  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.
