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The effect of modal truncation on the damping ratio and their derivatives with respect to an added mass is investigated for a simply supported, multispan beam with a linear quadratic Gaussian control system. It is found that both the damping ratios and derivatives converge slowly, but the derivatives converge more slowly than the damping ratios. However, it is shown that when Ritz vectors corresponding to static displacements due to actuator forces are added to the reduced model, the convergence of both the damping ratios and their derivatives is accelerated. It is also shown that the accuracy of the damping ratio predicted by a reduced-model control design can be improved significantly if the Ritz vectors are included in the design of the control system. Thus, it appears that Ritz vectors added to the reduced model of flexible structures can improve greatly the accuracy of both the design and analysis of the control system.

A method for determining the most effective locations for active struts on large space structures is developed and tested on the NASA CSI Evolutionary Model. Depending on the choice of weighting factors, the method can be used to maximize the maximum modal damping ratio or decay rate, or to maximize a mission-oriented measure of performance. Placement of 8, 16, and 32 active struts out of 1507 candidate truss elements is demonstrated. Preliminary estimates of damping enhancement are reported pending refined structural models and dynamic test results. The method can handle complicated FEM models wih a large number of truss elements and many target modes. It can treat each mode equally or it can emphasize the importance of selected modes. The method can eliminate some combinations of actuator locations based on topological constraints.

The paper is concerned with the on-line identification of system Markov parameters using observers. Two types of parameters are sought: the forward system parameters, which provide estimates of the outputs for given inputs, and inverse system parameters, which provide estimates of inputs for given outputs. A procedure is proposed which uses the inverse system to generate commands that produce a desired system response. The feasibility of the procedure is demonstrated using test results from an implementation of a truss structure fitted with piezoelectric actuators and collocated strain gauges.

It is well known that the Fourier series of discontinuous functions converge slowly and that the derivatives of the series may not converge at all. Since modal expansion of structural response is a generalization of the Fourier series, we may expect slow convergence of modal expansion when the applied loads exhibit discontinuities in time or space. Thus, in a structure controlled by point actuators, we may expect slow convergence of derivatives of structural response with respect to system parameters. To demonstrate this, the sensitivity of the closed-loop response to structural changes is calculated for a multispan beam with direct-rate feedback using colocated velocity sensors and point actuators. Reduced models based on the natural modes of the structure are formed, and derivatives of the damping ratios of the closed-loop eigenvalues are calculated. As expected, the convergence of the derivatives of the damping ratios with an increasing number of modes is slower than the convergence of the damping ratios themselves. The convergence is unproved when distributed actuators replace the point actuators. In transient response problems, it is known that complementing the vibration modes with a mode representing static response to the loads can improve convergence. Indeed, for the example studied, when Ritz vectors corresponding to static responses caused by unit loads at the actuators are added to the basis vectors, the convergence of the reduced-model derivatives is greatly enhanced.

The sensitivity of control system performance to structural changes is calculated for a multi-span beam with direct-rate feedback vibration control. Reduced models based on the natural modes of the structure are formed and derivatives of the damping ratios of the closed-loop eigenvalues are calculated. The convergence of the derivatives of the damping ratios with increasing number of modes is shown to be slower than the convergence of the damping ratios themselves. In particular, in some cases the convergence of finite-element approximations to the derivatives is much faster than the convergence of the modal approximations. The results indicate that the use of reduced models based on natural vibration modes may be ill-advised for calculating the sensitivity of control system performance to changes in the controlled structure.

Reference 1 investigated a multi-span beam, representative of a large space structure, with a direct-rate feedback control system using colocated sensors and actuators. It was shown that modal truncation can cause large errors in the derivatives of control performance with respect to structural parameters. The purpose of this study is to examine the effect of modal truncation on a similar structure with a linear-quadratic, optimal control system.