Stabilization with Prescribed Instant for High-Order Integrator Systems

This paper is a research related to finite-time stability. Different from traditional fixed-time, predefined-time, and prescribed time stability that more or less have some conservativeness, we manage to stabilize system states onto the equilibrium at an arbitrarily selected time instant irrespective of initial system states and parameters. In another word, the conservativeness of convergence time in our proposed control method is proved to be zero. Moreover, the control is bounded and also gradually goes to zero at the selected instant. It is obviously an improvement compared with the existing finite-time stabilization (FNTS), such as fixed-time stabilization (FTS), predefined-time stabilization (PDTS), and prescribed-time stabilization (PSTS).

The FNTS property is of great interest for scenarios where real-time constraints need to be satisfied, e.g., in missile guidance, the impact time control guidance laws require stabilization in a desired time. Our proposed PSIS can deal with the FNTS problems.

For other tasks with more accurate requirement on time, the FTS, the PDTS, and the PSTS are insufficient. For instant, two robot arms playing the piano. Music has its’ rhythm, each note is expected to appear at a specific time instant, or the music will sounds terrible. There are many other tasks that are easy for human beings but difficult for robots, e.g.. dancing and sports. The author think it is because human have rhythm feeling, while robots have not. Hence, it is important to develop such control methodologies.