Henri Gavin, Ph.D., P.E., Associate Professor
Methods and software for computing optimal control and state trajectories for structures with controllable dampers.
Structural Control and Health Monitoring,under revision
submitted manuscript, 2012
Control forces in semi-active control systems are constrained by the dynamics of actuators that regulate energy transmission through variable damping and/or stiffness mechanisms. The potential benefit of developing and implementing new semi-active control devices and applications can be determined by optimizing the controlled performance subject to the constraints of the dynamics of the system being controlled (given by the state equations), the constraints associated with the dynamics of the semi-active device, and the expected external forcing.
Performance optimization of semi-active control systems is a constrained two-point boundary value problem. This paper shows how this constrained problem can be transformed into an unconstrained problem and how to easily solve the related unconstrained problem with Matlab. The method is illustrated on the performance optimization of a simple semi-active tuned-mass-damper for a structure subjected to ground accelerations. Several possible extensions of this method and application are described in detail.
This paper presents a method to compute optimal open-loop trajectories for systems subject to state and control inequality constraints in which the cost function is quadratic and the state dynamics are linear. For the case in which inequality constraints are decentralized with respect to the controls, optimal Lagrange multipliers enforcing the inequality constraints may be found at any time through Pontryagin’s minimum principle. In so doing, the set of differential algebraic Euler-Lagrange equations are transformed into an unconstrained nonlinear two-point boundary-value problem for states and costates whose solution meets the necessary conditions for optimality. The optimal performance of inequality constrained control systems is calculable allowing for comparison to previous, sub-optimal solutions. The method is applied to the control of damping forces in a vibration isolation system subjected to constraints imposed by the physical implementation of a particular controllable damper.
The paper investigates the benefits of implementing the semiactive control systems in relation to passive viscous damping in the context of seismically isolated structures. Frequency response functions are compiled from the computed time history response to pulse-like seismic excitation. A simple semiactive control policy is evaluated in regard to passive linear viscous damping and an optimal non-causal semiactive control strategy. The optimal control strategy minimizes the integral of the squared absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied. The optimization procedure involves a numerical solution to the Euler-Lagrange equations.
The responses of two, low-rise, 2-degree-of-freedom base isolated structures with different isolation periods to a set of near-field earthquake ground motions are investigated under passive linear and nonlinear viscous damping, two pseudoskyhook semiactive control methods, and optimal semiactive control. The structures are isolated with a low damping elastic isolation system in parallel with a controllable damper. The optimal serniactive control strategy minimizes an integral norm of superstructure absolute accelerations subject to the constraint that the nonlinear equations of motion are satisfied and is determined through a numerical solution to the Euler-Lagrange equations. The optimal closed-loop performance is evaluated for a controllable damper and is compared to passive viscous damping and causal pseudoskyhook control rules. Results obtained from eight different earthquake records illustrate the type of ground motions and structures for which semiactive damping is most promising.
This material is based upon work supported by the National Science Foundation under grant No. NSF-CMMI-0900324. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.