A linear matrix inequality–based approach for estimating upper bound of settling time

Abstract

In this article, an upper bound on the settling time is provided for exponentially stable systems by the Lyapunov theory. In order to obtain a more accurate estimate on the upper bound of settling time, an optimization problem with a nonlinear objective function needs to be solved. Since this is a complex problem, a linear matrix inequality–based algorithm is proposed to solve it. Then, we address the stability analysis problem of linear systems with the aim to derive the upper bound on the settling time. This problem is extended to design a static feedback controller for linear systems under input saturation. By solving the optimization problem, the upper bound of the settling time for the closed-loop system is minimized. Finally, the validity and effectiveness of the proposed method are verified by the simulation results and comparisons.

Keywords

Settling time linear matrix inequality stability analysis control synthesis initial conditions

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References

Haddad

Nersesov

Finite-time stability for time-varying nonlinear dynamical systems. In: 2008 American control conference, Seattle, WA, 11–13 June 2008, pp.4135–4139. New York: IEEE.

Yin

Khoo

Man

Finite-time stability theorems of homogeneous stochastic nonlinear systems. Syst Contr Lett 2017; 100: 6–13.

Bhat

Bernstein

DS.

Continuous finite-time stabilization of the translational and rotational double integrators. IEEE Trans Automat Contr 1998; 43(5): 678–682.

Bhat

Bernstein

DS.

Finite-time stability of continuous autonomous systems. SIAM J Contr Optim 2000; 38(3): 751–766.

Liu

Yao

, et al. Distributed cooperative compound tracking control for a platoon of vehicles with adaptive NN. IEEE Trans Cybernet 2021; 52: 7039–7048.

Liu

Zuo

, et al. An overview of finite/fixed-time control and its application in engineering systems. IEEE/CAA J Automat Sinica 2022; 99: 1–15.

Liang

Huang

, et al. Neuroadaptive performance guaranteed control for multiagent systems with power integrators and unknown measurement sensitivity. IEEE Trans Neur Netw Learn Syst. Epub ahead of print 29 March 2022. DOI: 10.1109/TNNLS.2022.3160532.

Gao

Liu

Finite-time state-feedback stabilisation of non-holonomic systems with low-order non-linearities. IET Contr Theory Appl 2015; 9(10): 1553–1560.

Moulay

Dambrine

Yeganefar

, et al. Finite-time stability and stabilization of time-delay systems. Syst Contr Lett 2008; 57(7): 561–566.

10.

Yin

Khoo

Man

, et al. Finite-time stability and instability of stochastic nonlinear systems. Automatica 2011; 47(12): 2671–2677.

11.

Moulay

Perruquetti

Finite time stability of nonlinear systems. In: 42nd IEEE international conference on decision and control (IEEE Cat. No. 03CH37475), Maui, HI, 9–12 December 2003, vol. 4, pp.3641–3646. New York: IEEE.

12.

Chen

Yang

Pan

Finite time stability analysis of switched systems with stable and unstable subsystems. Asia J Contr 2014; 16(4): 1224–1228.

13.

Utkin

VI.

Scope of the theory of sliding modes. In: Sliding modes in control and optimization. Springer, 1992, pp.1–11, https://link-springer-com-443.web.bisu.edu.cn/chapter/10.1007/978-3-642-84379-2_1#citeas

14.

Yang

Sliding-mode control for systems with mismatched uncertainties via a disturbance observer. IEEE Trans Indus Electron 2012; 60(1): 160–169.

15.

Shtessel

Edwards

Fridman

, et al. Sliding mode control and observation, vol. 10. Berlin: Springer, 2014.

16.

Ding

Levant

Simple homogeneous sliding-mode controller. Automatica 2016; 67: 22–32.

17.

Polyakov

Efimov

Perruquetti

Finite-time and fixed-time stabilization: implicit Lyapunov function approach. Automatica 2015; 51: 332–340.

18.

Praly

Generalized weighted homogeneity and state dependent time scale for linear controllable systems. In: Proceedings of the 36th IEEE conference on decision and control, San Diego, CA, 12 December 1997, vol. 5, pp.4342–4347. New York: IEEE.

19.

Hong

Finite-time stabilization and stabilizability of a class of controllable systems. Syst Contr Lett 2002; 46(4): 231–236.

20.

Bhat

Bernstein

DS.

Geometric homogeneity with applications to finite-time stability. Math Contr Signal Syst 2005; 17(2): 101–127.

21.

Zuo

Tian

Defoort

, et al. Fixed-time consensus tracking for multi-agent systems with high-order integrator dynamics. IEEE Trans Automat Contr 2017; 63(2): 563–570.

22.

Yan

Fixed-time stability of stochastic nonlinear systems and its application into stochastic multi-agent systems. IET Contr Theory Appl 2021; 15: 12040.

23.

Aldana-López

Gómez-Gutiérrez

Jiménez-Rodrguez

, et al. Enhancing the settling time estimation of a class of fixed-time stable systems. Int J Robust Nonlin Contr 2019; 29(12): 4135–4148.

24.

Jiang

Fixed/preassigned-time synchronization of complex networks via improving fixed-time stability. IEEE Trans Cybernet 2020; 51(6): 2882–2892.

25.

Yan

Fixed-time stabilization of nonlinear system and its application into general neural networks. IEEE Access 2020; 8: 58171–58179.

26.

Chen

, et al. Fixed-time stability of dynamical systems and fixed-time synchronization of coupled discontinuous neural networks. Neural Net 2017; 89: 74–83.

27.

Heath

MT.

Scientific computing: an introductory survey (Revised 2nd ed.). Philadelphia, PA: SIAM, 2018.

28.

Boyd

El Ghaoui

Feron

, et al. Linear matrix inequalities in system and control theory. Philadelphia, PA: SIAM, 1994.

29.

Khalil

HK.

Nonlinear systems, 2002, https://ostad.hormozgan.ac.ir/ostad/UploadedFiles/863740/863740-8707186354456456.pdf

30.

Rout

Sahu

Panda

Design and analysis of differential evolution algorithm based automatic generation control for interconnected power system. Ain Shams Eng J 2013; 4(3): 409–421.

31.

Lofberg

YALMIP: a toolbox for modeling and optimization in MATLAB. In: 2004 IEEE international conference on robotics and automation (IEEE Cat. No. 04CH37508), Taipei, Taiwan, 2–4 September 2004, pp.284–289. New York: IEEE.

32.

ApS

The MOSEK optimization toolbox for MATLAB manual, version 9.0, 2019, http://docs.mosek.com/9.0/toolbox/index.html

33.

Gurumoorthy

Sanders

Controlling non-minimum phase nonlinear systems-the inverted pendulum on a cart example. In: 1993 American control conference, San Francisco, CA, 2–4 June 1993, pp.680–685. New York: IEEE.

34.

Huang

Nonlinear output regulation: theory and applications. Philadelphia, PA: SIAM, 2004.

35.

Lan

Wang

Structural design of composite nonlinear feedback control for linear systems with actuator constraint. Asia J Contr 2010; 12(5): 616–625.