Stabilization of second order nonlinear equations with variable delay
Leonid Berezansky, Elena Braverman, Lev Idels
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10 pages, 4 figures, published in 2015 in International Journal of Control, non-autonomous second order delay differential equations, stabilization, proportional feedback, a controller with damping, derivative control, sunflower equation
For a wide class of second order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal, allows to stabilize unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilization by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.
We develop a stabilization strategy of turning processes by means of delayed spindle control. We show that turning processes which contain intrinsic state-dependent delays can be stabilized by a spindle control with state-dependent delay, and develop analytical description of the stability region in the parameter space. Numerical simulations stability region are also given to illustrate the general results.
For vibrating systems, a delay in the application of a feedback control can destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end. We show that for certain feedback parameters the system is exponentially stable with constant delays of the form 4L/c, 8L/c, 12L/c ... Moreover, we show that the system is exponentially stable with piecewise constant delays that attain the values 4L/c and 8L/c.
We study the problem of mean-square exponential incremental stabilization of nonlinear systems over uncertain communication channels. We show the ability to stabilize a system over such channels is fundamentally limited and the channel uncertainty must provide a minimal Quality of Service (QoS) to support stabilization. The smallest QoS necessary for stabilization is shown as a function of the positive Lyapunov exponents of uncontrolled nonlinear systems. The positive Lyapunov exponent is a measure of dynamical complexity and captures the rate of exponential divergence of nearby system traject…
The paper is concerned with stabilization of a scalar delay differention equation $$ {\dot x}(t) - \sum_{k=1}^m A_k(t)x[h_k(t)] = 0,~t\geq 0,~ x(ξ)=φ(ξ), ξ<0, $$ by introducing impulses in certain moments of time $$ x(τ_j) = B_j x(τ_j -0), ~j=1,2, \dots ~. $$ Explicit stability results are presented both for the equation with positive coefficients and for the equation with $A_k$ being of arbitrary sign.
This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order that that of the nominal difference equation. In the general case, the controller consists of two combined parts, namely, the feedback nominal controller which stabilizes the nominal (i.e. perturbation - free) difference equation plus an incremental controller which completes the stabilization in the presence of dynamics in the uncontrolled difference equation. A stabilization variant co…
This paper investigates the robust stability and stabilization analysis of interval fractional-order systems with time-varying delay. The stability problem of such systems is solved first, and then using the proposed results a stabilization theorem is also included, where sufficient conditions are obtained for designing a stabilizing controller with a predetermined order, which can be chosen to be as low as possible. Utilizing efficient lemmas, the stability and stabilization theorems are proposed in the form of LMIs, which is more suitable to check due to various existing efficient convex opt…
Title: Global stabilization of control nonlinear system "inverted pendulum on a cart" using method of two Lyapunov functions Authors: B. L. Mazov (Nizhny Novgorod Technical University) Comments: 10 pages Subj-class: Dynamical Systems MSC-class: 34D23 (Primary); 93H10 (Secondary) In this work a mathematical apparatus for method of several global Lyapunov functions is applied to study the stability properties of nonlinear model of concrete singularly perturbed mechanical system "inverted pendulum on a cart" with discontinuous relay-type control. In this analysis, the concept of solution in sense…
We develop an eigenvalue-based approach for the stability assessment and stabilization of linear systems with multiple delays and periodic coefficient matrices. Delays and period are assumed commensurate numbers, such that the Floquet multipliers can be characterized as eigenvalues of the monodromy operator and by the solutions of a finite-dimensional non-linear eigenvalue problem, where the evaluation of the characteristic matrix involves solving an initial value problem. We demonstrate that such a dual interpretation can be exploited in a two-stage approach for computing dominant Floquet mul…
The problem of vibration suppression in helicopter rotors is the main topic of this paper. We revisit a strategy for the stabilization of such systems based on delayed feedback. By means of an improved analysis, made possible by recently developed mathematical tools, we present insights on how the delay can be used as stabilization tool for this problem. The results are supported with numerical simulations. For simplicity, only the case of hovering flight is considered.
This paper considers the problem of robust stability and stabilization for linear fractional-order system with nonlinear uncertain parameters, with fractional order 0<a<2. A dynamic output feedback controller, with predetermined order, for asymptotically stabilizing such uncertain fractional-order systems is designed. The derived stabilization conditions are in LMI form. Simulation results of two numerical examples illustrate that the proposed sufficient theoretical results are applicable and effective for tackling robust stabilization problems. Keywords: Fractional-order system, nonlinear unc…
We examine the phenomenon of nonlinear stabilization, exhibiting a variety of related examples and counterexamples. For Gâteaux differentiable maps, we discuss a mechanism of nonlinear stabilization, in finite and infinite dimensions, which applies in particular to hyperbolic partial differential equations, and, for Fréchet differentiable maps with linearized operators that are normal, we give a sharp criterion for nonlinear exponential instability at the linear rate. These results highlight the fundamental open question whether Fréchet differentiability is sufficient for linear exponential in…
Results on the problem of stabilizing a nonlinear continuous-time system by a finite number of control or measurement values are presented. The basic tool is a discontinuous version of the so-called semi-global backstepping lemma. We derive robust practical stabilizability results by quantized and ternary controllers and apply them to some significant control problems.
In this paper we present a general tool to handle the presence of zero dynamics which are asymptotically but not locally exponentially stable in problems of robust nonlinear stabilization by output feedback. We show how it is possible to design locally Lipschitz stabilizers under conditions which only rely upon a partial detectability assumption on the controlled plant, by obtaining a robust stabilizing paradigm which is not based on design of observers and separation principles. The main design idea comes from recent achievements in the field of output regulation and specifically in the desig…
