Nonlinear H-infinity control for underactuated systems: the Furuta pendulum example

The article applies nonlinear optimal (H-infinity) control to underactuated robotic systems using as a case study Furuta’s pendulum. The pendulum’s dynamic model is first transformed to an equivalent form after applying partial feedback linearization. The later description of the pendulum’s dynamics undergoes approximate linearization which takes place round a temporary operating point (equilibrium) recomputed at each iteration of the control algorithm. The linearization makes use of Taylor series expansion of the state-space model of the system and needs computation of the associated Jacobian matrices. For the approximately linearized model of the pendulum an H-infinity feedback controller is developed. The controller’s gain is computed through the repetitive solution of an algebraic Riccati equation which is also performed at each step of the control method. The stability analysis is based on Lyapunov’s method. First it is shown that the control loop satisfies the H-infinity tracking performance condition. Next, under moderate conditions it is also shown that the global asymptotic stability of the control loop can be assured.