Designing a terminal optimal control with an integral sliding mode component using a saddle point method approach: a Cartesian 3D-crane application

This paper proposes a new approach for designing a nonlinear optimal controller with an integral sliding mode component employing a generalization of the saddle point method which consists on controlling a controllable nonlinear system. Based on the initial and final conditions of the dynamical system, we consider an open-loop control such that the state of the system can be moved to a neighborhood of the equilibrium state corresponding to the given final condition. The implementation of the method for solving the problem involves a two-step iterated procedure: (i) The first step consists of a “prediction” which calculates the preliminary position approximation to the steady-state point, and (ii) the second step is designed to find a “basic adjustment” of the previous prediction. We apply the controller to a Cartesian 3D-crane. The formulation of the 3D-crane is in terms of nonlinear programming problems implementing the Lagrange principle. We transform the problem in a system of equations where each equation is itself an optimization problem. For designing the controller we suggest to employ an integral sliding mode method which suppress the model uncertainties consequence of moving the trolley and the bridge, lifting the cargo as well as external forces. As a result, the optimal controller will be simultaneously able to lift the cargo, suppressing the payload vibration, tracking the trolley and moving the bridge. A numerical example involving the simulation of a 3D-crane shows the effectiveness of the controller.