Robust vibration control of dynamical systems based on the derivative of the state

Summary

 Vibrations may be undesirable in dynamical systems for several reasons. They may affect the security, such as vibrations in primary cycle parts of (nuclear) power plants. They may decrease the quality and functionality of products, such as those manufactured by machine tools. And they may lower the comfort, such as vibrations in car wheel suspension systems or in power trains of cars. One possibility to attenuate these vibrations is by employing active suspension elements. Mounted at appropriate places inside the systems or with respect to their environment, they are able to interchange or dissipate kinetic and potential energy in an effective way with moderate control effort. Their effectiveness depends greatly on the control scheme applied to change damping and stiffness characteristics of the suspension elements. The control schemes, however, very often need information on the state variables involved in the mathematical modeling. On the other hand, it is mostly the acceleration or speed of certain parts that can be sensed reasonably and measured with sufficient accuracy. We propose here a control scheme which is solely based on the derivative of the state variables, provided that active suspension elements or actuators with the above-mentioned properties may be employed within the system. Furthermore, we only use control actions within a discrete set of possible values, which aids the real-time implementation of the designed control algorithms. And, last but not least, the number of control inputs (actuators) may be arbitrary, that is, the system may be mismatched. The scheme is based on the Lyapunov stability theory, which involves discontinuities of the Lyapunov function candidates along trajectories of the state derivative. The effectiveness and behavior of the control scheme is demonstrated on a two-DOF model of an active car seat suspension in order to enhance the driving comfort.