The Synthesis of Stabilizing Control of a Rigid Body with Attached Elastic Elements

摘要

We consider the problem of synthesis of nonlinear control law for a simulated mechanical system, which consists of a rigid carrying body and attached to it flexible beams. The system performs planar motion under controlling moment of forces applied to the carrying body. For obtaining a mathematical model in the form of a system of ordinary differential equations we make use of the modal method of discretization. The motion equations are presented in the Euler-Lagrange form with infinite number of elastic coordinates and a variable, which characterizes orientation of the carrying body. In order to obtain the nonlinear system we found control in the form of feedback, which provides asymptotical stability of the equilibrium state relative to definite combinations of elastic coordinates and orientation of the carrying body. For investigation we applied the approach of the Lyapunov control functions relative to a part of variables. We proved the Lyapunov stability of the complete system with respect to all phase variables. The question about realization of the obtained feedback for additional measurements of relative shears of a beam was studied. We show observability of the investigated system for different modes of motion. The results of numerical simulation are adduced.