Output Feedback Consensus for Networked Heterogeneous Nonlinear Negative-Imaginary Systems with Free Body Motion

Abstract

This paper presents a framework to address the robust output feedback consensus problem for networked heterogeneous nonlinear Negative-Imaginary (NI) systems with free body dynamics. The aim of this paper is to complete and extend the results in previous papers on robust output feedback consensus for multiple heterogeneous nonlinear NI systems so that the systems in the network are allowed to have free body motion. A subclass of NI systems called Output Strictly Negative-Imaginary (OSNI) systems are applied as controllers to ensure that the outputs of the nonlinear NI plants converge to the same limit trajectory. The definitions of nonlinear NI systems and nonlinear OSNI systems are extended and a new stability result is developed for the interconnection of a single nonlinear NI system and a single nonlinear OSNI system. Robust output feedback consensus is addressed by establishing a similar stability result for the interconnection of networked NI systems and networked OSNI systems.

Negative Imaginary State Feedback Equivalence for Systems of Relative Degree One and Relative Degree Two

“…Considering the emergence of the nonlinear negative imaginary systems theory [32], [33], it is worth investigating the feedback equivalence problem for nonlinear systems of relative degree one or two based on the nonlinear NI systems theory in the future. This future feedback nonlinear NI research is planned to complement the work done by Byrnes, Isidori and Williems in [19], which investigates the feedback passivity problem for a nonlinear system of relative degree one.…”

“…where K 1 is defined in (33), K 3 is defined in (32) and K 2 is defined in (36) with Y 2 also satisfying λ max (Y 2 ) < 1 γ . Proof: With the input (54) applied, the system (53) becomes a positive feedback interconnection of the system w = ∆(s)y and y = R(s)w, where R(s) is the transfer function of the state-space model (26) with v replaced by w. Therefore, according to Lemma 11, the state-feedback matrices K 1 , K 2 and K 3 as defined respectively in (33), (36) and (32) make R(s) negative imaginary. Also, we have…”

“…The assumptions in Lemma 14 that A 11 is Lyapunov stable and (A 11 , A 11 A 13 + A 12 ) is controllable are satisfied. According to (33), ( 36) and ( 32), choose the state-feedback matrices to be…”

Making Nonlinear Systems Negative Imaginary via State Feedback

“…Roughly speaking, a system is said to be nonlinear NI if it has a positive definite storage function and is dissipative with respect to the supply rate u T ẏ, where u and y are the input and output of the system, respectively. This definition is generalized in [22], to only require positive semidefiniteness of the storage function in order to allow for systems with poles at the origin; e.g., single and double integrators. Also introduced in [20] and [22] is the notion of nonlinear output strictly NI (OSNI) systems (see [11] and [23] for the definition of linear OSNI systems).…”

“…This definition is generalized in [22], to only require positive semidefiniteness of the storage function in order to allow for systems with poles at the origin; e.g., single and double integrators. Also introduced in [20] and [22] is the notion of nonlinear output strictly NI (OSNI) systems (see [11] and [23] for the definition of linear OSNI systems). Under the control of suitable nonlinear OSNI controllers, nonlinear NI systems can be asymptotically stabilized under mild assumptions.…”