Singular perturbation
|
|
Singular perturbation is a mathematical operation, which can be used on the class of linear/non-linear problems where two dynamics operating on different time scales exist. For instance, an electrically driven robot manipulator can have slower mechanical dynamics and faster electrical dynamics. In such cases, we can divide the systems into two subsystems one corresponding to faster dynamics and other corresponding to slower dynamics and then designing controllers for each one of them separately. Through singular perturbation technique, we can make these two subsystems independent of each other thereby simplifying the control problem.
Consider a class of system described by following set of equations:
- <math>\begin{matrix}
\dot{x}_1 &=& f_1(x,u) \\ \epsilon\dot{x}_2 &=& f_{21}(x_1)+f_{22}(x_1)x_2+g_2(x_1)u \end{matrix}<math>
where the state
- <math>x=[{x_1}^T {x_2}^T]^T<math>
is decomposed into two portions. In second equation, <math>\epsilon <\!\!< 1<math> indicates that the dynamics of <math>x_2<math> is faster than that of <math>x_1<math>.Template:Math-stub