State constrained predictive control of cart with inverted pendulum

This paper presents the design of predictive control with state constraints for the swing-up and stabilizing control of a cart with an inverted pendulum system where the state constraints are proposed on angle and velocity of pendulum, position and velocity of cart, Since the system has strong nonlinearity and inherent instability, a step of linearization is necessary to extract linear state space model. Then the discrete-time predictive controller is applied to the linearized system, where the general design objective of this one is to compute a trajectory of a future manipulated variable u in order to walk the cart along a setpoint trajectory while keeping the pendulum to the equilibrium position. Maintaining the pendulum in the balanced position is achieved by including state constrained in control objective where the basic idea is to translate the state constraints problems to linear inequalities then parameterize them using the same parameter vector ΔU as the only one used in the design of predictive control, in order to relate to the original model predictive control problem, then the optimal solution is found by solving a quadratic optimization problem. Simulation results show clearly the effect of taking the state constraints in the control design in keeping the pendulum at the balanced position while moving the cart in predefined trajectory.