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

DOI:

https://doi.org/10.7251/COMEN1301022D

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.

References

[1] S. L. Campbell, Singular Systems of Differential Equations, Pitman, Marshfield, MA, 1980.

[2] P. C. Müller, Stability of Linear Mechanical Systems with Holonomic Constraints, Applied Mechanics Review, Vol.4611 (1993) 160–164.

[3] D. Lj. Debeljković, D. H. Owens, On practical stability of singular systems, Proc. Melecon Conf. 85, Madrid, Spain 1985, 103105.

[4] D. Lj. Debeljković, V. B. Bajic, Z. Gajic, B. Petrovic, Boundedness and existence of solutions of regular and irregular singular systems, Automatic Control No.1 (1993) 69–78.

[5] I. Buzurovic, T. K. Podder, and Yan Yu, Prediction Control for Brachytherapy Robotic System, Journal of Robotics, Vol. 2010 (2010), 155165.

[6] I. Buzurovic, T. K. Podder, Y. Yu, Force Prediction and Tracking for Image-guided Robotic System using Neural Network Approach, IEEE Biomedical Circuits and Systems Conference, Bio CAS, Baltimore, Maryland, USA 2008, 4144.

[7] I. Buzurovic, D. Lj. Debeljkovic, Contact Problem and Controllability for Singular Systems in Biomedical Robotics, International Journal of Information and System Sciences, Volume 62 (2012) 128141.

[8] N. H. Mc-Clamroch, Singular Systems of Differential Equations as Dynamic Models for Constrained Robot Systems, Proc. of the 1986 IEEE International Conf. on Robotics and Automations, Vol.3 (1986) 2128.

[9] D. Lj. Debeljković, M.P. Lazarevic, Đ. Koruga, S. Tomaševic, Finite time stability of singular systems operating under perturbing forces: Matrix measure approach, Proc. AMSE Conference, Melbourne, Australia 1997, 447450.

[10] C. Y. Yang, Q.L. Zhang, Y.P. Lin, Practical Stability of Descriptor Systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series B(12.b) (2005) 4457.

[11] Y. Y. Nie, D. Lj. Debeljković, Non–Lyapunov Stability of Linear Singular Systems: A Quite new Approach in Time Domain, Dynamics of Continuous, Discrete and Impulsive Systems, Vol.11, Series A: Math. Analysis, No.5-6 (2004), 751760.

[12] C. Y. Yang, Q.L. Zhang, L. Zhou, Practical Stabilization and Controllability of Descriptor Systems, International Journal of Information and System Science, Vol. 1, No. 3-4 (2005), 455466.

[13] Y. Shen, W. Shen, Finite-Time Control of Linear Singular Systems with Parametric Uncertainties and Disturbances, Automatica (2006), 634637.

[14] C. Yang, Q. Zhang, Y. Lin, L. Zhou, Practical stability of descriptor systems with time-delay in terms of two measurements, Journal of the Franklin Institute, (2006), 635646.

[15] D. H. Owens, D. Lj. Debeljković, Consistency and Lyapunov Stability of Linear Descriptor Systems: a Geometric Analysis, IMA Journal of Math. Control and Information, No.2 (1985) 139151.

[16] V. B. Bajic, Lyapunov Direct Method in the Analysis of Singular Systems and Networks, Shades Technical Publications, Technicon Natal, RSA, (1992)

[17] S. Xu, P. V. Dooren, R. Stefan, J. Lam, Robust Stability and Stabilization for Singular Systems with State Delay and Parameter Uncertainty, IEEE Trans. Automat. Control, AC-47(7) (2002), 11221128.

[18] I. M. Buzurovic, Dynamic Model of Medical Robot Represented as Descriptor System, International Journal of Information and System Sciences, Vol.2, No.2 (2007) 316333.

Downloads

Published

2013-06-01

Issue

Vol. 4 No. 1 (2013): Contemporary Materials IV-1

Section

Articles