Input-Output Feedback Linearization Preserving Task Priority for Multivariate Nonlinear Systems Having Singular Input Gain Matrix

TL;DR

This paper extends input-output feedback linearization for multivariate nonlinear systems with singular input gain matrices, enabling prioritized control of tasks even at singular points, through a generalized normal form and multi-objective optimization.

Contribution

It introduces a prioritized input-output linearization method that handles singularities by generalizing the Byrnes-Isidori form and applying lexicographical optimization.

Findings

The proposed method achieves stable output tracking with task priority preservation.

It effectively manages singular input gain matrices without losing control performance.

Lyapunov analysis confirms boundedness and task achievement under the new framework.

Abstract

We propose an extension of the input-output feedback linearization for a class of multivariate systems that are not input-output linearizable in a classical manner. The key observation is that the usual input-output linearization problem can be interpreted as the problem of solving simultaneous linear equations associated with the input gain matrix: thus, even at points where the input gain matrix becomes singular, it is still possible to solve a part of linear equations, by which a subset of input-output relations is made linear or close to be linear. Based on this observation, we adopt the task priority-based approach in the input-output linearization problem. First, we generalize the classical Byrnes-Isidori normal form to a prioritized normal form having a triangular structure, so that the singularity of a subblock of the input gain matrix related to lower-priority tasks does not directly propagate to higher-priority tasks. Next, we present a prioritized input-output linearization via the multi-objective optimization with the lexicographical ordering, resulting in a prioritized semilinear form that establishes input output relations whose subset with higher priority is linear or close to be linear. Finally, Lyapunov analysis on ultimate boundedness and task achievement is provided, particularly when the proposed prioritized input-output linearization is applied to the output tracking problem. This work introduces a new control framework for complex systems having critical and noncritical control issues, by assigning higher priority to the critical ones.