Abstract: The general problem of this paper is the analysis of wave propagation in a bounded medium where the uncontrolled boundary obeys a coupled differential equation. More precisely, we study a one-dimensional wave equation with a nonlinear second-order dynamic boundary condition and a Neuman-type boundary control acting on the other extremity. A generic class of nonlinear collocated feedback laws is considered. Hadamard well-posedness is established for the closed-loop system, with initial data lying in the natural energy space of the problem. Moreover, we investigate an example of stabilization through a proportional controller.
Abstract: Preface.- 1. The Hahn-Banach Theorems. Introduction to the Theory of Conjugate Convex Functions.- 2. The Uniform Boundedness Principle and the Closed Graph Theorem. Unbounded Operators. Adjoint. Characterization of Surjective Operators.- 3. Weak Topologies. Reflexive Spaces. Separable Spaces. Uniform Convexity.- 4. L^p Spaces.- 5. Hilbert Spaces.- 6. Compact Operators. Spectral Decomposition of Self-Adjoint Compact Operators.- 7. The Hille-Yosida Theorem.- 8. Sobolev Spaces and the Variational Formulation of Boundary Value Problems in One Dimension.- 9. Sobolev Spaces and the Variational Formulation of Elliptic Boundary Value Problems in N Dimensions.- 10. Evolution Problems: The Heat Equation and the Wave Equation.- 11. Some Complements.- Problems.- Solutions of Some Exercises and Problems.- Bibliography.- Index.
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).
Abstract: Much of the boundary control of wave equations in one dimension is based on a single principle—passivity—under the assumption that control is applied through Neumann actuation on one boundary and the other boundary satisfies a homogeneous Dirichlet boundary condition. We have recently expanded the scope of tractable problems by allowing destabilizing anti-stiffness (a Robin type condition) on the uncontrolled boundary, where the uncontrolled system has a finite number of positive real eigenvalues. In this paper we go further and develop 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 system is “anti-stable” (exponentially stable in negative time). Using a conceptually novel integral transformation, we obtain extremely simple, explicit formulae for the gain functions. For the case with only boundary sensing available (at the same end with actuation), we design backstepping observers which are dual to the backstepping controllers and have explicit output injection gains. We then combine the control and observer designs into an output-feedback compensator and prove the exponential stability of the closed-loop system.
TL;DR: An adaptive output-feedback controller for a wave PDE in one dimension with actuation on one boundary and with an unknown anti-damping term on the opposite boundary, representative of a torsional stick–slip instability in drillstrings in deep oil drilling, as well as of various acoustic instabilities.
Abstract: We develop an adaptive output-feedback controller for a wave PDE in one dimension with actuation on one boundary and with an unknown anti-damping term on the opposite boundary. This model is representative of a torsional stick–slip instability in drillstrings in deep oil drilling, as well as of various acoustic instabilities. The key feature of the proposed controller is that it requires only the measurements of boundary values and not of the entire distributed state of the system. Our approach is based on employing Riemann variables to convert the wave PDE into a cascade of two delay elements, with the first of the two delay elements being fed by control and the same element in turn feeding into a scalar ODE. This enables us to employ a prediction-based design for systems with input delays, suitably converted to the adaptive output-feedback setting. The result’s relevance is illustrated with simulation example.
TL;DR: The asymptotic stability of the closed-loop nonlinear partial differential equations of the wave equation with a one-dimensional space variable that describes the dynamics of string deflection is proven by Lyapunov techniques.
Abstract: This paper deals with a wave equation with a one-dimensional space variable, which describes the dynamics of string deflection. Two kinds of control are considered: a distributed action and a boundary control. It is supposed that the control signal is subject to a cone-bounded nonlinearity. This kind of feedback laws includes (but is not restricted to) saturating inputs. By closing the loop with such a nonlinear control, it is thus obtained a nonlinear partial differential equation, which is the generalization of the classical 1D wave equation. The well-posedness is proven by using nonlinear semigroups techniques. Considering a sector condition to tackle the control nonlinearity and assuming that a tuning parameter has a suitable sign, the asymptotic stability of the closed-loop system is proven by Lyapunov techniques. Some numerical simulations illustrate the asymptotic stability of the closed-loop nonlinear partial differential equations.