Anti-chaos control of perturbed second-order systems: A disturbance observer-based H∞-control approach

Different real-life systems such as chemical, economic, and mechatronic systems may be required to exhibit a chaotic behavior for different applications. In these cases, such a behavior must be ensured despite the inherent disturbances that may appear when dealing with real-time applications. Hence, this paper develops a methodology ensuring that a second-order system behaves as a chaotic system. Besides, the novel scheme maintains the chaotic behavior despite matched and unmatched disturbances. The proposed chaotization method consists of a controller divided into two parts, a disturbance observer and a H∞ controller. The disturbance observer compensates for the effect of matched disturbances and the H∞ formalism achieves the chaotization of the second-order system while the matched and unmatched disturbances’s effect is attenuated. A complete mathematical development theoretically validates the proposed scheme. Furthermore, the performance of the proposed controller is validated through an extensive numerical study and is compared against a previously proposed anti-chaos control technique and a fixed-time sliding mode controller that compensates for matched and unmatched disturbances. The maximum Lyapunov exponent is used to demonstrate the existence of chaos in the closed-loop system. Then, the numerical results show the superior performance of the novel chaotization approach.