Finite time observation of nonlinear time-delay systems with unknown inputs

Abstract

Causal and non-causal observability are discussed in this paper for nonlinear time-delay systems. By extending the Lie derivative for time-delay systems in the algebraic framework introduced by Xia et al. (2002), we present a canonical form and give sufficient condition in order to deal with causal and non-causal observations of state and unknown inputs of time-delay systems.

Finite-time and asymptotic left inversion of nonlinear time-delay systems

“…Sufficient conditions for left invertibility are deduced based on the obtained canonical form. Before this, let us recall the following result stated in [39]. …”

“…In the context of constant time delay, the notions of Lie derivatives and relative degree are defined, and the differences between causal and non-causal invertibility are clarified in [38]. The canonical form of invertibility is also given in [39], and in [42] a method for estimating the unknown inputs is proposed. However, the algorithm for left invertibility proposed in [39] was only for system without internal dynamics (see also [4] and [13] and their references).…”

“…The canonical form of invertibility is also given in [39], and in [42] a method for estimating the unknown inputs is proposed. However, the algorithm for left invertibility proposed in [39] was only for system without internal dynamics (see also [4] and [13] and their references). For system with or without time-delay, the main difficulty when the internal dynamics appears is to estimate the state of such dynamics.…”

Causal observability of nonlinear time-delay systems with unknown inputs

“…And it is the unimodular characteristic of Γ over K(δ] which guarantees the causal reconstruction of unknown inputs. The non-causal case was studied in [20].…”

“…Based on the algebraic framework proposed in [19], the problem of causal and non-causal observability of the states and unknown inputs for nonlinear time-delay systems has been studied in [20] assuming that the systems can be transformed into a suitable canonical form. This condition is relaxed in this paper by introducing additional information depending on the available measurements.…”