Concurrent Feed-forward/feed-back Design for Flexible Structures

This paper discusses the concurrent design of a feedforward time-delay filter and a linear state feed-back controller. The optimization is carried out using a quadratic cost functional which weights both, disturbance rejection and control tracking performance. A minimax optimization scheme is employed to achieve robustness with respect to parameter deviations. Analytical gradients are provided for the cost function. This information can be exploited to expedite convergence of the optimization procedure. The proposed technique is illustrated on the Floating Oscillator benchmark problem. INTRODUCTION Vibration control has been subject to intense research, aiming at applications as diverse as maneuvering of large space structures, flexible arm robots, computer disk drives, and cranes. In most of these applications, the goal is to minimize the maneuver time, while ensuring quiescent final states. During this exhaustive research, multifarious constraints have been added to the problem formulation, such as limits on the fuel consumed, robustness to modelling uncertainties, and maximum permitted deformation. Traditionally, there are two approaches to the control of flexible structures, either avoiding excitations by shaping the control input appropriately or damping vibration by the application of a suitable feed-back control law. As separate entities, both of these approaches have been exhaustively investigated and a variety of solutions have been developed. These approaches will be briefly reviewed. One specific method of forming the control input which does not excite vibrations is by the use of timedelay filters. This technique was initially developed by Smith in 1957 and was termed “Posicast”. It is based on dividing a step input into a number of spaced excitations, which are chosen such that there is no remanent vibrational energy at the end of the maneuver. In this initial work, robustness issues had not been considered. In 1990, Singer and Seering suggested a practice to design input shapers addressing the sensitivity issues. All these techniques are based in the time domain. Singh and Vadali pointed out that a time delay filter which cancels the underdamped poles of a system results in the same control profile as the input shaping controller. Thus, input prefilters can also be designed using frequency domain techniques and properties. Recently, other techniques have been developed to address the issue of desensitizing the controller to modelling errors. Singh proposed a minimax formulation to desensitize the input-preshaping controller with respect to modelling errors. Muenchhof and Singh considered limitations on the time rate of change of the control input for both single-input and multi-input systems. They illustrate the reduction of energy in the power spectrum at the higher frequencies, which is desirable for flexible structures. The tutorial by Singh and Singhose summarizes the results obained in this area of research. A variety of design methods for feed-back control systems have been developed for both linear and nonlinear systems. For linear systems, the linear state feed-back is the prevailing control setup due to its ease of application and the multitude of applicable design methods such as pole placement or the linear quadratic regulator setting. The latter technique, abbreviated LQR, allows for the design of linear state feed-back controllers which minimize a quadratic cost function weighting both deviation of the states from the desired final value and the control effort expended. Many methods have been proposed which extend the basic idea behind the linear quadratic regulator. Among them are methods to robustify the regulator with respect to parametric uncertainties of the plant and