Fault‐Tolerant Stabilization in Discrete‐Time Multiple‐Integrator Networks with General Information Sharing

Abstract

The paper considers a network of agents with identical discrete-time multiple-integrator dynamics. The agents share information according to an arbitrary topology. The information is relative to the states corresponding to some of the highest integration levels. With reference to this setting, a decentralized stabilization problem is faced, under the further assumption that faults may occur in the communication apparatuses of one or several of the agents. Necessary and sufficient solvability conditions are presented for the problem, together with formulas for a class of least-order regulators.

Two fault‐tolerant control problems for multiple‐integrator networks

“…Quite unexpectedly, the same condition was found to be necessary and sufficient to solve the problems dealt with in [32] for n arbitrary, p D 1, and in [34] for n and p arbitrary, where only fault-tolerant stabilization, rather than pole-placement, was required. Finally, it coincides as well with the solvability condition for the fault-tolerant stabilization problem stated for chains of discrete-time integrators in [33], where n is arbitrary and p D 1, and in [35], where n and p are arbitrary. On the other hand, the regulator (23), (25)-(29), (32), (33) does depend on n and p, and, as already noticed, its order is the least possible one for any n and p.…”

“…where G n can be thought of as the known nominal topology matrix. Then, we consider Problem 3.1, where the conditions in points 1 and 2 have to be satisfied for G as in (35) and all l and r specified by (34).…”

“…We have already faced similar problems in . These papers make reference to the stabilization of continuous‐time and discrete‐time multiple‐integrator networks and propose the adoption of low‐gain regulators, that is, regulators that satisfy all the problems specifications provided that their gains are sufficiently low.…”

“…In both cases, the regulator is required to be decentralized, and the controlled network has to result tolerant with respect to faults in the communication apparatuses of the agents.We show that this problem can be solved by means of least-order regulators. These regulators are high-gain, that is, they satisfy all the problem specifications provided that their gains are sufficiently high.We have already faced similar problems in [32][33][34][35]. These papers make reference to the stabilization of continuous-time and discrete-time multiple-integrator networks and propose the adoption of low-gain regulators, that is, regulators that satisfy all the problems specifications provided that their gains are sufficiently low.…”

“…On the contrary, the states relative to the lowerorder integrals were not assumed as available for measurement. Here, we remove this assumption, as we did in [34,35].The second problem concerns consensus. We remind the reader that the consensus problems are those where the agents' states are required to asymptotically converge to a unique motion, which may be called agreement law, or else consensus function.…”