Funnel control for nonlinear systems with arbitrary relative degree using high-gain observers

Safety‐constrained robust adaptive control for a class of MIMO non‐linear systems

“…It can be noticed that the unstructured modelling uncertainties of the system d(x, u, t) are functions of x 1 , x 2 , and the state feedback controllers designed in (38) and (66) are function of x 1 , x 2 , x a 1 , x a 2 , x = x 1 , x 2 , x a 1 , x a 2 , x 1 * , ẋ 1 * , ẍ 1 * T ∈ ℜ 7n . Furthermore, using RBFNN, d(x, u, t) is approximated over the compact set…”

“…Using the proposed tracking controller in (38), the asymptotic stability of the closed-loop system is required to be shown. This requirement is fulfiled in the following theorem.…”

“…Theorem 2: Consider the auxiliary observer system (15) under the controller (38) (iii) The error e r i and its derivative e˙r i will remain bounded in the sets Ω κ i and Ω ζ i , respectively, where Ω κ i := e r i ∈ ℜ: − κ e i < e r i < κ e i and Ω ζ i := {e˙r i ∈ ℜ: − ζ e˙i < e˙r i < ζ e˙i } Proof:…”

“…The same control formulations in (38) and (66) can be used to design a symmetric state-constrained tracking controller by choosing κ i = κ i = κ i and ζ i = ζ i = ζ i , ∀i, respectively.…”

“…It can be mentioned here that other error transformation-based adaptive control designs are also available in the literature [33][34][35][36][37][38]. However, these controllers are used to enforce the error of the system within the prescribed bounds, thereby improving the transient performance.…”