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-tol < gapexact(p0,p1) <= gap
Create two plant models. One plant is unstable, first-order, with transfer function 1/( s -0.001). The other plant is stable and first-order 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 in the gap and nugap metrics.
[g,ng] = gapmetric(p1,p2)
g = 0.0029 ng = 0.0020
Intuitively, this result is obvious, because, for instance, the feedback controller K = 1 stabilizes both plants and renders the closed-loop systems nearly identical.
K = 1; H1 = loopsens(p1,K); H2 = loopsens(p2,K); subplot(2,2,1); bode(H1.Si,'-',H2.Si,'--'); subplot(2,2,2); bode(H1.Ti,'-',H2.Ti,'--'); subplot(2,2,3); bode(H1.PSi,'-',H2.PSi,'--'); subplot(2,2,4); bode(H1.CSo,'-',H2.CSo,'--');
Next, consider two stable plant models that differ by a first-order system. One plant is the transfer function 50/( s +50) and the other plant is the transfer function 50/( s +50) * 8/( s +8).
p3 = tf(,[1 50]); p4 = tf(,[1 8])*p3;
Although the two systems have similar high-frequency dynamics and the same unity gain at low frequency, the plants are modestly far apart in the gap and nugap metrics.
[g,ng] = gapmetric(p3,p4)
g = 0.6156 ng = 0.6147
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 (G0) and a perturbed system p1 (G1). 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(G1,K1) ≥ arcsin b(G0,K0) – arcsin δ(G0,G1) – arcsin δ(K0,K1)
The interpretation of this result is that if a nominal plant G0 is stabilized by controller K0, with "stability margin" b(G0,K0), then the stability margin when G0 is perturbed to G1 and K0 is perturbed to K1 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 W2GW1 and K by (similarly for G0, G1, K0 and K1). 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