# Observer-Based Consensus Control for Heterogeneous Multi-Agent Systems with Output Saturations
## 1 Introduction  
In the past decade, the consensus problem for multi-agent systems has received a lot of attention since it has wide applications such as formation control [1], and distributed filtering [2], cooperative control [3]. The goal of consensus is to achieve an agreement using local interactions between agents, and the consensus is one of the most studied methods to control the multi-agent systems in a distributed way. Most pioneering works have considered the consensus problem for homogeneous agents, which have identical dynamics, with single-integrator [1, 2, 4, 5], double-integrator [6, 7, 8], high-order linear [9, 10, 11, 12], and nonlinear dynamics [13, 14, 15]. However, in real applications, it is often unrealistic for all agents to have an identical model. Therefore, in recent years, consensus for heterogeneous agents, which have nonidentical dynamics, has been widely studied [16, 17, 18, 19]. Specifically, the necessary and sufficient condition for heterogeneous linear agents was studied in [16]. They showed that all agents must have a common internal model such that they can generate the same trajectory. Then, under this assumption, the observer-based consensus algorithm was constructed via the output regulation approach. In [17], the dynamic consensus protocol was proposed for leaderless and leader-following cases. In [18], the output consensus problem using the state feedback and the output feedbacks was studied. In [19], the dynamic event-triggered consensus protocol was developed based on input-to-state stability. In [20], heterogeneous oscillator were considered and the adaptive observer was developed. Most actuators and sensors in real systems have saturation constraints owing to their limited capacity. Since the saturations lead to poor performance of the system, control problems for systems under saturations are an important issue in real applications. Specifically, the consensus problem for agents with input saturations has been widely studied in recent years [21, 22, 23, 24, 25, 26, 27, 28]. For homogeneous agents with input saturations, semi-global consensus and global consensus were studied in [21, 22, 23, 24, 25]. For heterogeneous agents with input saturations, the semi-global output consensus was studied in [26, 27, 28]. Although there are many results dealing with input saturations in the consensus problem, the consensus problem under the output saturations has rarely been studied [29, 30, 31, 32]. Sincethe consensus algorithm uses relative information between two agents, consensus may not be realized when the outputs are saturated. In [29; 30; 31], the necessary and sufficient conditions for single-integrator agents with output saturations were investigated. They showed that the weighted average in a group should be bounded such that the consensus trajectory can be measured. In [32], leader-following consensus for homogeneous high-order linear agents with output saturations was studied. The authors developed an observer-based consensus algorithm inspired by the nonlinear observer [33] and analyzed the asymptotic convergence based on the Lyapunov stability theory. However, to the best of our knowledge, the existing works on the consensus problem under the output saturations have considered homogeneous agents. Motivated by the above observations, this paper studies the consensus problem for the heterogeneous agents with output saturations. We assume that the agents are neutrally stable and the communication graph is undirected. Then, we propose a distributed observer-based consensus algorithm based on the output regulation approach. The main contributions of this paper are summarized as follows. First, the output consensus problem for the heterogeneous agents with output saturations is investigated, and the leaderless and leader-following cases are considered. Therefore, this paper is a generalized version of the previous papers, which considered homogeneous agents [29; 30; 31; 32]. Second, we construct the observer-based algorithm considering the output saturations.