Tracking with prescribed transient performance for hysteretic systems

Tracking of reference signals (assumed bounded with essentially bounded derivative) is considered for a class of single-input, single-output, nonlinear systems, described by a functional differential equation with a hysteresis nonlinearity in the input channel. The first control objective is tracking, by the output, with prescribed accuracy: determine a feedback strategy which ensures that, for every reference signal and every system of the underlying class, the tracking error ultimately satisfies the prescribed accuracy requirements. The second objective is guaranteed output transient performance: the graph of the tracking error should be contained in a prescribed set (performance funnel). Under a weak sector boundedness assumption on the hysteresis operator, both objectives are achieved by a memoryless feedback which is universal for the underlying class of systems

Controller Gain Optimization for Position Control of an SMA Wire

There has been an increasing interest in the field of `smart structures' and `smart materials'. In constructing smart structures, a class of materials called smart materials are often used as sensors and actuators. An example of a smart material is shape memory alloy (SMA). A common actuator configuration uses an SMA wire with a constant load. The non-linear input-output behaviour of SMAs, known as hysteresis, made them difficult to model and control. The research in this thesis examines the effect of PID-controller gain optimization on SMA wire control at different frequencies of operation. A constant-load SMA wire actuator with a PID-controller is used in the study. Heat is applied to the wire using an input electric current. The system is cooled through convection with the surrounding area. The lack of active cooling prevents the system from operating at high frequencies. Three different cost functions are proposed for various applications. The Preisach model is chosen to model the hysteretic behaviour of the SMA wire contraction. Varying material properties such as electrical resistance and heat capacities are modelled to give a more accurate representation of the system's physical behaviour. Simulations show that by optimizing the controller gain values, the bandwidth of the system is improved. An interesting observation is made in the heating cycle of the SMA wire. In order to achieve faster cooling, overshoot is observed at low frequencies. This is a result of the system hysteresis. The system hysteresis allows different input signals to achieve the same output value. Since the rate of cooling is proportional to the temperature above ambient, better cooling is achieved by reaching a higher temperature. The error caused by the overshoot is compensated by the better cooling phase, which is not actively controlled

On the adaptive controls of nonlinear systems with different hysteresis model representations

The hysteresis phenomenon occurs in diverse disciplines ranging from physics to biology, from material science to mechanics, and from electronics to economics. When the hysteresis nonlinearity precedes a controlled system, the nonlinearity usually causes the overall closed-loop system to exhibit inaccuracies or oscillations, even leading to instability. Control techniques to mitigate the unwanted effects of hysteresis have been studied for decades and have recently once again attracted significant attention. In this thesis, several adaptive control strategies are developed for systems with different hysteresis model representations to guarantee the basic stability requirement of the closed-loop systems and to track a desired trajectory with a certain precision. These proposed strategies to mitigate the effects of hysteresis are as follows: i). With the classical Duhem model, an observer-based adaptive control scheme for a piezoelectric actuator system is proposed. Due to the unavailability of the hysteresis output, an observer-based adaptive controller incorporating a pre-inversion neural network compensator is developed for the purpose of mitigating the hysteretic effects; ii). With the Prandtl-Ishlinskii model, an adaptive tracking control approach is developed for a class of nonlinear systems in p-normal form by using the technique of adding a power integrator to address the challenge of how to fuse this hysteresis model with the control techniques to mitigate hysteresis, without necessarily constructing a hysteresis inverse; iii). With a newly proposed hysteresis model using play-like operators, two control strategies are proposed for a class of nonlinear systems: one with sliding mode control and the other with backstepping technique

PID Control of Systems with Hysteresis

Hysteresis is exhibited by many physical systems. Smart materials such as piezoelectrics, magnetostrictives and shape memory alloys possess useful properties, especially in the field of micropositioning, but the control of these systems is difficult due to the presence of hysteresis. An accurate model is required to predict the behaviour of these systems so that they can be controlled. Several hysteresis models including the backlash, elastic-plastic and Preisach operators are discussed in detail. Several other models are mentioned. Other control methods for this problem are discussed in the form of a literature review. The focus of this thesis is on the PID control of hysteretic systems. In particular, two systems experiencing hysteresis in their controllers are examined. The hysteresis in each system is described by different sets of assumptions. These assumptions are compared and found to be very similar. In the first system, a PI controller is used to track a reference signal. In the second, a PID controller is used to control a second-order system. The stability and tracking of both systems are discussed. An extension is made to the first system to include the dynamics of a first-order system. The results of the second system are verified to hold for a general first-order system. Simulations were performed with the extension to a first-order system using different hysteresis models

Inversion-based feedforward-feedback control: theory and implementation to high-speed atomic force microscope imaging

In this dissertation, a suite of inversion-based feedforward-feedback control techniques are developed and applied to achieve high speed AFM imaging. In the last decade, great efforts have been made in developing the inversion-based feedforward control as an effective approach for precision output tracking. Such efforts are facilitated by the fruitful results obtained in the stable-inversion theory, including, mainly, the bounded inverse of nonminimum-phase systems, the preview-based inversion method that quantified the effect of the future desired trajectory on the inverse input, the consideration of the model uncertainties in the system inverse, and the integration of inversion with feedback and iterative control. However, challenges still exist in those inversion-based approaches. For example, although it has been shown that the inversion-based iterative control (IIC) technique can effectively compensate for the vibrational dynamics during the output tracking in the repetitive applications, however, compensating for both the hysteresis effect and the dynamics effect simultaneously using the IIC approach has not been established yet. Moreover, the current design of the inversion-based feedforward feedback two-degree-of-freedom (2DOF) controller is ad-hoc, and the minimization of the model uncertainty effects on the feedforward control has not been addressed. Furthermore, although it is possible to combine system inversion with both iterative learning and feedback control in the so-called current cycle feedback iterative learning control (CCF-ILC) approach, the current controller design is limited to be casual and the use of such CCF-ILC approach for rejecting slowly varying periodic disturbance has not been explored. These challenges, as magnified in applications such as high-speed AFM imaging, motivate the research of this dissertation. Particularly, it is shown that the IIC approach can effectively compensate for both the hysteresis and vibrational dynamics effects of smart actuators. The convergence of the IIC algorithm is investigated by capturing the input-output behavior of piezo actuators with a cascade model consisting of a rate-independent hysteresis at the input followed by the dynamics part of the system. The size of the hysteresis and the vibrational dynamics variations that can be compensated for (by using the IIC method) has been quantified. Secondly, a novel robust-inversion has been developed for single-input-single-output (SISO) LTI systems, which minimized the dynamics uncertainty effect and obtained a guaranteed tracking performance for bounded dynamics uncertainties. Based on the robust-inversion approach, a systematic design of inversion-based two-degree-of-freedom (2DOF)-control was developed. Finally, the robust inversion- based current cycle feedback iterative learning control approach was developed for the rejection of slow varying periodic disturbances. The proposed CCF-ILC controller design utilizes the recently-developed robust-inversion technique to minimize the model uncertainty effect on the feedforward control, as well as to remove the causality constraints in other CCFILC approaches. It is shown that the iterative law converges, and attains a bounded tracking error upon noise and disturbances. In this dissertation, these techniques have been successfully implemented to achieve high-speed AFM imaging of large-size samples. Specifically, it is shown that precision positioning of the probe in the AFM lateral (x-y) scanning can be successfully achieved by using the inversion-based iterative-control (IIC) techniques and robust-inversion based 2DOF control design approach. The AFM imaging speed as well as the sample estimation can be substantially improved by using the CCF-ILC approach for the precision positioning of the probe in the vertical direction

Generalized Prandtl-Ishlinskii hysteresis model and its analytical inverse for compensation of hysteresis in smart actuators

Smart actuators such as piezoceramics, magnetostrictive and shape memory alloy actuators, invariably, exhibit hysteresis, which has been associated with oscillations in the open-loop system's responses, and poor tracking performance and potential instabilities of the close-loop system. A number of phenomological operator-based hysteresis models such as the Preisach model, Krasnosel'skii-Pokrovskii model and Prandtl-Ishlinskii model, have been formulated to describe the hysteresis nonlinearities and to seek compensation of the hysteresis effects. Among these, the Prandtl-Ishlinskii model offers greater flexibility and unique property that its inverse can be attained analytically. The Prandtl-Ishlinskii model, however, is limited to rate-independent and symmetric hysteresis nonlinearities. In this dissertation research, the unique flexibility of the Prandtl-Ishlinskii model is explored for describing the symmetric as well as nonlinear hysteresis and output saturation properties of smart actuators, and for deriving an analytical inverse for effective compensation. A generalized play operator with dissimilar envelope functions is proposed to describe asymmetric hysteresis and output saturation nonlinearities of different smart actuators, when applied in conjunction with the classical Prandtl-Ishlinskii model. Dynamic density and dynamic threshold functions of time rate of the input are further proposed and integrated in the classical model to describe rate-dependent symmetric and asymmetric hysteresis properties of smart actuators. A fundamental relationship between the thresholds of the classical and the resulting generalized models is also formulated to facilitate parameters identification. The validity of the resulting generalized Prandtl-Ishlinskii models is demonstrated using the laboratory-measured data for piezoceramic, magnetostrictive and SMA actuators under different inputs over a broad range of frequencies. The results suggest that the proposed generalized models can effectively characterize the rate-dependent as well as rate-independent hysteresis properties of a broad class of smart actuators with output saturation. The properties of the proposed generalized models are subsequently explored to derive its inverse to seek an effective compensator for the asymmetric as well as rate-dependent hysteresis effects. The resulting inverse is applied as a feedforward compensator and simulation results are obtained to demonstrate its effectiveness in compensating the symmetric as well as asymmetric hysteresis of different smart actuators. The effectiveness of the proposed analytical inverse model-based real-time compensator is further demonstrated through its implementation in the laboratory for a piezoceramic actuator. Considering that the generalized Prandtl-Ishlinskii model provides an estimate of the hysteresis properties and the analytical inverse is a hysteresis model, the output of the inverse compensation is expected to yield hysteresis, although of a considerably lower magnitude. The expected compensation error, attributed to possible errors in hysteresis characterization, is analytically derived on the basis of the generalized model and its inverse. The design of a robust controller is presented for a system preceded by the hysteresis effects of an actuator using the proposed error model. The primary purpose is to fuse the analytical inverse compensation error model with an adaptive controller to achieve to enhance tracking precision. The global stability of the chosen control law and the entire closed-loop system is also analytically established. The results demonstrated significantly enhanced tracking performance, when the inverse of the estimated Prandtl-Ishlinskii model is considered in the closed-loop control system