The problem of adaptive stabilization with respect to a set for a class of nonlinear parameterized systems in the presence of external disturbances is considered. A novel adaptive observer-based solution for the case of noisy measurements is proposed. The efficiency of proposed solution is demonstrated via example of swinging a pendulum with unknown parameters.

estimation and control for a class of

We propose a new method for the design of adaptation algorithms that guarantees a certain prescribed level of performance and applicable to systems with nonconvex parameterization. The main idea behind the method is two-fold. First, we augment the tuning error function and design the adaptation scheme in the form of ordinary differential equations. The resulting augmentation is allowed to depend on state derivatives. Second, we find a suitable realization of the designed adaptation scheme in an algebraic-integral form. Due to their explicit dependence on the state of the original system, such adaptation schemes are referred to as adaptive algorithms in finite form, in contrast to (conventional) algorithms in differential form. Sufficient conditions for the existence of finite form realizations are proposed. It is shown that our method to design algorithms in finite form is applicable to a broad class of nonlinear systems including systems with nonconvex parameterization and low-triangular systems.

Abstract: A new framework for adaptive regulation to invariant sets is proposed. Reaching the target dynamics (invariant set) is to be ensured by state feedback while adaptation to parametric uncertainties is provided by additional adaptation algorithm. We show that for a sufficiently large class of nonlinear systems it is possible to adaptively steer the system trajectories to the desired non-equilibrium state without requiring knowledge or existence of a specific strict Lyapunov function.