427 research outputs found
Global stabilization of a Korteweg-de Vries equation with saturating distributed control
This article deals with the design of saturated controls in the context of
partial differential equations. It focuses on a Korteweg-de Vries equation,
which is a nonlinear mathematical model of waves on shallow water surfaces. Two
different types of saturated controls are considered. The well-posedness is
proven applying a Banach fixed point theorem, using some estimates of this
equation and some properties of the saturation function. The proof of the
asymptotic stability of the closed-loop system is separated in two cases: i)
when the control acts on all the domain, a Lyapunov function together with a
sector condition describing the saturating input is used to conclude on the
stability, ii) when the control is localized, we argue by contradiction. Some
numerical simulations illustrate the stability of the closed-loop nonlinear
partial differential equation. 1. Introduction. In recent decades, a great
effort has been made to take into account input saturations in control designs
(see e.g [39], [15] or more recently [17]). In most applications, actuators are
limited due to some physical constraints and the control input has to be
bounded. Neglecting the amplitude actuator limitation can be source of
undesirable and catastrophic behaviors for the closed-loop system. The standard
method to analyze the stability with such nonlinear controls follows a two
steps design. First the design is carried out without taking into account the
saturation. In a second step, a nonlinear analysis of the closed-loop system is
made when adding the saturation. In this way, we often get local stabilization
results. Tackling this particular nonlinearity in the case of finite
dimensional systems is already a difficult problem. However, nowadays, numerous
techniques are available (see e.g. [39, 41, 37]) and such systems can be
analyzed with an appropriate Lyapunov function and a sector condition of the
saturation map, as introduced in [39]. In the literature, there are few papers
studying this topic in the infinite dimensional case. Among them, we can cite
[18], [29], where a wave equation equipped with a saturated distributed
actuator is studied, and [12], where a coupled PDE/ODE system modeling a
switched power converter with a transmission line is considered. Due to some
restrictions on the system, a saturated feedback has to be designed in the
latter paper. There exist also some papers using the nonlinear semigroup theory
and focusing on abstract systems ([20],[34],[36]). Let us note that in [36],
[34] and [20], the study of a priori bounded controller is tackled using
abstract nonlinear theory. To be more specific, for bounded ([36],[34]) and
unbounded ([34]) control operators, some conditions are derived to deduce, from
the asymptotic stability of an infinite-dimensional linear system in abstract
form, the asymptotic stability when closing the loop with saturating
controller. These articles use the nonlinear semigroup theory (see e.g. [24] or
[1]). The Korteweg-de Vries equation (KdV for short)Comment: arXiv admin note: text overlap with arXiv:1609.0144
A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems
This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version
Global stabilization of a Korteweg-de Vries equation with a distributed control saturated in L 2 -norm
International audienceThis article deals with the design of saturated controls in the context of partial differential equations. It is focused on a Korteweg-de Vries equation, which is a nonlinear mathematical model of waves on shallow water surfaces. The aim of this article is to study the influence of a saturating in L 2-norm distributed control on the well-posedness and the stability of this equation. The well-posedness is proven applying a Banach fixed point theorem. The proof of the asymptotic stability of the closed-loop system is tackled with a Lyapunov function together with a sector condition describing the saturating input. Some numerical simulations illustrate the stability of the closed-loop nonlinear partial differential equation
Application of Lyapunov matrix inequality based unsymmetrical saturated control to a multi-vectored propeller airship
The problem of the design of a controller for a multi-vectored propeller airship is addressed. The controller includes anti-windup that takes into account unsymmetrical actuator constraints. First, a linear transformation is applied to transform the unsymmetrical constraints into symmetric constraints with an amplitude-bounded exogenous disturbance. Then, a stability condition based on a quadratic Lyapunov function for the saturated closed-loop system is proposed. The condition considers both amplitude-bounded and energy-bounded exogenous disturbances. Thus, the controller design problem is transformed into a convex optimization problem expressed in a bilinear matrix inequality form. Two controller design methods were applied: one-step controller and traditional anti-windup controller. The one-step method obtains the controller and the anti-windup compensator in one step while the anti-windup controller method separates this process into the linear controller design and the compensator design. Simulation results showed that both controllers enlarge the stability zone of the saturation system and have good tracking performance. It is shown that the anti-windup controller design method not only has a larger region of stability, but the demanded actuator output exceeds the constraints less and has a smaller anti-windup coefficient matrix compared to the one-step method
Application of Lyapunov matrix inequality based unsymmetrical saturated control to a multi-vectored propeller airship
The problem of the design of a controller for a multi-vectored propeller airship is addressed. The controller includes anti-windup that takes into account unsymmetrical actuator constraints. First, a linear transformation is applied to transform the unsymmetrical constraints into symmetric constraints with an amplitude-bounded exogenous disturbance. Then, a stability condition based on a quadratic Lyapunov function for the saturated closed-loop system is proposed. The condition considers both amplitude-bounded and energy-bounded exogenous disturbances. Thus, the controller design problem is transformed into a convex optimization problem expressed in a bilinear matrix inequality form. Two controller design methods were applied: one-step controller and traditional anti-windup controller. The one-step method obtains the controller and the anti-windup compensator in one step while the anti-windup controller method separates this process into the linear controller design and the compensator design. Simulation results showed that both controllers enlarge the stability zone of the saturation system and have good tracking performance. It is shown that the anti-windup controller design method not only has a larger region of stability, but the demanded actuator output exceeds the constraints less and has a smaller anti-windup coefficient matrix compared to the one-step method
Feedback Control Stabilization of the No-Motion State of a Fluid Confined in a Horizontal Porous Layer Heated From Below
We consider a horizontal three-dimensional saturated porous layer, confined in an upright cubic box, heated from below and cooled from above. In the absence of a controller, the fluid maintains a no-motion state for subcritical Rayleigh numbers R \u3c Rc, where Rc, depends on the box’s aspect ratio. Once this critical number is exceeded, fluid motion ensues. We demonstrate that, with the use of feedback control strategies which suppress flow instabilities, one can maintain a stable no-motion state for Rayleigh numbers far exceeding the classical critical one for the onset of convection. To preserve the equilibrium no-motion state of the classical problem, the controller alters the system’s dynamics so as to stabilize an otherwise non-stable state
Local stabilization of an unstable parabolic equation via saturated controls
We derive a saturated feedback control, which locally stabilizes a linear
reaction-diffusion equation. In contrast to most other works on this topic, we
do not assume the Lyapunov stability of the uncontrolled system and consider
general unstable systems. Using Lyapunov methods, we provide estimates for the
region of attraction for the closed-loop system, given in terms of linear and
bilinear matrix inequalities. We show that our results can be used with
distributed as well as scalar boundary control, and with different types of
saturations. The efficiency of the proposed method is demonstrated by means of
numerical simulations
Finite-Time H
This paper investigates the finite-time control problem for discrete-time Markov jump systems subject to saturating actuators. A finite-state Markovian process is given to govern the transition of the jumping parameters. The finite-time H∞ controller via state feedback is designed to guarantee that the resulting system is mean-square locally asymptotically finite-time stabilizable. Based on stochastic finite-time stability analysis, sufficient conditions that ensure stochastic control performance of discrete-time Markov jump systems are derived in the form of linear matrix inequalities. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach
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