Sets Of Nonlinear Systems Research Articles

Probabilistic ultimate bounds and invariant sets in nonlinear systems

This article extends the notions of probabilistic ultimate bounds (PUB) and probabilistic invariant sets (PIS) to nonlinear continuous-time systems providing tools for their characterization and for the usage of these concepts in control design. Two design strategies are proposed that allow finding a nonlinear control law which ensures that the closed loop system is probabilistically ultimate bounded to a desired region. These strategies are based on Lyapunov and stochastic feedback linearization, respectively.

Fuzzy modeling and control of a class of non‐differentiable multi‐input multi‐output nonlinear systems

AbstractThis paper presents a new approach in modeling and control of multi‐input multi‐output (MIMO) systems that have non‐differentiable operating points. A circle criterion is introduced at the non‐differentiable operating points to divide the entire operating region into two parts. Takagi‐Sugeno fuzzy models are developed in each part, and a switching framework is introduced to model the operating region. Accordingly, a sliding mode controller (SMC) is developed. The proposed modeling and controller are implemented on the benchmark quadruple tank process (QTP). It is demonstrated that the proposed modeling and controller design provides simplicity and universality in a set of nonlinear systems; it is robust with respect to various internal and external disturbances and model uncertainties which allows for accurate regulation and tracking; and it converges in finite time, is capable of controlling nonlinear systems with coupling dynamic, and enables fuzzy modeling of continuous‐time systems without imposing differentiability.

On Asymptotics of Small-Time Reachable Sets for Nonlinear Systems with Isoperimetric Constraints

We study the problem of approximate representation of reachable sets at small time intervals for nonlinear control-affine systems with isoperimetric control constraints. By isoperimetric constraints we mean integral constraints on the control input and state of the system. It is shown that under certain conditions imposed on the controllability Gramian of the linearized system the reachable set turns out to be convex and asymptotically close in shape to an ellipsoid in the state space for a sufficiently small time interval. This ellipsoid is a reachable set of the system linearized along the trajectory generated by zero control input.

Smooth controller design for non‐linear systems using multiple fixed models

Multiple model adaptive control (MMAC) with second-level adaptation is a recently proposed methodology for dealing with systems where the parametric uncertainty is large. Compared with the multiple model switching scheme, the new scheme can lead to significant improvements in performance. Some research has been conducted using the new scheme, but all of the results concern linear systems with an adaptive identification model set. In this study, MMAC with second-level adaptation scheme is extended to non-linear systems in strict feedback form, and the fixed identification model set is under consideration. This is motivated by the fact that a smooth controller can lead to smooth performance and the fixed identification model set gains potential advantages over the adaptive identification model set, especially for the case that the parameters of the system change over the time. Design details are presented and the stability of MMAC with second-level adaptation using a fixed identification model set for non-linear systems is given, which has not been discussed before. Finally, two simulations are performed to show that this scheme performs much better than conventional schemes, including adaptive control and multiple-model switching schemes, in terms of convergence speed and transient performance.

Adaptive time stepping in pseudo-dynamic testing of earthquake driven structures

The problem of assessing errors in implementing time-marching algorithms in the context of pseudo-dynamic seismic testing of structures is considered. These errors occur in implementing the numerical and experimental steps of the test procedure. The study investigates how a linearized variational equation can be augmented with the governing equation of motion to track the effect of the errors, and, accordingly, adjust the step size of integration adaptively to keep a global error norm within specified limits. The governing augmented equations are integrated using an explicit operator splitting scheme. Additional efforts, in terms of evaluation of the tangent stiffness matrix, are shown to become necessary while modelling the errors. Illustrative examples include numerical studies on a set of nonlinear systems and an experimental study on a geometrically nonlinear two-storied building frame. The experimental results from pseudo-dynamic test are shown to compare reasonably well with pertinent results from an effective force test.

Reduction theorems for stability of closed sets with application to backstepping control design

We present a solution to the following reduction problem for asymptotic stability of closed sets in nonlinear systems. Given two closed, positively invariant subsets of the state space of a nonlinear system, Γ1⊂Γ2, assuming that Γ1 is asymptotically stable relative to Γ2, find conditions under which Γ1 is asymptotically stable. We also investigate analogous reduction problems for stability and attractivity. We illustrate the implications of our results on the stability of sets for cascade-connected systems and on a hierarchical control design problem. For upper triangular control systems, we present a reduction-based backstepping technique that does not require the knowledge of a Lyapunov function, and mitigates the problem of controller complexity arising in classical backstepping design.

Estimating the Parameters of the Negative-Lindley Distribution using Broyden-Fletcher-Goldfarb-Shanno

Problem statement: The Maximum Likelihood Estimation (MLE) technique is the most efficient statistical approach to estimate parameters in a cross-sectional model. Often, MLE gives rise to a set of non-linear systems of equations that need to be solved iteratively using the Newton-Raphson technique. However, in some situations such as in the Negative-Lindley distribution where it involves more than one unknown parameter, it becomes difficult to apply the Newton-Raphson approach to estimate the parameters jointly as the second derivatives of the score functions in the Hessian matrix are complicated. Approach: In this study, we propose an alternate iterative algorithm based on the Broyden-Fletcher-Goldfarb-Shanno (BFGS) approach that does not require the computation of the higher derivatives. Conclusion: To assess the performance of BFGS, we generate samples of overdispersed count with various dispersion parameters and estimate the mean and dispersion parameters. Results: BFGS estimates the parameters of the Negative-Lindley model efficiently.

LOCALIZING BOUNDS FOR COMPACT INVARIANT SETS OF NONLINEAR SYSTEMS POSSESSING FIRST INTEGRALS WITH APPLICATIONS TO HAMILTONIAN SYSTEMS

In this paper, we study the localization problem of compact invariant sets of nonlinear systems possessing first integrals by using the first order extremum conditions and positive definite polynomials. In the case of natural polynomial Hamiltonian systems, our results include those in [Starkov, 2008] as a special case. This paper discusses the application to studies of the generalized Yang–Mills Hamiltonian system and the Hamiltonian system describing dynamics of hydrogenic atoms in external fields.

Sampled-data control and stability of sets for nonlinear systems

This paper is devoted to the investigation of the relation between stability properties of nonlinear closed-loop control systems and those of families of approximations with respect to a given set. Firstly, such relation is derived for continuous-time closed-loop systems and the associated sampled-data systems. Secondly, stability results are presented for parametrized families of exact and approximate discrete-time models considered with parametrized family of feedbacks. In both cases, the sampling period is allowed to vary from step to step, and the stability is investigated with respect to nonempty closed (not necessarily compact) sets. Necessary and sufficient conditions are presented on the stability of approximations ensuring stability of the approximated systems.

A parametric approach for eigenvalue assignment by static output feedback

A well-known method of eigenvalue assignment by static output feedback is improved. The main result is a parametric expression for the output feedback controller gain matrix explicitly characterized by the set of non-linear system of equations obtained for the state feedback design and the set of linear equations resulting from static output feedback consideration. In practice, it is shown that all the possible controllers can be generated for exactly assigning the prescribed eigenvalues of the nominal plant by appropriate software for solving the set of non-linear system of equations thus obtained. This in turn makes it possible to select the output feedback matrix with minimum norm or other constraints. Some numerical examples are presented to illustrate the design technique.

Fault tolerant nonlinear control with applications to an automated transit bus

This paper presents an integrated structure for a passive and active fault tolerant control (FTC) design approach in the framework of a robust nonlinear control technique called Dynamic Surface Control (DSC). As motivated by the automated vehicle application, we consider two categories of possible faults: pre-specified (a priori) and non-specified faults. It is first shown that DSC can be considered as a passive FTC approach in the sense that it gives simultaneous robust stability to a set of nonlinear systems even in the presence of model uncertainties and the pre-specified faults. Then, the non-specified fault is classified depending on the fault’s impact on the closed-loop system and isolatability from a fault detection and diagnosis (FDD) system. If a fault is both intolerable and isolatable, an active FTC approach is taken which includes FDD and controller reconfiguration. More specifically, trajectory reconfiguration is considered to accommodate the actuator fault, i.e., to compensate for the performance loss due to the fault within the framework of a switched hierarchical structure. Finally, the integrated structure for the longitudinal control of an automated transit bus is designed through the proposed method. Simulation results of the fault tolerant controller are shown for both single and multiple multiplicative faults. This controller was implemented on the California PATH transit buses in a demonstration of automated public transportation technology in San Diego, California in August of 2003.

Excitable dynamics and threshold sets in nonlinear systems.

Following our previous work [J. Zagora et al., Faraday Discuss. 120, 313 (2001)], we present a quantitative definition of a threshold that separates large-amplitude excitatory responses and small-amplitude nonexcitatory responses to a perturbation of an excitable system with a single globally attracting steady state. For systems with two variables, finding the threshold set is formulated as a boundary value problem supplemented by a condition of a maximum separation rate. For this highly nonlinear problem we formulate a numerical method based on the use of multiple shooting and continuation methods. The threshold phenomena are examined by using an example dynamical system with chemical reaction--the bromate-sulfite-ferrocyanide system. In a model of this reaction we find the threshold set, construct a bifurcation diagram and discuss how excitability can vanish. These results are compared with recent experiments. We also discuss relevance of other definitions of the excitability threshold including the concept of nullclines.

Stabilization of Invariant Sets in Nonlinear Systems: Chetaev's Bundles and Speed-Gradient

Abstract The current results related to stabilization of sets based on SpeedGradient method and the notion of V -detectability are overviewed. Extensions to nonaffine and cascaded systems are presented.

Frequency criteria for the stability and instability in the large of stationary sets of nonlinear systems of differential equations with a single monotone nonlinearity

Frequency criteria for the stability and instability in the large of stationary sets of nonlinear systems of differential equations with a single monotone nonlinearity

Global Controllability of Nonlinear Systems

This paper examines the relationship between the structure of the reachable set for nonlinear systems and the properties of the Lie algebras of vector fields associated with nonlinear systems. An expression for the reachable set at time t is obtained for a large class of nonlinear systems using unbounded controls.