Neural network control of unknown systems

We show that for all unknown multi-input (MI) nonlinear systems that are affected by external disturbances, it is possible to construct a semi-global state-feedback stabilizer when the only information about the unknown system is that (A1) the system is robustly stabilizable, (A2) the state dimension of the system is known, (A3) the system vector-fields are at least C¹. The proposed stabilizer uses linear-in-the-weights neural networks whose synaptic weights are adaptively adjusted. Robust control Lyapunov functions (RCLF) and switching adaptive derivative feedback control are used. Using Lyapunov stability arguments, we show that the closed-loop system is stable and the state vector converges arbitrarily close to zero, provided that the controller´s neural networks have a sufficiently large number of regressor terms, and that the controller parameters are appropriately chosen. It is worth noticing, that no growth conditions are imposed on the unknown system nonlinearities:also, the proposed approach does not require knowledge of the RCLF of the system. Moreover, although the proposed controller is a discontinuous one, the closed-loop system does not enter in sliding motions. However, the proposed controller might be a very conservative one and may result in very poor transient behavior and/or very large control inputs