Hassan K. Khalil

Output feedback stabilization of fully linearizable systems

An observer-based controller is designed to stabilize a fully linearizable nonlinear system. The system is assumed to be left-invertible and minimum-phase. The controller is robust to uncertainties in modelling the nonlinearities of the system. The design of the controller and the stability analysis employs the techniques of singular perturbations. A new ‘Tikhonov-like’ theorem is presented and used to analyse the system when the control is globally bounded.

Adaptive control of a class of nonlinear discrete-time systems using neural networks

Layered neural networks are used in a nonlinear self-tuning adaptive control problem. The plant is an unknown feedback-linearizable discrete-time system, represented by an input-output model. To derive the linearizing-stabilizing feedback control, a (possibly nonminimal) state-space model of the plant is obtained. This model is used to define the zero dynamics, which are assumed to be stable, i.e., the system is assumed to be minimum phase. A linearizing feedback control is derived in terms of some unknown nonlinear functions. A layered neural network is used to model the unknown system and generate the feedback control. Based on the error between the plant output and the model output, the weights of the neural network are updated. A local convergence result is given. The result says that, for any bounded initial conditions of the plant, if the neural network model contains enough number of nonlinear hidden neurons and if the initial guess of the network weights is sufficiently close to the correct weights, then the tracking error between the plant output and the reference command will converge to a bounded ball, whose size is determined by a dead-zone nonlinearity. Computer simulations verify the theoretical result. >

Robust servomechanism output feedback controllers for feedback linearizable systems

Abstract We consider a single-input-single-output nonlinear system which has a uniform relative degree equal to the dimension of the state vector. The system can be transformed into a normal form with no zero dynamics. We allow the systems equation to depend on bounded uncertain parameters which do not change the relative degree. Disturbances are assumed to satisfy a strict-feedback condition which allows us to use a time-varying, disturbance-dependent transformation to transform the system into an error space where disturbances and uncertainties satisfy the matching condition. The nonlinear functions in the error space are used to choose an internal model which is augmented with the system, and a robust state feedback control is designed to drive the error to a positively invariant set that contains the origin. We then show the existence of a zero-error manifold inside this set which attracts all trajectories inside the set. To implement this control using output feedback, we saturate the state feedback control outside a compact set of interest and estimate the state using a high-gain observer. The output feedback controller recovers the robustness and asymptotic tracking properties of the state feedback controller.

High-gain observers in the presence of measurement noise: A switched-gain approach

This paper considers output feedback control using high-gain observers in the presence of measurement noise for a class of nonlinear systems. We study stability in the presence of measurement noise and illustrate the tradeoff when selecting the observer gain between state reconstruction speed and robustness to model uncertainty on the one hand versus amplification of noise on the other. Based on this tradeoff we propose a high-gain observer that switches between two gain values. This scheme is able to quickly recover the system states during large estimation error and reduce the effect of measurement noise in a neighborhood of the origin of the estimation error. We argue boundedness and ultimate boundedness of the closed-loop system under switched-gain output feedback.

Separation results for the stabilization of nonlinear systems using different high-gain observer designs

High-gain observers have been used in the design of output feedback controllers due to their ability to robustly estimate the unmeasured states while asymptotically attenuating disturbances. The available techniques for the design of high-gain observers can be classified into three groups: pole-placement algorithms, Riccati equation-based algorithms, and Lyapunov equation-based algorithms. In [1], we presented separation results for globally bounded stabilizing state feedback controllers when the high-gain observer is designed using pole-placement so as to create a closed-loop system with two-time-scale structure. In this paper, we show that the separation results of [1] hold for the other observer designs.

Performance Recovery of Feedback-Linearization-Based Designs

We consider a tracking problem for a partially feedback linearizable nonlinear system with stable zero dynamics. The system is uncertain and only the output is measured. We use an extended high-gain observer of dimension n+1, where n is the relative degree. The observer estimates n derivatives of the tracking error, of which the first (n-1) derivatives are states of the plant in the normal form and the nth derivative estimates the perturbation due to model uncertainty and disturbance. The controller cancels the perturbation estimate and implements a feedback control law, designed for the nominal linear model that would have been obtained by feedback linearization had all the nonlinearities been known and the signals been available. We prove that the closed-loop system under the observer-based controller recovers the performance of the nominal linear model as the observer gain becomes sufficiently high. Moreover, we prove that the controller has an integral action property in that it ensures regulation of the tracking error to zero in the presence of constant nonvanishing perturbation.

Asymptotic regulation of minimum phase nonlinear systems using output feedback

We consider a single-input/single-output (SISO) nonlinear system which has a well-defined normal form with asymptotically stable zero dynamics. We allow the systems equation to depend on constant uncertain parameters and disturbance inputs which do not change the relative degree. Our goal is to design an output feedback controller which regulates the output to a constant reference. The integral of the regulation error is augmented to the system equation, and a robust output feedback controller is designed to bring the state of the closed-loop system to a positively invariant set. Once inside this set, the trajectories approach a unique equilibrium point at which the regulation error is zero. We give regional as well as semiglobal results.

Nonlinear output-feedback tracking using high-gain observer and variable structure control

Abstract We consider a single-input-single-output nonlinear system which can be represented by an input-output model. The system, which can be transformed into the normal form, is required to be minimum phase. The model contains unknown bounded disturbances. We assume that the reference signal and its derivatives are bounded. A high-gain observer is used to estimate derivatives of the tracking error while rejecting the effect of the disturbances. We design a globally bounded output-feedback variable structure controller that ensures tracking of the reference signal in the presence of unknown time-varying disturbances and modeling errors. We give regional as well as semiglobal results. We do not require exponential stability of the zero dynamics nor global growth conditions.

Discrete-time implementation of high-gain observers for numerical differentiation

High-gain observers have been used in non-linear control to estimate derivatives of the output. In this paper, we study discrete-time implementation of high-gain observers and their use as numerical differentiators, in noise-free as well as noisy measurements. We show that discretization using the bilinear transformation method gives better results than other discretization methods. We also show that many of the available numerical differentiators are special cases of the bilinear discrete-time equivalents of full-order or reduced-order high-gain observers.

Performance recovery under output feedback sampled-data stabilization of a class of nonlinear systems

This paper studies sampled-data output feedback control of a class of nonlinear systems. It is shown that the performance of a stabilizing continuous-time state feedback controller can be recovered by a sampled-data output feedback controller when the sampling period is sufficiently small. The output feedback controller uses a deadbeat discrete-time observer to estimate the unmeasured states. Two schemes are proposed to overcome large initial transients when the controller is switched on.