Nonlinear H-infinity control for the rotary pendulum

The rotary (Furuta's) pendulum is used to analyzed the performance of a new nonlinear optimal (H-infinity) control for underactuated robotic systems. After applying partial feedback linearization, the pendulum's dynamic model is first transformed to an equivalent form. 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 of computation of the associated Jacobian matrices. For the approximately linearized model of the pendulum an H-infinity feedback controller is developed. Through the repetitive solution of an algebraic Riccati equation which is also performed at each step of the control method, the controller's gain is computed. The stability features of the control loop are proven with Lyapunov analysis. 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.