Finite gain [formula omitted] stabilization requires analog control

Abstract

A causal feedback map, taking sequences of measurements and producing sequences of controls, is denoted as finite set if, within any finite time horizon, its range is in a finite set. Bit-rate constrained or digital control are particular cases of finite-set feedback. In this paper, we show that the finite gain (FG) l p stabilization, with 1 ⩽ p ⩽ ∞ , of a discrete-time, linear and time-invariant unstable plant is impossible by finite-set feedback. In addition, we show that, under finite-set feedback, weaker (local) versions of FG l p stability are also impossible. These facts are not obvious, since recent results have shown that input to state stabilization is viable by bit-rate constrained control. In view of such existing work, this paper leads to two conclusions: (1) even when input to state stability is attainable by finite-set feedback, small changes in the amplitude of the external excitation may cause, in relative terms, a large increase in the amplitude of the state (2) FG l p stabilization requires logarithmic precision around zero. Since our conclusions hold with no assumption on the feedback structure, they cannot be derived from existing results. We adopt an information theoretic viewpoint, which also brings new insights into the problem of stabilization.