Flat Control Systems with Delay

Abstract

The notion of flatness was introduced in the control theory in [1], where the dynamic feedback method was developed for solving control problems for flat dynamical systems. It turns out that numerous nonlinear control systems in various fields of engineering are flat and control problems for these systems can be solved by this method (see the bibliography in [1]). Flatness conditions were obtained as well [2, 3], and a numerical method for computing the flat output was suggested, which proves effective in some cases. In the present paper, we generalize the notion of flatness to the case of delay systems. In this connection, we use Fliess’s remark [4] that the flat output in this case can depend not only on the values of the variables “in the past” but also on the values of the variables “in the future.” (Therefore, the set G introduced below contains both positive and negative numbers.) We show how to solve terminal control problems and stabilization problems for flat systems with delay. We prove Theorem 1 on the relationship between autonomous flat systems with delay and flat dynamical systems. By using this theorem, we find a flat output of a delay system describing the model of a diesel engine [5]. We show how to solve the stabilization problem for this system. We construct an infinite-dimensional geometric model of delay systems in the spirit of [6] and use it to prove the theorems stated.