Regulation control of an underactuated mechanical system with discontinuous friction and backlash

Rascón, R. ; Rosas, D. ; Hernandez-Balbuena, D.

International Journal of Applied Mathematics and Computer Science, Tome 27 (2017) no. 4, pp. 785-797.

Résumé

In this work, the problem of position regulation control is addressed for a 2DOF underactuated mechanical system with friction and backlash. For this purpose, a method combining sliding mode and H∞ control is developed. We prove that the application of the method to the nonlinear model considered results in an asymptotically stable equilibria set. Moreover, it is possible to achieve a sufficiently small and bounded steady-state position error even in the presence of disturbances by employing the proposed technique. That is, the developed controller is able to account not only for unmatched external perturbations and model discrepancies of the test rig considered, but also for matched bounded perturbations. The control methodology is presented from both the theoretical and experimental angles to demonstrate the good performance of the proposed controller.

Détail

Keywords: underactuated mechanical system, sliding mode control, H infinity control, discontinuous observer
Mots-clés : sterowany układ mechaniczny, sterowanie ślizgowe, obserwator nieciągły

Bibliographie

[1] Adetola, V., DeHaan, D. and Guay, M. (2009). Adaptive model predictive control for constrained nonlinear systems, Systems Control Letters 58(5): 320–326.

[2] Aguilar, L., Orlov, Y. and Acho, L. (2003). Nonlinear H∞ control of nonsmooth time varying systems with application to friction mechanical manipulators, Automatica 39(9): 1531–1542.

[3] Brahim, A.B., Dhahri, S., Hmida, F.B. and Sellami, A. (2015). An H∞ sliding mode observer for Takagi–Sugeno nonlinear systems with simultaneous actuator and sensor faults, International Journal of Applied Mathematics and Computer Science 25(3): 547–559, DOI: 10.1515/amcs-2015-0041.

[4] Branicky, M. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Transactions on Automatic Control 43(4): 475–482.

[5] Brogliato, B. (1999). Nonsmooth Mechanics, Springer, London.

[6] Brogliato, B., Niculescu, S.-I. and Orhant, P. (1997). On the control of finite-dimensional mechanical systems with unilateral constraints, IEEE Transactions on Automatic Control 42(2): 200–215.

[7] Castaños, F. and Fridman, L. (2006). Analysis and design of integral sliding manifolds for systems with unmatched perturbations, IEEE Transactions on Automatic Control 51(5): 853–858.

[8] Castaños, F. and Fridman, L. (2011). Dynamic switching surfaces for output sliding mode control: An H∞ approach, Automatica 47(9): 1957–1961.

[9] Chang, Y.-C. and Lee, C.-H. (1999). Robust tracking control for constrained robots actuated by DC motors without velocity measurements, IEE Proceedings—Control Theory and Applications 146(2): 147–156.

[10] Chiu, C.-S., Lian, K.-Y. and Wu, T.-C. (2004). Robust adaptive motion/force tracking control design for uncertain constrained robot manipulators, Automatica 40(12): 2111–2119.

[11] Christophersen, F.J. (2007). Optimal Control of Constrained Piecewise Affine Systems, Lecture Notes in Control and Information Sciences, Springer, Berlin/Heidelberg.

[12] Doyle, J.C., Glover, K., Khargonekar, P.P. and Francis, B.A. (1989). State-space solutions to standard H2 and H∞ control problems, IEEE Transactions on Automatic Control 34(8): 831–847.

[13] Durmaz, B., Özgören, M. K. and Salamci,M. U. (2011). Sliding mode control for non-linear systems with adaptive sliding surfaces, Transactions of the Institute of Measurement and Control 34(1): 56–90.

[14] Fallaha, C.J., Saad, M., Kanaan, H.Y. and Al-Haddad, K. (2011). Sliding-mode robot control with exponential reaching law, IEEE Transactions on Industrial Electronics 58(2): 600–610.

[15] Filippov, A.F. (1988). Differential Equations with Discontinuous Right-Handsides, Kluwer, Dordercht.

[16] Ghafari-Kashani, A., Faiz, J. and Yazdanpanah, M. (2010). Integration of non-linear H∞ and sliding mode control techniques for motion control of a permanent magnet synchronous motor, IET Electric Power Applications 4(4): 267–280.

[17] Isidori, A. (2000). A tool for semiglobal stabilization of uncertain non minimum-phase nonlinear systems via output feedback, IEEE Transactions on Automatic Control 48(10): 1817–1827.

[18] Isidori, A. and Astolfi, A. (1992). Disturbance attenuation and H∞-control via measurement feedback in nonlinear systems, IEEE Transactions on Automatic Control 37(9): 1283–1293.

[19] Kazerooni, H. (1990). Contact instability of the direct drive robot when constrained by a rigid environment, IEEE Transactions on Automatic Control 35(6): 710–714.

[20] Khalil, H. (2002). Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ.

[21] Kwakernaak, H. (1993). Robust control and H∞-optimization–tutorial paper, Automatica 29(2): 255–273.

[22] Leine, R. I. and Van de Wouw, N. (2010). Stability and Convergence of Mechanical Systems with Unilateral Constraints, Springer, Berlin.

[23] Lian, J. and Zhao, J. (2010). Robust H-infinity integral sliding mode control for a class of uncertain switched nonlinear systems, Journal of Control Theory and Applications 8(4): 521–526.

[24] Lian, K.-Y. and Lin, C.-R. (1998). Sliding-mode motion/force control of constrained robots, IEEE Transactions on Automatic Control 43(8): 1101–1103.

[25] Luo, N., Tan, Y. and Dong, R. (2015). Observability and controllability analysis for sandwich systems with backlash, International Journal of Applied Mathematics and Computer Science 25(4): 803–814, DOI: 10.1515/amcs-2015-0057.

[26] Mansard, N. and Khatib, O. (2008). Continuous control law from unilateral constraint, IEEE International Conference on Robotics and Automation, ICRA 2008, Pasadena, CA, USA, pp. 3359–3364.

[27] Menini, L. and Tornambe, A. (2001). Dynamic position feedback stabilisation of multidegrees-of-freedom linear mechanical systems subject to nonsmooth impacts, IEE Proceedings—Control Theory and Applications 148(6): 488–496.

[28] Orlov, Y., Aoustin, Y. and Chevallereau, C. (2011). Finite time stabilization of a perturbed double integrator. I: Continuous sliding mode-based output feedback synthesis, IEEE Transactions on Automatic Control 56(3): 614–618.

[29] Paden, B. and Sastry, S. (1987). A calculus for computing Filippov’s differential inclusion with application to the variable structure control of robot manipulators, IEEE Transactions on Circuits and Systems 34(1): 73–81.

[30] Perez, M., Jimenez, E. and Camacho, E. (2010). Design of an explicit constrained predictive sliding mode controller, IET Control Theory Applications 4(4): 552–562.

[31] Potini, A., Tornambe, A., Menini, L., Abdallah, C. and Dorato, P. (2006). Finite-time control of linear mechanical systems subject to non-smooth impacts, 14th Mediterranean Conference on Control and Automation, MED’06, Ancona, Italy, pp. 1–5.

[32] Rascón, R., Alvarez, J. and Aguilar, L. (2012). Sliding mode control with H∞: Attenuator for unmatched disturbances in a mechanical system with friction and a force constraint, 12th International Workshop on Variable Structure Systems (VSS), Mumbai, Maharashtra, India, pp. 434–439.

[33] Rascón, R., Alvarez, J. and Aguilar, L.T. (2014). Control robusto de posici´on para un sistema mec´anico subactuado con fricción y holgura elástica, Revista Iberoamericana de Automática e Informática Industrial 11(3): 275–284.

[34] Rascón, R., Alvarez, J. and Aguilar, L.T. (2016). Discontinuous H∞ control of underactuated mechanical systems with friction and backlash, International Journal of Control, Automation and Systems 14(5): 1213–1222.

[35] Sabanovic, A., Elitas, M. and Ohnishi, K. (2008). Sliding modes in constrained systems control, IEEE Transactions on Industrial Electronics 55(9): 3332–3339.

[36] Safonov, M.G., Limebeer, D.J.N. and Chiang, R.Y. (1989). Simplifying the H∞ theory via loop-shifting, matrix-pencil and descriptor concepts, International Journal of Control 50(6): 2467–2488.

[37] Shevitz, D. and Paden, B. (1994). Lyapunov stability theory of nonsmooth systems, IEEE Transactions on Automatic Control 39(9): 1910–1914.

[38] Tseng, C.-S. (2005). Mixed H2/H∞ adaptive tracking control design for uncertain constrained robots, Asian Journal of Control 7(3): 296–309.

[39] Utkin, V. (1978). Sliding Modes and Their Applications, Mir, Moscow.

[40] Utkin, V. (1992). Sliding Modes in Control Optimization, Springer, Berlin.

[41] Virgin, L.N. (2000). Introduction to Experimental Nonlinear Dynamics: A Case Study in Mechanical Vibration, Cambridge University Press, New York, NY.

[42] Zhu, J. and Khayati, K. (2015). A new approach for adaptive sliding mode control: Integral/exponential gain law, Transactions of the Institute of Measurement and Control 38(4): 385–394.