Using frozen-time control and tangent linearised control systems for the feedback control of nonlinear systems

Motivated by recent results on tangent linearised control systems and continuation methods for solving nonlinear optimisation problems we propose a method to design open-loop and closed-loop controls for nonlinear systems. Frozen-time linear control techniques are applied to the tangent linearised system and the obtained linearised control is “lifted” to a control of the underlying nonlinear system. We present a concise proof that the deviation of the actual state from the target state is decaying exponentially with time if a certain control is applied to the nonlinear system. This proof includes an explicit design instruction for the desired nonlinear control. Two different possibilities for carrying out this instruction are discussed and applied to a highly nonlinear unstable example system. A short comparison of two usual nonlinear control methods to the proposed method concludes the paper