Optimal bounded control of stochastically excited strongly nonlinear vibro-impact system

Abstract

An optimal bounded control strategy for strongly nonlinear vibro-impact systems under stochastic excitations with actuator saturation is proposed. First, the impact effect is incorporated in an equivalent equation by using a nonsmooth transformation. Under the assumption of light damping and weak random perturbation, the system energy is a slowly varying process. By using the stochastic averaging of envelope for strongly nonlinear systems, the partially averaged Itô stochastic differential equation for system energy can be derived. The optimal control problem is transformed from the original optimal control problem for the state variables to an equivalent optimal control problem for the system energy, which decreases the dimensions of the optimal control problem. Then, based on stochastic maximum principle, an adjoint equation for the adjoint variable and the maximum condition of partially averaged control problem are established. For infinite time-interval ergodic control, the adjoint variable is assumed to be a stationary process and the adjoint equation can be further simplified. Finally, the probability density function of the system energy and other statistics of the optimally controlled system are derived by calculating the associated Fokker–Plank–Kolmogorov equation. For comparison, the bang–bang control is also investigated and the control results are compared to show the advantages of the developed control strategy.

Keywords

Nonlinear vibro-impact system stochastic averaging stochastic optimal control stochastic maximum principle Fokker–Plank–Kolmogorov equation

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References

Agrawal

Yang

(2000) Optimal polynomial control of seismically excited linear structures. Journal of Engineering Mechanics 122: 753–761.

Andreaus

Placidi

Rega

(2010) Numerical simulation of the soft contact dynamics of an impacting bilinear oscillator. Communications in Nonlinear Science and Numerical Simulation 15(9): 2603–2616.

Babitsky

(1998) Theory of Vibro-Impact System and Applications. Berlin: Springer.

Crespo

Sun

(2002) Stochastic optimal control of nonlinear systems via short-time gaussian approximation and cell mapping. Nonlinear Dynamics 28: 323–342.

Crespo

Sun

J-Q

(2003) Stochastic optimal control via Bellman’s principle. Automatica 39: 2109–2114.

Dankowicz

Nordmark

(2000) On the origin and bifurcations of stick-slip oscillations. Physica D: Nonlinear Phenomena 136: 280–302.

Dimentberg

Gaidai

Naess

(2009) Random vibrations with strongly inelastic impacts: response pdf by the path integration method. International Journal of Non-Linear Mechanics 44: 791–796.

Dimentberg

Iourtchenko

(1999) Towards incorporating impact losses into random vibration analyses: a model problem. Probabilistic Engineering Mechanics 14(4): 323–328.

Dimentberg

Iourtchenko

(2004) Random vibrations with impacts: a review. Nonlinear Dynamics 36: 229–254.

10.

Feng

Liu

(2015) Chaotic dynamics of the vibro-impact system under bounded noise perturbation. Chaos, Solitons & Fractals 73: 10–16.

11.

Feng

Wang

(2008) Stochastic responses of vibro-impact duffing oscillator excited by additive Gaussian noise. Journal of Sound and Vibration 309: 730–738.

12.

Fleming

Soner

(1992) Controlled Markov Process and Viscosity Solutions. New York: Springer.

13.

Zhu

(2014a) Optimal bounded control of quasi-nonintegrable Hamiltonian systems using stochastic maximum principle. Nonlinear Dynamics 76: 1051–1058.

14.

Zhu

(2014b) Time-delayed stochastic optimal control of strongly non-linear systems with actuator saturation by using stochastic maximum principle. International Journal of Non-Linear Mechanics 58: 199–205.

15.

Zhu

(2012) Stochastic optimal control of quasi non-integrable Hamiltonian systems with stochastic maximum principle. Nonlinear Dynamics 70: 779–787.

16.

Ibrahim

(2009) Vibro-Impact Dynamics: Modeling, Mapping and Applicationss. Berlin: Springer.

17.

Jing

H-S

Sheu

K-C

(1990) Exact stationary solutions of the random response of a single-degree-of-freedom vibro-impact system. Journal of Sound and Vibration 141: 363–373.

18.

Kovaleva

(1999) Asymptotic solution of the optimal control problem for non-linear oscillations in the neighbourhood of resonance. Prikl Math Mekh 62(6): 913–922.

19.

Kushner

(1978) Optimality conditions for the average cost per unit time problem with a diffusion model. SIAM Journal on Control and Optimization 16: 330–346.

20.

Lizzy

Balachandran

Trujillo

(2017) Controllability of nonlinear stochastic fractional neutral systems with multiple time varying delays in control. Chaos, Solitons & Fractals 102: 162–167.

21.

Shen

Zhao

(2008) On numerical approximations of forward–backward stochastic differential equations. SIAM Journal on Numerical Analysis 46: 2636–2661.

22.

Yong

(1999) Forward–backward Stochastic Differential Equations and their Applications. Berlin: Springer.

23.

Milstein

Tretyakov

(2006) Numerical algorithms for forward–backward stochastic differential equations. SIAM Journal of Scientific Computing 39: 1535–1546.

24.

Nayak

(1972) Contact vibrations. Journal of Sound and Vibration 22: 297–322.

25.

Shaw

Homes

(1983) A periodically forced impact oscillator with large dissipation. ASME, Journal of Applied Mechanics 50: 840–857.

26.

Spedicato

Notarstefano

(2017) An optimal control approach to the design of periodic orbits for mechanical systems with impacts. Nonlinear Analysis: Hybrid Systems 23: 111–121.

27.

Spencer

Krok

(1986) Linear quadratic gaussian control of a nonlinear drive system. In: American control conference, 18–20 June 1986, Seattle, WA, pp. 1230–1235. Washington, DC: IEEE.

28.

Wong

Zakai

(1965) On the relation between ordinary and stochastic differential equations. International Journal of Engineering Science 3: 213–229.

29.

Zhu

(2008) Stationary response of multi-degree-of-freedom vibro-impact systems to Poisson white noises. Physics Letters A 372: 623–630.

30.

Yin

Qin

Zou

(2007) Transient responses of repeated impact of a beam against a stop. International Journal of Solids and Structures 44: 7323–7339.

31.

Yong

Zhou

(1999) Stochastic Controls, Hamiltonian Systems and HJB Equations. New York: Springer.

32.

Zhu

Lin

(1991) Stochastic averaging of energy envelope. Journal of Engineering Mechanics 117: 1890–1905.

33.

Zhu

Huang

Suzuki

(2001a) Response and stability of strongly non-linear oscillators under wide-band random excitation. International Journal of Non-Linear Mechanics 36: 1235–1250.

34.

Zhu

Ying

Soong

(2001b) An optimal nonlinear feedback control strategy for randomly excited structural systems. Nonlinear Dynamics 24: 31–51.

35.

Zhuravlev

(1976) A method for analyzing vibration-impact systems by means of special functions. Mechanics of Solids 11: 23–27.