Adaptive control of nonlinear systems with a triangular structure

In this paper, we introduce two distinct types of nonlinear dynamical systems, 𝒯₁ and 𝒯₂, both of which possess a triangular structure. It is shown that all systems belonging to 𝒯₁ can be made stable and that if they belong to a subclass 𝒯_1s, the stability holds globally. A precise characterization of the general class of nonlinear systems transformable to 𝒯₁ is carried out. The second class, 𝒯₂, corresponds to a set of second-order nonlinear differential equations and is motivated by problems that occur in mechanical systems. It is shown that global tracking can be achieved for all systems in 𝒯₂. A constructive approach is used in all cases to develop the adaptive controller, and both stabilization and tracking are shown to be realizable. Simple examples are given to illustrate the different classes of nonlinear systems as well as the idea behind the approach used to stabilize them