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Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary

机译：在不受控制的边界上具有反阻尼的反稳定波动方程的边界控制

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摘要

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.
机译：一维波动方程的边界控制大部分是基于单一原理的无源性，即假设通过在一个边界上通过Neumann致动施加控制，而另一个边界满足齐次Dirichlet边界条件，则这一假设是成立的。最近，我们通过允许不受控制的边界上的反刚度（Robin型条件）不稳定，从而扩大了可解决问题的范围，其中不受控制的系统具有有限数量的正实特征值。在本文中，我们进一步研究了不受控制的边界条件具有反阻尼的情况的方法，该方法使不受控制的系统的所有特征值的实部都为正且任意高，即系统是“反稳定的”。 ”（在负数时间内呈指数稳定）。使用概念上新颖的积分变换，我们可以获得增益函数的极其简单，明确的公式。对于只有边界检测可用的情况（与促动在同一端），我们设计了反步观察器，该观察器对反步控制器是双重的，并且具有明显的输出注入增益。然后，我们将控制和观察器设计组合到一个输出反馈补偿器中，并证明了闭环系统的指数稳定性。

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