^{1}

^{2}

^{1}

^{2}

We propose a generic spatial domain control scheme for a class of nonlinear rotary systems of variable speeds and subject to spatially periodic disturbances. The nonlinear model of the rotary system in time domain is transformed into one in spatial domain employing a coordinate transformation with respect to angular displacement. Under the circumstances that measurement of the system states is not available, a nonlinear state observer is established for providing the estimated states. A two-degree-of-freedom spatial domain control configuration is then proposed to stabilize the system and improve the tracking performance. The first control module applies adaptive backstepping with projected parametric update and concentrates on robust stabilization of the closed-loop system. The second control module introduces an internal model of the periodic disturbances cascaded with a loop-shaping filter, which not only further reduces the tracking error but also improves parametric adaptation. The overall spatial domain output feedback adaptive control system is robust to model uncertainties and state estimated error and capable of rejecting spatially periodic disturbances under varying system speeds. Stability proof of the overall system is given. A design example with simulation demonstrates the applicability of the proposed design.

Rotary systems play important roles in various industry applications, for example, packaging, printing, assembly, fabrication, semiconductor, robotics, and so forth. Design of control algorithm for a motion system often comes up with nonlinearities and uncertainties. Nonlinearities are either inherent to the system or due to the dynamics of actuators and sensors. Uncertainties are mainly caused by unmodeled dynamics, parametric uncertainty, and disturbances. For dealing with nonlinearities, common techniques, for example, feedback linearization and backstepping, are to utilize feedback to cancel all or part of the nonlinear terms. On the other hand, design techniques for conducting disturbance rejection or attenuation in control systems mostly originate from the internal model principle [

This paper extends the work of Chen and Yang [

Recently, there have been emerging design techniques based on adaptive fuzzy control (AFC), which may cope with nonlinearities and uncertainties with unknown structures [

This paper is organized as follows. Reformulation of a generic nonlinear rotary system with respect to angular displacement will be presented in Section

In this section, we show how a generic NTI model can be transformed into an NPI model by choosing an alternate independent variable (angular displacement instead of time) and defining a new set of states (or coordinates) with respect to the angular displacement. Note that the transformation described here is equivalent to a nonlinear coordinate transformation or a diffeomorphism. The NPI model will be used for the subsequent design and discussion,

Drop the

Equation (

The output disturbance is sufficiently smooth, that is,

The reference command

To apply adaptive backstepping method, we firstly rewrite the derivative of output

In view of designing output feedback backstepping with

With (

With respect to the new set of coordinates (

With respect to the new set of coordinates (

Substituting (

The control configuration for the proposed spatial domain adaptive control system.

Consider the control law of (

Denote by

For realistic simulation, we set up a simulation configuration as shown in Figure

The configuration for numerical simulation.

The output disturbance and the corresponding frequency spectrum.

The tracking command for a variable speed control task.

Comparison of tracking performance.

This paper presents the design of a new spatial domain adaptive control system, which can be applied to rotary systems operating at varying speeds and subject to spatially periodic and band-limited disturbances and structured/unstructured parametric uncertainties. The proposed design integrates two control paradigms, that is, adaptive backstepping and internal model control. The overall output feedback adaptive control system can be shown to be stable and have bounded state estimated error and output tracking error. Feasibility and effectiveness of the proposed design are further justified by a numerical example. Future effort will be dedicated to implementation and verification of the proposed control design to a practical rotary system, for example, a brushless dc-motor-driven control system.