Effects of Input Redundancy on Time Optimal Control

Due to the popularity of the systems with input redundancy, this paper focuses on the problems with input redundancy, where we concern about the effects of adding new input redundancy into the controllable systems. Time optimal control problems are discussed, where such effects are evaluated by the optimal time. Based on the assumption of the existence and uniqueness of the optimal control, the paper proves that increasing the number of input redundancy will result in a strict reduction of the optimal time from the same initial state if there exists non-idle channel among the redundant input channels. Moreover, if the problem is normal, then all of the redundant input channels are used to shorten the optimal time. On the other hand, without the assumption of normality, the optimal time will also be smaller for the redundant system as comparing to the original system if at least one of these redundant input channels is completely controllable. Finally, two numerical examples are deployed to demonstrate the main results of this paper.