# NON-LYAPUNOV STABILITY OF SINGULAR SYSTEMS: CLASSICAL AND MODERN APPROACHES WITH APPLICATION TO AUTOMATIC DRUG DELIVERY

## Authors

• Dragutin Lj. Debeljković University of Belgrade, Faculty of Mechanical Engineering, Department of Control Engineering, Belgrade, Serbia
• Ivan Buzurović homas Jefferson University, Philadelphia, USA
• Lidija Matija University of Belgrade, Faculty of Mechanical Engineering, Innovation Center, Belgrade, Serbia
• Đuro Koruga 1University of Belgrade, Faculty of Mechanical Engineering, Department of Control Engineering, Belgrade, Serbia

## Abstract

In this paper sufficient conditions for both practical and finite time stability of linear singular continuous time delay systems were introduced. The singular and singular time delay systems can be mathematically described as  and , respectively. Analyzing finite time stability, the new delay independent and delay dependent conditions were derived using the approaches based on Lyapunov-like functions and their properties on the subspace of consistent initial conditions. These functions do not need to be positive on the whole state space and to have negative derivatives along the system trajectories. When the practical stability was analyzed, the approach was combined with classical Lyapunov technique to guarantee the attractivity property of the system behavior. Furthermore, an LMI approach was applied to obtain less conservative stability conditions. The proposed methodology was applied and tested on a medical robotic system. The system was designed for different insertion tasks playing important roles in automatic drug delivery, biopsy or radioactive seeds delivery. In this paper we have summarized different techniques for adequate modeling, control and stability analysis of the medical robots. The model of the robotic system, with the tasks described above, the entire system can be decomposed to the robotic subsystem and the environment subsystem. Modeling of the system by the method mentioned has been proved to be suitable when the force appears as a result of the interaction of the two subsystems. The mathematical model of the system has a singular characteristic. The singular system theory could be applied to the case described. It is well known that all mechanical systems have some delay. In that case a theory of singular systems with delayed states may be applied, as well. For the second phase in which there is no interaction, the dynamic behavior can be analyzed by the classic theory.

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