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Stable extension control of nonlinear systems

A design method for an extension controller is developed. The proposed adaptive extension control resulting from the direct adaptive approach is employed to directly adapt the gain parameter of the extension controller. Then the constructed controller can be best approximated to a given optimal control. Unlike a fuzzy controller, only one linguistic-like level is needed in the extension controller. An appropriate control law can be adapted to the output gain of the developed extension controller. The best merits of the proposed controller are that: (a) the number of adaption parameters is small; (b) the design algorithm is easily implemented. In addition, a maximum control is established to guarantee the system robust stability. The derivation shows that the proposed extension controller is stable in the sense of Lyapunov. Finally, a nonlinear system simulation example is applied to verify the effectiveness and ability of the proposed adaptive extension controller.