Above is the controlled pendulum, assuming all parameters are known exactly. On the phase plane, the controller drives the system to the sliding surface (designed to be ) in finite time and maintains it there. On the sliding surface, the system is exponentially stable. The system approaches the designed equilibrium position at an exponential rate of 1.
This is the closed loop system to the robust controller designed. The parameter is not known exactly, only an upper bound. For this simulation, the values were used. The robust controller performs exactly as desired, despite the uncertainty.
This is the closed loop system to the robust controller designed. The parameter is not known exactly, only an upper bound. For this simulation, the values were used. The uncertainty in this case if very large! (The a conservative bound is 5 times the actual value.) Nonetheless, the robust controller performs exactly as desired, despite the very large uncertainty.
