טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentBeilkin Lea
SubjectFractional Order Control of Continuous Flexible
Structures
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Yoram Halevi
Full Thesis textFull thesis text - English Version


Abstract

Continuous flexible structures are employed in various advanced engineering applications, such as unmanned space or aerial vehicles, deep well drilling and medical instruments. Although flexibility has a major advantage, as it yields light weight structures, it has adverse effects on the dynamic response. Even slow tracking maneuvers or disturbance signals excite undesired vibrations. Since passive rigidization results in heavier structures, active rigidization via control algorithms is preferred. 
We consider finite length structures that have no resistance to bending, which includes taut strings and membranes in transverse vibration, rods in axial or torsional vibration, and more. Systems of this class are governed by different forms of the wave equation, the full form of which we denote by the generalized wave equation (GWE). The GWE is a second order hyperbolic partial differential equation (PDE). Investigating the free response of the GWE, we devote special attention to boundary conditions (BC) of viscous damping. The literature falls short from providing closed form solutions for this case. We derive the solution in a traveling wave form, extending the classical d'Alembert formula to finite length systems of this kind. We also obtain the full and explicit modal series solution by deriving a new orthogonality condition for the damped eigenfunctions. Since PDEs are not a standard starting point for common control methods, the prevailing approach is finite dimension approximation. However, the system physical characteristics, such as propagating waves and the associated delays are then lost and cannot be utilized for controllers design. 
The main result of this research is developing a control algorithm for the GWE, based on exact, infinite dimensional transfer function (TF) models. The TFs include exponents that have different square root arguments of the complex variable, and are thus of fractional order. We find that the time domain equivalents of such exponents are different kinds of Bessel functions that represent non-pure delays. We show how they exhibit the wave propagation in the system, including wave shape evolution during motion, the effect of reflection from the boundaries, etc. The control algorithm stems from this model and consists of three building blocks.  The first is a controller that eliminates wave reflections from the system. Mathematically, it yields a tracking TF with a single fractional order exponent at the numerator. An additional controller cancels that exponent and places the poles in any desired locations. The latter is a fractional order delay compensator, which so far existed only for integer order systems. Finally, a pre-filter is designed to produce a rational tracking TF with a pure delay, independent of the original system fractional order and the physical parameters. Since the closed loop system has no residual vibration, this method is denoted by the absolute vibration suppression (AVS) control. The original AVS method was developed in a previous research for the classical wave equation, which is a particular, non-fractional case of the GWE. Therefore, the current work may be also regarded as the extension of the AVS approach to the fractional order realm.