# gapmetric

Gap metric and Vinnicombe (nu-gap) metric for distance between two systems

## Description

`[`

computes the gap and Vinnicombe (`gap`

,`nugap`

] = gapmetric(`P1,P2`

)*ν*-gap) metrics for the distance between
dynamic systems `P1`

and `P2`

. The gap metric values
satisfy 0 ≤ `nugap`

≤ `gap`

≤ 1. Values close to zero imply that any controller that stabilizes
`P1`

also stabilizes `P2`

with similar closed-loop
gains.

## Examples

### Compute Gap Metrics for Stable and Unstable Plant Models

Create two plant models. One plant, `P1`

, is an unstable first-order system with transfer function 1/(*s*–0.001). The other plant, `P2`

, is stable, with transfer function 1/(*s* +0.001).

P1 = tf(1,[1 -0.001]); P2 = tf(1,[1 0.001]);

Despite the fact that one plant is unstable and the other is stable, these plants are close as measured by the `gap`

and `nugap`

metrics.

[gap,nugap] = gapmetric(P1,P2)

gap = 0.0021

nugap = 0.0020

The gap is very small compared to 1. Thus a controller that yields a stable closed-loop system with `P2`

also tends to stabilize `P1`

. For instance, the feedback controller `C = 1`

stabilizes both plants and renders nearly identical closed-loop gains. To see this, examine the sensitivity functions of the two closed-loop systems.

C = 1; H1 = loopsens(P1,C); H2 = loopsens(P2,C); subplot(2,2,1); bode(H1.Si,'-',H2.Si,'r--'); subplot(2,2,2); bode(H1.Ti,'-',H2.Ti,'r--'); subplot(2,2,3); bode(H1.PSi,'-',H2.PSi,'r--'); subplot(2,2,4); bode(H1.CSo,'-',H2.CSo,'r--');

Next, consider two stable plant models that differ by a first-order system. One plant, `P3`

, is the transfer function 50/(*s*+50), and the other plant, `P4`

, is the transfer function [50/(*s*+50)]*8/(*s*+8).

P3 = tf(50,[1 50]); P4 = tf(8,[1 8])*P3; figure bode(P3,P4)

Although the two systems have similar high-frequency dynamics and the same unity gain at low frequency, by the `gap`

and `nugap`

metrics, the plants are fairly far apart.

[gap,nugap] = gapmetric(P3,P4)

gap = 0.6148

nugap = 0.6147

### Compute Gap Metric and Stability Margin

Consider a plant and a stabilizing controller.

P1 = tf([1 2],[1 5 10]); C = tf(4.4,[1 0]);

Compute the stability margin for this plant and controller.

b1 = ncfmargin(P1,C)

b1 = 0.1961

Next, compute the gap between `P1`

and the perturbed plant, `P2`

.

P2 = tf([1 1],[1 3 10]); [gap,nugap] = gapmetric(P1,P2)

gap = 0.1391

nugap = 0.1390

Because the stability margin `b1 = b(P1,C)`

is greater than the gap between the two plants, `C`

also stabilizes `P2`

. As discussed in Gap Metrics and Stability Margins, the stability margin `b2 = b(P2,C)`

satisfies the inequality `asin(b(P2,C)) ≥ asin(b1)-asin(gap)`

. Confirm this result.

b2 = ncfmargin(P2,C); [asin(b2) asin(b1)-asin(gap)]

`ans = `*1×2*
0.0997 0.0579

## Input Arguments

`P1,P2`

— Input systems

dynamic system models

Input systems, specified as dynamic system models. `P1`

and
`P2`

must have the same input and output dimensions. If
`P1`

or `P2`

is a generalized state-space model
(`genss`

or `uss`

) then
`gapmetric`

uses the current or nominal value of all control design
blocks.

## Output Arguments

`gap`

— Gap between `P1`

and `P2`

scalar in [0,1]

Gap between
`P1`

and `P2`

, returned as a scalar in the range
[0,1]. A value close to zero implies that any controller that stabilizes
`P1`

also stabilizes `P2`

with similar closed-loop
gains. A value close to 1 means that `P1`

and `P2`

are
far apart. A value of 0 means that the two systems are identical.

`nugap`

— Vinnicombe gap (*ν*-gap) between `P1`

and `P2`

scalar in [0,1]

Vinnicombe gap
(*ν*-gap) between `P1`

and `P2`

,
returned as a scalar value in the range [0,1]. As with `gap`

, a value
close to zero implies that any controller that stabilizes `P1`

also
stabilizes `P2`

with similar closed-loop gains. A value close to 1
means that `P1`

and `P2`

are far apart. A value of 0
means that the two systems are identical. Because 0 ≤ `nugap`

≤ `gap`

≤ 1, the *ν*-gap can provide a more stringent test for
robustness as described in Gap Metrics and Stability Margins.

## More About

### Gap Metric

For plants *P*_{1} and
*P*_{2}, let $${P}_{1}={N}_{1}{M}_{1}^{-1}$$ and $${P}_{2}={N}_{2}{M}_{2}^{-1}$$ be right normalized coprime factorizations (see `rncf`

). Then the
gap metric *δ _{g}* is given by:

$${\delta}_{g}\left({P}_{1},{P}_{2}\right)=\mathrm{max}\left\{{\overrightarrow{\delta}}_{g}\left({P}_{1},{P}_{2}\right),{\overrightarrow{\delta}}_{g}\left({P}_{2},{P}_{1}\right)\right\}.$$

Here, $${\overrightarrow{\delta}}_{g}\left({P}_{1},{P}_{2}\right)$$ is the *directed gap*, given by

$${\overrightarrow{\delta}}_{g}\left({P}_{1},{P}_{2}\right)=\underset{\text{stable}Q\left(s\right)}{\mathrm{min}}{\Vert \left[\begin{array}{c}{M}_{1}\\ {N}_{1}\end{array}\right]-\left[\begin{array}{c}{M}_{2}\\ N2\end{array}\right]Q\Vert}_{\infty}.$$

### Vinnicombe Gap Metric

For *P*_{1} and
*P*_{2}, the Vinnicombe gap metric is given by

$${\delta}_{\nu}\left({P}_{1},{P}_{2}\right)=\underset{\omega}{\mathrm{max}}{\Vert {\left(I+{P}_{2}{P}_{2}^{*}\right)}^{-1/2}\left({P}_{1}-{P}_{2}\right){\left(I+{P}_{1}{P}_{1}^{*}\right)}^{-1/2}\Vert}_{\infty},$$

provided that $$\mathrm{det}\left(I+{P}_{2}^{*}{P}_{1}\right)$$ has the right winding number. Here, * denotes the conjugate (see `ctranspose`

).
This expression is a weighted difference between the two frequency responses
*P*_{1}(*jω*) and
*P*_{2}(*jω*). For more information,
see Chapter 17 of [2].

### Gap Metrics and Stability Margins

The gap and *ν*-gap metrics give a numerical value
*δ*(*P*_{1},*P*_{2})
for the distance between two LTI systems. For both metrics, the following robust performance
result holds:

arcsin
*b*(*P*_{2},*C*_{2})
≥ arcsin
*b*(*P*_{1},*C*_{1})
– arcsin
δ(*P*_{1},*P*_{2})
– arcsin
δ(*C*_{1},*C*_{2}),

where the stability margin *b* (see `ncfmargin`

), assuming negative-feedback architecture, is given by

$$b(P,C)={\Vert \left[\begin{array}{c}I\\ C\end{array}\right]{(I+PC)}^{-1}\left[\begin{array}{cc}I& P\end{array}\right]\Vert}_{\infty}^{-1}={\Vert \left[\begin{array}{c}I\\ P\end{array}\right]{(I+CP)}^{-1}\left[\begin{array}{cc}I& C\end{array}\right]\Vert}_{\infty}^{-1}.$$

To interpret this result, suppose that a nominal plant
*P*_{1} is stabilized by controller
*C*_{1} with stability margin *b*(*P*_{1},*C*_{1}). Then, if *P*_{1} is perturbed to
*P*_{2} and
*C*_{1} is perturbed to
*C*_{2}, the stability margin is degraded by no more
than the above formula. For an example, see Compute Gap Metric and Stability Margin.

The *ν*-gap is always less than or equal to the gap, so its predictions
using the above robustness result are tighter.

The quantity
*b*(*P*,*C*)^{–1}
is the signal gain from disturbances on the plant input and output to the input and output
of the controller.

### Gap Metrics in Robust Design

To make use of the gap metrics in robust design, you must introduce
weighting functions. In the robust performance formula, replace *P* by
*W*_{2}*P**W*_{1},
and replace *C* by $${W}_{1}^{-1}C{W}_{2}^{-1}$$. You can make similar substitutions for
*P*_{1},
*P*_{2}, *C*_{1}
and *C*_{2}. This form makes the weighting functions
compatible with the weighting structure in the *H*_{∞}
loop shaping control design procedure used by functions such as `loopsyn`

and `ncfsyn`

.

## References

[1] Georgiou, Tryphon T. “On the
Computation of the Gap Metric.” *Systems & Control Letters* 11, no. 4
(October 1988): 253–57. https://doi.org/10.1016/0167-6911(88)90067-9.

[2] Zhou, K., Doyle, J.C.,
*Essentials of Robust Control*. London, UK: Pearson, 1997.

## Version History

**Introduced before R2006a**

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