Practical Output Tracking of Nonlinear Systems with Uncontrollable Unstable Linearization: an Alternative Adaptive Mechanism

A compensator is proposed to achieve globally and practically adaptive output tracking control for a class of linearly parameterized nonlinear systems with uncontrollable unstable linearization under the framework Immersion and Invariance (I&I) tool. As illustrated in [9], the globally asymptotic output tracking of a time-varying reference signal for this class of nonlinear systems is usually not possible even locally by any smooth state feedback, because the linearized system has uncontrollable modes whose eigenvalues are on the right half plane. In this paper the practically adaptive output-tracking control problem will be addressed. Compared with the available adaptive results in the literature, the proposed adaptive approach does not necessarily need to satisfy the certainty equivalence principle which relies on Immersion and Invariance (I&I) tool developed in recent years. By using this I&I mechanism and a modified adding a power integrator approach, we construct an alternative adaptive state feedback controller that ensures all signals of closed-loop systems are globally bounded while practically keeping the output tracking error to a arbitrary small neighborhood of the origin simultaneously. It is interesting to note that, viewed from a Lyapunov perspective, the proposed method provides a procedure to add cross terms between the parameter estimates and the system states in every design step. Finally, an example illustrates the feasibility of the proposed adaptive method.