Arbitrary Decay Rate Research Articles

Feedback Stabilization of a Two-Fluid Surface Tension System Modeling the Motion of a Soap Bubble at Low Reynolds Number: The Two-Dimensional Case

The aim of this paper is to design a feedback operator for stabilizing in infinite time horizon a system modeling the interactions between a viscous incompressible fluid and the deformation of a soap bubble. The latter is represented by an interface separating a bounded domain of mathbb {R}^2 into two connected parts filled with viscous incompressible fluids. The interface is a smooth perturbation of the 1-sphere, and the surrounding fluids satisfy the incompressible Stokes equations in time-dependent domains. The mean curvature of the surface defines a surface tension force which induces a jump of the normal trace of the Cauchy stress tensor. The response of the fluids is a velocity trace on the interface, governing the time evolution of the latter, via the equality of velocities. The data are assumed to be sufficiently small, in particular the initial perturbation, that is the initial shape of the soap bubble is close enough to a circle. The control function is a surface tension type force on the interface. We design it as the sum of two feedback operators: one is explicit, the second one is finite-dimensional. They enable us to define a control operator that stabilizes locally the soap bubble to a circle with an arbitrary exponential decay rate, up to translations, and up to non-contact with the outer boundary.

Robust output regulation of a class of smart actuators described by a minimum phase LPV dynamics

This paper presents a new control strategy for a class of minimum phase linear parameter-varying (LPV) systems representative of smart material actuators. In position control applications of such devices, one is typically interested in achieving output regulation with an arbitrary exponential decay rate. In principle, this problem can be tackled by assigning an exponential decay rate to the full system state, by means of well-established linear matrix inequalities (LMI) methods. For the class of systems under investigation, however, this approach leads to excessively large controller gains in case the desired decay rate is faster than the open-loop zero dynamics. In addition, the existence of unmeasurable state variables related to the material microstructure makes it not possible to directly implement full state feedback laws which are commonly adopted in LPV theory. To overcome these issues, a new design strategy is proposed. The key idea relies on arbitrarily shaping the output exponential decay rate without requiring fast convergence of the full state vector. This goal is achieved by means of a partial state feedback control law which solely depends on the measurable system states. A LMI algorithm is also developed to systematically address the design of the partial state feedback controller. The method is validated by means of simulations on generic examples, as well as via experiments on a mechatronic positioning system based on a dielectric elastomer membrane. In case the desired output convergence speed is faster than the open-loop zero dynamics, it is shown that the new approach leads to better transient behaviors and significantly smaller controller gains than standard LPV design techniques.

Internal stabilization of three interconnected semilinear reaction-diffusion PDEs with one actuated state

This work deals with the exponential stabilization of a system of three semilinear parabolic partial differential equations (PDEs), written in a strict feedforward form. The diffusion coefficients are considered distinct and the PDEs are interconnected via both a reaction matrix and a nonlinearity. Only one of the PDEs is assumed to be controlled internally, thereby leading to an underactuated system. Constructive and efficient control of such underactuated systems is a nontrivial open problem, which has been solved recently for the linear case. In this work, these results are extended to the semilinear case, which is highly challenging due the coupling introduced by the semilinearity. Modal decomposition is employed, where due to the semilinearity, the finite-dimensional part of the solution is coupled with the infinite-dimensional tail. A transformation is then employed to map the finite-dimensional part into a target system, which allows for an efficient design of a static linear proportional state-feedback controller. Furthermore, a high-gain approach is employed in order to compensate for the semilinear terms. Lyapunov stability analysis is performed, leading to LMI conditions guaranteeing exponential stability with an arbitrary decay rate. The LMIs are shown to always be feasible, provided the number of actuators and the value of the high gain parameter are large enough. Numerical examples demonstrate the proposed approach.

On exact controllability and complete stabilizability for linear systems

In this paper we consider linear systems with control described by the equation $\dot x = \mathcal A x +\mathcal B u$ where functions u and x take values in U and X respectively. For such object, a short review of results concerning relations between exact controllability and complete stabilizability (stabilizability with arbitrary decay rate) is given. The analysis is done in various situations: bounded or unbounded state and control operators $\mathcal A$ and $\mathcal B$, Banach or Hilbert spaces U and X. The well known equivalence between complete controllability and pole assignment in the situation of finite dimensional spaces is no longer true in general in infinite dimensional spaces. Exact controllability is not sufficient for complete stabilizability if U and X are Banach spaces. In Hilbert space setting this implication holds true. The converse also is not so simple: in some situations, complete stabilizability implies exact controllability (Banach space setting with bounded operators), in other situation, it is not true. The corresponding results are given with some ideas for the proofs. Complete technical development are indicated in the cited literature. Several examples are given. Special attention is paid to the case of infinite dimensional systems generated by delay systems of neutral type in some general form (distributed delays). The question of the relation between exact null controllability and complete stabilizability is more precisely investigated. In general there is no equivalence between the two notions. However for some classes of neutral type equations there is an equivalence. The question how the equivalence occurs for more general systems is still open. This is a short and non exhaustive review of some research on control theory for infinite dimensional spaces. Our works in this area were initiated by V. I. Korobov during the 70th of the past century in Kharkov State University.

Feedback stabilization of a 3D fluid flow by shape deformations of an obstacle

We consider a fluid flow in a time dependent domain Ωf(t)=Ω\Ωs(t)̅⊂ℝ3, surrounding a deformable obstacle Ωs(t). We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in Ωf(t), t > 0. We prove that, for any arbitrary exponential decay rate ω > 0, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary ∂Ωs(t) can be chosen so that the fluid flow is stabilized to rest, and the obstacle to its initial shape and its initial location, with the exponential decay rate ω > 0.

On the controllability and stabilization of the Benjamin equation on a periodic domain

The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain T. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in Hps(T), with s≥0. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32].

Stabilization with Arbitrary Convergence Rate for the Schrödinger Equation Subjected to an Input Time Delay

The stabilization problem for the Schrodinger equation with an input time delay is considered from the view of system equivalence. First, a linear transform from the original system into an exponentially stable system with arbitrary decay rate, also called “target system”, is introduced. The linear transform is constructed via a kind of Volterra-type integration with singular kernels functions. As a result, a feedback control law for the original system is obtained. Secondly, a linear transform from the target system into the original closed-loop system is derived. Finally, the exponential stability with arbitrary decay rate of the closed-loop system is obtained through the established equivalence between the original closed-loop system and the target one. The authors conclude this work with some numerical simulations giving support to the results obtained in this paper.

A Sliding Mode Fault Compensation Scheme for a Coupled Rigid-Flexible System in PDE-ODE Form

This paper develops a new fault-tolerant control scheme for a coupled rigid-flexible system to handle control-matched uncertain faults, using backstepping based sliding mode control technique. The system dynamics is modeled by an Euler-Bernoulli beam-ODE (EBB-ODE) system. A distinct model refinement is utilized to map this EBB-ODE cascade into a Schrödinger-like PDE-ODE cascade. In order to surmount the obstacles when deriving the kernels in a one-step backstepping control design, a distinct backstepping control procedure is employed by constructing an exponentially stable target system with arbitrary decay rate. Based on the relative degrees of the selected sliding mode surfaces, a novel fault-tolerant controller is developed to drive faulty boundaries to zero in finite time and to avoid chattering effectively. A modified Lyapunov-based Riesz basis frame is adopted to prove the exponential stability and the well-posedness of closed-loop system. A numerical simulation is presented to validate the effectiveness of the developed control scheme.

Polynomial decay rate for a new class of viscoelastic Kirchhoff equation related with Balakrishnan-Taylor dissipation and logarithmic source terms

In this paper, a polynomial decay rate of Kirchhoff’s nonlinear viscoelastic viscoelastic equation solution related with Balakrishnan-Taylor dissipation solution and logarithmic source terms is obtained, where we obtain the result of energy decay of solutions without impose the usual relationship between certain relaxation functions and their derivatives. This result generalizes the previous results to an arbitrary decay rate, that is not necessarily of the polynomial or exponential decay.

Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation

In this paper, we study the exponential stabilization of a shock steady state for the inviscid Burgers equation on a bounded interval. Our analysis relies on the construction of an explicit strict control Lyapunov function. We prove that by appropriately choosing the feedback boundary conditions, we can stabilize the state as well as the shock location to the desired steady state in [Formula: see text]-norm, with an arbitrary decay rate.

Boundary feedback stabilization of hydraulic jumps

In an open channel, a hydraulic jump is an abrupt transition between a torrential (supercritical) flow and a fluvial (subcritical) flow. In this article hydraulic jumps are represented by discontinuous shock solutions of hyperbolic Saint-Venant equations. Using a Lyapunov approach, we prove that we can stabilize the state of the system in H2-norm as well as the hydraulic jump location, with simple feedback boundary controls and an arbitrary decay rate, by appropriately choosing the gains of the feedback boundary controls.

Decay Estimates for a Kind of Linear Wave Equations

We consider one kind of dissipative wave equations with exponential speed of propagation. An arbitrary power decay rate for the L2 -norm and energy is obtained by using the multiplier method.

Delayed point control of a reaction–diffusion PDE under discrete-time point measurements

We consider stabilization problem for reaction–diffusion PDEs with point actuations subject to a known constant delay. The point measurements are sampled in time and transmitted through a communication network with a time-varying delay. To compensate the input delay, we construct an observer for the future value of the state. Using a time-varying observer gain, we ensure that the estimation error vanishes exponentially with a desired decay rate if the delays and sampling intervals are small enough while the number of sensors is large enough. The convergence conditions are obtained using a Lyapunov–Krasovskii functional, which leads to linear matrix inequalities (LMIs). We design output-feedback point controllers in the presence of input delays using the above observer. The boundary controller is constructed using the backstepping transformation, which leads to a target system containing the exponentially decaying estimation error. The in-domain point controller is designed and analysed using an improved Wirtinger-based inequality. We show that both controllers can guarantee the exponential stability of the closed-loop system with an arbitrary decay rate smaller than that of the observer’s estimation error.

Decay rates for stabilization of linear continuous-time systems with random switching

For a class of linear switched systems in continuous time a controllability condition implies that state feedbacks allow to achieve almost sure stabilization with arbitrary exponential decay rates. This is based on the Multiplicative Ergodic Theorem applied to an associated system in discrete time. This result is related to the stabilizability problem for linear persistently excited systems.

Exact null controllability, complete stabilizability and continuous final observability of neutral type systems

Abstract For abstract linear systems in Hilbert spaces we revisit the problems of exact controllability and complete stabilizability (stabilizability with an arbitrary decay rate), the latter property being related to exact null controllability. We also consider the case when the feedback is not bounded. We obtain a characterization of complete stabilizability for neutral type systems. Conditions for exact null controllability of neutral type systems are discussed. By duality, we obtain a result about continuous final observability. Illustrative examples are given.

Observer Design and Exponential Stabilization for Wave Equation in Energy Space by Boundary Displacement Measurement Only

In this technical note, we consider finite-time and exponential stabilization for one-dimensional wave equations by boundary displacement measurement only. We limit ourselves in the energy state space where the usual observability inequality is not valid anymore. However, in the optimal state space, the boundary displacement is indeed exactly observable. This motivates us to design observer via displacement output only for these systems. We first discuss a simple case as a motivation. An observer is designed and an output feedback control is then synthesized to make system finite-time stable in energy space. In this same spirit, we consider the same problem for an unstable wave equation and the finite-time stability is also achieved by displacement output feedback. Finally, we consider an anti-stable wave equation. A direct delayed output feedback control can achieve exponential stability with arbitrary decay rate. Simulation results are presented to validate the theoretical conclusions.

Arbitrary decay rates of energy for a von Karman equation of memory type

In this paper we consider a von Karman equation of memory type utt+Δ2u−∫0tg(t−s)Δ2u(s)ds=[u,F(u)] with clamped boundary condition. We establish a decay result of solutions without imposing the usual relation between a kernel function g and its derivative. This result generalizes earlier ones to an arbitrary rate of decay, which is not necessarily of an exponential or polynomial decay.

Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages

This paper is concerned with the observer design for one-dimensional linear parabolic partial differential equations whose output is a weighted spatial average of the state over the entire spatial domain. We focus on the backstepping approach, which provides a systematic procedure to design an observer gain for systems with boundary measurement. If the output is not a boundary value of the state, the backstepping approach is not directly applicable to obtaining an observer gain that stabilizes the error dynamics. Therefore, we attempt to convert the error system into another system to which backstepping is applicable. The conversion is successfully achieved for a class of weighting functions, and the resultant observer realizes exponential convergence of the estimation error with an arbitrary decay rate in terms of the L2 norm. In addition, an explicit expression of the observer gain is available in a special case. The effectiveness of the proposed observer is also confirmed by numerical simulations.

Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.

This paper is the first part of a work which consists in proving the stabilization to zero of a fluid-solid system, in dimension 2 and 3. The considered system couples a deformable solid and a viscous incompressible fluid which satisfies the incompressible Navier-Stokes equations. By deforming itself, the solid can interact with the environing fluid and then move itself. The control function represents nothing else than the deformation of the solid in its own frame of reference. We there prove that the velocities of the linearized system are stabilizable to zero with an arbitrary exponential decay rate, by a boundary deformation velocity which can be chosen in the form of a feedback operator. We then show that this boundary feedback operator can be obtained from an internal deformation of the solid which satisfies the linearized physical constraints that a self-propelled solid has to satisfy.

Backstepping observer using weighted spatial average for 1-dimensional parabolic distributed parameter systems

AbstractThis paper is concerned with designing a backstepping-based observer for a class of 1-dimensional parabolic distributed parameter systems whose output is a weighted spatial average of the state. We first show that an integral transformation converts the original system into another parabolic system with boundary observation if the weighting function satisfies a parameterized ordinary differential equation. Then the backstepping observer for the transformed system is available and an estimate of the original state is obtained through the inverse transformation. We also show that the estimation error exponentially converges to 0 in terms of the L2 norm with arbitrary decay rate. Furthermore, a closed-form expression of the observer is provided for systems described by linear reaction-diffusion equations.