H∞, optimisation for L1 performance objectives

The problem of non-conservative minimisation of the maximum amplitude of error signals in the time domain for multivariable systems subject to uncertain inputs is considered. It is shown that this problem, which is usually tackled using L₁ control theory, can also be addressed in the framework of H∞ theory, via the introduction of some new signal uncertainty sets, and a slight modification of the standard H∞ optimisation problem. The resulting problem formulation provides some new insights into the relationship between H∞ and L₁ control, while also capturing some of the more attractive features of both theories.