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Compute upper bounds on Vinnicombe `gap`

and `nugap`

distances
between two systems

[gap,nugap] = gapmetric(p0,p1) [gap,nugap] = gapmetric(p0,p1,tol)

`[gap,nugap] = gapmetric(p0,p1)`

calculates upper bounds on the `gap`

and `nugap`

(Vinnicombe)
metric between systems `p0`

and `p1`

.
The `gap`

and `nugap`

values lie
between 0 and 1. A small value (relative to 1) implies that any controller
that stabilizes `p0`

will likely stabilize `p1`

,
and, moreover, that the closed-loop gains of the two closed-loop systems
will be similar. A `gap`

or `nugap`

of
0 implies that `p0`

equals `p1`

,
and a value of 1 implies that the plants are far apart. The input
and output dimensions of `p0`

and `p1`

must
be the same.

`[gap,nugap] = gapmetric(p0,p1,tol)`

specifies a relative accuracy for calculating the `gap`

metric
and `nugap`

metric. The default value for `tol`

is
0.001. The computed answers are guaranteed to satisfy

gap-tol < gapexact(p0,p1) <= gap

`gap`

and `nugap`

compute
the gap and ν gap metrics between two LTI objects. Both quantities
give a numerical value δ(`p0`

,`p1`

)
between 0 and 1 for the distance between a nominal system `p0`

(*G*_{0})
and a perturbed system p1 (*G*_{1}).
The gap metric was introduced into the control literature by Zames
and El-Sakkary 1980, and exploited by Georgiou and Smith 1990. The
ν gap metric was derived by Vinnicombe 1993. For both of these
metrics the following robust performance result holds from Qui and
Davidson 1992, and Vinnicombe 1993

arcsin *b*(*G*_{1},*K*_{1})
≥ arcsin *b*(*G*_{0},*K*_{0})
– arcsin δ(*G*_{0},*G*_{1})
– arcsin δ(*K*_{0},*K*_{1})

where

$$b(G,K)={\Vert \left[\begin{array}{c}I\\ K\end{array}\right]{(I-GK)}^{-1}\left[\begin{array}{cc}G& I\end{array}\right]\Vert}_{\infty}^{-1}$$

The interpretation of this result is that if a nominal plant *G*_{0} is
stabilized by controller *K*_{0},
with “stability margin” *b*(*G*_{0},*K*_{0}),
then the stability margin when *G*_{0} is
perturbed to *G*_{1} and *K*_{0} is
perturbed to *K*_{1} is degraded
by no more than the above formula. Note that 1/*b*(*G*,*K*)
is also the signal gain from disturbances on the plant input and output
to the input and output of the controller. The ν `gap`

is
always less than or equal to the `gap`

, so its predictions
using the above robustness result are tighter.

To make use of the gap metrics in robust design, weighting functions
need to be introduced. In the above robustness result, *G* needs
to be replaced by *W*_{2}*GW*_{1} and *K* by $${W}_{1}^{-1}K{W}_{2}^{-1}$$(similarly for *G*_{0}, *G*_{1}, *K*_{0} and *K*_{1}).
This makes the weighting functions compatible with the weighting structure
in the *H*_{∞} loop shaping
control design procedure (see `loopsyn`

and `ncfsyn`

for
more details).

The computation of the gap amounts to solving 2-block *H*_{∞} problems
(Georgiou, Smith 1988). The particular method used here for solving
the *H*_{∞} problems is
based on Green et al., 1990. The computation of the `nugap`

uses
the method of Vinnicombe, 1993.

Georgiou, T.T., “On the computation of the gap metric,
” *Systems Control Letters,* Vol. 11, 1988,
p. 253-257

Georgiou, T.T., and M. Smith, “Optimal robustness in
the gap metric,” *IEEE Transactions on Automatic Control,* Vol.
35, 1990, p. 673-686

Green, M., K. Glover, D. Limebeer, and J.C. Doyle, “A
J-spectral factorization approach to *H*_{∞} control,” *SIAM
J. of Control and Opt.,* 28(6), 1990, p. 1350-1371

Qiu, L., and E.J. Davison, “Feedback stability under
simultaneous gap metric uncertainties in plant and controller,” *Systems
Control Letters,* Vol. 18-1, 1992 p. 9-22

Vinnicombe, G., “Measuring Robustness of Feedback Systems,” PhD Dissertation, Department of Engineering, University of Cambridge, 1993.

Zames, G., and El-Sakkary, “Unstable systems and feedback:
The gap metric,” *Proceedings of the Allerton Conference,* October
1980, p. 380-385

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