Strictly speaking, Newton's second law of motion is only an approximation of the so-called relativistic dynamics, i.e., Einstein's modification of the second law based on his theory of special relativity. Although the approximation is almost exact when the velocity of the dynamical system is far less than the speed of light, the difference will become larger and larger (and will eventually go to infinity) as the velocity approaches the speed of light. Correspondingly, feedback control of such dynamics should also take this modification into consideration (though it will render the system nonlinear), especially when the velocity is relatively large. Towards this end, we start this note by studying the state-space representation of the relativistic dynamics. We then investigate on how to employ the feedback linearization approach for such relativistic dynamics, based upon which an additional linear controller may then be designed. As such, the feedback linearization together with the linear controller compose the overall relativistic feedback control law. We also provide discussions on, e.g., controllability, state feedback and output feedback, as well as PID control, in the relativistic setting.
翻译:严格地说,牛顿的第二个运动法只是所谓的相对论动态的近似值,即爱因斯坦根据他的特殊相对论理论对第二项法律的修改。尽管当动态系统的速度远低于光速时,近似几乎精确,但随着速度接近光速,差异会越来越大,(并最终会变得无限化),因此,这种动态的反馈控制也应该考虑到这一修改(尽管它将使系统变得非线性),特别是在速度相对较大的情况下。为此,我们首先研究相对论动态的状态-空间代表。然后我们研究如何对此种相对论动态采用反馈线性直线化方法,在此基础上再设计一个线性控制器。因此,反馈线性控制与线性控制器一起构成总体相对性反馈控制法。我们还提供了关于控制性、状态反馈和输出反馈的讨论,以及确定PID的相对性控制。