Almost Invariance and Noninteracting Control: A Frequency-Domain Analysis

We solve a number of feedback synthesis problems in the context of noninter-acting control or block-diagonal decoupling for fin&dimensional linear time-invariant systems. We consider a plant that, apart from a control input and a measurement output, has a given number of exogenous input vectors and the same number of exogenous output vectors. The decoupling problem studied is to find dynamic compensators from the plant measurement output (which in this paper will be assumed to be the full plant state) to the plant control input in such a way that the following requirements are met: (1) the closed-loop transfer matrix is block-diagonal, (2) the remaining diagonal blocks are stable with respect to an u priori given first stability set, and (3) the closed-loop system is internally stable with respect to an a priori given second (in general larger) stability set. In addition, we study the “almost ” version of the above problem. In the latter the requirement of exact decoupling is replaced by a requirement of approximate decoupling in the sense that the compensators to be designed should yield off-diagonal blocks in the closed-loop transfer matrix that are arbitrarily small in P-norm. Necessary and sufficient conditions for the existence of such dynamic compensators are formulated in terms of controlled invariant and almost controlled invariant subspaces. 1