Robust partial pole assignment for vibrating systems with aerodynamic effects

[[abstract]]This note proposes a novel algorithm for robust partial eigenvalue assignment (RPEVA) problem for a cubic matrix pencil arising from modeling of vibrating systems with aerodynamic effects. The RPEVA problem for a cubic pencil is the one of choosing suitable feedback matrices to reassign a few (say k < 3n) unwanted eigenvalues while leaving the remaining large number (3n - k) of them unchanged, in such a way that the the eigenvalues of the closed-loop matrix are as insensitive as possible to small perturbation of the data. The latter amounts to minimizing the condition number of the closed-loop eigenvector matrix. The problem is solved directly in the cubic matrix polynomial setting without making any transformation to a standard first-order state-space system. This allows us to take advantage of the exploitable structures such as the sparsity, definiteness, bandness, etc., very often offered by large practical problems The major computational requirements are: i) solution of a small Sylveste; equation, ii) QR factorizations, and iii) solution of a standard least squares problem. The least-squares problem result from matrix rank-two update techniques used in the algorithm for reassigning complex eigenvalues. The practical effectiveness of the method is demonstrated by implementational results on simulated data provided by the Boeing company.[[fileno]]2010223010004[[department]]數學

Active vibration control in linear time-invariant and nonlinear systems

Active vibration control techniques are widely used in linear time-invariant and nonlinear systems. However, there still exist many difficulties in the application of conventional active vibration control techniques, including the following: (1) In application, some of the degrees of freedom may not be physically accessible to actuation and sensing simultaneously; (2) large flexible structures are difficult in terms of isolating one substructure from the vibration of another; (3) the incomplete understanding of the effects of softening nonlinearity may put conventional active controllers at risk; and (4) global stability of under-actuated nonlinear aeroelastic systems, resulting from actuator failure or motivated by weight and cost constraints imposed on next-generation flight vehicles, is extremely challenging, especially in the case of uncertainty and external disturbances. These intellectual challenges are addressed in this research by linear and nonlinear active control techniques. A new theory for partial pole placement by the method of receptances in the presence of inaccessible degrees of freedom is proposed. By the application of a new double input control and orthogonality conditions on the input and feedback gain vectors, partial pole placement is achieved in a linear fashion while some chosen degrees of freedom are free from both actuation and sensing. A lower bound on the maximum number of degrees of freedom inaccessible to both actuation and sensing is established. A theoretical study is presented on the feasibility of applying active control for the purpose of simultaneous vibration isolation and suppression in large flexible structures by block diagonalisation of the system matrices and at the same time assigning eigenvalues to the chosen substructures separately. The methodology, based on eigenstructure assignment using the method of receptances, is found to work successfully when the open-loop system, with lumped or banded mass matrix, is controllable. A comprehensive study of the effects of softening structural nonlinearity in aeroelastic systems is carried out using the simple example of a pitch-flap wing, with softening cubic nonlinearity in the pitch stiffness. Complex dynamical behaviour, including stable and unstable limit cycles and chaos, is revealed using sinusoidal-input describing functions and numerical integration in the time domain. Bifurcation analysis is undertaken using numerical continuation methods to reveal Hopf, symmetry breaking, fold and period doubling bifurcations. The effects of initial conditions on the system stability and the destabilising effects of softening nonlinearity on aerodynamic responses are considered. The global stability of an under-actuated wing section with torsional nonlinearity, softening or hardening, is addressed using a robust passivity-based continuous sliding-mode control approach. The controller is shown to be capable of stabilising the system in the presence of large matched and mismatched uncertainties and large input disturbance. With known bounds on the input disturbance and nonlinearity uncertainty, the continuous control input is able to globally stabilise the overall system if the zero dynamics of the system are globally exponentially stable. The merits and performance of the proposed methods are exemplified in a series of numerical case studies