Notes on Recent Achievements in Proving Stability using KeYmaeraX

KeYmaeraX is a Hoare-style theorem prover for hybrid systems. A hybrid system can be seen as an aggregation of both discrete and continuous variables, whose values can change abruptly or continuously, respectively. KeYmaeraX supports only variables having the primitive type bool or real. Due to the mixture of discrete and continuous system elements, one promising application area for KeYmaeraX are closed-loop control systems. A closed-loop control system consists of a plant and a controller. While the plant is basically an aggregation of continuous variables whose values change over time accordingly to physical laws, the controller can be seen as an algorithm formulated in a classical programming language. In this paper, we review some recent extensions of the proof calculus applied by KeYmaeraX that make formal proofs on the stability of dynamic systems more feasible. Based on an example, we first introduce to the topic and prove asymptotic stability of a given system in a hand-written mathematical style. This approach is then compared with a formal encoding of the problem and a formal proof established in KeYmaeraX. We also discuss open problems such as the formalization of asymptotic stability.

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The Impact of Visual Feedback Type on the Mastery of Visuo-Motor Transformations

Zeitschrift für Psychologie ◽

10.1027/2151-2604/a000084 ◽

2012 ◽

Vol 220 (1) ◽

pp. 3-9 ◽

Cited By ~ 4

Author(s):

Sandra Sülzenbrück

Keyword(s):

Empirical Research ◽

Visual Feedback ◽

Closed Loop ◽

Motor Planning ◽

Visual Input ◽

Closed Loop Control ◽

Loop Control ◽

Feedback Type ◽

Effective Use ◽

The Impact

For the effective use of modern tools, the inherent visuo-motor transformation needs to be mastered. The successful adjustment to and learning of these transformations crucially depends on practice conditions, particularly on the type of visual feedback during practice. Here, a review about empirical research exploring the influence of continuous and terminal visual feedback during practice on the mastery of visuo-motor transformations is provided. Two studies investigating the impact of the type of visual feedback on either direction-dependent visuo-motor gains or the complex visuo-motor transformation of a virtual two-sided lever are presented in more detail. The findings of these studies indicate that the continuous availability of visual feedback supports performance when closed-loop control is possible, but impairs performance when visual input is no longer available. Different approaches to explain these performance differences due to the type of visual feedback during practice are considered. For example, these differences could reflect a process of re-optimization of motor planning in a novel environment or represent effects of the specificity of practice. Furthermore, differences in the allocation of attention during movements with terminal and continuous visual feedback could account for the observed differences.

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