Sliding mode control of flexible structures (Q2725705)

This very important and useful paper presents the theoretical fundamentals of the sliding mode control. The authors consider and actively control a nonlinear base isolated structure, whose dynamic behaviour is described by the following model composed of two coupled subsystems:NEWLINENEWLINENEWLINEMain structure: NEWLINE\[NEWLINE{\mathbf M}\ddot {\mathbf q}_r+ {\mathbf C}\dot {\mathbf q}_r+ {\mathbf {Kq}}_r= \rho(q_c,\dot q_c),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\rho(q_c,\dot q_c):= \rho_1\dot q_c+ \rho_2 q_c,\quad \rho_1= [c_1,0,\dots,0]^T,\quad \rho_2= [k_1,0,\dots, 0]^T.NEWLINE\]NEWLINE Base isolation: NEWLINE\[NEWLINEm_0\ddot q_c+ (c_0+ c_1)\dot q_c+ (k_0+ k_1) q_c- c_1\dot q_{r1}- k_1 q_{r1}+ f= u,NEWLINE\]NEWLINE NEWLINE\[NEWLINEf:= -c_0\dot d- k_0 d+ f_N(q_c,\dot q_c, d,\dot d),NEWLINE\]NEWLINE where \({\mathbf q}_r= [q_{r1}, q_{r2},\dots, q_{rn}]^T\in \mathbb{R}^n\) and \(q_c\in \mathbb{R}\) represent the horizontal displacements of each floor and the structural base with respect to an inertial frame, respectively, and \(u\) is the control force. The coupling effect between the main structure and the base isolation subsystems is described by a vector function \(\rho(q_c,\dot q_c)\in \mathbb{R}^n\). A robust control scheme based on the sliding mode principle is chosen to drive the state variables \((q_c,\dot q_c)\) of the base to zero exponentially so that the coupling term \(\rho(q_c,\dot q_c)\) between the main structure subsystem and the base isolation subsystem vanishes exponentially in the presence of unknown seismic excitation and parametric uncertainties.NEWLINENEWLINENEWLINEMain result: The sliding function \(s\) and the state variables \((q_c,\dot q_c)\) of the base are bounded for all \(t\geq 0\) and converge to zero exponentially as \(t\to\infty\) for any bounded initial conditions. Finally, two examples related to active control of base-isolated structures and cable stayed bridges are presented to show the usefulness of the sliding mode control system in structural control.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00020].