Almost invariance and non interacting control : a frequency domain analysis

In this paper we shall solve a number of feedback synthesis problems in the context of non interacting control or block diagonal decoupling for finite dimensional linear time invariant systems. We shall consider a plant that, apart from a control input and a measurement output, has a given numher of exogenous input vectors and the same numher of exogenous output vectors. The decoupling problem that will he studied here is to find dynamic compensators from the plant measurement output (which in this paper will he assumed to be the full plant state) to the plant control input in such a way that the following requirements are met: (i) the closed loop transfer matrix is block diagonal, (ii) the remaining diagonal blocks are stable with respect to an a priori given first stability set and (iii) the closed loop system is internally stable with respect to an a priori given second, in general larger, stability set In addition, we will study the "almost" version of the above problem. In the latter the requirement of exact decoupling will he replaced by a requirement of approximate decoupIing in the sense that the to-be-designed compensators should yield off-diagonal blocks in the closed loop transfer matrix that are arbitrarily small in $H^\infty$-norm. Necessary and sufficient conditions for the existence of such dynamic compensators will he formulated in terms of controlled invariant and almost controlled invariant subspaces.