Classical, Optimal-Quadratic and Nonlinear Control Design In application of linear control theory to flexible orbiting systems, three procedures have been commonly used to develop control laws for large flexible structures: (a) decoupling techniques; (b) pole placement; and (c) optimal linear regulator theory. The decoupling technique can be applied in two situations where: (i) the linear state equations are decoupled using state variable feedback; and (ii) the open loop linear equations are first transformed into a decoupled set in the modal coordinates and then control laws are developed independently for each mode. Thus it becomes necessary to transform the control laws as expressed in modal coordinates to the actual control in the original coordinates. In the pole clustering method the overall transient requirements of the system are considered instead of concentrating on the behaviour of the individual coordinates. The linearized equations of motion are recast in the state space form and the feedback control law selected. The linear regulator theory allows one to set, a priori, distinct penalty weighting functions on the control effort as well as the state variables. The feedback control law is selected such that a quadratic performance criterion is minimized. Both the linear regulator problem and the pole clustering method can result in some of the closed loop frequencies being orders of magnitude greater than those of the uncontrolled system. These higher frequencies may also correspond to die frequencies of higher modes not included in the previously truncated model. To account for such effects the order of the original system model will have to be increased in order to avoid the effects of spillover. On the other hand, these methods have the advantage of being applicable even in situations where the number of actuators is less than the number of modes in the mathematical model, in contrast to the the decoupling methods. Bainum et al. have provided considerable insight into the behaviour of complex large space systems by modelling simple systems such as flexible beams and plates in orbit. Many more studies towards control of flexible Spacecraft have been documented in the literature such as the works by Denman and Jeon (eigenvalue relocation), Ih et al. (adaptive control), Meirovitch et al. (perturbation approach). Young (decentralized control) and Williams and Juang (pole/zero cancellation), to name a few. The linear control optimization procedures, based either on the Bellman principle of optimality or the Pontryagin maximum principle, have served as efficient algorithms to develop control tools and strategies in the design of a large class of dynamical systems. However, in real situations, the nonlinear character of the system may be important so as to warrant linearization unadequate. The mathematical theory of bilinear systems and the general nonlinear controllability and observability theory have been investigated via certain aspects of the differential geometric theory such as Lie algebras to yield some understanding of the behaviour of these nonlinear control systems. Dwyer and Batten have proposed an attitude motion controller for a rigid Spacecraft based on the inversion of a nonlinear input-output map. Singh and Bossart[ attempted a linear representation of the nonlinear dynamics of the rigid Space Station using feedback linearization theory. More recently, Karray and Modi extended the application of the feedback linearization technique to flexible systems. The nonlinear control strategy based on this technique has the advantage over controller designs based on linearized dynamics that a linearized model is only approximate, and valid only near operating points. Robust Control Design A recently developed (during the last decade) control-theoretic approach that holds promise for addressing control problems for Large Space Structures (LSS) and others similarly challenging, is the so-called "Robust Control" approach. This methodology produces control designs which are less sensitive than conventional ones to inaccuracies in modeling and varying and/or unknown parameters, at the expense of being more computationally intensive. Application papers and experiments in aerospace and other areas have been extensively produced. It must be mentioned that the LSS control problems have helped stimulate these theoretical developments, and happen to be very much suited for them.We can mention, among others, the following robust control techniques: 1) The H-infinity approach. This is basically an optimal control theory which uses a different measure for the "size" of the sub-systems involved ("infinity" norm instead of quadratic norm),
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