Control system design based on orbit attractor

What is attractor?

A system evolving in time is called a dynamical system, and it is represented by a differential equation in continuous time domain as;

　　

a difference equation in discrete time domain as;

　　

If we design the system to converge to a certain trajectory as time progresses, this corresponds to designing a dynamical system with an orbital attractor.
The state valiable xk+1 in time k+1 is decided from the current value of xk. Even though the motion of x is changed by the external force, it continues its motion, which means the motion of x does not have time constraint. This is because the attractor based controller is suitable for human-machine collaborative control.

1. Dynamics design via a functional approximation of vector field

Set the reference orbit.
Define the vector field that converges to the orbit on some points near to the orbit.
The defined vector field is approximated by a function of x.
The dynamics that has an attractor on the given orbit is design.

2. Attractor dsign with global stability

The method in 1 realizes small basin of attractor while the following method designs global stable attractor.
The given orbit is approximated by a function of parameter t

　　

Find the nearest point on the orbit to the current xh.
As shown in the figure, the convergence vector and the flow vector are defined,
and the next point is defined as follows;

　　

We can design a dynamics with global stability.


e-mail : .m.aa@m.titech.ac.jp