State estimation and optimal control of an inverted pendulum on a cart system with stochastic approximation approach

In this paper, optimal control of an inverted pendulum on a cart system is studied. Since the nonlinear structure of the system is complex, and in the presence of random disturbances, optimization and control of the motion of the system become more challenging. For handling this system, a discrete-time stochastic optimal control problem for the system is described, where the external force is considered as the control input. By defining a loss function, namely, the mean squared errors to be minimized, the stochastic approximation (SA) approach is applied to estimate the state dynamics. In addition, the Hamiltonian function is defined, and the first-order necessary conditions are derived. The gradient of the cost function is determined so that the SA approach is employed to update the control sequences. For illustration, considering the values of the related parameters in the system, the discrete-time stochastic optimal control problem is solved iteratively by using the SA algorithm. The simulation results show that the state estimation and the optimal control law design are well performed with the SA algorithm, and the motion of the inverted pendulum cart is addressed satisfactorily. In conclusion, the efficiency of the SA approach for solving the inverted pendulum on a cart system is verified.