Polytopic best-mean performance analysis

In Boyarski and Shaked [(2005), ‘Robust H∞ Control Design for Best Mean Performance Over an Uncertain-parameters Box’, Systems and Control Letters, 54, 585–595], a novel best-mean approach to robust analysis and control over uncertain-parameters boxes was presented. This article extends the results of Boyarski and Shaked (2005) to convex uncertainty polytopes of arbitrary shape and arbitrary number of vertices, without an underlying parameters model. The article addresses robust, polytopic, probabilistic H∞ analysis of linear systems and focuses on the performance distribution over the uncertainty region (rather than just on the performance bound, as is customary in robust control). It is assumed that all system instances over the uncertainty polytope may occur with equal probability; this uniform distribution assumption is common in robust statistical analysis and is known to be conservative. The proposed approach considers different disturbance attenuation levels (DALs) at the vertices of the uncertainty polytope. It is shown that, under the latter assumption, the mean DAL over the polytope is the algebraic average of the DALs at the polytope's vertices. Thus, the mean DAL over the polytope can be optimised by minimising the sum of the DALs at the vertices. The standard deviation of the DAL over the uncertainty polytope is also addressed, and a method to minimise this standard deviation (in order to enforce uniform performance over the polytope) is shown. The example utilises a state-feedback synthesis theorem presented in Boyarski and Shaked (2005). A Monte-Carlo analysis verifies correct statistics of the resulting closed-loop ‘pointwise’ H∞-norms over the uncertainty region, and highlights the differences from a corresponding bound minimisation: a marginally higher bound, but much better mean and variance.