Robust Output Regulation in Distributed Control

• Robust output regulation is a control method that ensures agents asymptotically track a reference signal and reject disturbances in distributed networks.
• It employs a distributed internal model principle by embedding minimal replicas of the exosystem dynamics into each agent for resilient performance.
• Dynamic state or output feedback controllers with delay compensation stabilize the overall system amid uncertainties and communication imperfections.

Robust output regulation is a central paradigm in distributed control, concerned with achieving asymptotic tracking and disturbance rejection for networks of dynamical systems—multi-agent systems (MAS)—in the presence of both exogenous signals and model uncertainties. The robust output regulation framework extends classical output regulation to a cooperative, networked setting and encompasses diverse system classes: continuous-time and discrete-time, finite-dimensional and infinite-dimensional (PDE/PIDE), with or without delays, and under varying communication topologies. Core to all rigorous solutions is the internal model principle, generalized to distributed architectures and robustified against uncertainties, delays, and communication imperfections.

1. Problem Formulation and System Setting

Robust output regulation requires that each agent (typically called a follower) in a network tracks a reference signal and rejects disturbances generated by a (typically autonomous) exosystem (leader), despite plant and environment uncertainties. The general nonlinear equation for a network of $N$ uncertain linear agents with both input and communication delay is, for each $i=1,...,N$: $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$ with input delay $\tau_d$, and exosystem: $\dot v(t) = S\,v(t),\quad y_0(t) = -F\,v(t)$ where $S$ encodes the dynamics of the reference and disturbance, assumed neutrally stable ($\sigma(S)\subset j\mathbb{R}$).

The objective is to design distributed controllers using only local and neighbor information, such that the regulation error at each agent,

$e_i(t) = y_i(t) - y_0(t) = C\,x_i(t) + F\,v(t)$

tends to zero asymptotically, uniformly over admissible uncertainties and delays. Communication among agents is represented by a weighted digraph $\bar{\mathcal G}$, with its induced Laplacian $H$, required to contain a directed spanning tree rooted at the leader.

This networked robust output regulation (ROR) framework subsumes classical leader-following consensus, output synchronization, disturbance rejection, and dynamic coupling schemes in MAS control (Lu et al., 2015).

2. Distributed Internal Model Principle

Central to robust output regulation is the internal model principle (IMP), classically establishing that robust regulation is only possible if the controller incorporates an "internal model" of the exosystem dynamics. In the distributed multi-agent context, this entails embedding, within each agent, a minimal $i=1,...,N$0-copy ($i=1,...,N$1 = regulated output dimension) of the exosystem model $i=1,...,N$2. The standard realizations use a pair $i=1,...,N$3, where $i=1,...,N$4's characteristic polynomial is the minimal polynomial of $i=1,...,N$5, and $i=1,...,N$6 is controllable: $i=1,...,N$7 Here, $i=1,...,N$8 has $i=1,...,N$9 and $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$0 is controllable.

The internal model is driven by a "virtual" error signal based on the relative regulated outputs in the network: $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$1 with coupling weights $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$2 from the digraph. The internal model states evolve as

$$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$3

accommodating communication delay $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$4. This construction robustifies tracking and rejection against exosystem signals in the presence of network coupling, uncertainties, and delays (Lu et al., 2015).

3. Controller Synthesis: State and Output Feedback with Delays

Robust output regulation is guaranteed through dynamic state or output feedback controllers that augment each agent's local dynamics with its internal model. Two principal schemes are:

State-feedback (with delayed network signals): $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$5

$$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$6

where $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$7 are designed so the overall augmented time-delay system is stabilized.

Output-feedback (full distributed, delayed observer): $$\dot x_i(t) = A\,x_i(t) + B\,u_i(t-\tau_d) + E_i\,v(t), \;\;\; y_i(t) = C\,x_i(t)$$8

[ \dot\xi_i(t) = A\,\xi_i(t) +