Robust Cooperative Output Regulation of Discrete-Time Heterogeneous Multi-Agent Systems

This article considers robust cooperative output regulation of discrete-time uncertain heterogeneous (in dimension) multi-agent systems (MASs). We show that the solvability of this problem with an internal model-based distributed control law reduces to the existence of a structured control gain that makes the nominal closed-loop system matrix of the MAS Schur. Accordingly, this article focuses on global and agent-wise local sufficient conditions for the existence and design of such a structured control gain. Based on a structured Lyapunov inequality, we present a convexification that yields a linear matrix inequality (LMI), whose feasibility is a global sufficient condition for the existence and design. Considering the individual nominal dynamics of each agent, the existence is also ensured if each agent solves a structure-free control problem. Its convexification yields LMIs that allow each agent to separately design its structure-free control gain. Lastly, we study the relationships between the sets of control gains emerging from both global and local perspectives.

💡 Research Summary

This paper addresses the robust cooperative output regulation problem (RCORP) for discrete‑time heterogeneous multi‑agent systems (MAS) subject to parametric uncertainties. Each follower is modeled as a linear discrete‑time system with possibly different state dimensions, and the exogenous signal generated by an autonomous exosystem provides both reference and disturbance signals. Communication among agents is described by a time‑varying directed graph, and a leader node is added to model the availability of the exosystem’s output to a subset of followers.

The authors propose a distributed dynamic state‑feedback control law that incorporates a p‑copy internal model of the exosystem. The control law does not require exchange of controller states between neighboring agents; instead, each agent uses its own tracking error and relative errors with respect to its neighbors. The closed‑loop dynamics can be written compactly as

 x⁺ = A x + B u, u = K x,

where the gain matrix K must respect a sparsity pattern imposed by the communication graph. The central solvability condition (Theorem 1) states that, under standard graph connectivity, exosystem eigenvalue, internal‑model, and stabilizability assumptions (Conditions 1‑3), the RCORP is solved if the nominal closed‑loop matrix A_g = A + B K is Schur (all eigenvalues lie strictly inside the unit circle). While stabilizability of (A, B) is necessary, it is not sufficient because the imposed sparsity may prevent the existence of a suitable K (as illustrated by Example 1).

To guarantee the existence of a structured K, the paper develops two complementary design approaches, both cast as linear matrix inequalities (LMIs).

1. Global design – A structured Lyapunov inequality P > 0, A_gᵀ P A_g − P < 0 is restricted to the prescribed sparsity of K. By applying the projection lemma and suitable variable changes, the inequality is convexified into a single LMI whose feasibility yields a globally valid K for the entire network. This method treats the MAS as a single large system; consequently, it is computationally more demanding but less conservative, as proved by Corollary 3 (the set of gains obtained globally contains those obtained locally).

2. Agent‑wise local design – Each agent i solves a “structure‑free” control problem for its own dynamics (A_i, B_i) together with its internal‑model matrices (G_{1i}, G_{2i}). The resulting LMIs involve only the local matrices and produce gains K_{1i}, K_{2i} that automatically satisfy the global sparsity pattern when assembled. This approach is highly scalable because the LMIs are of fixed size independent of the total number of agents, allowing parallel computation. However, it is more conservative; some feasible global solutions may be missed, as demonstrated in Example 3.

The paper also establishes that under Conditions 1–5 the pair (A, B) is stabilizable (Lemma 5), but stabilizability alone does not guarantee a feasible structured K. The relationship between the two design methods is examined in Section VI, where it is shown that the global design is a superset of the local design and therefore less conservative, while the local design offers superior scalability.

Compared with prior work on continuous‑time heterogeneous MAS and on discrete‑time homogeneous MAS, this contribution is novel in three respects: (i) it handles heterogeneous agent dimensions in discrete time, where the closed‑loop matrix lacks a convenient block‑triangular structure; (ii) it provides polynomial‑time LMI‑based synthesis for both global and local designs despite the NP‑hard nature of structured gain synthesis; (iii) the control law relies only on relative outputs, making it applicable even when a communication network for controller‑state exchange is unavailable.

In summary, the authors deliver a complete theoretical framework for robust cooperative output regulation of uncertain discrete‑time heterogeneous MAS. They supply constructive LMI conditions for both a globally optimal (less conservative) gain and a scalable agent‑wise gain, discuss their trade‑offs, and illustrate the results with several examples. The work opens avenues for extensions to switching topologies, event‑triggered implementations, and more general uncertainty models.