CONVERGENCE SPEED INCREASE OF A FINITE-TIME STABLE CONTROL SYSTEM

This paper is devoted to tuning of the finite-time control algorithm in order to reduce the transient process time of a closed-loop system. The control object is a chain of consecutively connected integrators with a state vector available for measurement. It is assumed that the control object operates under conditions of deterministic external disturbances and parametric uncertainties of the system. The advantage of the modifiable algorithm in comparison with previously presented results is the absence of the need for additional computational procedures. However, a lower convergence rate of the system trajectories to the equilibrium position can be noticed as a drawback. The presented tuning of the finite control law makes it possible to reduce the time of the transient process of a closed system. The tuned finite-time control algorithm is developed by combination of the method of implicitly defined Lyapunov function with the theory of weighted homogeneous systems. Analytic expressions are obtained that give the possibility to estimate the convergence time of the state vector to the equilibrium position, as well as the possibility of its decrease at the proposed approach application. Computer simulation of the developed control algorithm was performed on the triple-integrator system with deterministic external disturbances in control channel. It has confirmed the effectiveness of the presented theoretical results.

 1.      Bhat S.P., Bernstein D.S. Finite-time stability of continuous autonomous systems. SIAM Journal of Control and Optimization, 2000, vol. 38, no. 3, pp. 751–766. doi: 10.1137/s0363012997321358

2.      Moulay E., Perruquetti W. Finite-time stability and stabilization: state of the art. Lecture Notes in Control and Information Sciences, 2006, vol. 334, pp. 23–41. doi: 10.1007/11612735_2

3.      Orlov Y. Finite time stability and robust control synthesis of uncertain switched systems. SIAM Journal of Control and Optimization, 2004, vol. 43, no. 4, pp. 1253–1271. doi: 10.1137/s0363012903425593

4.      Haimo V.T. Finite time controllers. SIAM Journal of Control and Optimization, 1986, vol. 24, no. 4, pp. 760–770. doi: 10.1137/0324047

5.      Utkin V.I., Guldner J., Shi J. Sliding Mode Control in Electro-Mechanical Systems. 2^nded. CRCPress, 2009,503 p.

6.      Zimenko K.A., Polyakov A.E., Efimov D.V., Kremlev A.S. Finite-time stability of system of series-connected integrators. Journal of Instrument Engineering, 2015, vol. 58, no. 9, pp. 681–686. (In Russian) doi: 10.17586/0021-3454-2015-58-9-681-686

7.      Zimenko K.A., Polyakov A.E., Efimov D.V. Stabilization of chain of integrators with arbitrary order in finite-time. Proc. 54^th IEEE Conf. on Decision and Control, CDC, 2016, pp. 4637–4641. doi: 10.1109/cdc.2015.7402942

8.      Polyakov A., Efimov D., Perruquetti W. Robust stabilization of MIMO systems in finite/fixed time. International Journal of Robust and Nonlinear Control, 2015, vol. 26, no. 1, pp. 69–90. doi: 10.1002/rnc.3297

9.      Polyakov A., Efimov D., Perruquetti W. Finite-time and fixed-time stabilization: implicit Lyapunov function approach. Automatica, 2015, vol. 51, pp. 332–340. doi: 10.1016/j.automatica.2014.10.082

10.   Roxin E. On finite stability in control systems. Rendiconti del Circolo Matematico di Palermo, 1966, vol. 15, no. 3, pp. 273–282. doi: 10.1007/bf02844106

11.   Zubov V.I. A.M. Lyapunov Methods and Their Application. Leningrad, LSU Publ., 1957, 242 p. (In Russian)

12.   Zubov V.I. On systems of ordinary differential equations with generalized-homogeneous right-hand sides. Izvestiya Vuzov. Matematika, 1958, no. 1, pp. 80–88. (In Russian)

13.   Bacciotti A., Rosier L. Lyapunov Functions and Stability in Control Theory. Springer, 2005, 236 p.

14.   Korobov V.I. Solution of the synthesis problem using the controllability function. Doklady AN SSSR, 1979, vol. 248, no. 5, pp. 1051–1055. (In Russian)

15.   Adamy J., Flemming A. Soft variable-structure controls: a survey. Automatica, 2004, vol. 40, no. 11, pp. 1821–1844. doi: 10.1016/j.automatica.2004.05.017

16.   Levant A., Dvir Y. Accelerated high-order MIMO sliding mode control. Proc. 13^th Int. Workshop on Variable Structure Systems, 2014, pp. 4753–6. doi: 10.1109/vss.2014.6881095

Efimov D., Levant A., Polyakov A., Perruquetti W. Supervisory acceleration of convergence for homogeneous systems. International Journal of Control, 2017,pp. 1–11

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License