Simple Adaptive Control Applications to Large Flexible Structures

What is it about?

When tackling challenging control problems (in particular, in the presence of uncertainty or changing plant parameters), one often considers the application of adaptive control. In particular, control of flexible structures presents a few difficult challenges. Even if the number of relevant flexible modes is finite, this number is usually larger than the admissible dimension of any realistic controller. Besides, the frequency of the higher harmonics becomes more and more uncertain as the frequency increases and may vary in various environments. When actuators and sensors are collocated, it has been shown that oscillation damping can be obtained using output velocity feedback and that both oscillation damping and position control can be obtained when both output position and velocity feedback are used. Nevertheless, designing efficient feedback controllers requires good knowledge of the plant parameters and may also require varying the controller parameters when the environmental or operational conditions vary. The aforementioned arguments seem to point to the need of adaptive controllers that would fit the controller parameters to the specific (and eventually varying) environmental and operational conditions without requiring exact knowledge of plant parameters. In particular, the so-called simplified adaptive control (SAC) methodology seems appropriate for control of the flexible structure as its structure is low order, and it does not require knowledge of plant parameters or even the number of flexible modes. However, application of adaptive control in general, and the SAC in particular, require the controlled system to be almost passive and its transfer function almost strictly positive real (ASPR). In spite of successful implementations of SAC, for a long time, these basic conditions have been considered difficult to satisfy and have actually remained rather obscure. As we present in this Note, it was finally shown that the basic almost passivity conditions required in order to guarantee stability with SAC are equivalent to requiring that the plant {A, B,C} be minimum phase and that the product CB be positive definite symmetric (PDS). Furthermore, it was recently shown that the symmetry condition for CB can be mitigated if the non-symmetric CB is diagonalizable and has real and positive eigenvalues. References shows that, by using parallel feedforward, these conditions can be achieved in systems that do not inherently satisfy them, and thus can allow using adaptive control methodologies even in unstable and nonminimum phase systems. The Note shows that these conditions are satisfied in flexible structures, and examples show SAC efficiency in control of a flexible structure with unknown parameters.

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