Inverted Pendulum 2
A state-feedback fuzzy controller is designed to stabilise/regulate an inverted pendulum on a cart based on 2-rule T-S fuzzy model with the consideration of system stability and performance. The feedback gains is obtained by solving a feasible solution to a set of stability and performance conditions in terms of linear matrix inequalities. The following movie clip shows a simulation result (the first 8 second is in slow motion (0.02x) and the rest is in normal speed). The initial condition for all cart is the same: pendulum angle = 75 degrees, pendulum angular velocity = 0 degrees per second, cart displacement = 0 m eter and cart velocity = 0 meters per second.
Details can be found in the following paper:
- H.K. Lam and Mohammad Narimani, “Stability analysis and performance deign for fuzzy-model-based control system under imperfect premise matching,” IEEE Trans. Fuzzy Systems, vol. 17, no. 4, pp. 949-961, Aug. 2009.
Three fuzzy controllers are designed to stabilise the inverted pendulum on a cart subject to different performance indices. The performance index of the upper cart is to minimise the pendulum angle and cart displacement. The fuzzy controller is able to drive the pendulum and the cart to the origin in the shortest time among the three carts. The one in the bottom is to minimise the pendulum angle, cart displacement and the control energy. This fuzzy controller is still able drive the pendulum and the cart to the origin but a longer time is required and the control energy consumption is lower compared with the upper one. The one in the middle puts a heavy weight on the minimisation of control energy. It consumes the least control energy among the three fuzzy controllers. However, it requires the longest time to return the cart to the origin (around 200 seconds but the movie only shows up to 38 seconds).