Nonlinear model predictive acceleration synchronization technique for rigid-model-based tracking control of multi-DOF flexible systems

Abstract

In this paper, the problem of stabilization and tracking control of uncertain flexible multivariable continuum mechanism systems with additional appendage considering boundedness of undesired vibrational effects are investigated. It is proposed that rather than controlling the entire dynamical system through installation of additional actuators on appendage, it is more desirable to stabilize the core system while considering undesired vibrational effect in the appendage as optimization cost. To this end, the nonlinear programming problem corresponding to a predictive control scheme is presented based on a cost defined according to magnitude of accelerations of undesired vibrations in the appendage. Stabilization of control algorithm is conducted based on demonstration of existence of feasible Lyapunov functions. Candidate Lyapunov functions are defined based on sliding functions analysis. The associated sliding functions are defined for core subsystems which correspond to degrees of freedom of dynamical system that are associated with reference signals. Undesired vibrations caused by other degrees of freedom corresponding to appendage are considered as segments of control cost in the optimization problem. To ensure the feasibility of control model in real-time applications, the control algorithm is obtained in discrete-time basis. Based on investigation of monotonically decreasing Lyapunov functions and expression of sliding functions as a combination of dynamical terms, reference inputs and control input terms, stabilizing constraints of closed-loop system are calculated. Similarly, based on combination of sliding surfaces with considered control and vibrational constraints, additional constraints for boundedness of vibrational terms in nonlinear programming problem are obtained. Numerical comparisons with a sliding control algorithm for core mechanical system and a reaching law method indicate improvements regarding reduction of undesired vibrational effects in appendage. Furthermore, regulation of appendage dynamics is conducted without installation of additional actuators, which reduces the implementation cost in comparison with a full-control scheme for core system and appendage.

Keywords

Nonlinear predictive control optimal control active vibrations control sliding functions analysis acceleration synchronization

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References

Golovin

Palis

. Robust control for active damping of elastic gantry crane vibrations. Mech Syst Signal Process 2019; 121: 264–278.

Chen

Shao

. Dynamic behavior of time-delayed acceleration feedback controller for active vibration control of flexible structures. J Sound Vib 2014; 333: 4789–4809.

Issa

. Vibration absorbers for simply supported beams subjected to constant moving loads. Proc IMechE, Part K: J Multi-body Dynamics 2012; 226: 398–404.

Fang

. Impact analysis and vibration reduction design of spiral bevel gears. Proc IMechE, Part K: J Multi-body Dynamics 2019; 233: 668–676.

Raei

Dardel

. Tuned mass damper and high static low dynamic stiffness isolator for vibration reduction of beam structure. Proc IMechE, Part K: J Multi-body Dynamics 2020; 234: 95–115.

Gawronski

. Advanced structural dynamics and active control of structures, New York, NY: Springer New York, 2004.

Ben Zarouala

El Mjabber

. Sliding mode based control for a flexible wind turbine. Energy Proc 2017; 141: 399–404.

Cai

Tian

Wang

, et al. Development of a piezo-driven 3-DOF stage with T-shape flexible hinge mechanism. Robot Comput Integr Manuf 2016; 37: 125–138.

Trindade

Benjeddou

. Hybrid active-passive damping treatments using viscoelastic and piezoelectric materials: review and assessment. Journal Vib Control 2002; 8: 699–745.

10.

Kattimani

Ray

. Vibration control of multiferroic fibrous composite plates using active constrained layer damping. Mech Syst Signal Process 2018; 106: 334–354.

11.

Verbaan

CAM

Peters

GWM

Steinbuch

. Linear viscoelastic fluid characterization of ultra-high-viscosity fluids for high-frequency damper design: with application to industrial high-precision motion stages. Rheol Acta 2015; 54: 667–677.

12.

Vinyas

. Vibration control of skew magneto-electro-elastic plates using active constrained layer damping. Compos Struct 2019; 208: 600–617.

13.

Cao

Liu

. Optimal trajectory control for a two-link rigid-flexible manipulator with ODE-PDE model. Optim Control Appl Methods 2018; 39: 1515–1529.

14.

Zhang

Geng

. Fault-tolerant sliding mode control for uncertain active suspension systems against simultaneous actuator and sensor faults via a novel sliding mode observer. Optim Control Appl Methods 2018; 39: 1728–1749.

15.

Xiong

Zhu

. A stochastic optimal control strategy for viscoelastic systems with actuator saturation. Probabilistic Eng Mech 2016; 45: 44–51.

16.

Skaar

Michel

Miller

. Stability of viscoelastic control systems. IEEE Trans Automat Contr 1988; 33: 348–357.

17.

Yuan

Zhang

Liu

, et al. Sliding mode control of vibration in single-degree-of-freedom fractional oscillators. J Dyn Syst Meas Control 2017; 139: 114503–114503.

18.

Yaqubi

Dardel

Daniali

. Nonlinear dynamics and control of crank-slider mechanism with link flexibility and joint clearance. Proc IMechE, Part C: J Mechanical Engineering Science 2016; 230: 737–755.

19.

Yaqubi

Dardel

Daniali

, et al. Modeling and control of crank–slider mechanism with multiple clearance joints. Multibody Syst Dyn 2016; 36: 143–167. DOI: 10.1007/s11044-015-9486-3.

20.

Zhong

. State estimation based on fractional order sliding mode observer method for a class of uncertain fractional-order nonlinear systems. Signal Process 2016; 127: 168–184.

21.

Slotine

. Applied nonlinear control, Englewood Cliffs, NJ: Prentice Hall, 1990.

22.

Zhang

Zong

Zhao

Z-S

. Higher order sliding mode control with self-tuning law based on integral sliding mode. IET Control Theory Appl 2010; 4: 1282–1289.

23.

Homaeinezhad

Tahbaz-zadeh Moghaddam

Khakpour

, et al. Short-time linear quadratic form technique for estimating fast-varying parameters in feedback loops. Asian J Control 2015; 17: 2289–2302.

24.

Chen

MZQ

, et al. Chattering-free discrete-time sliding mode control. Automatica 2016; 68: 87–91.

25.

Homaeinezhad

Yaqubi

. Discrete-time sliding-surface based control of parametrically uncertain nonlinear systems with unknown time-delay and inaccessible switching mode detection. Int J Control 2019. DOI: 10.1080/00207179.2019.1605205.

26.

Homaeinezhad

Shahhosseini

. Discrete-time sliding mode control of uncertain non-commensurate linear time invariant MIMO fractional order dynamic systems. Math Model Methods Appl Sci 2019. DOI: 10.1016/j.cnsns.2020.105200.

27.

Wen

Liu

. Optimal fast-response sliding-mode control for flexible-substrate measurement. Control Eng Pract 2013; 21: 593–603.

28.

Yin

Gao

Qiu

, et al. Adaptive fault-tolerant control for nonlinear system with unknown control directions based on fuzzy approximation. IEEE Trans Syst Man, Cybern Syst 2017; 47: 1909–1918.

29.

Homaeinezhad

Yaqubi

Fotoohinia

. FEA based discrete-time sliding mode control of uncertain continuum mechanics MIMO vibrational systems. J Sound Vib 2019; 460: 114902–114902.

30.

Homaeinezhad

Yaqubi

Gholyan

. Control of MIMO mechanical systems interacting with actuators through viscoelastic linkages. Mech Mach Theory 2020; 147: 103763–103763.

31.

Gao

, et al. Optimal control of discrete-time interval type-2 fuzzy-model-based systems with D-stability constraint and control saturation. Signal Process 2016; 120: 409–421.

32.

Allan

Bates

Risbeck

, et al. On the inherent robustness of optimal and suboptimal nonlinear MPC. Syst Control Lett 2017; 106: 68–78.

33.

Ding

. An efficient offline implementation for output feedback min-max MPC. Int J Robust Nonlinear Control 2019; 29: 492–506.

34.

Mayne

Falugi

. Stabilizing conditions for model predictive control. Int J Robust Nonlinear Control 2019; 29: 894–903.

35.

Cheng

Cannon

Kouvaritakis

. The design of dynamics in the prediction structure of robust MPC. Int J Control 2013; 86: 2096–2103.

36.

Bossi

Rottenbacher

Mimmi

, et al. Multivariable predictive control for vibrating structures: an application. Control Eng Pract 2011; 19: 1087–1098.

37.

Hairer

Wanner

Nørsett

. Solving ordinary differential equations I: nonstiff problems, Berlin: Springer, 1993.

38.

Lofberg J. YALMIP : a toolbox for modeling and optimization in MATLAB. In: 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508). New Jersey: IEEE, pp.284–289.

39.

Devika

Thomas

. Sliding mode controller design for MIMO nonlinear systems: a novel power rate reaching law approach for improved performance. J Franklin Inst 2018; 355: 5082–5098.