Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts. (English) Zbl 1319.93036

Naldi, Roberto; Sanfelice, Ricardo G.

Automatica 49, No. 5, 1104-1116 (2013).

Summary: Motivated by applications of systems interacting with their environments, we study the design of passivity-based controllers for a class of hybrid systems in which the energy dissipation may only happen along either the continuous or the discrete dynamics. A general definition of passivity, encompassing the said special cases, is introduced and, along with detectability and solution conditions, linked to stability and asymptotic stability of compact sets. The proposed results allow us to take advantage of the passivity property of the system at flows or at jumps and are employed to design passivity-based controllers for the class of hybrid systems of interest. Two applications, one pertaining to a point mass physically interacting with a wall and another about controlling a ball bouncing on an actuated surface, illustrate the definitions and results throughout the paper.

MSC:

93C30	Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems)
93D20	Asymptotic stability in control theory
93D99	Stability of control systems
70Q05	Control of mechanical systems

Keywords:

hybrid systems; passivity; passivity-based control; mechanical systems

References:

[2]	Bartolo, D.; Boudaoud, A.; Narcy, G.; Bonn, D., Dynamics of non-Newtonian droplets, Physical Review Letters, 99, 174502 (2007)
[3]	Brogliato, B., Nonsmooth mechanics model, dynamics and control (1996), Springer · Zbl 0861.73001
[4]	Brogliato, B., Some perspectives on the analysis and control of complementarity systems, IEEE Transactions on Automatic Control, 48, 6, 918-935 (2003) · Zbl 1364.93356
[5]	Brogliato, B.; Lozano, R.; Egeland, O., Dissipative systems analysis and control (2007), Springer
[6]	Brogliato, B.; Rio, A. Z., On the control of complementary-slackness juggling mechanical systems, IEEE Transactions on Automatic Control, 45, 2, 235-246 (2000) · Zbl 0982.70018
[8]	Galeani, S.; Menini, L.; Potini, A.; Tornambé, A., Trajectory tracking for a particle in elliptical billiards, International Journal of Control, 81, 2, 189-213 (2008) · Zbl 1152.93381
[9]	Glilardi, G.; Sharf, I., Literature survey of contact dynamics modeling, Journal of Mechanism and Machine Theory, 37, 10, 1213-1239 (2002) · Zbl 1062.70553
[10]	Goebel, R.; Sanfelice, R. G.; Teel, A. R., Hybrid dynamical systems, IEEE Control Systems Magazine, 29, 2, 28-93 (2009) · Zbl 1395.93001
[11]	Goebel, R.; Sanfelice, R. G.; Teel, A. R., Hybrid dynamical systems modeling, stability, and robustness (2012), Princeton University Press · Zbl 1241.93002
[12]	Goebel, R.; Teel, A. R., Solutions to hybrid inclusions via set and graphical convergence with stability theory applications, Automatica, 42, 4, 573-587 (2006) · Zbl 1106.93042
[14]	Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983), Springer · Zbl 0515.34001
[16]	Haddad, W. M.; Chellaboina, V.; Nersesov, S. G., Impulsive and hybrid dynamical systems: stability, dissipativity, and control (2006), Princeton University Press · Zbl 1114.34001
[17]	Haddad, W. M.; Hui, Q., Energy dissipating hybrid control for impulsive dynamical systems, Nonlinear Analysis, 69, 3232-3248 (2008) · Zbl 1181.34018
[18]	Hunt, K.; Crossley, F., Coefficient of restitution interpreted as damping in vibroimpact, ASME Journal of Applied Mechanics, 42, 440-445 (1975)
[19]	Isidori, A., (Nonlinear control systems II. Nonlinear control systems II, Communication and control engineering series (1998), Springer-Verlag: Springer-Verlag London)
[20]	Khalil, H. K., Nonlinear systems (1996), Prentice Hall
[21]	Kokotovic, P.; Sussman, H., A positive real condition for global stabilization of nonlinear systems, Systems & Control Letters, 13, 2, 125-133 (1989) · Zbl 0684.93066
[22]	Leine, R. I.; Heimsch, T. F., Global uniform symptotic attractive stability of the non-autonomous bouncing ball system, Physica D: Nonlinear Phenomena, 241, 22, 2029-2041 (2012)
[23]	Leine, R. I.; van de Wouw, N., Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact, Nonlinear Dynamics, 51, 4, 551-583 (2007) · Zbl 1170.70340
[24]	Lin, W.; Byrnes, C. I., Passivity and absolute stabilization of a class of discrete-time nonlinear systems, Automatica, 31, 2, 263-267 (1995) · Zbl 0826.93060
[25]	Menini, L.; Tornambe, A., Asymptotic tracking of periodic trajectories for a simple mechanical system subject to nonsmooth impacts, IEEE Transactions on Automatic Control, 46, 7, 1122-1126 (2001) · Zbl 1008.93061
[26]	Morărescu, I. C.; Brogliato, B., Passivity-based switching control of flexible-joint complementarity mechanical systems, Automatica, 46, 1, 160-166 (2010) · Zbl 1214.93040
[27]	Morărescu, I. C.; Brogliato, B., Trajectory tracking control of multiconstraint complementarity Lagrangian systems, IEEE Transactions on Automatic Control, 55, 6, 1300-1313 (2010) · Zbl 1368.93253
[28]	Or, Y.; Ames, A. D., Stability and completion of zeno equilibria in Lagrangian hybrid systems, IEEE Transactions on Automatic Control, 56, 6, 1322-1336 (2011) · Zbl 1368.93476
[29]	Ortega, R.; Garcia-Canseco, E., Interconnection and damping assignment passivity-based control: a survey, European Journal of Control, 10, 432-450 (2004) · Zbl 1293.93615
[30]	Ortega, R.; van der Schaft, A.; Mareels, I.; Maschke, B., Putting energy back in control, IEEE Control Systems Magazine, 21, 2, 18-33 (2001)
[33]	Sanfelice, R. G.; Goebel, R.; Teel, A. R., Invariance principles for hybrid systems with connections to detectability and asymptotic stability, IEEE Transactions on Automatic Control, 52, 12, 2282-2297 (2007) · Zbl 1366.93554
[35]	Sepulchre, R.; Jankovic, M.; Kokotovic, P., Constructive nonlinear control (1997), Springer · Zbl 1067.93500
[36]	Spong, M. W.; Holm, J. K.; Lee, D., Passivity-based control of bipedal locomotion, IEEE Robotics & Automation Magazine, 14, 2, 30-40 (2007)
[37]	Stronge, W. J., Impact mechanics (2000), The University Press: The University Press Cambridge · Zbl 0961.74002
[38]	Tarn, T.; Wu, Y.; Xi, N.; Isidori, A., Force regulation and contact transition control, IEEE Control Systems Magazine, 32-40 (1996)
[40]	Tornambé, A.; Menini, L., Control of (otherwise) uncontrollable linear mechanical systems through non-smooth impacts, Systems & Control Letters, 49, 4, 311-322 (2003) · Zbl 1157.93390
[41]	van der Schaft, A., L2-gain and passivity techniques in nonlinear control (2000), Springer · Zbl 0937.93020
[42]	Villani, L.; De Shutter, J., Force control, (Siciliano, B.; Khatib, O., Springer handbook of robotics (2008), Springer)
[43]	Westervelt, E. R.; Grizzle, J. W.; Chevallereau, C.; Choi, J. H.; Morris, B., Feedback control of dynamical bipedal robot locomotion (2007), Taylor & Francis
[44]	Willems, J. C., Dissipative dynamical systems, i. general theory, Archive for Rational Mechanics and Analysis, 45, 5, 321-352 (1972) · Zbl 0252.93002
[45]	Zavala-Rio, A.; Brogliato, B., On the control of a one degree-of-freedom juggling robot, Dynamics and Control, 9, 67-91 (1999) · Zbl 0922.93029
[47]	Zhang, X.; Vu-Quoc, L., Modeling the dependence of the coefficient of restitution on the impact velocity in elasto-plastic collisions, International Journal of Impact Engineering, 27, 317-341 (2002)
[48]	Zhao, J.; Hill, D. J., Passivity and stability of switched systems: a multiple storage function method, Systems & Control Letters, 57, 2, 158-164 (2008) · Zbl 1137.93051