gapmetric

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

Syntax

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

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

Description

[gap,nugap] = gapmetric(P1,P2) computes the gap and Vinnicombe (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.

[gap,nugap] = gapmetric(P1,P2,tol) specifies a relative accuracy for calculating the gaps.

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.

tol — Relative accuracy

0.001 (default) | positive scalar

Relative accuracy for computing the gap metrics, specified as a positive scalar. If gap_actual is the true value of the gap (or the Vinnicombe gap), the returned value gap (or nugap) is guaranteed to satisfy

|1 – gap/gap_actual| < tol.

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 P₁ and P₂, 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:

$${\delta }_g\left(P_1,P_2\right)=\max \left\{{\overrightarrow{\delta }}_g\left(P_1,P_2\right),{\overrightarrow{\delta }}_g\left(P_2,P_1\right)\right\}.$$

Here, $${\overrightarrow{\delta }}_g\left(P_1,P_2\right)$$ is the directed gap, given by

$${\overrightarrow{\delta }}_g\left(P_1,P_2\right)=\underset{\text{stable }Q\left(s\right)}{\min }{\Vert \left[\begin{array}{c}{M}_1\\ N_1\end{array}\right]-\left[\begin{array}{c}{M}_2\\ N2\end{array}\right]Q\Vert }_{\infty }.$$

For more information, see [1] and Chapter 17 of [2].

Vinnicombe Gap Metric

For P₁ and P₂, the Vinnicombe gap metric is given by

$${\delta }_{\nu }\left(P_1,P_2\right)=\underset{\omega }{\max }{\Vert {\left(I+P_2{P}_2^{*}\right)}^{-1/2}\left(P_1-P_2\right){\left(I+P_1{P}_1^{*}\right)}^{-1/2}\Vert }_{\infty },$$

provided that $$\det \left(I+P_2^{*}{P}_1\right)$$ has the right winding number. Here, * denotes the conjugate (see ctranspose). This expression is a weighted difference between the two frequency responses P₁(jω) and P₂(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(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

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

To interpret this result, suppose that a nominal plant 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.

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 W₂PW₁, and replace C by $$W_1^{-1}C{W}_2^{-1}$$. You can make similar substitutions for P₁, P₂, C₁ 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 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.