一种基于轨迹的离散时间平坦化方法
A Trajectory-Based Approach to Discrete-Time Flatness
摘要：
对于离散时间系统，平坦度通常是通过用前向位移代替众所周知的连续时间定义的时间导数来定义的。有了这个定义，这类平坦系统正好对应于一类系统，它可以线性化的离散时间内生动态反馈，因为它是在文献中提出的。最近，可验证的必要和充分的微分几何条件，这一性质已经得到。在本文中，我们试图考虑后移。这种扩展方法的动机是一对一的对应的解决方案的平坦系统的解决方案的平凡系统，因为它是已知的连续时间的情况下。如果我们把这个想法转移到离散时间的情况下，这将导致一种方法，也允许向后移位。为了区别经典的前向平移定义和本文的方法，我们称前者为前向平坦。我们发现，平坦系统（在向后移位的扩展意义上）仍然具有许多有益的性质的向前平坦系统。特别是，它们仍然是可达/可控的，允许直接规划轨迹，并且可以通过动态反馈的某个子类线性化。
英文摘要：
For discrete-time systems, flatness is usually defined by replacing the time-derivatives of the well-known continuous-time definition by forward-shifts. With this definition, the class of flat systems corresponds exactly to the class of systems which can be linearized by a discrete-time endogenous dynamic feedback as it is proposed in the literature. Recently, verifiable necessary and sufficient differential-geometric conditions for this property have been derived. In the present contribution, we make an attempt to take into account also backward-shifts. This extended approach is motivated by the one-to-one correspondence of solutions of flat systems to solutions of a trivial system as it is known from the continuous-time case. If we transfer this idea to the discrete-time case, this leads to an approach which also allows backward-shifts. To distinguish the classical definition with forward-shifts and the approach of the present paper, we refer to the former as forward-flatness. We show that flat systems (in the extended sense with backward-shifts) still share many beneficial properties of forward-flat systems. In particular, they still are reachable/controllable, allow a straightforward planning of trajectories and can be linearized by a certain subclass of dynamic feedbacks.
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