Comment on 'Robustness of solutions to a benchmark control problem'

Abstract

R EFERENCE lisa comparative study of a number of solutions to a common benchmark problem. Stochastic robustness analysis is used to assess the performance and stability robustness of a number of controllers in the presence of plant model error. This error is assumed to take the form of random parameter variations with specified probability density functions. Monte Carlo methods are used to estimate the probability of instability P/. It is found in Ref. 1 that the gain and phase margins (GM and PM) of the nominal systems are not good predictors of P/. Although increasing parameter uncertainty usually increases P/, there are no consistent trends with GM and PM. This important and surprising result contradicts extensive experience that GM and PM are reliable measures of stability robustness for single-input/single-output (SISO) systems. It is important to understand the extent to which the result is applicable outside the immediate context of the benchmark problem. The purpose of this comment is to supplement the authors' discussion of this point. The authors attribute their findings mainly to the fact that the shape of the Nyquist plot of the open-loop transfer function varies in a complicated way as a function of the plant uncertainties. The implication is that placing bounds on the closeness of approach of the Nyquist plot to the critical point at the two cross-over frequencies provides little assurance that it will not cross the critical point at some other frequency. Of course this is true; but this Note also argues that one would not expect GM and PM to be reliable robustness measures for a class of systems that includes the example used in Ref. 1. Let the open-loop transfer function be L(s), and let the s-plane contour on which L(s) is evaluated, in applying the Nyquist stability criterion, be C. Normally C runs along the imaginary s axis with closure to the right at infinity. GM and PM are measures of how much L(s) can change on C before the encirclement count, and hence the number of closed-loop poles inside C is altered. Values of GM and PM should not be quoted blindly unless there is some confidence that the magnitude and phase errors on C will actually be smaller than the margins over the relevant frequency range. If L(s) has poles or zeros near to C, then even small variations in their locations due to model error can give rise to an error in L(s) on C that is much larger than that covered by the margins. A simple example is benchmark problem 1 (BP-1) of Ref. 1. The plant model can be written as G(s) = p/2s(s + p) where 1 < p < 2 and the nominal value is ^2. The loop transfer function is given by L (s) = G(s)K(s), where K(s) is the controller. Even if the portion of C from s = j to s = 2j is indented infinitesimally to the left or right so that L(s) is defined over all of the contour for all values of p, the magnitude and phase errors on and near this part of the contour, due to the displacement of poles from their nominal positions, are too large for meaningful conclusions to be drawn about the stability robustness from GM 2.5