Inversion of state-dependent delay

Abstract

The exact determination of the state of a finite dimensional linear system where a variable of interest, say y(t), and its delayed versions appear in an implicit relation with the delays themselves is addressed. It is assumed that the observed signal is not this variable of interest, but rather the delay (or multiple delay vector) itself. This implies that this delay must be state dependent (through y). An implicit relation with known F : RN+1 → RN is assumed. If x(t) ε Rn is the state of the linear system, the observability problem is to determine this state x(t) from the knowledge of the system input u(t) and the delays τ(t). This differs form thewell known observability problem where x(t) is to be determined from u(t) and y(t). In the problem at hand, an inversion is involved, to obtain y(t) from τ(t), rendering the problem nonlinear. Such problems are relevant when dealing with sonar, pertinent in robotics, where mobile systems must avoid hitting walls, and in underwater vehicles, for instance the soft "landing" problem on an ocean floor. The requisite observability/invertibility conditions are derived. The relevance of the restrictions τi < 1 for the problem to be well posed is illuminated from the physical context in the problem. In addition, the inversion of a special 'autoregressive' relation (in iterated function sense) obeyed by a delay, is solved. This is of interest in singular perturbation approaches to systems with state dependent delay.