An algorithm to select an output for desired vector relative degree, which is not restricted to be uniform (the same for all outputs), and zero dynamics in linear time-varying (LTV) multi-input multi-output (MIMO) systems is proposed. The proposed algorithm is developed under reasonable assumptions of uniform (with respect to time) and lexicography-fixed controllability (with some choice of fixed controllability indices) and is inspired by a generalization of the Ackermann’s formula. One application of the proposed algorithm is the context of higher-order sliding-mode control for perturbed LTV MIMO systems, achieving finite-time stability. The developed algorithm also provides an extension of the classical Ackermann–Utkin formula for outputs of uniform relative degree one and includes the design of flat outputs for LTV MIMO systems. Hence other application is the feedback controller design for trajectory tracking task. Finally, a complete HOSM methodology is provided and tested in an LTV model of the non-holonomic car.

提出了一种选择期望矢量相对度的输出的算法，该算法不限于统一的（对于所有输出都相同），并且在线性时变（LTV）多输入多输出（MIMO）系统中为零动态。所提出的算法是在统一（相对于时间）和字典法固定的可控性（可以选择固定的可控性指标）的合理假设下开发的，并受到阿克曼公式的推广的启发。所提出算法的一种应用是对扰动的LTV MIMO系统进行高阶滑模控制，从而实现了有限时间的稳定性。所开发的算法还为经典的一致度一的输出提供了经典Ackermann-Utkin公式的扩展，并且包括了LTV MIMO系统的平面输出的设计。因此，其他应用是用于轨迹跟踪任务的反馈控制器设计。最后，提供了完整的HOSM方法并在非完整汽车的LTV模型中进行了测试。