Kirsten Morris

Survey paper: Transfer functions of distributed parameter systems: A tutorial

In this tutorial article the rich variety of transfer functions for systems described by partial-differential equations is illustrated by means of several examples under various boundary conditions. An important feature is the strong influence of the choice of boundary conditions on the dynamics and on system theoretic properties such as pole and zero locations, properness, relative degree and minimum phase. It is sometimes possible to design a controller using the irrational transfer function, and several such techniques are outlined. More often, the irrational transfer function is approximated by a rational one for the purpose of controller design. Various approximation techniques and their underlying theory are briefly discussed.

Passivity-based stability and control of hysteresis in smart actuators

The past decade has seen an increase in the use of smart materials in actuator design, notably for inclusion in active structures such as noise-reducing paneling or vibration-controlled buildings. Materials such as shape memory alloys (SMAs), piezoceramics, magnetostrictives and others all offer exciting new actuation possibilities. However, all of these materials present an interesting control challenge due to their nonlinear hysteretic behavior in some regimes. We look at the energy properties of the Preisach hysteresis model, widely regarded as the most general hysteresis model available for the representation of classes of hysteretic systems. We consider the ideas of energy storage and minimum energy states of the Preisach model, and derive a passivity property of the model. Passivity is useful in controller design, and experimental results are included showing control of a differential shape memory alloy actuator using a passivity-based rate controller.

Preisach model identification of a two-wire SMA actuator

In recent years, the Preisach hysteresis model (1935) has emerged as the model of choice for the behaviour of many smart materials, such as shape memory alloys (SMA). This research treats the identification of Preisach models for a differential SMA actuator. The traditional identification technique is applied, and several models are derived for the actuator. It is seen that the classical Preisach model is able to model its behaviour. However, the results raise questions concerning the robustness of the traditional identification technique, as well as the conservative nature of current passivity-based control results for the Preisach model.

Dynamics of an Inverted Pendulum with Delayed Feedback Control

We consider an experimental system consisting of a pendulum, which is free to rotate 360 degrees, attached to a cart. The cart can move in one dimension. We describe a model for this system and use it to design a feedback control law that stabilizes the pendulum in the upright position. We then introduce a time delay into the feedback and prove that for values of the delay below a critical delay, the system remains stable. Using a center manifold reduction, we show that the system undergoes a supercritical Hopf bifurcation at the critical delay. Both the critical value of the delay and the stability of the limit cycle are verified experimentally. Our experimental data is illustrated with plots and videos.

Linear-Quadratic Optimal Actuator Location

In control of vibrations, diffusion and many other problems governed by partial differential equations, there is freedom in the choice of actuator location. The actuator location should be chosen to optimize performance objectives. In this paper, we consider linear quadratic performance. Two types of cost are considered; the choice depends on whether the response to the worst initial condition is to be minimized; or whether the initial condition is regarded as random. In practice, approximations are used in controller design and thus in selection of the actuator locations. The optimal cost and location of the approximating sequence should converge to the exact optimal cost and location. In this work conditions for this convergence are given in the case of linear quadratic control. Examples are provided to illustrate that convergence may fail when these conditions are not satisfied.

Well-Posedness of Boundary Control Systems

Continuity of the input/output map for boundary control systems is shown through the system transfer function. Our approach transforms the question of continuity of the input/output map of a boundary control system into boundedness of the solution to a related elliptic problem. This is shown for a class of boundary control systems with Neumann control or Robin boundary control.

Friction and the Inverted Pendulum Stabilization Problem

We consider an experimental system consisting of a pendulum, which is free to rotate 360 deg, attached to a cart. The cart can move in one dimension. We study the effect of friction on the design and performance of a feedback controller, a linear quadratic regulator, that aims to stabilize the pendulum in the upright position. We show that a controller designed using a simple viscous friction model has poor performance-small amplitude oscillations occur when the controller is implemented. We consider various models for stick slip friction between the cart and the track and measure the friction parameters experimentally. We give strong evidence that stick slip friction is the source of the small amplitude oscillations. A controller designed using a stick slip friction model stabilizes the system, and the small amplitude oscillations are eliminated.

An algorithm for LQ optimal actuator location

The locations of the control hardware are typically a design variable in controller design for distributed parameter systems. In order to obtain the most efficient control system, the locations of control hardware as well as the feedback gain should be optimized. These optimization problems are generally non-convex. In addition, the models for these systems typically have a large number of degrees of freedom. Consequently, existing optimization schemes for optimal actuator placement may be inaccurate or computationally impractical. In this paper, the feedback control is chosen to be an optimal linear quadratic regulator. The optimal actuator location problem is reformulated as a convex optimization problem. A subgradient-based optimization scheme which leads to the global solution of the problem is used to optimize actuator locations. The optimization algorithm is applied to optimize the placement of piezoelectric actuators in vibration control of flexible structures. This method is compared with a genetic algorithm, and is observed to be faster and more accurate. Experiments are performed to verify the efficacy of optimal actuator placement.

H ∞ -Optimal Actuator Location

In many control applications, there is freedom to place the actuators. The actuator location should be chosen to optimize certain performance objectives. In this paper, H∞-performance with state-feedback is considered. That is, both the controller and the actuator locations are chosen to minimize the effect of disturbances on the output. A framework for calculating H∞-optimal actuator locations is developed. Conditions for well-posedness of the H∞-optimal actuator location problem are presented. Many optimal actuator problems involve systems modelled by partial differential equations and conditions under which approximations yield reliable results are given. A derivative-free optimization algorithm to calculate H∞-optimal actuator locations is described. The results are illustrated using several examples.

Review and Comparison of Hysteresis Models for Magnetostrictive Materials

The modeling of magnetization in magnetostrictive materials is studied in this article. Magnetostrictive materials elongate in the presence of a magnetic field, and can be useful as actuators. These materials are highly nonlinear, and hence, difficult to control. Accurate models are important to the development of stabilizing controllers with good performance. Here, Terfenol-D, a commonly used magnetostrictive material, is studied. A setup is designed to measure magnetic flux density and stress applied to a Terfenol-D sample. Displacement, electrical current sent to a magnet generating the requested magnetic field, and temperature at different locations are measured. Using experimental data, the Preisach, homogenized energy, and Jiles—Atherton models are evaluated. For each model, the parameters are identified for Terfenol-D. The ease of use and accuracy of these models in the prediction of Terfenol-D behavior are compared.