Optimal Output Consensus of Second-Order Uncertain Nonlinear Systems on Weight-Unbalanced Directed Networks

Abstract

This paper investigates the distributed optimal output consensus problem of second-order uncertain nonlinear multi-agent systems over weight-unbalanced directed networks. Under the standard assumption that local cost functions are strongly convex with globally Lipschitz gradients, a novel distributed dynamic state feedback controller is developed such that the outputs of all the agents reach the optimal solution to minimize the global cost function which is the sum of all the local cost functions. The controller design is based on a two-layer strategy, where a distributed optimal coordinator and a reference-tracking controller are proposed to address the challenges arising from unbalanced directed networks and uncertain nonlinear functions respectively. A key feature of the proposed controller is that the nonlinear functions containing the uncertainties and disturbances are not required to be globally Lipschitz. Furthermore, by exploiting adaptive control technique, no prior knowledge of the uncertainties or disturbances is required either. Two simulation examples are finally provided to illustrate the effectiveness of the proposed control scheme.

Distributed Optimal Output Consensus of Uncertain Nonlinear Multi-Agent Systems over Unbalanced Directed Networks via Output Feedback

“…Then the proposed optimal coordinator is embedded into the feedback loop to convert the DOOC problem to a reference-tracking problem, which will be addressed in the next subsection. Specifically, inspired by [15], [20], the optimal coordinator for each agent i is designed as follows,…”

“…where α1 and α2 are two positive constants, y r i ∈ R represents the generated reference signal for agent i, ζi ∈ R and ξi ∈ R N are auxiliary variables, with ξ k i being the k-th component of ξi and initial value ξi(0) satisfying ξ i i (0) = 1, otherwise ξ k i (0) = 0 for all k = i. With these choices, it is shown in [15] that ξ i i (t) > 0 for all t ≥ 0, which means that the algorithm ( 4) is well defined.…”

“…It should be emphasized that, under Assumption 2, it can be proved that limt→∞ ξ i i = ri, where ri represents the i-th component of the left eigenvector r corresponding to the zero eigenvalue of the Laplacian matrix [15]. Note that the optimal coordinator of each agent only requires information from its neighbors and itself, and thus is distributed.…”

“…It is worth noting that the controller developed in [14] is based on a twolayer framework, which consists of an optimal coordinator generating the optimal solution and a decentralized output feedback controller driving each agent to track its individual optimal coordinator. Then, in our preliminary work [15], the two-layer framework is extended to solve the DOOC problem of disturbed second-order nonlinear systems but over unbalanced directed networks. The DOOC problem of more general nonlinear multi-agent systems in the normal form over undirected or balanced directed networks are addressed in [16], [17], where the inverse dynamics of agents are unfortunately required to be input-to-state stable (ISS).…”

“…1) In contrast to linear agent dynamics or nonlinear agent dynamics in output feedback form considered in [12]- [15], [19]- [21], the nonlinear agent dynamics in the normal form are more general and include the above-mentioned agent dynamics as special cases [22]. Furthermore, in virtue of a high-gain observer, only agent outputs are needed for controller design in this note, which greatly enhances its applicability in practice.…”