Control of the Lorenz chaos by the exact linearization
In a recent review of the book by \textit{G. Chen} and \textit{X. Dong}, From chaos to order (1998; Zbl 0908.93005), we mentioned that the only control method unused was the exact linearization. The paper is concerned with the exact linearization method to stabilize the Lorenz chaos near some off-center equilibrium point. The standard method based on the Lie bracket of the vector field gives \(\dot z_1= z_2\), \(\dot z_2 =z_3\), \(z_3=v\) and the linear feedback control law is \(v=\alpha_1z_1+ \alpha_2z_2 +\alpha_3 z_3\). Simulation results are given. To implement this control law, all three state components should be known and the initial state should be near the off-center equilibrium point. The Lorenz system has a smooth nonlinearity which is used for this Lie approach. On the other hand, the Chua system with piecewise nonlinearity gives a possibility for a state control law to stabilize it near any of the three equilibria. This, according to our knowledge, was never implemented because of limited possibilities of practical realization.
scientific article; zbMATH DE number 1197095