# Robustness Analysis for Multi-Agent Consensus Systems with Application to DC Motor Synchronization
## 1 Introduction  
Multi-agent coordination control has been a research field of interest during the last few decades. A wide variety of multi-agent control methods can be found in the literature. These control approaches are classified as (a) virtual structure [1], where each agent is considered a component of a bigger structure; (b) behavioral control [2], which deals with conflicting tasks such as obstacle avoidance and move to goal, for example; and (c) leader-follower [3; 4], where one of the agents works as a group leader and the others are designated as followers. The consensus is one of the most studied strategies to control multi-agent systems [4; 5; 6; 7], and its applications include power systems [8], robotics [3; 9], flight formation control [10], to mention a few. Arguably, the most common actuator in robotics and industrial applications is the DC motor. It is well known that motor synchronization plays an important role in the quality of manufactured products. For example, in rolling mill machines, the conveyor belt's tension must lie in an allowable tension to guarantee the thickness of the rolled steel [11]. As discussed in [12], many manufacturing processes require coordinated motion of two or more motion axes to achieve the desired performance, e.g., CNC machines, paper machines. Poor motor synchronization in such industrial applications would result in low-quality products that might be considered unusable in the worst case. The problemof designing control algorithms for motor synchronization has been considered in different research works. In [13], the authors presented some results on the synchronization of two motors' speed using the leader-follower approach, where leader speed was used as a reference for the follower system, and the error was controlled by an LQR controller. A control scheme based on an adaptive sliding mode control technique was employed into the synchronization of the control system for four motors in [14]. In [15], a synchronization control strategy of multiple induction motors was proposed. The approach is based on an adjacent cross-coupling control structure, and then it can use a total sliding mode control method. A stator flux regulation method is used in [16] to apply an adaptive time-delay feedback control and the direct torque control to synchronize the chaotic speed of induction motors in a master-slave scheme. Three common synchronization methods for chaotic systems, namely an SMC control, a PI control, and an adaptive controller, were used in [17] to synchronize two brushless DC motor models in a master-slave configuration and were compared by using well-known performance indices. Recently, the theory and applications of robust control in the presence of uncertainty and time delays have been addressed in the literature using a variety of control approaches [18; 19; 20; 21], for example. The problem of the robustness of a multi-agent consensus control in the presence of uncertainty and time delays has been studied using various approaches [22; 23; 24; 25]. In [22], a robust controller using algebraic Ricatti equations is presented. The agents are assumed to have linear dynamics, while uncertainty is treated as additive perturbations, which are assumed to be stable and \(\mathcal{H}_{\infty}\) bounded. Sliding mode control is proposed in [23] to deal with uncertainty as additive perturbations in a leader-follower multi-agent system. In a similar way than [22], it is assumed that additive perturbation is stable and \(\mathcal{H}_{\infty}\) bounded. In [24], a stochastic Lyapunov function to solve the consensus-seeking problem in networks where the agents can only obtain noisy measurements of its neighbor's state is proposed. In these previously mentioned papers, dynamic uncertainty is considered, which is introduced to the system as an additive disturbance, and, in other cases, it is additive noise in contrast to this paper where uncertainty is considered in each of the parameters.