Parameter identification and control of distributed-parameter systems

Two methods, one for the identification and one for the control implementation of distributed-parameter systems are presented. The methods are designed to identify and control the actual distributed system, without resorting to discretization. They are implemented using discrete sensors and actuators.

The identification process is carried out in two steps. First, the eigensolution of the distributed system is identified. The lowest frequencies and associated eigenfunctions are identified using an extension of a time-domain approach developed for discrete systems. The extension to distributed systems is carried out in this dissertation. To this end, the sensors output is interpolated to identify the eigenfunctions. Next, the parameters contained in the equations of motion are identified. The motion of distributed-parameter systems is described in terms of partial differential equations, so that these parameters are in general continuous functions of the spatial variables. For vibrating systems, these parameters ordinarily represent the mass, stiffness and damping distributions. These distributions are expanded in terms of finite series of known functions of the spatial variables multiplied by undetermined coefficients. Then, using the identified eigensolution and assuming that the general nature of the equation of motion is known, use is made of the least squares method, in conjunction with the eigenfunctions orthogonality to compute the undetermined coefficients, thus identifying the actual distributed system.

The control system design is based on the concept of independent modal-space control. Implementation of the independent modal-space control method requires that the number of actuators be equal to the number of controlled modes. Because the actuators are discrete elements, control spillover into the uncontrolled modes is experienced. The effect of control spillover is to pump part of the energy imparted to the distributed system into the uncontrolled modes. It is shown that when the independent modal-space control method is used, the energy required to control the controlled modes does not depend on the actuators locations, so that the placement of the actuators does not represent a serious problem, as it can for coupled controls.

A new concept in extracting modal coordinates from the system output, namely modal filters, is introduced. Modal filters extract the modal quantities from the sensors data by interpolating the output of the sensors to obtain continuous displacement patterns and by performing certain weighted integrations over the distributed domain. If the interpolation functions are chosen following the same guidelines as in the finite element method, the integrations can be carried out as offline computations, which facilitates the control implementation. It is shown that when modal filters are used, control of the actual distributed system is possible and no spatial discretization is necessary. In addition, observation spillover, a possible significant problem when observers are used, is eliminated.

Two numerical examples are presented to illustrate the identification and control methods. The methods described in this dissertation are in terms of vibrating systems, with special emphasis on large flexible structures. However, these methods are applicable to any distributed-parameter system.