of a chain of odd power integrators by Holder continuous output feedback. The other one is the local stabilization using nonsmooth output feedback for a wide class of nonlinear systems in the Hessenberg form studied in a previous paper, where global stabilizability by nonsmooth state feedback was already proved to be possible

A method for solving a boundary problem for a nonlinear controlled system

An algorithm is proposed for constructing a control function that transfers a wide class of nonlinear systems of ordinary differential equations from an initial state to an arbitrarily small neighborhood of a given terminal state. The algorithm is convenient for numerical implementation. Taking into account the restrictions on the control and phase coordinates, a constructive criterion is obtained for choosing terminal states for which this transfer is possible. The problem of an interorbital flight is considered and modeled numerically.

CHAOS SYNCHRONIZATION IN THE MULTIFEEDBACK MACKEY–GLASS MODEL

We investigate synchronization between two unidirectionally linearly coupled chaotic multifeedback Mackey–Glass systems and find the existence and stability conditions for complete synchronization. Numerical simulations fully support the theory. We also present generalization of the approach to the wider class of nonlinear systems.

An investigation into the characteristics of non-linear frequency response functions. Part 1: Understanding the higher dimensional frequency spaces

The characteristics of generalized frequency response functions (GFRFs) of non-linear systems in higher dimensional space are investigated using a combination of graphical and symbolic decomposition techniques. It is shown how a systematic analysis can be achieved for a wide class of non-linear systems in the frequency domain using the proposed methods. The paper is divided into two parts. In Part 1, the concepts of input and output frequency subdomains are introduced to give insight into the relationship between one dimensional and multi-dimensional frequency spaces. The visualization of both magnitude and phase responses of third order generalized frequency response functions is presented for the first time. In Part 2 symbolic expansion techniques are introduced and new methods are developed to analyse the properties of generalized frequency response functions of non-linear systems described by the NARMAX class of models. Case studies are included in Part 2 to illustrate the application of the new methods.

Space curves, anholonomy and nonlinearity

Using classical differential geometry, we discuss the phenomenon of anholonomy that gets associated with a static and a moving curve. We obtain the expressions for the respective geometric phases in the two cases and interpret them. We show that there is a close connection between anholonomy and nonlinearity in a wide class of nonlinear systems.

Nonlinear MPC and self-concordant barrier functions

The theory of self-concordant barriers was introduced by Nesterov and Nemirovskii (1994) in the context of interior-point methods for convex optimisation. Their development is general, elegant and enjoys widespread implementation in state-of-the-art algorithms. In this paper we exploit the theory of self-concordant functions with application to nonlinear MPC. In particular we construct an invariant terminal constraint set via properties of self-concordant functions. We also extend earlier results on recent red barrier function MPC to nonlinear MPC (model predictive control) with state constraints. We show nominal closed-loop stability for a wide class of nonlinear systems under full state feedback.

Stabilization in probability of nonlinear stochastic systems with guaranteed region of attraction and target set

We deal with nonlinear dynamical systems, consisting of a linear nominal part perturbed by model uncertainties, nonlinearities and both additive and multiplicative random noise, modeled as a Wiener process. In particular, we study the problem of finding suitable measurement feedback control laws such that the resulting closed-loop system is stable in some probabilistic sense. To this aim, we introduce a new notion of stabilization in probability, which is the natural counterpart of the classical concept of regional stabilization for deterministic nonlinear dynamical systems and stands as an intermediate notion between local and global stabilization in probability. This notion requires that, given a target set, a trajectory, starting from some compact region of the state space containing the target, remains forever inside some larger compact set, eventually enters any given neighborhood of the target in finite time and remains thereinafter, all these events being guaranteed with some probability. We give a Lyapunov-based sufficient condition for achieving stability in probability and a separation result which splits the control design into a state feedback problem and a filtering problem. Finally, we point out constructive procedures for solving the state feedback and filtering problem with arbitrarily large region of attraction and arbitrarily small target for a wide class of nonlinear systems, which at least include feedback linearizable systems. The generality of the result is promising for applications to other classes of stochastic nonlinear systems. In the deterministic case, our results recover classical stabilization results for nonlinear systems.

Exponential stabilization for nonlinear systems with applications to nonholonomic systems

The paper proposes a general framework to study the exponential stabilization problem for a wide class of nonlinear systems. By combining the concept of dilation with the method of σ-processing, a simple stability criterion is given in terms of the stability property of an augmented system. Moreover, the global ρ-exponential stability of the closed-loop system can be achieved by employing continuous controllers. A well-known nonholonomic system is given to illustrate the effectiveness of the proposed results.

On a switching control scheme for nonlinear systems with ill-defined relative degree

This paper discusses the applicability of a switching control scheme for a nonlinear system with ill-defined relative degree. The control scheme switches between exact and approximate input–output linearisation control laws. Unlike a linear system under a switching control scheme, the equilibria of a nonlinear system may change with the switching. It is pointed out that this is not sufficient to cause instability. When the region of the approximate linearisation control law is attractive to the exact zero dynamics, it is possible that the closed-loop system under the switching control scheme is still stable. The results in this paper shows that the switching control scheme proposed in Tomlin and Sastry (Systems Control Lett. 35(3) (1998) 145) is applicable for a wider class of nonlinear systems.

Stabilization in Probability of Nonlinear Stochastic Systems with Guaranteed Cost

We deal with nonlinear dynamical systems, consisting of a linear nominal part plus model uncertainties, nonlinearities, and both additive and multiplicative random noise, modeled as a Wiener process. In particular, we study the problem of finding suitable measurement feedback control laws such that the resulting closed-loop system is stable in some probabilistic sense and a given cost functional is minimized. We give a Lyapunov-based separation result which splits the control design into a state feedback problem and a filtering problem. Finally, we point out constructive algorithms for solving the state feedback and filtering problems with arbitrarily large region of attraction for a wide class of nonlinear systems, which at least include feedback linearizable systems.