Neural stabilizers for unknown systems

In this paper, we show that for the unknown single-input nonlinear system x˙=f(x)+g(x)u, xεℜⁿ, uεℜ (0.1), it is possible to construct a semi-global state-feedback stabilizer when the only information about the unknown system is that: the system is stabilizable; the state dimension of the system is known; and the system vector-field are at least C¹. The proposed stabilizer uses linear-in-the-weights neural networks whose synaptic weights are adaptively adjusted, control Lyapunov functions. 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 sufficiently large number of regressor terms, and that the controller parameters are appropriately chosen. Although the proposed controller is a discontinuous one, the closed-loop system does not enter in sliding motions. However, the proposed controller can be a very conservative one and may result in very poor transient behavior and/or very large control inputs