Geometric approach for observability and accessibility of discrete‐time non‐linear switched impulsive systems

Abstract

This study is concerned with observability and accessibility of discrete-time non-linear switched impulsive systems, for which the problem is more complicated than that for general discrete systems and has novel features. A new method from the combination of differential geometric theory and Lie group analysis is developed. Specifically, under the geometric framework of the system, infinitesimal invariance for the observability and accessibility is investigated, respectively. Based on the invariance investigation, explicit criteria for local observability and local accessibility are derived. Numerical examples are discussed to show the effectiveness of the proposed methods, and the features of observability and accessibility for discrete switched impulsive systems. NomenclatureLet Z + be the set of positive integer. x(k), y(k) and u(k) are n x −, n y − and n u −dimensional vectors, respectively. Denote J = {τ k |k ∈ Z + } ⊆ Z + to be the sequence of impulsive instants satisfying τ 1 < τ 2 < · · · < τ k < · · · , τ k+1 − τ k ≥ 2 and lim k→∞ τ k = ∞. Let J 1 = {1, 2, . . . , N }.

Full‐order impulsive observers for impulsive systems with unknown inputs

“…By Theorems 4.1‐ 4.2, the impulsive observer () can let the error $$$ e(t) $$$ completely trend to zero even if $$$ \lim \underset{t\to \infty }{\sup}\left\Vert v(t)\right\Vert &gt;0 $$$. Hence, the impulsive observer (), which has nonidentical structure with (), can let the estimated state $$$ \hat{x}(t) $$$ completely track the observed system's state $$$ x(t) $$$, not just track the state within an error bound as in Zhao and Sun [30] and LIO () and (). (ii) The unknown input $$$ v $$$ no longer exists in the error system ().…”

“…Here, the impulsive observers are designed for both stabilizing and destabilizing impulses. Thirdly, the impulsive observer (e.g., previous studies [24–30, 32, 34, 36]) usually requires both the observed system and the observer to have identical dynamic structure. Here, in the second type of observers, they are allowed to have nonidentical dynamic structures.…”

Controller synthesis for constrained discrete-time switched positive linear systems

“…Assume the system state does not jump at the switching instants. Switched systems with state jumps are called impulsive systems or switched impulsive systems in the literature [38][39][40][41].…”

Functional observer for switched discrete‐time singular systems with time delays and unknown inputs

“…Although plentiful results have been established in observability and observers design for switched systems without UIs (see, e.g. [24–28] and the references therein), the problem of observers design for switched systems with UIs is much less mature. In [29], a structured state observer was designed for discrete‐time switched systems with UIs under arbitrary switching, and the global convergence of the error dynamics was analysed via switched Lyapunov function method and linear matrix inequality (LMI) technique.…”