Periodic solution of the selector-linear differential inclusion and estimation of invariant regions in the wheeled robot control problem

Abstract

In this paper the extremum periodic trajectory of the two-dimensional selector-linear differential inclusion (SLDI) is used to estimate boundary of the invariant set of the nonlinear time-varying system arising in the stability analysis of the wheeled robot control. The motion is supposed to be planar without a lateral slippage. The control goal is to drive the target point of the robot plaform to the specified trajectory and to stabilize the motion along it. The trajectory consists of line segments and circular arcs. The current curvature of the trajectory of the target point is taken as control. The control must satisfy two-sided constraints. Given control law, the attraction domain estimation problem is considered. The attraction domain must be inscribed into the certain parallelepiped of the 'distance to the trajectory-orientation' phase space. Time-varying curvature of the target trajectory is considered as arbitrary varying function which takes values from the specified interval. The feedback linearization scheme is used for synthesis of the control low. The 'saturation function' is then used to take into account control constraints. The closed loop system takes form of the nonlinear system with parametric disturbances. The absolute stability approach is explored for stability analysis. Some nonlinearities take values from the interval. Other nonlinearities satisfy sector constraints. Along with the nonlinear time-varying system the uncertain linear time varying system is considered. Every solution of the nonlinear system is also solution of the time varying system for certain set of time-varying disturbances. To estimate the attraction domain of the nonlinear closed loop system, the Lyapunov function for SLDI is constructed. A convex invariant function is known to exist at the boundary of the absolute stability of SLDI. The extremum trajectory, corresponding to the boundary of the absolute stability in the second order case belongs to the level set of the invariant function and is the periodic solution. The periodic solution has finite number of switches on the period. It circumscribes the boundary of the attraction domain estimate. The illustrative example is given.