A direct adaptive state-feedback controller is proposed for highly nonlinear systems. We consider uncertain or ill-defined nonaffine nonlinear systems and employ a neural network (NN) with flexible structure, i.e., an online variation of the number of neurons. The NN approximates and adaptively cancels an unknown plant nonlinearity. A control law and adaptive laws for the weights in the hidden layer and output layer of the NN are established so that the whole closed-loop system is stable in the sense of Lyapunov. Moreover, the tracking error is guaranteed to be uniformly asymptotically stable (UAS) rather than uniformly ultimately bounded (UUB) with the aid of an additional robustifying control term. The proposed control algorithm is relatively simple and requires no restrictive conditions on the design constants for the stability. The efficiency of the proposed scheme is shown through the simulation of a simple nonaffine nonlinear system.

In this brief, a new adaptive neurocontrol algorithm for a single-input-single-output (SISO) strict-feedback nonlinear system is proposed. Most of the previous adaptive neural control algorithms for strict-feedback nonlinear systems were based on the backstepping scheme, which makes the control law and stability analysis very complicated. The main contribution of the proposed method is that it demonstrates that the state-feedback control of the strict-feedback system can be viewed as the output-feedback control problem of the system in the normal form. As a result, the proposed control algorithm is considerably simpler than the previous ones based on backstepping. Depending heavily on the universal approximation property of the neural network (NN), only one NN is employed to approximate the lumped uncertain system nonlinearity. The Lyapunov stability of the NN weights and filtered tracking error is guaranteed in the semiglobal sense.

A novel adaptive neural output-feedback controller for SISO nonaffine pure-feedback nonlinear systems is proposed. The majority of the previously described adaptive neural controllers for pure-feedback nonlinear systems were based on the dynamic surface control (DSC) or backstepping schemes. This makes the control law as well as the stability analysis highly lengthy and complicated. Moreover, there has been very limited research till date on the output-feedback neural controller for this class of the systems. The proposed controller evades adopting adaptive backstepping or DSC scheme through reformulating the original system into the Brunovsky form, which considerably simplifies the control law. Combining a high-order sliding mode observer and single radial-basis function network with universal approximation property, it is shown that the controller guarantees closed-loop system stability in the Lyapunov sense.

A novel output-feedback controller for uncertain SISO nonaffine nonlinear systems is proposed using high-order sliding mode (HOSM) observer which is a robust exact finite-time convergent differentiator. The proposed controller utilizes (n+1)th-order HOSM observer to cancel the uncertainty and disturbance where $n$ is the relative degree of the controlled system. As a result, the control law has an extremely simple form of linear in the differentiator states. It is required no separate sliding-mode controller or universal approximators such as neural networks or fuzzy logic systems that are adaptively tuned online. The proposed controller guarantees finite time (FT) stability of the output tracking error.

A novel differentiator-based approximation-free output-feedback controller for uncertain nonautonomous nonlinear pure-feedback systems is proposed. Using high-order sliding mode observer, which is a finite-time exact differentiator, the time-derivatives of the signal generated using tracking error and filtered input are directly estimated. As a result, the proposed non-backstepping control law and stability analysis are drastically simple. The tracking error vector is guaranteed to be exponentially stable in finite time regardless of the nonautonomous property in the considered system. It does not require neural networks or fuzzy logic systems, which are typically adopted to capture unstructured uncertainties intrinsic in the controlled system. As far as the authors know, there are no research results on the output-feedback controller for the uncertain nonautonomous pure-feedback nonlinear systems. The results of the simulation show clearly the performance and compactness of the control scheme proposed.