**Nikiforov**

Vladimir O.

D.Sc., Prof.

Vladimir O.

D.Sc., Prof.

DOI: 10.17586/2226-1494-2016-16-1-68-75

DOI: 10.17586/2226-1494-2016-16-1-68-75

# RESEARCH OF FREE MOTION TRAJECTORIES FEATURES OF CONTINUOUS SYSTEM DEFINED AS A CONSECUTIVE CHAIN OF IDENTICAL FIRST-ORDER APERIODIC LINKS

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**Article in**Russian

**For citation:**Vunder N.A., Nuyya O.S., Pescherov R.O., Ushakov A.V. Research of free motion trajectories features of continuous system defined as a consecutive chain of identical first-order aperiodic links.

*Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2016, vol. 16, no. 1, pp. 68–75.

**Abstract**

It is stated that the model of desired behavior has found a widespread usage in the theory and practice of control system design, with the state matrix having a binomial Newton placement of eigenvalues. A structural representation of these systems in the case of the transfer functions approach application leads to a system defined as a consecutive chain of identical first-order aperiodic links. Such model of the desired system behavior has the transient response of the system, which is characterized by the absence of overshoot, that is particularly valuable in the unique technological equipment control. Situation varies considerably when the control system with a binomial placement of eigenvalues has a nonzero initial state. Such situation may arise in the case of an unexpected power fail interrupt of the system electrical components followed by its recovery. This problem is especially important for remote online control of continuous plant in the case of the normal functioning disruption of the channel environment and its restoration in the future. The system in a form of consecutive chain of identical first-order aperiodic links mathematically has a three-parametric set as a module of the negative real eigenvalue, its multiplicity equal to the system dimension and aperiodic link gain. It was found that the three-parametric system may have trajectory emissions at any of negative eigenvalue module. The paper results are illustrated by the computer experiment.

**Keywords:**identical first-order aperiodic links, state matrix, modified canonical Jordan form, multiplicity of eigenvalues, free motion, norm, overshoot

**Acknowledgements.**This work was supported by the Government of the Russian Federation, Grant 074-U01 and the Ministry of Education and Science of the Russian Federation (Project 14. Z50.31.0031)

**References**

1. Dudarenko N.A., Polinova N.A., Ushakov A.V. Fundamental matrix of linear continuous system in the problem of estimating its transport delay. *Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2014, no. 5 (93), pp. 32–37. (In Russian)

2. Liholetova E.S., Nuiya O.S., Peshcherov R.O., Ushakov A.V. Factors of the channel medium, problem of digital remote control of continuous technological resources. *Proc. 3rd Int. Conf. on Circuits, Systems, Communications, Computers and Applications. *Florence, 2014, pp. 68–72.

3. Burke J.V., Lewis A.S. and Overton M.L. Optimal stability and eigenvalue multiplicity. *Foundations of Computational Mathematics*, 2001, vol. 1, no. 2, pp. 205–225. doi: 10.1007/s102080010008

4. Ushakov A.V., Akunov T.A., Dudarenko N.A., Polinova N.A. Factor multiplicity of the eigenvalues of the state matrix in the system dynamics. *Proc. 18th Int. Conf. on Applied Mathematics. *Budapesht, Hungary, 2013, pp. 58–62.

5. Akunov T.A., Dudarenko N.A., Polinova N.A., Ushakov A.V. Process oscillativity study in aperiodic continuous systems, generated by eigenvalues multiplication factor.* Scientific and Technical Journal of Information Technologies, Mechanics and Optics*, 2013, no. 3 (85), pp. 55–61. (In Russian)

6. Polinova N.A., Akunov T.A., Dudarenko N.A. Eigenvalues multiplicity of the aperiodic system state as a causal factor in appearance of emission on trajectories to the norm of the state vector of the system of free movement and degeneration. *Trudy XII Vserossiiskogo Soveshchaniya po Problemam Upravleniya, VSPU-2014 [Proc. XII All-Russian Conference on Control Problems VSPU-2014]. *Moscow, 2014, pp. 173–182. (In Russian)

7. Polyak B.T., Tremba A.A. Analytical solution of linear differential equation with the same roots of the characteristic polynomial. *Trudy XII Vserossiiskogo Soveshchaniya po Problemam Upravleniya, VSPU-2014 [Proc. XII All-Russian Conference on Control Problems VSPU-2014]. *Moscow, 2014, pp. 212–217. (In Russian)

8. Polyak B.T., Smirnov G.V. Large deviations in continuous-time linear single-input control systems. *Proc. 19th IFAC World Congress on International Federation of Automatic Control, IFAC 2014. *Cape Town, South Africa, 2014, pp. 5586–5591.

9. Bellman R. Introduction to Matrix Analysis. NY, McGraw-Hill, 1970.

10. Arnol'd V.I. Obyknovennye Differentsial'nye Uravneniya [Ordinary Differential Equations]. 4th ed. Moscow, MTsNMO Publ., 2012, 380 p.

11. Moler C., Van Loan C. Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later. *SIAM Review*, 2003, vol. 45, no. 1, pp. 3–49.

12. Gantmakher F.R. Matrix Theory. Moscow, Nauka Publ., 1973, 575 p. (In Russian)

13. Fikhtengol'ts G.M. Kurs Differentsial'nogo i Integral'nogo Ischisleniya [Course of Differential and Integral Calculus]. 8th ed. Moscow, Fizmatlit Publ., 2003, vol. 1, 680 p.

14. Golub G.H., Van Loan C.F. Matrix Computations. Baltimore, Johns Hopkins University Press, 1996, 728 p.

15. Kaluzhnin L.A. Vvedenie v Obshchuyu Algebru [Introduction to General Algebra]. Moscow, Nauka Publ., 1973, 448 p.