Optimal Control of a Double Inverted Pendulum on a Cart

In this report a number of algorithms for optimal control of a double inverted pendulum on a cart (DIPC) are investigated and compared. Modeling is based on Euler-Lagrange equations derived by specifying a Lagrangian, difference between kinetic and potential energy of the DIPC system. This results in a system of nonlinear differential equations consisting of three 2-nd order equations. This system of equations is then transformed into a usual form of six 1-st order ordinary differential equations (ODE) for control design purposes. Control of a DIPC poses a certain challenge, since unlike a robot, the system is underactuated: one controlling force per three degrees of freedom (DOF). In this report, problem of optimal control minimizing a quadratic cost functional is addressed. Several approaches are tested: linear quadratic regulator (LQR), state-dependent Riccati equation (SDRE), optimal neural network (NN) control, and combinations of the NN with the LQR and the SDRE. Simulations reveal superior performance of the SDRE over the LQR and improvements provided by the NN, which compensates for model inadequacies in the LQR. Limited capabilities of the NN to approximate functions over the wide range of arguments prevent it from significantly improving the SDRE performance, providing only marginal benefits at larger pendulum deflections.