The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Piezo-actuated stages have become more and more promising in nanopositioning applications due to the excellent advantages of the fast response time, large mechanical force, and extremely fine resolution. Modeling and control are critical to achieve objectives for high-precision motion. However, piezo-actuated stages themselves suffer from the inherent drawbacks produced by the inherent creep and hysteresis nonlinearities and vibration caused by the lightly damped resonant dynamics, which make modeling and control of such systems challenging. To address these challenges, various techniques have been reported in the literature. This paper surveys and discusses the progresses of different modeling and control approaches for piezo-actuated nanopositioning stages and highlights new opportunities for the extended studies.

Modeling and Control of Piezo-Actuated Nanopositioning Stages: A Survey

A survey on hysteresis modeling, identification and control

Smart actuators employed in micropositioning are known to exhibit strong hysteresis nonlinearities, which may be asymmetric and could adversely affect the positioning accuracy. In this paper, the analytical inverse of a generalized Prandtl-Ishlinskii model is formulated to compensate for hysteresis nonlinearities of smart actuators. The generalized model was modified to ensure its continuity, and its validity in characterizing different hysteresis properties is briefly demonstrated by comparing the model responses with the measured data for the magnetostrictive, shape memory alloys (SMA), and piezo micropositioning actuators. Since the proposed generalized model is a mere extension of the analytically invertible classical Prandtl-Ishlinskii model, an inverse of the generalized model is formulated using the inverse of the classical model together with those of the envelope functions of the generalized play operator. The effectiveness of the inverse of the generalized model in compensating for the symmetric and asymmetric saturated hysteresis effects is subsequently investigated through simulations for a magnetostrictive and a SMA actuators, and through preliminary experiments performed on a piezo micropositioning stage. The simulation results suggest that the inverse of the generalized Prandtl-Ishlinskii model can be conveniently applied as a feedforward compensator to effectively mitigate the effects of the asymmetric and saturated hysteresis in magnetostrictive and SMA actuators. The experimental results further revealed that the proposed generalized analytical inverse model can be conveniently implemented as a real-time feedforward compensator to compensate for hysteresis nonlinearities of a piezo micropositioining stage.

An Analytical Generalized Prandtl–Ishlinskii Model Inversion for Hysteresis Compensation in Micropositioning Control

This paper proposes an improved sliding mode control with perturbation estimation (SMCPE) featuring a PID-type sliding surface and adaptive gains for the motion tracking control of a micromanipulator system with piezoelectric actuation One advantage of the proposed controller lies in that its implementation only requires the online estimation of perturbation and control gains without acquiring the knowledge of bounds on system uncertainties The dynamic model of the system with Bouc-Wen hysteresis is established and identified through particle swarm optimization (PSO) approach, and the controller is designed based on Lyapunov stability analysis A high-gain observer is adopted to estimate the full state from the only measurable position information Experimental results demonstrate that the performance of proposed controller is superior to that of conventional SMCPE in both set-point regulation and motion tracking control Moreover, a submicron accuracy tracking and contouring is achieved by the micromanipulator with dominant hysteresis compensated for a low magnitude level, which validates the feasibility of the proposed controller in the field of micro/nano scale manipulation as well

Adaptive Sliding Mode Control With Perturbation Estimation and PID Sliding Surface for Motion Tracking of a Piezo-Driven Micromanipulator

This paper presents the classical Preisach hysteresis modeling and tracking control of a curved pre-stressed piezoceramic patch actuator system with severe hysteresis The actuator is also flexible with very small inherent damping It has potential applications in active antennas A series of tests are conducted to study the hysteresis properties of the piezoceramic actuator system The numerical expressions of the classical Preisach model for different input variations are presented The classical Preisach model is applied to simulate the static hysteresis behavior of the system Higher order hysteresis reversal curves predicted by the classical Preisach model are verified experimentally The good agreement found between the measured and predicted curves showed that the classical Preisach model is an effective mean for modeling the hysteresis of the piezoceramic actuator system Subsequently, the inverse classical Preisach model is established and applied to cancel the hysteresis the piezoceramic actuator system for the real-time microposition tracking control In order to improve the control accuracy and to increase damping of the actuator system, a cascaded PD/lead-lag feedback controller is designed with consideration of the dynamics of the actuator In the experiments, two cases are considered, control with major loop hysteresis compensation, and control with minor loop hysteresis compensation Experimental results show that RMS tracking errors are reduced by 50% to 70% if the hysteresis compensation is added in the feedforward path in both cases Therefore, hysteresis compensation with the feedback controller greatly improves the tracking control accuracy of the piezoceramic actuator

Tracking control of a piezoceramic actuator with hysteresis compensation using inverse Preisach model

A partial differential equation for the lateral motion of a web conveyance system is derived by modeling the web as a viscoelastic beam under axial tension. This model treats the web position between rollers as a function of both time and space, assumes that there is no slip between the web material and the rollers, and incorporates the web material's viscoelastic damping property. A finite-difference approximation of the model is used to simulate a typical two-span web system. The finite-difference approximation is validated by comparing its frequency responses with those of an analytical frequency domain model. The analytical frequency domain model is used to design feedback compensation strategies that make the two-span web system less sensitive to upstream disturbances. The results show that, using a transverse vibration model incorporating viscoelasticity to design even simple classical controllers, it is possible to make the web system less sensitive to upstream disturbances at the sensor location.

Control of lateral motion in moving webs

A balancing-based algorithm for obtaining nonlinear reduced-order models of nonlinear systems is presented. This is done by considering the input/output behavior of the system around its equilibrium point. The algorithm is shown to be particularly useful in the real-time simulation of stiff nonlinear systems. >

On the computation of reduced order models of nonlinear systems using balancing technique

A model reduction method based on the least squares algorithm is derived. This method calculates a low order autoregressive moving average (ARMA) predictor equation from a high order ARMA equation. The low order ARMA equation minimizes the sum of the squares of the prediction errors when the input is white noise. This is almost equivalent to minimizing the sum of the squares of the error in the impulse response function. Transfer function models can also be used and a steady-state gain constraint can be incorporated into the procedure. The merits of this method of model order reduction are shown with three examples. In these examples, the proposed method produced results that compared favorably with highly regarded existing model reduction techniques which require many more computations.

Least squares model reduction

A simple and powerful method for unstable model reduction has been developed in which the approach is based on the fact that translation transformations in the s-plane preserve the input-output properties of a system. Using translation transformations in the frequency domain it is possible to change the stability of the system without losing input-output information. Although balancing requires that the model be asymptotically stable, it reduces the model depending only on the information of input to state and state to output. The stability requirement comes from the computation of the controllability and observability gramians which are used for characterizing the contribution of the states to the input-output map. It has been shown in this paper that it is possible to use balancing to reduce the models of unstable systems by transforming them into the stable models, reducing the model order, and then transforming the models back. The method has been demonstrated by case studies.

J.A. De Abreu-Garcia

Papers

Tracking control of a piezoceramic actuator with hysteresis compensation using inverse Preisach model

Control of lateral motion in moving webs

On the computation of reduced order models of nonlinear systems using balancing technique

Least squares model reduction

Model reduction of unstable systems