Complexity of stability and controllability of elementary hybrid systems

Caption title.Includes bibliographical references (p. 16-18).Supported by ARO. DAAL-03-92-G-0115 Supported by NATO. CRG-961115Vincent D. Blondel, John N. Tsitsiklis

On feedback stabilization of linear switched systems via switching signal control

Motivated by recent applications in control theory, we study the feedback stabilizability of switched systems, where one is allowed to chose the switching signal as a function of $x(t)$ in order to stabilize the system. We propose new algorithms and analyze several mathematical features of the problem which were unnoticed up to now, to our knowledge. We prove complexity results, (in-)equivalence between various notions of stabilizability, existence of Lyapunov functions, and provide a case study for a paradigmatic example introduced by Stanford and Urbano.Comment: 19 pages, 3 figure

Revisiting the Complexity of Stability of Continuous and Hybrid Systems

We develop a framework to give upper bounds on the "practical" computational complexity of stability problems for a wide range of nonlinear continuous and hybrid systems. To do so, we describe stability properties of dynamical systems using first-order formulas over the real numbers, and reduce stability problems to the delta-decision problems of these formulas. The framework allows us to obtain a precise characterization of the complexity of different notions of stability for nonlinear continuous and hybrid systems. We prove that bounded versions of the stability problems are generally decidable, and give upper bounds on their complexity. The unbounded versions are generally undecidable, for which we give upper bounds on their degrees of unsolvability

Equivalences of discrete-event systems and of hybrid systems

no abstract

A Survey on Continuous Time Computations

We provide an overview of theories of continuous time computation. These theories allow us to understand both the hardness of questions related to continuous time dynamical systems and the computational power of continuous time analog models. We survey the existing models, summarizing results, and point to relevant references in the literature

Stability and controllability of planar bimodal linear complementarity systems

The object of study of this paper is the class of hybrid systems consisting of so-called linear complementarity (LC) systems, that received a lot of attention recently and has strong connections to piecewise affine (PWA) systems. In addition to PWA systems, some of the linear or affine submodels of the LC systems can ‘live’ at lower-dimensional subspaces and re-initializations of the state variable at mode changes is possible. For LC systems we study the stability and controllability problem. Although these problems received for various classes of hybrid systems ample attention, necessary and sufficient conditions, which are explicit and easily verifiable, are hardly found in the literature. For LC systems with two modes and a state dimension of two such conditions are presented