| The tracking problem in systems with hysteresis has become an important topic of research in the past two decades, due in large part to advances in smart material actuators. In particular, applications like Scanning Probe Microscopy require high performance from hysteretic smart material actuators. Servocompensators, or internal model controllers, have been used successfully in many varieties of tracking problems for both linear and nonlinear systems; therefore, their application to systems with hysteresis is considered in this dissertation.;The use of Multi-Harmonic Servocompensators (MHSC) is first proposed to simultaneously compensate for hysteresis and enable high-bandwidth tracking in systems with hysteresis, such as nanopositioners. With the model represented by linear dynamics preceded with a Prandtl-Ishlinskii hysteresis operator, the stability and periodicity of the closed-loop system's solutions are established when hysteresis inversion is included in the controller. Experiments on a commercial nanopositioner show that, with the proposed method, tracking can be achieved for a 200 Hz reference signal with 0.52% mean error and 1.5% peak error over a travel range of 40 μm. Additionally, the proposed method is shown at high frequencies to be superior to Iterative Learning Control (ILC), a common technique in nanopositioning control.;The theoretical and practical weaknesses of the proposed approach are then addressed. First, the design of a novel adaptive servocompensator specialized to systems with hysteresis is presented, based on frequency estimation coupled with slow adaptation, and the stability in cases with one, two, or n unknown frequencies are established. Next, a condition in the form of a Linear Matrix Inequality is presented proving the stability of the proposed MHSC when hysteresis inversion is not used. It is then experimentally demonstrated that removing hysteresis inversion further reduces the tracking error achievable by the MHSC. Finally, the properties of self-excited limit cycles are studied for an integral-controlled system containing a play operator. A Newton-Raphson algorithm is formulated to calculate the limit cycles, and linear relationships between the amplitude and period of these limit cycles and system parameters are obtained. |
