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Local analysis of hybrid systems on polyhedral sets with state-dependent switching

Leth J., Wisniewski R.

International Journal of Applied Mathematics and Computer Science

2014

Vol. 24, no. 2

341--355

This paper deals with stability analysis of hybrid systems. Various stability concepts related to hybrid systems are introduced. The paper advocates a local analysis. It involves the equivalence relation generated by reset maps of a hybrid system. To establish a tangible method for stability analysis, we introduce the notion of a chart, which locally reduces the complexity of the hybrid system. In a chart, a hybrid system is particularly simple and can be analyzed with the use of methods borrowed from the theory of differential inclusions. Thus, the main contribution of this paper is to show how stability of a hybrid system can be reduced to a specialization of the well established stability theory of differential inclusions. A number of examples illustrate the concepts introduced in the paper.

Hard and soft sub-time-optimal controllers for a mechanical system with uncertain mass

Kulczycki P., Wisniewski R., Kowalski P., Krawiec K.

Control and Cybernetics

2004

Vol. 33, no 4

573-587

An essential limitation in using the classical optimal control has been its limited robustness to modeling inadequacies and perturbations. This paper presents the concepts of two practical control structures based on the time-optimal approach, a hard and soft one. The hard structure is defined by the parameters selected in accordance with the rules of the statistical decision theory: however, the soft structure allows additionally for elimination of rapid changes in control values. The object is a basic mechanical system, with uncertain (also non-stationary) mass treated as a stochastic process. The methodology proposed here is of a universal nature and may easily be applied with respect to other elements of uncertainty of time-optimal controlled mechanical systems.