Neural stabilizers for unknown systems

In this paper, we show that for the unknown single-input nonlinear system x/spl dot/=f(x)+g(x)u, x/spl epsiv//spl Rfr//sup n/, u/spl epsiv//spl Rfr/ (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 C/sup 1/. 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.