Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary
TL;DR: This paper develops a methodology for the case where the uncontrolled boundary condition has anti-damping, which makes the real parts of all the eigenvalues of the uncontrolled system positive and arbitrarily high, i.e., the plant is “anti-stable” (exponentially stable in negative time).
About: This article is published in Systems & Control Letters.The article was published on 2009-08-01. It has received 129 citations till now. The article focuses on the topics: Mixed boundary condition & Robin boundary condition.
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413 citations
TL;DR: This work considers a cascade of a diffusion–convection PDE with an ODE, where the convection direction is “away” from the ODE and relies on the diffusion process to propagate the control signal through the PDE towards the Ode, to stabilize the ODR.
Abstract: We extend several recent results on full-state feedback stabilization and state estimation of PDE–ODE cascades, where the PDEs are either of heat type or of wave type, from the previously considered cases where the interconnections are of Dirichlet type, to interconnections of Neumann type. The Neumann type interconnections constrain the PDE state to be subject to a Dirichlet boundary condition at the PDE–ODE interface, and employ the boundary value of the first spatial derivative of the PDE state to be the input to the ODE. In addition to considering heat-ODE and wave-ODE cascades, we also consider a cascade of a diffusion–convection PDE with an ODE, where the convection direction is “away” from the ODE. We refer to this case as a PDE–ODE cascade with “counter-convection.” This case is not only interesting because the PDE subsystem is unstable, but because the control signal is subject to competing effects of diffusion, which is in both directions in the one-dimensional domain, and counter-convection, which is in the direction that is opposite from the propagation direction of the standard delay (transport PDE) process. We rely on the diffusion process to propagate the control signal through the PDE towards the ODE, to stabilize the ODE.
208 citations
13 Sep 2012
TL;DR: This work develops an ADRC to attenuate the disturbance of a one-dimensional anti-stable wave equation subject to boundary disturbance and shows that this strategy works well when the derivative of the disturbance is also bounded.
Abstract: In this technical note, we are concerned with the boundary stabilization of a one-dimensional anti-stable wave equation subject to boundary disturbance. We propose two strategies, namely, sliding mode control (SMC) and the active disturbance rejection control (ADRC). The reaching condition, and the existence and uniqueness of the solution for all states in the state space in SMC are established. The continuity and monotonicity of the sliding surface are proved. Considering the SMC usually requires the large control gain and may exhibit chattering behavior, we then develop an ADRC to attenuate the disturbance. We show that this strategy works well when the derivative of the disturbance is also bounded. Simulation examples are presented for both control strategies.
168 citations
TL;DR: A feedback law based on the backstepping method is designed and exponential stability of the closed-loop system with a desired decay rate is proved and a new Lyapunov function is produced for the classical wave equation with passive boundary damping.
Abstract: We consider the problem of boundary stabilization of a 1-D (one-dimensional) wave equation with an internal spatially varying antidamping term. This term puts all the eigenvalues of the open-loop system in the right half of the complex plane. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate. For plants with constant parameters the control gains are found in closed form. Our design also produces a new Lyapunov function for the classical wave equation with passive boundary damping.
158 citations
TL;DR: Capabilities of the proposed synthesis and its effectiveness are supported by numerical studies made for three coupled systems with distinct diffusivity parameters and for underactuated linearized dimensionless temperature-concentration dynamics of a tubular chemical reactor, controlled through a boundary at low fluid superficial velocities when convection terms become negligible.
120 citations
References
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Book•
01 Jan 1987
TL;DR: In this article, the authors considered the boundary feedback stabilization of Kirchhoff plates subject to weak viscoelastic damping, and the asymptotic stability of the limit systems.
Abstract: Preface 1. Introduction: orientation Background Connection with exact controllability 2. Thin plate models: Kirchhoff model Mindlin-Timoshenko model von Karman model A viscoelastic plate model A linear termoelastic plate model 3. Boundary feedback stabilization of Mindlin-Timoshenko plates: Orientation: existence, uniqueness, and properties of solutions Uniform asymptotic stability of solutions 4. Limits of the Mindlin-Timoshenko system and asymptotic stability of the limit systems: Orientation The limit of the M-T system as KE 0+ The limit of the M-T system as K Study of the Kirchhoff system Uniform asymptotic stability of solutions Limit of the Kirchhoff system as 0+ 5. Uniform stabilization in some nonlinear plate problems: Uniform stabilization of the Kirchhoff system by nonlinear feedback Uniform asymptotic energy estimates for a von Karman plate 6. Boundary feedback stabilization of Kirchhoff plates subject to weak viscoelastic Damping: formulation of the boundary value problem Existence, uniqueness, and properties of solutions Asymptotic energy estimates 7. Uniform asymptotic energy estimates for thermoelastic plates: Orientation Existence, uniqueness, regularity, and strong stability Uniform asymptotic energy estimates Bibliography Index.
624 citations
TL;DR: A problem of boundary stabilization of a class of linear parabolic partial integro-differential equations in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts.
Abstract: In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.
523 citations
"Boundary control of an anti-stable ..." refers methods in this paper
...Our control design is based on the method of “backstepping” [6], [5], [3], which results in explicit formulae for the gain functions....
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TL;DR: Observer gain (output injection function) is shown to satisfy a well-posed hyperbolic PDE that is closely related to the hyperbolics PDE governing backstepping control gain for the state-feedback problem.
435 citations
"Boundary control of an anti-stable ..." refers methods in this paper
...Our control design is based on the method of “backstepping” [6], [5], [3], which results in explicit formulae for the gain functions....
[...]
413 citations
Additional excerpts
...(9) Matching all the terms we get the following PDE for k(x, y): kxx(x, y) = kyy(x, y), 0 < y < x < 1, (10) ky(x, 0) = 0, x ∈ [0, 1), (11) d dx k(x, x) = 0, x ∈ [0, 1), (12) and two coupled PDEs for s(x, y) andm(x, y): sxx(x, y) = syy(x, y), 0 < y < x < 1, (13) sy(x, 0) = qmy(x, 0)− qk(x, 0), x ∈ [0, 1), (14) d dx s(x, x) = 0, x ∈ [0, 1) (15) and mxx(x, y) = myy(x, y), 0 < y < x < 1, (16) m(x, 0) = qs(x, 0), x ∈ [0, 1), (17) d dx m(x, x) = 0, x ∈ [0, 1)....
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...(21) The solution to (10)–(12), (20) is simply k(x, y) ≡ 0....
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Book•
15 Aug 1999
TL;DR: In this article, the authors present a general framework for system passivity and a framework for dynamic boundary control of Vibration Systems based on passivity, as well as a series of properties of nonlinear semigroups of contractions.
Abstract: 1 Introduction.- 1.1 Overview and examples of infinite dimensional systems.- 1.2 Organization and brief summary.- 1.3 Remarks on notation.- 1.4 Notes and references.- 2 Semigroups of Linear Operators.- 2.1 Motivation and definitions.- 2.2 Properties of semigroups.- 2.3 Generation theorems for semigroups.- 2.4 Relation with the Laplace transform.- 2.5 Differentiability and analytic semigroups.- 2.6 Compact semigroups.- 2.7 Abstract Cauchy problem.- 2.7.1 Homogeneous initial value problems.- 2.7.2 Inhomogeneous initial value problems.- 2.7.3 Lipschitz perturbations.- 2.8 Integrated semigroups.- 2.9 Nonlinear semigroups of contractions.- 2.10 Notes and references.- 3 Stability of C0-Semigroups.- 3.1 Spectral mapping theorems.- 3.2 Spectrum-determined growth condition.- 3.3 Weak stability and asymptotic stability.- 3.4 Exponential stability - time domain criteria.- 3.5 Exponential stability - frequency domain criteria.- 3.6 Essential spectrum and compact perturbations.- 3.7 Invariance principle for nonlinear semigroups.- 3.8 Notes and references.- 4 Static Sensor Feedback Stabilization of Euler-Bernoulli Beam Equations.- 4.1 Modeling of a rotating beam with a rigid tip body.- 4.2 Stabilization using strain or shear force feedback.- 4.3 Damped second order systems.- 4.4 Exponential stability and spectral analysis.- 4.4.1 Exponential stability.- 4.4.2 Spectral analysis.- 4.5 Shear force feedback control of a rotating beam.- 4.5.1 Well-posedness and exponential stability.- 4.5.2 Asymptotic behavior of the spectrum.- 4.6 Stability analysis of a hybrid system.- 4.6.1 Well-posedness and exponential stability.- 4.6.2 Spectral analysis.- 4.7 Gain adaptive strain feedback control of Euler-Bernoulli beams.- 4.8 Notes and references.- 5 Dynamic Boundary Control of Vibration Systems Based on Passivity.- 5.1 A general framework for system passivity.- 5.1.1 Uncontrolled case.- 5.1.2 Controlled case.- 5.2 Dynamic boundary control using positive real controllers.- 5.2.1 Positive real controllers and their characterizations.- 5.2.2 Stability analysis of control systems with SPR controllers.- 5.3 Dynamic boundary control of a rotating flexible beam.- 5.3.1 Stabilization problem using SPR controllers.- 5.3.2 Orientation problem using positive real controllers.- 5.4 Stability robustness against small time delays.- 5.5 Notes and references.- 6 Other Applications.- 6.1 A General linear hyperbolic system.- 6.2 Stabilization of serially connected vibrating strings.- 6.3 Two coupled vibrating strings.- 6.4 A vibration cable with a tip mass.- 6.5 Thermoelastic system with Dirichlet - Dirichlet boundary conditions.- 6.6 Thermoelastic system with Dirichlet - Neumann boundary conditions.- 6.7 Renardy's counter-example on spectrum-determined growth condition.- 6.8 Notes and references.
350 citations