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Finite-time stability and finite-time stabilization

机译：有限时间稳定和有限时间稳定

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摘要

While non-Lipschitzian effects such as Coulomb friction abound in nature, most of the available techniques for feedback stabilization yield closed-loop systems with Lipschitzian dynamics. The convergence in such systems is at best exponential with infinite settling time. In this dissertation, we are interested in finite-settling-time behavior, that is, finite-time stability. The object of this dissertation is to provide a rigorous foundation for the theory of finite-time stability of continuous autonomous systems and motivate a closer examination of finite-time stability as a possible objective in control design.;Accordingly, the notion of finite-time stability is precisely formulated and properties of the settling-time function are studied. Lyapunov and converse Lyapunov results involving scalar differential inequalities are obtained. It is shown that, under certain conditions, finite-time-stable systems possess better disturbance rejection and robustness properties.;As an application of these ideas, we consider the finite-time stabilization of the translational and rotational double integrators. In the case of the rotational double integrator, the topology of the cylindrical state space renders continuous global stabilization impossible and hence the closed-loop system possesses saddle points. Because of the non-Lipschitzian character of the feedback law, these saddle points are, in fact, finite-time repellers--equilibria from which solutions can spontaneously and unpredictably depart.;It is generally believed that models obtained from classical dynamics are completely deterministic. We present a counterexample to this widely held notion. The counterexample consists of a particle moving along a nonsmooth (once, but not twice, differentiable) constraint in a uniform gravitational field. The equation of motion obtained by applying classical Lagrangian dynamics contains non-Lipschitzian terms and, consequently, the dynamics of the system exhibit a finite-time saddle. The system possesses multiple solutions starting at the finite-time saddle. Every such solution corresponds to the particle spontaneously departing from the equilibrium position.;Homogeneity is introduced as a tool for analyzing finite-time stability of higher dimensional systems. For this purpose a coordinate-free notion of homogeneity is formulated. Our main result is that a homogeneous system is finite-time stable if and only if it is asymptotically stable and has negative degree of homogeneity. Under the assumption of homogeneity, results related to finite-time stability can be strengthened. Finally, homogeneity is exploited to obtain a finite-time stabilizing controller for a chain of integrators and hence prove that every controllable linear control system can be finite-time stabilized through a continuous feedback law.
机译：尽管自然界中存在大量非立普兹效应，例如库仑摩擦，但大多数可用的反馈稳定技术会产生具有立普兹动力学的闭环系统。在这种系统中，收敛时间最多是无限的建立时间。在本文中，我们对有限建立时间行为，即有限时间稳定性感兴趣。本文的目的是为连续自治系统的有限时间稳定性理论提供严格的基础，并激发对有限时间稳定性的仔细研究作为控制设计中可能的目标。精确地确定了稳定性，并研究了稳定时间函数的性质。获得了涉及标量微分不等式的Lyapunov和逆Lyapunov结果。结果表明，在一定条件下，有限时间稳定系统具有较好的抗扰性和鲁棒性。作为这些思想的应用，我们考虑了平动和旋转对偶积分器的有限时间稳定。在旋转双积分器的情况下，圆柱状态空间的拓扑使连续的全局稳定成为不可能，因此闭环系统具有鞍点。由于反馈定律具有非李普希兹特性，因此这些鞍点实际上是有限时间排斥器-平衡点，解决方案可以自发地，不可预测地偏离平衡点;通常认为，从经典动力学获得的模型是完全确定性的。我们提出了一个反对这种广泛持有的概念的例子。反例由一个粒子在均匀的引力场中沿着不平滑（一次，但不是两次，可微分）约束移动。通过应用经典拉格朗日动力学获得的运动方程包含非Lipschitz项，因此，系统的动力学表现出有限时间的鞍形。该系统拥有从有限时间鞍开始的多种解决方案。每个这样的解决方案都对应于自发离开平衡位置的粒子。均一性被引入作为分析高维系统的有限时间稳定性的工具。为此，提出了无坐标的均匀性概念。我们的主要结果是，当且仅当它是渐近稳定的并且具有负均匀性时，齐次系统才是有限时间稳定的。在同质性的假设下，与有限时间稳定性有关的结果可以得到加强。最后，利用同质性获得了积分器链的有限时间稳定控制器，因此证明了每个可控制的线性控制系统都可以通过连续反馈定律进行有限时间稳定。

著录项

  • 作者

    Bhat, Sanjay Purushottam.;

  • 作者单位

    University of Michigan.;

  • 授予单位 University of Michigan.;
  • 学科 Aerospace engineering.;Mathematics.
  • 学位 Ph.D.
  • 年度 1997
  • 页码 111 p.
  • 总页数 111
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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