Local Feedback Stabilization Research Articles

Local Output Feedback Stabilization of a Nonlinear Kuramoto–Sivashinsky Equation

Local Output Feedback Stabilization of a Nonlinear Kuramoto–Sivashinsky Equation

Local Output Feedback Stabilization of a Reaction–Diffusion Equation With Saturated Actuation

This article is concerned with the output feedback stabilization of a reaction–diffusion equation by means of bounded control inputs in the presence of saturations. Using a finite-dimensional controller composed of an observer coupled with a finite-dimensional state-feedback, we derive a set of conditions ensuring the stability of the closed-loop plant while estimating the associated domain of attraction in the presence of saturations. This set of conditions is shown to be always feasible for an order of the observer selected large enough. The stability analysis relies on Lyapunov functionals along with a generalized sector condition classically used to study the stability of linear finite-dimensional plants in the presence of saturations.

Local output feedback stabilization of reaction–diffusion PDEs with saturated measurement

AbstractThis paper addresses the topic of output feedback stabilization of general one-dimensional reaction–diffusion partial differential equations (PDEs) in the presence of a saturation in the measurement. The boundary control and the second boundary condition take the form of Dirichlet/Neumann/Robin boundary conditions. The measurement is selected as a boundary Dirichlet trace. The boundary measurement, as available for feedback control, is assumed to be subject to a saturation. In this context, we achieve the local exponential stabilization of the reaction–diffusion PDE while estimating a subset of the domain of attraction of the origin.

Local output feedback stabilization of reaction-diffusion PDEs with saturated measurement

Local output feedback stabilization of reaction-diffusion PDEs with saturated measurement

Decentralized vibration control of smart constrained layer damping plate

The traditional centralized control strategy to vibration suppression of large-scale thin plate structures may increase the design difficulty of the controller. In this article, a decentralized vibration active control method is proposed to suppress the vibration of the thin plate structure with smart constrained layer damping treatment. First, the dynamics model of the smart constrained layer damping plate is established based on the finite element method, and the characteristics of viscoelastic materials with temperature and frequency are described by Golla-Hughes-McTavish damping model. Subsequently, a decentralized subsystem control model is obtained from the balanced model reduction method and complex mode truncation method. The modal test proves that the theoretical model is accurate. Then, the particular emphasis is placed on the stability and vibration attenuation of a decentralized system, which is composed of multiple subsystems. The local state feedback stabilization, using interaction of local state feedback and output feedback, is introduced to achieve system stability. To solve the practical problem of local state feedback, a decentralized controller with an observer is developed by adopting the pole placement method. Finally, the numerical simulation and hardware-in-the-loop experiment under different excitation are performed to investigate the effectiveness of decentralized vibration active control. The results demonstrate that the decentralized controller can effectively suppress the vibration, especially under mixed periodic signal and Gauss white noise signal.

Oblique projection local feedback stabilization of nonautonomous semilinear damped wave-like equations

The stabilization of a class of nonlinear weakly damped wave equations by means of a finite-dimensional feedback control is investigated. The stabilizing control is constructed based on an appropriate oblique projection and it enters as a time-dependent linear combination of a finite number of suitable indicator functions supported in small regions. Firstly, it is shown that an oblique projection feedback is able to globally exponentially stabilize linear nonautonomous weakly damped wave equations. Then, relying on this result, the local stabilization for semilinear equations is proven for a suitable class of nonlinearities. Finally, numerical experiments are given which validate the theoretical results.

Local feedback stabilization of time-periodic evolution equations by finite dimensional controls

We study the feedback stabilization around periodic solutions of parabolic control systems with unbounded control operators, by controls of finite dimension. We prove that the stabilization of the infinite dimensional system relies on the stabilization of a finite dimensional control system obtained by projection and next transformedviaits Floquet-Lyapunov representation. We emphasize that this approach allows us to define feedback control laws by solving Riccati equations of finite dimension. This approach, which has been developed in the recent years for the boundary stabilization of autonomous parabolic systems, seems to be totally new for the stabilization of periodic systems of infinite dimension. We apply results obtained for the linearized system to prove a local stabilization result, around periodic solutions, of the Navier-Stokes equations, by finite dimensional Dirichlet boundary controls.

Feedback stabilization with one simultaneous control for systems of parabolic equations

In this work controlled systems of semilinear parabolic equations are considered. Only one control is acting in both equations and it is distributed in a subdomain. Local feedback stabilization is studied. The approach is based on approximate controllability for the linearized system and the use of an appropriate norm obtained from a Lyapunov equation. Applications to reaction-diffusion systems are discussed.

Control of Hopf bifurcations of nonlinear infinite-dimensional systems: application to axial flow engine compressor

We study the local feedback stabilization of Hopf bifurcations for nonlinear systems of infinite dimensions in the case where the linearized vector field has a pair of simple nonzero imaginary eigenvalues and all its other eigenvalues lie strictly in the left half-plane. Discussing the normal form of nonlinear systems obtained by making use of the integral averaging method, we obtain sufficient and necessary condition for controlling the stability of the systems even if the critical modes are uncontrollable. As an application, we apply the obtained results to the control of axial flow engine compressor.

Sufficient Conditions for Local Asymptotic Stability and Stabilization for Discrete-Time Varying Systems

The purpose of this paper is to establish sufficient conditions for local asymptotic stability and feedback stabilization for discrete-time systems with time depended dynamics. Our main results constitute generalizations of those developed by same authors in a recent paper, published in same journal, for the case of continuous-time systems.

Resilient decentralized stabilization of interconnected time‐delay systems with polytopic uncertainties

AbstractDecentralized delay‐dependent local stability and resilient feedback stabilization methods are developed for a class of linear interconnected continuous‐time systems. The subsystems are time‐delay plants which are subjected to convex‐bounded parametric uncertainties and additive feedback gain perturbations while allowing time‐varying delays to occur within the local subsystems and across the interconnections. The delay‐dependent local stability conditions are established at the subsystem level through the construction of appropriate Lyapunov–Krasovskii functional. We characterize decentralized linear matrix inequalities (LMIs)‐based delay‐dependent stability conditions by deploying an injection procedure such that every local subsystem is delay‐dependent robustly asymptotically stable with an γ‐level ℒ︁2‐gain. Resilient decentralized state‐feedback stabilization schemes are designed, which takes into account additive gain perturbations such that the family of closed‐loop feedback subsystems enjoys the delay‐dependent asymptotic stability with a prescribed γ‐level ℒ︁2‐gain for each subsystem. The decentralized feedback gains are determined by convex optimization over LMIs. All the developed results are tested on representative examples. Copyright © 2010 John Wiley & Sons, Ltd.

Local asymptotic feedback stabilization to a submanifold: Topological conditions

We consider the problem of local asymptotic feedback stabilization of a control system x ˙ = f ( x , u ) defined in R n to a compact, connected, oriented, embedded codimension one submanifold P of R n using a continuous feedback law. This generalizes the problem of local asymptotic feedback stabilization to a point which has been previously considered in the control theory literature. It is natural to expect the topology of P to play a role in deciding whether or not local asymptotic stabilization to P of the system x ˙ = f ( x , u ) is feasible via continuous feedback, and our aim in this paper is precisely to outline necessary conditions on the topology of P for stabilization via continuous feedback to be achievable.

Local separation principle for a special class of nonlinear systems

In this paper, we establish the local separation principle for the class of nonlinear systems which is approximately error linearizable and locally asymptotically stabilizable. For this class of nonlinear systems, it is first shown that there exists a local nonlinear observer whose estimation error converges to zero exponentially. In order to achieve local output feedback stabilization, the proposed nonlinear observer is employed in conjunction with the locally asymptotically stabilizing controller. As a result, we identify the class of nonlinear systems for which the local separation principle holds.

Feedback stabilization via designed planar centre manifold

AbstractThis paper addresses the problem of local state feedback stabilization of a class of nonlinear systems, which have planar centre manifold. A new technique for designing centre manifold has been developed. Using this approach, a sufficient condition for stabilization of a class of nonlinear system is obtained. The approach has been specified for two particular critical cases: ‘zero centre’ with zero eigenvalue of multiplicity 2 and ‘oscillatory centre’ with a pair of pure imaginary eigenvalues. The control law, which stabilizes the overall system, is also presented. Some examples are presented to indicate how the present theorem may be implemented. Copyright © 2003 John Wiley & Sons, Ltd.

Robust Feedback Stabilization of Limit Cycles in PWM DC-DC Converters

Local feedback stabilization of limit cycles in PWM DC-DC converters isconsidered, using recently developed general sampled-data models. Thepaper focuses on converters for which the nominal operating conditionhas lost stability due to off-design operation. The results apply tostabilization of the nominal periodic operating condition. In addition,the same approach can be used to stabilize other limit cycles such asthose embedded in a possible chaotic trajectory. Two feedbackstabilization schemes are proposed and studied in detail. The firststabilization technique uses voltage reference compensation and thesecond uses dynamic ramp compensation. Both employ discrete-time washoutfilters to ensure preservation of the size and shape of the limit cycle.Washout filters ensure that the nominal operating branch is unaffectedby the control, without the need for accurate knowledge of the limitcycle.

Stabilization of nonlinear systems via designed center manifold

This paper addresses the problem of local state feedback stabilization of a class of nonlinear systems with nonminimum phase zero dynamics. A new technique, namely, the Lyapunov function with homogeneous derivative along solution curves was developed to test the approximate stability of the dynamics on the center manifold. A set of convenient sufficient conditions are provided to test the negativity of the homogeneous derivatives. Using these conditions and assuming the zero dynamics has stable and center linear parts, a method is proposed to design controls such that the dynamics on the designed center manifold of the closed-loop system is approximately stable. It is proved that using this method, the first variables in each of the integral chains of the linearized part of the system do not affect the approximation order of the dynamics on the center manifold. Based on this fact, the concept of injection degree is proposed. According to different kinds of injection degrees certain sufficient conditions are obtained for the stabilizability of the nonminimum phase zero dynamics. Corresponding formulas are presented for the design of controls.

Bifurcation Stabilization with Local Output Feedback

Local output feedback stabilization with smooth nonlinear controllers is studied for parameterized nonlinear systems for which the linearized system possesses either a simple zero eigenvalue or a pair of imaginary eigenvalues and the bifurcated solution is unstable at the critical value of the parameter. It is assumed that the unstable mode corresponding to the critical eigenvalue of the linearized system is not linearly controllable. Results are established for bifurcation stabilization using output feedback where the critical mode can be either linearly observable or linearly unobservable. The stabilizability conditions are characterized in explicit forms that can be used to synthesize stabilizing controllers. The results obtained in this paper are applied to rotating stall control for axial flow compressors as an application example.

Application of center manifold reduction to nonlinear system stabilization

The Center Manifold Theorem is applied to the local feedback stabilization of nonlinear systems in critical cases. The paper addresses two particular critical cases of which the system linearization at the equilibrium point of interest is assumed to possess either a simple zero eigenvalue or a complex conjugate pair of simple and pure imaginary eigenvalues. In either case, the noncritical eigenvalues are taken to be stable. The results on stabilizability and stabilization are given explicitly in terms of the nonlinear model of interest in its original form, i.e., before reduction to the center manifold. Moreover, the formulation given in this paper uncovers connections between results obtained using the center manifold reduction and those of an alternative approach.

Output feedback stabilization of SISO nonlinear systems via dynamic sliding modes

Previous work on asymptotic stabilization of SISO nonlinear systems using dynamic sliding mode control to produce dynamic state feedback has been developed further to produce dynamic output feedback. All the states in the feedback controller are replaced with estimated states which come from a semi-high gain observer. An upper bound of the semi-high gain parameter is explicitly given. It is proved that if the given differential I-O system is minimum phase and proper, local uniform asymptotic output feedback stabilization can be achieved. Pertinent examples are given to show how the method works for both differential inputoutput models and state space models.

On Stabilization of Nonlinear Systems

The local feedback stabilization problem for a nonlinear time-dependent control system is studied from a viewpoint of Lyapunov's direct method. Sufficient conditions of asymptotic stability for a general nonlinear nonautonomous system of ordinary differential equations in the critical case are given. It is achieved through the construction of an appropriate generalized Lyapunov function. The algorithm of this construction is given for the nonautonomous system whose linearization at an equilibrium point has a constant matrix which possesses 2m pure imaginary eigenvalues (m ≥ 1) and the remaining eigenvalues lying in the open left half of the complex plane.