Gap metric and Vinnicombe (nu-gap) metric for distance between two systems
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,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
nugap metrics, the plants are fairly far apart.
[gap,nugap] = gapmetric(P3,P4)
gap = 0.6148
nugap = 0.6147
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 = 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
P1,P2— Input systems
Input systems, specified as dynamic system models.
P2 must have the same input and output dimensions. If
P2 is a generalized state-space model
gapmetric uses the current or nominal value of all control design
gap— Gap between
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
far apart. A value of 0 means that the two systems are identical.
nugap— Vinnicombe gap (ν-gap) between
returned as a scalar value in the range [0,1]. As with
gap, a value
close to zero implies that any controller that stabilizes
P2 with similar closed-loop gains. A value close to 1
P2 are far apart. A value of 0
means that the two systems are identical. Because 0 ≤
gap ≤ 1, the ν-gap can provide a more stringent test for
robustness as described in Gap Metrics and Stability Margins.
The gap and ν-gap metrics give a numerical value δ(P1,P2) for the distance between two LTI systems. For both metrics, the following robust performance result holds:
arcsin b(P2,C2) ≥ arcsin b(P1,C1) – arcsin δ(P1,P2) – arcsin δ(C1,C2),
where the stability margin b (see
ncfmargin), assuming negative-feedback architecture, is given by
To interpret this result, suppose that a nominal plant P1 is stabilized by controller C1 with stability margin b(P1,C1). Then, if P1 is perturbed to P2 and C1 is perturbed to C2, 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.
To make use of the gap metrics in robust design, you must introduce
weighting functions. In the robust performance formula, replace P by
and replace C by . You can make similar substitutions for
and C2. This form makes the weighting functions
compatible with the weighting structure in the H∞
loop shaping control design procedure used by functions such as
 Zhou, K., Doyle, J.C., Essentials of Robust Control. London, UK: Pearson, 1997.