Vinnicombe Gap
The Vinnicombe gap is a metric which is used in the context of robust control in order to show the distance between two transfer functions and is defined as follows:
Vinnicombe gap is equal to the infinity norm of y(P1,P2) if a condition on the winding number of (1+P2* P1) and the number of unstable poles of P1 and P2 is satisfied. y(P1,P2) is defined as follows:
y(P1,P2)=(1+P2* P2)1/2(P1-P2)(1+P1* P1)1/2
where P* is the complex conjugate of P. The Vinnicombe gap is equal to 1 if:
where h(P) is the number of unstable poles of P and h0(P) is the number of poles on the unit circuit.
The infinity norm of y(P1,P2) is computed using the grid method and the number of frequency in which the function is evaluated is related to the inverse of "tol". Then for more precise calculation of the infinity norm, "tol" should be enough small.
See : Controller order reduction module .
(file vgap.htm)